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ER
express
is
a V q E b 3 p Ea
ER)) t R : : p E TC(a) such t h a t p . < M , . We u s e t h e 0 1 n 1- 1 u s u a l n o t a t i o n s ( p ) . f o r p . , l h ( p ) f o r n + l ; and we use t h e n o t a t i o n (borrowed 1 cy' : new}. The : p\/-A(y)},then this axiom :pII-A(y)] exponentiation axiom.) : plI-(B(y) tion. €a' : p((-(B(y) & yea)}. B : p//-C) such that rz' and Ea'. Now Ez' belonging to some u with C a'. This set can be formed in B, using the union axiom to take a union over Rng(a). Pairing: Take x' = Exponentiation: : plI-(Fcn(y) is to give in by separation. : 3n V q S E y' Vu Vq 6 p( * , P a p a t t e r n , D a d r e s s i n g of t h e same l e n g t h ; t h e l e n g t h of P and D i s c a l l e d t h e Zengtk of t h e r e s t r i c t i o n . The " i n t e n s i o n a l " information a t s t a g e x concerning p o s s i b l e c o n t i n u a t i o n s of f i n i t e sequences of d r e s s e d c a r r i e r s i s completely given by a r e s t r i c t i o n . For a f i n i t e sequence of d r e s s e d c a r r i e r s W e s h a l l use D , D ' , D 1 , , , D = <mo,ml, ,
A -formula. Note that with the aid of the foundation axiom, we P can prove that for each transitive y , if Q(R,y) m d Q(S,y) , then R and S
Note that Q S is a
are extensionally equal relations, i.e. transfinite induction to prove this.)
E R
++
fZ S
.
(We don't need
14
M. BEESON
Suppose W
is a fixed transitive set; then for
x G W , TC(x)
exists. What axioms
does it take to prove \ d x C W 3 R Q(R,TC(x)) ? It seems to require collection and transfinite induction, as well as union. If we know this to be provable, we could go ahead and define the model; setting x
%
y
iff
< x , ~ > E R ) ,or more precisely, < x , u > w
R(Q(R,Tc({x,~)))
the same condition holds, where the
TC's occuring are to be extracted from
u,v
.
, which has collection, transfinite induction, T4
This model allows us to handle
and full separation. In order to handle the weaker theories, we must refine the model more. The first thing that occurs to one is to restrict attention to those "sets" (x,TC(x))
such that there is an
Let us call pairs <x,y>
<x,y>
such that for all
M -sets 1
gR(Q(R,TC({x,y})
=
1
M -set 1
, there is an R such that Q(R.TC({x,a})).
&
%
y
<x,y>& R)
iff
(more precisely) <x,TC(x)>
iff \dR(Q(R,TC(ix,y))
work for T ,T2,TJ , and incidentally for 1
instead of <x,TC(x)>
x
Q(R,TC(x)). This is not enough.
TC(x) , M -sets. Consider an
2
as above, x
%
such that y
we call M -sets. These are the "sets" of our next model.
Such sets <x,y> We define
R
such that
for
T4
-+
%
<x,y> E R).
as well. For simglicity we write
M -sets. Suppose x and y have the same E-mem1 x is equivalent to a member of y and vice-
bers, that is, every element of versa. We have to show that
x
and
y
E-members of
are equivalent (hence are
x and y are M -sets, there is an R such that 2 If z 6 TC(ix,y}) , and Q(R*,TC(z)) , then Rc is a subset of
the same set). Since Q(R,TC(Ix,y})).
R; hence every member of using
R
x
is equivalent to a member of <x,y> E R
for the equivalence; hence
A -separation. If a
We now check
;
is a fixed
y
and vice-versa,
this verifies extensionality.
M -set, and 2
A -formula,
is a
A
M -set x such that for all M -sets z , z E x iff 2 2 (Here A* is the interpretation of A . ) Since E is not given by
we have to find an z
a
E
a
&
A*(z).
A -formula, and neither is
not obvious how to produce on the complexity of
A
. We
M2, the range of the quantified variables, it is
, using only
x
there are two possibilities, either pose
A
is
formed using
z €
b. Take
x
A -separation. We proceed by induction
first have to handle the case of atomic is
A
to be { z : z f a
0
is
b € z
a
and
. This time
or
3 y & b(y
&
is
A %
{
A -separation, since we have
lable to use as a parameter, since Next suppose A
C b
z
b
take
&
z)}
b
. This
yeb
&
z
tion
steps in which
A
set can be Q
yj avai-
x
M -sets. 2 to be { z (r a: l b * E z(b'
R
%
b)}.
such that
Q(R,TC(a)). It is easy to check that these two sets A
here
are
This set can be formed, using as a parameter some rslation This takes care of the case
A ;
z . First sup-
x are actually M -sets. 2 atomic. Similar arguments take care of the induc-
is of the form
B
&
C
,B
Cr C
, or B
-f
C.
CONTINUITY IN INTUITIONISTIC SET THEORIES Next. observe that an easy induction on the complexity of x
1,
y
A*(x)
&
which
+
A*(y)
is provable, for each fixed
member of
is \dw & y B(z,w,y). Since every
A
.
A
15
shows
A
that
Now consider the case in y
is equivalent to some
€-member, the remark just made shows that it suffices to quantify over
c-members;
that is the key to the verification in this case, which we now give in more detail. is an
By the induction hypothesis, we have that v w G yj?(P (<w,z> that
P
E
-+
V w G y
3
c- ~T ( ...) ~ V P 3~w tTy ( ...
a set but an
M -set, if 2 transitive closure of T
).AS a matter of fact,
T
T
such
is not only
M -set. To check this, first note that the 2 can be obtained by taking the union of the transitive
{T}
;
b
a ETC(T)
and
T
i
M1-set, we have to get R
&
is an
y
closures of the elements of relation
M -set 2
z E a & B*(z,w,y))). By strong collection, there is a set
:
a
%
&
thus
M 1-set; if
is an
T &
b e TC(T'))
T'
is Some
to exist. This
is just the union of the corresponding relations for the members of
T (which can be formed using collection), union the set union
{
is an
M -set. Now form the set
:
a G TC(T)
a
&
?I
2
be formed using
{
:
b E TC(T') &T'"b),
, both of which are easily defined. Hence T { z E a : v w G y]?eT <w,z> E ?I , which can
T') S =
A -separation. We have 0
z
E
S
z E
-+
a
w
&
E
.
y B*(z,w,y)
This completes the verification of the case in which
A
is formed by bounded uni-
versal quantification. The case of bounded existential quantification is much easier, and we leave it to the rcader. This completes the verification of
A -sepa-
ration. We turn our attention to the axiom of infinity. Here the only difficult part of the proof is to prove that n
is an integer, then
is an
<w,w>
M -set; we first have to prove that if 2 M 2-set. (Each integer is its own transitive
is an
closure). This is, we have to prove that
n E wwtransitive x2R(
-+
p 6 n
&
q E x
The obvious way to prove this is by induction on
&
)
n ; however, in some of the weak
theories we do not have full induction available, and it seems to require at least A -induction. Fortu:iately,wecan prove it without induction. Fix an integer 1
a R
transitive set
= @
and
Rk+l
then
x
;
=
[
R p E
can be taken to be n
V b C q 3 a t' p
&
q E x
&
e'a
t Rk}
C
Rn
, where
p3b E q
ERk
.
Now the desired property of R can be proved by a bounded induction.
&
n and
M. BEESON
16
We next undertake to verify the sound interpretation of the axiom of exponen-
In our non-extensional set theories, this axiom takes the following
tiation. form:
Note
It
1s not necessary to wut in that
because if E X
is a set of functions from
X
A
to
B,
A -separation can alway be applied to get
Fcn(f)
:
Dom(f)
&
A
=
Rng(f)
&
B).
Now we have to use this axiom to verify that the 2rdinary exponentiation axiom is satisfied in the model. The first ooint to make is that if the model, then so is
BA
{and indeed any
X
A
and
are sets of
B
I*) ) . This is because any
as in
descending G-chain has one of the forms
... t ... E ... &
x1 E x2 x t x 1
2
x 1 & x2
a C {a) 6 t f t X a t [a,bi
H+nce the transltive closure
TC(X)
not g
.
(*)
however the
Fix
X"
B
Eq
a'
E
(iii) F q
&
B
&3a'3 b'
Now
f
a
E
A
&
can he defined in
E
b
t
f %
A
& E
E
B
q
+
b'
Next suppose
&
!>I
9,
b
E f
for some
:
&
E q 8. E q
.
E
f
f. Then A
(This
X
is
later). Suppose
to
c'
a
B.
&
a
2.
a'
shown above,
&
b
%
, %
and
E
b']
A
to
a" , b %
c'
B. Suppose & b c B
aEA
so
b' ;
, a
hence
, 2.
a'
b
'ii
c
E
, c
.
con-
.
W
a'
so
W
restricted
%
can be relativized to
Hence, by (iii) b'
satisfied to be a function from
.
(a)
x*
we can fix a transitive set
3 a 1 3b '
%
Eq
is satisfied to be a function from a
are sets.
we give this
-
b'
B-ext , because
are sets and the quantifiers
a'
TC(B)
be given by axiom
A,B,and q , and then as we have
both
We claim then
:
.
in the model; that is
(*)
E
E
and
(*)
-*
E = {
W
X
q
Let
taining
B. Let
Aab'
Val
(ii)
to
and
A
TC(A)
we choose to verify
satisfies the hypothesis of (i)
X
E
can be defined as the union over all members
of such sequences, using the fact that
Now to verify
f
E
b e ia,biC b f C X
A %
&
f
E
b'
E
;
B.
c' with
Hence
f
is
CONTINUITY IN INTUITIONISTIC SET THEORIES Moreover, \da
f
A\db,c
E
b' where
(iii), ass"
imnlies
%
b
&
f
E
%
a
a'
satisfies v x € A B(
E
(f(x) = q(x)), i.e.
&
q
E
+
b
E. f ,
c".b"
since we have
a
so
, since if
c)
%
;
a"
%
f
&
we get
B I , where
e
lation
%
on
Note that
{[b]
Ib:l
.
B :
b
is the equivalence class of
B}
& '
can be formed in
by exponentiation we can form the set
EH' Take
X*
{H
=
A
A
in
X
(*)
B , f
to
to ?-he set
in the equivalence re-
by the abstraction axiom; hence A
to {[b]
:
b t BT
, let Hff be defined by
H E S
Now if
B-ext
b
of all functions from
S
b%c.
which is to work for
X*
is satisfied to be a function from
may not actually be a function; but it induces a function from { [b] : b
f , then
E
b' ?,b" where < a " , b " X g , and by
b".b'%b"%c,
After these preliminaries we can give the set in the model. Note that if
li
:
.
H t S}
[b]€
t+
H (a)
can be produced using the abstraction axiom of
X*
B-ext. T o see this exDlicitly, we write X* = {{: 3
Note also that
t {[b!:
2
b t B l (
exists, so
TC(X*)
H
E
is
X*
b
&
t z ) ]
:
H
e
.
S]
a "set" of the model.
Now supoose the model satisfies
As above, we can produce to
B
f* =
see so
b'
%
=
q(x))
!: a E A
&
b E B
f
'L
a'
%
f* , note that a (A
.
b"
b'
&
in
&
f
{ D
G B
.
B
€*
.
Hence
f* c X*
.
Define
Technically,
:
{[bl: b E B] , and But
Hlf t X*
claim
, and if
f
f
"u
H
f*
E %
H
fl
.
by H(a)
f
In any case
, in fact
by definition of
f*
X*
f*
E
X* , and
E f , then
for some
a'
a" and b"
E
f
A
with
of
f
f
&
%
A
X*. Let
E
f*. To
b' E B a'
%
A''
, &
is equivalent
. H(a)
=
[f(a)]
, the equivalence class
is not a function (though it is satisfied to be
Actually, looking at the definition of =
is satisfied to be a function from
. We
t f}
f*&
a function) so we really must define H(a)
f
b E B & t g
of
Finally we prove f(a)
%
such that
holds in the model. It remains to show
E f also; thus every member
Hence
toamember
of
f
and t/x(f(x)
;
f*
hence
{b t B : 3 b ' C B
=
t f 8, b' %b}.
it is enough to take H
and
is a function from H
f* E XI.
A
to
have exactly the same members.
M. BEESON
18
S i n c e we have proved
f*
?cn& f *
E
f
?1
,
it follows t h a t
f
XI. Hence t h e conclu-
E
s i o n of t h e e x p o n e n t i a t i o n axiom i s v e r i f i e d . This l e a v e s p a i r i n g , union, c o l l e c t i o n , f o u n d a t i o n ,
i n d u c t i o n , dependent c h o i c e and
t r a n s f i n i t e i n d u c t i o n s t i l l t o check. Because of l i m i t a t i o n s o f s p a c e , we o m i t t h e d e t a i l s of t h e s e v e r i f i c a t i o n s . Thus we t a k e i t a s proved t h a t o u r i n t e r p r e t a t i o n
i s sound f o r
T 1 , T 2 , T 3 and
g
leaves only
Tq
.
We have a l r e a d y done
ZF-
+
RE€
and
2
, which
t o consider.
B, ,
The above model u s e s c o l l e c t i o n q u i t e h e a v i l y ; however, i n t h e c a s e of
we
w + w
s h o u l d be a b l e t o d e s c r i b e t h e model c o n s i s t i n g of s e t s o f rank l e s s t h a n
q u i t e e x p l i c i t l y ( i n c i d e n t a l l y g i v i n g a n o t h e r i n t e r p r e t a t i o n t h a t works f o r Zermel o ) . Define a s e t x
from
x
w
-
+ n
w
S(x)
i f every descending € - c h a i n
(This i s not e x a c t l y t h e usual
steps.
n
is convenient). Write
d e f i n i t i o n , b u t i: than
t o b e o f rank l e s s t h a n
t e r m i n a t e s i n an i n t e g e r i n
i f f o r some
n,x
Again we s h a l l c o n s i d e r t h e " s e t s " o f t h e model t o be p a i r s
A.
is
S(x)
in
x
and
cending t - c h a i n s ) . L e t that
is
Namely, R 1+1
=
TCfx) (we need
R
, where
TC(x)
(x,TC(x))
,
where
Note t h a t
5
w + n
.
We prove
i s a s above.
Q
and
Ro = {
{<x,y> c W 2 : q a E x 3 b c= y
.
t o b e a b l e t o q u a n t i f y o v e r des-
be a f i x e d t r a n s i t i v e s e t of rank
W
is a % e t , t h a t i s , 3 R G(R,W)
21
R.
i s of rank l e s s
+ n .
< a , b > E R . & v b C- y 3 a t x 1
f=
R.1 1
A s w e d i s c u s s e d n e a r t h e b e g i n n i n g o f t h i s p r o o f , t h i s i s a l l we need t o make t h e
x E y
i n t e r o r e t a t i o n work. We i n t e r p r e t
We i n t e r p r e t s e t s , a s mentioned, a s w a i r s
32
E y(z
<x,y>
with
as
t o v e r i f y t h a t t h e i n t e r r j r e t a t i o n i s sound f o r
,
Z
is
x
We l e a v e t h e r e a d e r
Ta and
Tb, where
is
Ta
and
T ; t o f i n i s h t h e p r o o f , we have t o v e r i f y t h a t t h e e x t r a o f
Ta
and
, T2 , T 3
Tb
a r e soundly i n t e r p r e t e d .
or
T4
We have t o t e l l how t h e c o n s t a n t
a
%
T
TI
a s for
just
.
S(x)
z
u s e t h e same i n t e r p r e t a t i o n f o r
,
Tb
is as
x ) , where
.
Next we t u r n t o t h e c a s e of t h e a u x i l i a r y t h e o r i e s
ZF-+ RDC
%
a
. We
(or
b ) i s t o be i n t e r p r e t e d ; of course it
( i n t h e c a s e o f t h e i n t e r p r e t a t i o n t h a t works f o r t h e
i n t h e case of t h e i n t e r p r e t a t i o n t h a t w o r k s f o r
depends o n t h e f a c t t h a t e v e r y member o f
NN
i s d e f i n a b l e fr-om
TC(a)
(remember
a
a
.
ZF-+ RDC
and
T i ) and Z
.
This
One e a s i l y v e r i f i e s t h a t
i s a member o f a s u b s e t of
NN) h a s a t r a n s i t i v e
CONTINUITY IN INTUITIONISTIC SET THEORIES closure definable from
a
note that each integer is its own transitive closure.
;
NN
Next we have to verify that each member of
M -set; that is, if
closure) an
x
2
i
G
q G. x
&
19
p
2r
is actually (with its transitive
is any transitive set, we can form
ql. (More precisely, p and q should be paired with
their transitive closure). This is easy, once we know that each integer is an M -set, which we have alr->?dydiscussed. Finally, we have to verify that the axiom 2 a & X and the axioms b(n) = where m = f(n) , for some fixed f in X , are
m
soundly interpreted. Recall from
52
universal condition on the values of
that membership in
X
is given by a purely
a. Below we give a proof that the interpre-
tation preserves arithmetic sentences; the same proof apDlies to show that it preserves the axioms in question
.
We have now viven the interpretation A*
for each of the set theories discussed
in this paper, and proven the soundness of the interpretation. Next we prove that
Tk A
++
xity of
, for bo-formulae A
A*
A
;
.
to be quite precise, A
lae with free variables, say
x
-
This is established by induction on the compleA*
is only for closed formulae
T1,T ,T ,T
(Here we are considering the intemretation that works for is
S(<x,y>)
&
S()
(<x,y>
-f
;
for formu-
).
The basis
, we should say
E
+
2
3
4
x E a), which can be established using
the foundation axiom; extensionality is used here. The induction step proceeds
M -sets determine
smoothly, using the fact that members of
2
details to the reader. Note that we cannot seem to get A
but only for
A -formulae. (For ZF-+ RDC
A*
M -sets; we leave the 2 A for all formulae
t*
we can get it for all formulae, be-
cause the interpretation does not require that t.ransitive closures be t;.cked on.) The same argument works for the internretation of than
w
+
w ;
the members of sets of rank less than than
ili
+
w
.
using sets of rank less
here the induction step over a bounded quantifier uses the fact that
+
w
w
are themselves sets of rank less
A similar induction works for the interpretation used for Zermelo
set theory. This completes the proof of
(i) of the theorem.
We next consider the question of which sentences are preserved by the interpretations; it is for these sentences that we get a conservative extension result. First arithmetic sentences are preserved. This is shown by induction on the complexity of an arithmetic formula; actually, as above we have to Drove <m,X> € -t
0
&
n e w
G
<m,x>
(A (<m,x>) tf A(n))
%
(this time without extensionality)
M. BEESON
20
Note that every integer has a transitive closure, namely itself; in fact, every :I -set.
integer is (part of) an Here
A
2
is a formula of set theory translating a formula of arithmetic, which we
also call
A ; the induction is on the comnlexity of the arithmetic formula. The
details are easy but cumbersome; we leave them to the reader. Next note that every f
in
NN
has a transitive closure; this allows us to extend the above indiction
to formulae involving quantifiers over such objects. Actually, we must verify that each such
f
is (part of)
each transitive set
an
M -set; to do this, we must be able to form for
x , the set
2
!: a E TC(f)
pi
b 6 x
a
&
1
b).
This boils
down to the faut that we can form the corresponding set with an integer place of
f
, in other words that each inteqer is an
above. This completes the oroof of part
m
in
M -set, a fact alluded to 2
(ii) of the theorem. Part
(iii) of the theo-
rem, which sdys that the first two Darts are Drovable in arithmetic, is proved by examining the above proof, and noticing that only arithmetic is needed. In other words, we proved by induction on (Godel numbers of) oroofs in pretation of the last formula of the oroof is provable in
T
T-ext
that the inter-
.
This completes
the proof of theorem 3.1.
54. ?.eaiizability for Set Theories In this section, we
give a variant of
q-realizability adapted to set theories.
This type of realizability has been used before for arithmetic and the theory of species to obtain explicit definability theorems [Tr].
Here we extend this program
to set theories. The extension to set theories without extensionality is relatively straightforward, but there seems to be no simple way to handle set theories with extensionality. (Myhill gave [ M l I
a comnlicated realizability for his extensional
set theory; but it cannot be made to work for our puruoses.) For this reason, even if we want to obtain eerived rules only for extensional theories, we have to consider the non-extensional ones and use the results of the previous secti.,~. The plan of the present is to give the realizability interoretation we need and prove its soundness both for t.he basic set theories Tb-ext la
A
.
T-ext
and for
Ta-ext
Our definition of realizability will Droceed by associating to each formu-
another formula
e r A
theorems of the form, if
T k A
("e realizes
A"). We will then Drove soundness
, then for some e ,
T k
e r
a
.
e
Here
is an
inteqer; a l l our realizing objects are integers, not arbitrary sets. (We use e , n , m
and
*?ti'.to indicate variables whose ranqe is restricted to
(LI
.)
CONTINUITY IN INTUITIONISTIC SET THEORIES W e b e g i n by a s s i g n i n g t o e a c h set v a r i a b l e d i s c u s s e d i n [Bl].
t h e free variables of
x
another variable
e r A
The f r e e v a r i a b l e s o f
21
are
e,x
x*
, i n t h e manner
x* , where
and
are
x
.
A
(Our c o n v e n t i o n i s t h a t a s i n g l e l e t t e r c a n d e n o t e a f i n i t e l i s t o f v a r i a b l e s . ) We e r A
now g i v e t h e c l a u s e s d e f i n i n g
, f o r t h e n o t i o n o f r e a l i z a b i l i t y t h a t works
f o r theories without extensionality:
is
x* = Y*
is
<e,x,x*> f y *
e r(XRB)
is
(elo r A & (e ) r B 1
e r(AVB)
is
e r x e r x
= y
EY
( ( e ) o= 0
e r(A
+
a r A
is
B)
+
( ( o ) 9~ 0
fu
da(a
A
r A
+
&
e r v x A
is
qx.x* e r A
e r’3x A
1s
=J x , x *
(A R e
In o r d e r t o c o m p l e t e t h e d c f i n i t i o n . we h a v e t o d e f i n e volvinq t p r m s
t
A
(ell r A
& A)
( e ) ] r B & B)
-f
i e l ( a ) r B ; o r more x e c i s e l y -f
3 1 - ( T ( e , a , k ) & U(k) r B )
r A)
e r A
f o r atomic
in-
A
o f t h e n o n - e x t e n s i o n a l s e t t h e o r i e s . T h i s c a n bp d o n e b y t h e same
c l a u s e s as a b o v e , a s s o o n as we a s s o c i a t e t o e a c h term {<e,x,x*>: e r x
which i n t u i t i v e l y d e f i n e s
E
t).
t
term
another
t*
These t e r m s
t*
w i l l be given
in t h e c o u r s e of t h e s o u n d n e s s p r o o f b e l o w ; t h e y c o u l d b e l i s t e d h e r e , b u t would b e u n i r , t e l l i c i b l c F o r i n s t a n c e we d e f i n e ple,
if
t = {y P a : B(y) 1, then
hi*=
fCn,‘n,n>>:n€hi}
t* = { ‘ e , y , y * > :
’
.
A s a n o t h e r rciim-
(e)o,y,y*:,Ca* R
(e)l r B(y)}.
We g i v e t h i s e x a m p l e i n o r d e r t o c l a r i f y t h e f o l l o w i n g p o i n t : T h e r e i s n o v i c i o u s circle in the fact that
e r x E t
t * , which must
a u D e a r s i n t h e d e f i n i t i o n of
( e ) l r B(y)
precede t h e d e f i n i t i o n of
; f o r , a s discussed above, t h e d e f i n i t i o n s
- f o r m u l a e a n d t e r m s u r o c e e d by s i m u l t a n e o u s i n d u c t i o n , so t h a t
of
t a i n s only less-complicated
t e r m s than
t
.
B(y)
c o u l d a s s i g n a measure o f c o m p l e x i t y t o b o t h t e r m s and f o r m u l a e , say
C ( t ) , giving
a t o m i c f o r m u l a e w i t h o u t compound t e r m s c o m n l e x i t y z e r o , a n d a t o m i c f o r m u l a e or
tc
s
t h e comulexity
t=s
m a x ( C ( t ) , C ( s l ) ; l e t u r o p o s i t i o n a l c o n n e c t i v e s and q u a n t i -
f i e r s i n c r e a s e t h e c o m p l e x i t y by complexity
con-
To make t h i s c o m p l e t e l y p r e c i s e , we
1
, a n d l e t s e p a r a t i o n terms
‘xea:
B(xl\
have
l + m a x ( C ( a ) , C ( B ) ) ; s i m i l a r l y f o r u n i o n , p a i r , a n d c h o i c e t e r m s . Then o u r
d e f i n i t i o n of
e r A
p r o c e e d s b y i n d u c t i o n on t h e c o m p l e x i t y o f
A
.
22
M. BEESON
Remarks : (1 )
If we were doing 1945-realizability (see LTrl
),
we would not need the extra
variables with stars, but could avoid them by defining e r x C a to be <e,x>Ey. Trying to do something similar for q-realizability is more trouble than it is wort-h. ( 2 ) One cannot define e r x E y to be <e,x>€y*
, though this may seem tempting.
In this case, all the axioms except dependent choice will be realized (including extensionality ), but one will not be able to get anything realized to be a function.
Consider Fcn(f ) which says
In order to get y=w realized, there will have to be some relation between y* and w* , which we cannot get from having the antecedent realized, with this definition of realizability. This is somewhat interesting because it points up the absolute necessity of the axiom of choice in proving the existence of
functions. ( 3 ) The motivation behind the definition of e r x E y is that y* is thought of
as the set of <e,x> such that e proves, or verifies, or realizes, that xcy
.
Remember that Kleene’s original motivation for realizability was that realizing numbers were thought of as like proofs.
It is no wonder that extensionality
gives trouble, because one can have x and y extenslonally equal, without any relationship at all between x * and y’
;
yet if v a ( x e a
++
y t a ) is to be
realized, there has to be some relationship between x* and y*. Theorem 4.1. (soundness of q-realizability ). Let T be any of the set theories considered in this paper, without extensionality.
Then for the notion of realizability just given, if T
some number e, we have ‘I/-
e
r A.
I-
A, then for
CONTINUITY IN INTUITIONISTIC SET THEORIES Proof:
23
AS usual for realizability soundness theorems, we proceed by induction on
the length of the proof of
, proving that the universal closures of all state-
A
ments in the proof are realized. Thus we have to verify that the universal closures of all statements in the uroof are realized. Thus we have to verify that the universal closures of all the axioms are realized, and the rules of inference preserve realizability. The logical axioms and rules of inference are handled in the usual
.
way (see [Tr] )
We have to check the non-logical axioms. r'3 xvy(y E x
(Pairing).
B.
Take
x*
=
{<e,y,y*>
e t
:
This set can be formed in
-f
(y=a v y=b)
y E {a,bi
0 &
y* E {a*,b"l
&
without extensionality. Take
@
&
e r(y=a V y=b)l.
x
=
{a,b}.
(Infinity).
C.
0 E
0
c v y c- w (y
Take Then
w*
=
u {y? e
u
\ d z ( 0 G z & b y c z(y
.
Then, "irst of all, ducr
w
5
[yj C z )
0 E
{PI
c
z(y
u {yl
E 2 ) -* w
c_
2)
z &
is realized and true.
w)
+
w
5
is realized and true. Suppose
z)
tfy E z(y U {y1 E z )
w C
is true and realized, z
realized, we intro-
by the recursion theorem to satisfy the equation a
{PI
(0)
{PI
(Y + 1)
Then we prove by induction that
.
&vy
z is true; in order to get
a recursive function
<{p}(y),y,y*> E z*
z
{
z* are given, so that
say by
& V Z ( 0 c-
0 E w & b y E w (y U {y} C
We have to show that
z and
W)
=
=
{b} ({PI (y))
{ u } (y) r y E
z
;
that is,
(What we are proving by induction has a free variable
y*.)
Note that only the restricted induction axiom is needed. D. (Union).
take
t*
u
y 6
to be
zc a
t*
03z(y
v y(yCt(a)
t--)
.
If
t
is the term
z ze a
t
z & z E a)];
this can be formed in
, we
1.
be the function symbol such that the following is an
(y E a
&
B(y1). To get this realized, we define a function
by t.*(a)
t*
G z & z f2 a)
{<e,y,y*> : e r q z ( y
E. (Separation). Let axiom:
z
=
{<e,y,y*>
can be proved to exist in
:
<(e)o,y,y*> E a*
5-ext, since
u r B
is a
&
(e)l r ~ ( y ) } A -formula if
B
is.
24
M. BEESON
Then
t*(a)
that
A -separation suffices to interpret
{<e,y,y*>: e r (y E a
=
B(y))}; this finishes the verification. Note
&
A -separation, and full separation for
full separation. F. (strong collection). a N x G a3y A Suppose
a
Let
QX,,*=
vx
a ijx*
G
and
+ Iz (‘dx E a 3 y c
A
z
&
Vy 6 z lx
a A) ) .
E
are given, and suppose p r v x C- a 3 y A, and \dx C- a 3 y A.
a*
{c: c r x E a) = {c:
Rng(a*)tlc E Ox,x+3 Y(A
Rng
get the existence of some
{p}(c) r A)
&
\dx G a v x * L Rng
Rng(a*)
v c C- Qx,,+I
Also, applying collection to V x c- a 3 y A, we get some
. Take
A & v y E z1 3 x C- a A
V x C - a 3 y E z1 v x g a3 y C z
A
&
3x
Vy E z
applying collection, we
;
such that
z
z
u
z
=
. ..
y C z
such that
z1
Then
zl.
t a A, i.e. the conclusion of axiom ? is true.
Note that this works because we have strong collection, not just plain collection;
...
the extra conclusion indicated by
in the choice of
is needed. we need to
z
show that this conclusion is not only true but realized. Define z* =
t<<e,c,x,x”>,y>: c r x E a
&
x E a
First we show that v x E a 1 y t z A on
p 1 . SuDpose
(pl(c) r A(x,y)
c r x E a
. Hence
and
1’ & z
&
x*
&
C Rng
Rng(a*)
e r A(x,y))
&
is realized (by a number depending recursively x t a
.
Then for some
<<{P)(c) ,c,x,x*>,y>t z*
y
, so v x
in
zo,A(x,y) and
G a3y C z A
is
realized. Similarly, we have to show v y € Z then
b
xE a
.
has the form Hence \dy E
~
E Xa A
<<e,c,x,x*>,y> where
23 x e
a A
is realized. Suppose
b r y E z
c r x E a
e rA(x,y) and
;
and
is realized, by a simple combination of unpairing
functions. G. (foundation). va,b(a G b
Suppose
z,z*
&
are given, and
9 r V Y t x (Y & z
+
b E x
+
a t x) & v y E x(y
p rva,b(a 6 b
&
b
E
x
-*
y C z ) , and the formulae realized by
Introduce a recursive function
if)
5z
-*
y t z)
-*
xC
z
a E x) , and p
and
by the recursion theorem
q
so
are true. that
.
CONTINUITY IN INTUITIONISTIC SET THEORIES (Here A u h
is an index of
,
u h
so
25
{ A u h(u,v)} ( u ) = h(u,v) ; this is an
old and useful notation of Kleene.) We claim
f r x c
z
, the conclusion of the
foundation axiom. (Which will finish the proof, since the conclusion of the axiom is true, because we have assumed the hypothesis.) We must show
,
y C' x
that is, whenever
and
this by transfinite induction on
f rt(y(y(.
, we have {f}(e) r y 6
e r yE x
show
.
r y G z
{f}(e)
p
hypothesis on e)
, (e r a
Rng (y*)
.
.
If} (e) r a E z
t y
Suppose
a E y
and
e r y E x
we must
;
, we have {f}((p} ()) r a
u r af y
and
23;
, then {p)() r a E x by our {PI () for
u r a € y
Note that if
That is, A u{f}(ip}()) f
+
+ ye
Applying our induction hypothesis (substituting
, we see that if a
definition of
G x)
x
We prove
(then later show how to get by with only
the foundation axiom). Our induction hypothesis is that for all a* E Rny
.
z
r y E z . Now, applying the hypothesis on
.
E z
q, and the
, we reach the desired conclusion, that {f}(e) r y E
.
z
This
completes our proof by transfinite induction; now we have to show how to get by with only foundation. The foundation axiom amounts to proof by transfinite induction, where what is proved is membership in some set. Here the set in question is
(Recasting the induction on the pair is
left to the reader.) This completes the verification of foundation.
H. (Exponentiation). q x v y ( y C x
Fcn(y)
+ i
&
Dom(y1 = a
.
Rng(y) C_ b)
&
(Eventhe strong version of the exponentiation axiom is realized.) Suppose
, and b*
a,b,a*
are given; we have to produce
the problem is to produce
x*
that if
a
Fcn(y)
&
Dom(y)
=
.
&
Rng(y) & b
"a priori bound" on the complexity of {<e,y,a*> E w Since Fcn(y)
x
x
0 : e r (Fcn(y1
x
&
{<e,y,y*> w
=
&
Dom(y)
=
a
&
X
x
X
&
.
x*
is realized, then
y*.)
Let
x
be
y* E (I
.
ba
Dom(y) = a
&
;
such
(An
Then we would like to take
x*
to be
Rng(y) & b) . ]
Dom(y)
=
a
&
.
However, we can instead take
Q: e r (Vs g aVt,t' E b(<s,t> C y
Rng(y) C _ b)} (here =
From a member of this Fcn(y)
and
involves an unbounded quantifier, it is not immediate that this
set can be formed using the axioms of x*
x
Suppose for the moment that we had a set Q
x*
&
<s,t'> E y
+
t =t'
is extensional equality).
, we can recursively pass to a realizer of
Rng(y) C _ b
and vice versa, so that the exponentiation axiom
will be realized. The difficult part, namely producing the set Q
, is still ahead
of us. At first, this seems to be a serious problem. A typical element of will be
)
<e,<s,t>,<s*,t*>> with
t=y(s) : but
t*
is not uniquely determined
M. BEESON
26
(even extensionally) so we cannot seem to use the exaonentiation axiom to get a set in which gument
y*
must lie, and power set is not available. However, the following ar-
gets us around the difficulty. First, set 2
1s realized. Thus
{<x*,y*> G RngRng(b*)
we can form in
the equivalence classes
Set
B
x* [t*]
%
w*
<x,z>€ y
.
t*
x*
5
c- b(zc
under the relation
are realized, and Fcn(y)
B ~r
X-ZG
.
y)
Now,
and the
is realized then
is not uniquely determined, the equivalence class
y*
is such that
Fcn(y)
&
Dom(y) = a
Rng(y) C b
&
is realized, then
has the form
has the form
f
from
a
{{q}({p}(e)):
x
RngRng(a*)
Bo
to
<.e,s,s*>E a*}
and some set
, for some q
the power set axiom to form the set of all such q
iffvz
can be formed in
y*]
Lx*]
y*
(using the abstraction axiom). Next, observe
<x,w> E y
Hence, although
for some function P
RngRng(b*)
and
:
5
is uniquely determined (extensionally). Suppose
Then if y*
[-X*-I: X* E
=
that if
B
x*
P
;
P C_ w
, where
w. We do not need
in
quantification over integers
is enough. Using exponentiation, we can form the set of all such functions f;
0
thus the set
of all such
y*
can be formed; this completes the verification
of the exponentiation axiom. I. (Bounded dependent choice). \dx 6 a 3 y t. a P + v x G a l f (Icn(f) & m g ( f )C-a &
Suppose
Dom(f)
p rVx
=
w
&
a3 y & a
f ( 0 ) = x &\dn E w P(f(n),f(n+l))), with P
and
e
h
.
by
by
Now, we will prove the existence of a (set-theoretic) function {e}(n) r (f(n) E a
A.
s r x C- a , and these formulae are true, as
well as realized. Define a recursive function
and introduce
P
&
f
such
that
P(f(n),f(n+l)). First note that the Drlnciple of countable
27
C O N T I N U I T Y I N I N T U I T I O N I S T I C S E T THEORIES
independent choice
\d(n
w
3y
Then prove by induction on (*)
t. a
This can be done in
can be derived from dependent choice
that
n
,...,y n E
\dnE:3yo
.. .
3
a
yo* ,...,y n * & R n g R n g ( a * ) v j
since the formula being proved by
-
A -formula. Apply countable choice to get two functions
-
f(n) f
=
and
h
-
and
h(n) =
themselves can be defined. Let
Then we will show that, with
f
and
f*
f*
induction i s
f
as in
I n
and (*).
h
a
such that
Then the functions
be defined by
substituted in, the conclusion of the
dependent choices axiom is true and realized: that is,
is true ind realized. First we show \ J n 6 w P(f(n),f(n+l)) is true and realized. The truth follows from the last clause in the formula we have to use the definition contains Now,
w*
P(f(n),f(n+l))
i s a pair
then
(s)
=
is true and realized; Fcn(f) -t
y
=
w
.
Suppose p
n
nC w
.
y = w = f(n)
w
= y
and
w*
Dom(f)
= ui
is realized since if
y*
=
w*
, we have y
=
h(n)
w
=
;
and
'j,
c.
f*
;
here equality i s extensional. But since
realized. Hence
s r n E w
is
realizes the left side of this; then
hence
= y*
For the realizability,
is actually an abbreviation for
Fcn(f)
.
f
, which tells us that a realizer of
n , specifically, if s r n E o
Next we show p
used to define
(*)
of
, then
f*, so 3 y
Fcn(f)
n = (s)
a y
=
f(n)
i s realized.
and
M. BEESON
28
f(0) =
is realized. Finally,
x
is realized, because
h(0)
=
x*
and
f(0)
=
x.
This completes the verification of the axiom of bounded dependent choices.
J. (induction). The verification of this axiom is standard.
K. (relativized dependent choice RDC). Like bounded dependent choice, except that we use full InductLon and separation instead of bounded induction and separation. L. (transfinite induction). Like foundation, except that transfinite induction must be used to make the verification, instead of foundation. M.
(full separation). Like
N.
( p o w e r set). 3 x
t/ y(y
x*
=
Y C. a Suppose
y
and
y*
A -separation.
C_ a
y 6 x ) . Take
-f
{<e,y,y*>
e r Y
&
B w
y g a
p(a)
g(w
x
and
Rng(a*)
:
is realized, say by
is a subset of
y*
x
=
a)
are given and
Then we have tocheck that
@(a)
x
x
Since
y c a
E Rnq(a*). This completes the verification of power set.
0.
(abstraction). For
A
,
and
y c a.
cp,z,z*> E y*
, we have <po,z,z*> E a*; hence
po
is realized, for some
e
w xRng(a*). Suppose
A -formula,
a
~ X ~ z \ d w ( ~ € z f - * ~ y ( y C x & \ d w+-+A(u,y) u ( u C &uEx))).
Take
z = {{u t
x: A(u,y)}
the existence of
z*
:
<e,w,w*>E z* *3y,y*(e
To prove that iff w
z*
P
%
=
E
x
1,
formed by abstraction. We want to prove
r (y t x & t F u ( u E w
* A(U,Y)
u E XI)).
exists, we again need to use the equivalence relation, w*
extensionally
Note that
y
such that
=
v
v*
is realized iff
is given by a
A -formula. Now, by abstraction, we can form the set
{ i < q , u , u * > : q r(A(u,y)
&
u E x)} ;
CONTlNUITY I N INTUITIONISTIC SET THEORIES
Now define
z*
{<e,w,w*>;3v* c P(v*
=
In order to show that the
%
w*
v*
in
then
P
W*
2,
, v* v*
%
.
.
w*
Let
v*
&
so defined is the
z*
prove that if 3y,y*(e r ( y E x & v u ( uE w =
I E w
we are seeking, we have to
z*
* A(u,y)
E
, then for some
u € x)))
Rng(x*)
x
<(e)o,y,y*> E x* , we have
Since
29
:
q r(A(u,y)
v* E P
.
u E x)l
&
i
This completes the
verification of abstraction, and with it, the proof of the soundness theorem 4.1. We turn now to the auxiliary theories
Ta
and
Tb, and discuss the notion of
realizability appropriate for them. These theories, as defined in a particular definable metric space
X , which is a subset
of
in
§? , the new constant
5 or
"standard form" as discussed in
an element of
X , that is, an element of
N N
in
with a metric
9
stands for
satisfying an additional condition.
Instead of recursive functions, we use functions case may be) to realize the theory
, depend on
$2
NN
5 (or & , as the
recursive in
Ta (or Tb). The theory of functions recursive
& can be formalized in Tb , and the verification that all the set-theoretical b {el-
axioms are realized proceeds exactly as in theorem 4.1, using
, place of in
{e) throughout. This leads to Theorem 4.2
9,
If q-realizability is defined using functions recursive in pretation
is sound for
Tb
;
similarly for
.
Ta
(Here T
then the inter-
is any non-extensional
set theory considered in this paper.) Proof: Actually, we first have to give a complete description of the interpretation. We have to explain what to be
<e,b,&*>t x*
e r and
bE x and e e x E '1 are. We shall take e r & E x e r x E & to be <e.x,x*>E b* . Here g* is a particu-
lar set (more precisely, a particular term of o u r non-extensional set theory to be explicit, b*
=
{
=
ml. Thus (e r
b(n)
=
Tbl;
b ( n ) = m.
m)-
All the logical and set-theoretical axioms can now be verified exactly as before. It remains only to check the extra axioms involving 5 the axiom Fcn(5)
&
5 C X
vnE
w
. This has the form 3 m C w(
or
n,m C w(p(an,am) < l'/m
may be taken to be some recursive function, a5 discussed in these three clauses is realized essentially by sionally equal to
IeF
9.
First, consider
+ l/n)
;
here
P
5 2 . The second of
5 itself; of course 5 is exten-
for a certain n u d e r e.
M. BEESON
30 Next, consider the axioms
g(n)=mfor b(n)=m.
(Remember that Tb is based on a par-
ticular function b, while Ta is not based on any particular a.)
- -
(e r b(n)=m)
-+
-
We have
-
b(n)=m, so that these axioms also are realized in Tb.
This com-
pletes the proof of Theorem 4.2. 55.
Explicit Definability
In this section we consider the old metamathematical property, if T
I-
then for some n, T property.
P(n).
1-
3 n Ew P(n),
We call this the "numerical explicit definability"
Our goal is to derive various formalized versions of this property for
set theories T and the auxiliary theories Ta and Tb, which will suffice to get the desired continuity rules.
A few general remarks are in order.
The numerical ex-
plicit definability property should be compared and contrasted with the set explicit definabillty property, if t T
I-
P(2).
1-3~ P(x)
then for some explicitly defined
2,
(One might give different meanings to the words "explicitly defined"
here; but for example, any set given by a term of our non-extensional set theories is explicitly defined.) These explicit definability properties are already known for certain intuitionistic set theories (See [Fr 31 theories have replacement instead of collection.
, [Ml] , and [M3]
.)
These
(However, the double-negation
interpretation has not been made to work for ZF with replacement, but has been made to work for ZF with collection; see [Frl]
.)
Friedman and Myhill use a vari-
ant of Kleene's "slash", which becomes quite complicated because extensionality is dealt with directly.
This realizability is not enough for the needs of the
present paper, because it is not recursive, and it is not easily formalized. Numerical explicit definability results for the auxiliary theories Ta and Tb provide information generalizing what is usually known as "Church's rule", which says that if V n l m A(n,m) is provable, then for some e, v n A(n,{e)(n)) vable.
is pro-
If we take the complete separable metric space X to be the integers N,
then to say V n 3 m A(n,m) say that Ta proves In the case X=NN
3m
is provable (the hypothesis of Church's rule) is
or X=ZN
to
(the hypothesis of explicit definability for Ta)
A(a,m)
.
(not to mention the reals or certain function spaces)
we get other interesting information. The exact form of these results will be given below.
We begin with the most straightforward explicit definability
theorem. Theorem 5.1
Let T be one of the non-extensional set theories considered in this
paper.
If T l-3xEwP(x)
Proof:
Suppose T
3 x Ew
then T
tion of
e1
i.
. Then, by the soundness of q-realizability for I- e r 3 x G . w P(x). That is, T proves
P(x)
T, thero is some e such that T
3x,x*(eo r x C W &
P f n ) for some numeral
r P(x)
&
u*,T proves ( s r x E w
P(x)), where e -+
x=(s) ) .
=
<e ,el>. According to the defini-
Hence T proves P((e
)
3 0
1.
NOW since
31
CONTINUITY IN INTUITIONISTIC SET THEORIES
-
-
T contains arithmetic, T proves (e) = n , where n=(e)
.
-
Hence T proves P(n), which
completes the proof. Now consider explicit definability for extensional theories. Our methods yield numerical explicit definability, not for all formulae P, but only for P of the form x E Q , where Q is a specific definable set.
We take "definable set" to mean
"set given by one of the terms of the non-extensional set theory T". What we would ideally want is a larger system of terms, adequate to prove the set explicit definability theorem. In trying to get such a system of terms, there is a problem in that the choice and collection axioms assert the existence of a set, without there being any obvious definable one.
This is why the set explicit definablity
property is known only for theories with replacement, and not for theories with collection. Although this is an interesting phenomenon, we regard it as a side issue, since our focus here is on continuity rules.
We therefore restrict our
attention to sets defined by terms. Actually, we could include exponentiation terms as well; if this is done, the definable sets seem to encompass most sets needed for mathematical practice. Lemma 5.1
Let T be one of the extensional set theories considered in this paper.
Let A* be the interpretation of A in the non-extensional set theory T-ext, given in Theorem 3.1.
Let Q be a definable set in T.
Then T
I-
(xCQ)*++ XIE Q.
Proof: First we must explain precisely what is meant by (xbQ)*.
Here Q
is a
term, which belongs to T-ext, but not to T; so xtQ must be interpreted as the formula of T obtained by writing out the definitions of the terms composing Q. Secondly, if A has a free variable x, then A* has two free variables, x and y, where y is supposed to "witness" that x is a set. (Technically x is interpreted as the pair <x,y>.) Thus &
( x G Q +3y(xBQ)*(xry)).
of the term Q.
(xeQ)*
t--f
( x ~ Q really ) means ((xtQ)*(x,y)
x€Q)
Now, the proof proceeds by induction on the complexity
For instance, if Q is {xed: P(x)),
we have (xEQ)*(x,y)
+
S(<x,y>)
t+
(xed)*
&
&
defining the sets of the model of Section 3 . have S(<x,y>) -t (P(x,y)t--fP*(x,y) By induction hypothesis (xEa)*
++
;
so
where a is a term and P is A
According to Theorem 3.1(i), we
(xEQ)*(x,y)-
S(<x,y>)
&
(xEa)*&P(x,y).
x E a , since we can prove by induction on terms
that the transitive closure of a definable set is definable, so that S() for some term b. f-f
S(<x,y>)
Thus (xEQ)*(x,y)++ &
xgQ.
S(<x,y>)
&
xt'a
&
P(x,y), i.e.
But x t Q implies 3 y S(<x,y>), since TC(x) can be defined if
x is known to belong to some transive set, and as we have just mentioned, the transitive closure of Q is definable. (xEQ)*(x,y)
+
xCQ.
in the induction on Q.
,
P*(x,y), where S is the formula
Thus x E Q
+3y(x& Q)*(x,y)
and
For reasons of space limitation we omit the other cases
M. BEESON
32 Theorem 5.2
Let T be one of the set theories discussed in this paper, including Suppose T I-3xtw(xbQ), where Q is a definable set in T.
extensionality. for some numeral
n,
T
I- ic Q.
Proof: Suppose TI- g x C w ( x 6 Q ) . Hence T-ext
1-3.E
Then, by Theorem 3.1, T-ext 1-(3x~w(x€Q))*.
w ( x CQ)*, since x e w
is equivalent to its *-interpretation, by
Theorem 3.1 (ii). By Theorem 5.1, T-ext /-(nEQ)*, for some numeral Lemma 5.1, T
I-:
Then
n.
Hence, by
EQ. This completes the proof,
Next we turn to explicit definability results for the auxiliary theories Ta and Tb. Theorem 5.3
Then
(i) Suppose Tb /-3xEwP(x), where T is without extensionality.
Tb /-P(n), for some numeral
n.
If T has extensionality, then the same result holds
for P of the form x € Q , where Q is a definable set in T. (ii) Suppose Ta 1 - 3 x t w P(x), where T is without extensionality. Then for some numeral
e,
Ta
I- {el"(@)E w
&
P({e}"(o)).
If T has extensionality, the same result
holds for P of the form x C Q , where Q is a definable set in T. Proof: .
Exactly like Theorems 5.1 and 5.2, appealing to the realizability used in
Theorem 4.2 instead of 4.1. For (i), we also have to observe that in Tb, if - b { , l b ( 0 ) = n is provable. This {e+(O) E w is provable, then for some numeral
n,
is proved just like the corresponding result for T; it consists in observing that the axioms of Tb suffice to formalize the computations of a Turing machine; when a value b(n) is called for in the course of a computation,one of the axioms of Tb is there to formalize the step in which the "oracle" answers.
Of course, this
cannot be carried out in Ta, which is why the theorem takes the form it does. This completes the proof of the theorem. Formalized Explicit _ _ _ Definability ~ We have to discuss the formalization of the preceding results on explicit definability. [Bl])
They cannot be formalized as they stand (see the general discussion in
, but instead we have to show that there is a sequence of subsystems T of
each set theory T,such that the explicit definability theorems for T
can be
proved in T, for each fixed integer n . This may not be possible for systems T which have only restricted induction. Here we carry it o u t for the other theories considered in this paper, which have full induction. The complexity of a formula of set theory is an integer defined by induction
SO
that prime formulae have complexity zero, and the complexity increases by one at each logical connective and quantifier.
We can, for each fixed n, introduce a
truth-definition Tr (a formula of two free variables; one of which is a number n
variable, i.e.
CONTINUITY IN INTUITIONISTIC SET THEORIES
33
technically precise, we have to worry about the fact that A can have more than one free variable (x can be a list of variables), and code these variables into the single variable on the left, so that we should actually say T
I-
.
Trn('A) .~)++A((X)~,..(x),).
We neglect this distinction where it is safe to do so.
The construction of Tr
is standard; Tr ('A',x) is a disjunction, according to the finitely many possible forms of A.
If T is one of our set theories, let T be T with all proofs restricted to contain are those axioms of T which are formulae of complexity In; and the axioms of T n of complexity and occur among the first n axioms of T in some natural enumera-
zn
tion.
Thus T has finitely many axioms.
Note that T
is not a formal system in
the usual sense, since a formula of complexity In might be provable from axioms of complexity In, but only through intermediate steps of greater complexity. Nevertheless, T
is useful for our purposes (chiefly because it saves us from
having to use formalized cut-elimination theorems, which in some cases are not even proved yet).
By the reflection principle for S , we mean
, for all formulae A.
PK ('A') + A
S
Lemma 5.2
Let T be one of the set theories considered in this paper with full
-
+
induction (i.e. T2, T3, T4, Z, ZF ciple for T Proof: We on j.
RDC).
Then T proves the reflection prin-
, for each fixed n.
have Tr ('A') ++ A; and we prove
Prfn(j,'P')
+
Tr ('A') by induction
It seems that bounded induction will not suffice.
By the 1-consistency of a theory S, (terminology due tc Kreisel and Levy) we mean, for A recursive with one free variable, P r S ( 3 n E w A(n)) For such A, we have A(n)
+
3 n € w Prs('A(i)').
-
Prs('A(n)'), if S contains a modicum of arithmetic,
-+
and indeed this fact itself is provable in any theory which proves that S contains a modicum of arithmetic.
Hence 1-consistency follows from the reflection princi-
ple for S, for all the S we have reason to consider. Next we discuss the formalization of the soundness theorems for q-realizabillty. Let us first discuss what goes wrong with a straightforward attempt to formalize the theorem, Pr('A') + 3 e Pr('e r
A').
One would try to prove this by induction
on the length of the proof of A; the induction step involves proving ~ r ( l ir (A + B)')
&
~ r ( l br A') + 3 m Pr('m r B').
M. BEESON
34 Now from Pr('a r (A -f
6)') &
Pr('b r A'), we easily get Pr(']n(Tabn
&
U(n) r 6 ) ' ) .
To pass from that to the desired conclusion of the induction step requires the 1-consistency of the theory. However, this is the only obstacle to the straightforward formalization of the proof.
In other words, if we have the I-consistency
of a theory S, and we can prove the axioms of S are realized, then we can prove the soundness theorem for
S,
using nothing more complicated than bounded induction.
Thus we obtain (Formalized realizability). Let T be one of the non-extensional set
Lemma 5.3
theories discussed in this paper, and let T be as in Lemma 5.2. fixed n, there is some n* T
where Pr Remark:
1-
Then for each
such that
3e
(Prn('A*)-t
Prn,(le
is the provability predicate of T
r A')),
.
By writing Pr('e r A'), to be perfectly explicit, we mean
Pr (Sub(Num(e),'x r A'), where Num is a primitive recursive function producing from e a G d e l number of the numeral
e,
and Sub is a function producing a a d e l
number of P(t) from Godel numbers of a term t and a formula P.
__ Proof: As sketched above, we go by induction on the length of the proof of A, using 1-consistency in the induction step for modus ponens; it is provided by Lemma 5.2.
The use of n* on the right in place of n is necessary because the
Complexity of e r A is usually greater than that of A. that for each axiom A of T
,
T proves g e Prn,(e
enough, this will be a true L'T arithmetic, hence in T.
r A).
sentence, by Theorem 4.1;
We also have to check If n*
is chosen large
therefore provable in
(Here we use that T has only finitely many axioms.)
This completes the proof of the lemma. Lemma 5.4
(Formalized explicit definability).
sional) set theories discussed in this paper. For each fixed n, there is an n* (i) T
I-
(Prn('A') + 3 e Prn,('e
such that
Let T be one of the (non-extenLet Ta and Tb be as in Lemma 5.2 n n
r A ' ) ) , where Pr
is the provability predicate
of T . b (ii) TI- (Pr ('A') -t 3 e Prn,('{e)a(0) predicate of Ta
.
r A')), where Pr
is the provability
35
CONTINUITY IN INTUITIONISTIC SET THEORIES
Proof: Like Lemma 5.3, appealing to Theorem 4.2 instead of Theorem
4.1.
Theorem 5.4 (Formalized explicit definability). Let T be any of the set theories discussed in this paper.
Then T proves numerical
, Tb , and Ta , for each fixed n. To be precise, if n n are altered by changing T, Tb, and Ta to T , T b , n n
explicit definability for T Theorems 5.1, 5.2, and 5 . 3 and Ta
in the conclusion, then the in the hypothesis and to Tn*, Tbn+, and Ta n* depending on n.
resulting statements are provable in T, for some n*
Proof: We first choose n* so large that TnX will prove
s r x<w
+
( S ) ~ = X .
Now
the proof of Theorem 5.1 can be formalized directly, appealing to Lemma 5.3 where the soundness of q-realizability is used. Next, note that the proof of Lemma 5.1 can be Eormalized in T (with T replaced by T statement proved by induction there is A right because, as noted in Theorem 3.1 can be proved in arithmetic; so if n* provable in T
n
*.
.
in the conclusion), since the
The appeal to Theorem 3.1 is all
(iii), the relevant part of Theorem 3.1 j s
chosen large enough, Theorem 3.1 will be
Now the proof of Theorem 5.2 can be directly formalized, appeal-
ing to Lemma 5.4 where the souridriess of q-realizability for Ta and Tb is needed. This completes the proof of Theorem 5.4.
M. BEESON
36 56.
Uniform C o n t i n u i t y
and F o r c i n g .
The r e s u l t s of t h e p r e v i o u s s e c t i o n s a r e s u f f i c i e n t t o e s t a b l i s h t h e d e r i v e d r u l e s c o n c e r n i n g l o c a l c o n t i n u i t y , b u t n o t t h o s e c o n c e r n i n g l o c a l uniform c o n t i nuity.
I t i s worth r e v i e w i n g t h e r e a s o n s why t h e p r e c e d i n g r e s u l t s a r e n o t s u f f i -
cient.
What we need t o e s t a b l i s h is c o n d i t i o n (iii)o f
[Bl],
which s a y s roughly
t h a t each p r o v a b l y r e c u r s i v e f u n c t i o n from a compact m e t r i c s p a c e X t o t h e i n t e g e r s N i s p r o v a b l y u n i f o r m l y c o n t i n u o u s (hence p r o v a b l y bounded). c u s s e d in[Bl],
Now, a s d i s -
we c a n n o t hope t o p r o v e a l l f u n c t i o n s from X t o N a r e u n i f o r m l y
c o n t i n u o u s i n any t h e o r y c o n s i s t e n t w i t h C h u r c h ' s t h e s i s , b e c a u s e t h e r e i s a r e c u r s i v e f u n c t i o n a l d e f i n e d on a l l r e c u r s i v e members of 2N, b u t n o t u n i f o r m l y continuous there. y(O),y(l)
(To compute t h i s f u n c t i o n a l a t an argument y , examine t h e v a l u e s
...u n t i l
you come t o y ( n ) such t h a t i n n s t e p s of computation, you can
v e r i f y t h a t y c a n n o t b e a s e p a r a t i o n o f two f i x e d r e c u r s i v e l y i n s e p a r a b l e r . e . s e t s ; t h e n s e t the o u t p u t e q u a l t o y ( n + l ) . )
T h i s i s our f i r s t o b s e r v a t i o n .
Our second o b s e r v a t i o n i s t h a t any p r o v a b l y r e c u r s i v e f u n c t i o n a l can b e proved t o b e c o n t i n u o u s , by t h e d e r i v e d r u l e s which f o l l o w from t h e r e s u l t s a l r e a d y p r o v e d ; h e n c e , c l a s s i c a l l y , it i s u n i f o r m l y c o n t i n u o u s , s i n c e X i s compact.
However,
t h i s i s n o t enough; w e want t o know t h a t i t i s p r o v a b l y u n i f o r m l y c o n t i n u o u s . Our s o l u t i o n t o t h i s problem l i e s i n u s i n g f o r c i n g t o add a g e n e r i c r e a l t o t h e u n i v e r s e ; any f u n c t i o n which i s d e f i n e d on a l l members of a compact s p a c e , i n c l u d i n g g e n e r i c o n e s , w i l l have t o b e u n i f o r m l y c o n t i n u o u s .
We used f o r c i n g i n [Bl]
t o e s t a b l i s h t h e s e r u l e s f o r F e f e r m a n ' s t h e o r i e s ; h e r e we a p p l y a s i m i l a r t e c h nique t o Friedman's t h e o r i e s .
I t t u r n s o u t t o be r a t h e r complicated t o give a
s u i t a b l e d e f i n i t i o n of f o r c i n g t h a t works f o r t h e e x p o n e n t i a t i o n axiom, a l t h o u g h f o r t h e o r i e s c o n t a i n i n g power s e t i t i s s t r a i g h t f o r w a r d . Suppose t h e compact s p a c e X , whose members a r e t h e members x of N f o r some f i x e d r e c u r s i v e sequence M
n
,
N
is f i x e d once and f o r a l l .
with x ( n ) M n , L e t C be t h e
s e t of f i n i t e sequences of i n t e g e r s p=
1
from f o r c i n g ) p 2 q t o mean t h a t q i s an i n i t i a l segment of p ( s o p g i v e s more i n formation than 9 ) .
No harm w i l l r e s u l t from u s i n g @ t o d e n o t e t h e empty sequence.
We u s e p , q , and r f o r members of C ; t h u s v p means V p E C .
We a r e g o i n g t o as-
s i g n t o each formula A of a s e t t h e o r y Ta ( w i t h an e x t r a c o n s t a n t o f X), a formula pll-A of T ( w i t h o u t
a),which
5 f o r a memher
i s r e a d "p f o r c e s A " .
The f r e e va-
r i a b l e s of pl1-A a r e p t o g e t h e r w i t h x ' , where x a r e t h e f r e e v a r i a b l e s of A . (The u s e of x ' h e r e i s p u r e l y f o r i n t e l l i g i b i l i t y ; we may t e c h n i c a l l y assume x '
i s t h e same v a r i a b l e a s x . ) We w r i t e p,nll-A t o a b b r e v i a t e v q - ( ( L h ( q ) n + l h ( p ) + qll-A).
We a r e now ready t o g i v e t h e c l a u s e s d e f i n i n g t h e f o r c i n g i n t e r p r e t a t i o n
t h a t works f o r t h e o r i e s c o n t a i n i n g t h e power s e t axiom (below we s h a l l d i s c u s s
37
CONTINUITY IN INTUITIONISTIC SET THEORIES t h e m o d i f i c a t i o n s needed t o t r e a t t h e o r i e s w i t h t h e e x p o n e n t i a t i o n axiom).
PII-AVB
PII-Av~/I-B
is
~ / / - A & B i s p l / - ~& p l / - ~
p l l - 3 x ~ is 3 x ' p l l - ~
is vx'3n(p,nll-A)
pl[-vxA
is v e ( q 1 j - A +3n(q,nll-B))
plI-(A+B)
p 1 l - x ~ ~ is
VW(WEX'
is
pII-x=y
w ~ y ' ) ;p
++
11-
I is I
These l a s t c l a u s e s w i l l a l s o s e r v e t o d e f i n e what it means f o r p t o f o r c e a n a t o mic formula c o n t a i n i n g terms of t h e n o n - e x t e n s i o n a l s e t t h e o r i e s , once we a s s o c i a t e t o e a c h such term t a n o t h e r term t ' t o u s e i n t h e s e c l a u s e s .
As i n t h e
soundness proof f o r r e a l i z a b i l i t y , t h e c h o i c e of t ' w i l l be a p p a r e n t i n t h e c o u r s e of t h e soundness proof f o r f o r c i n g , and we postpone t h e d e f i n i t i o n s of t h e terms
t ' u n t i l t h e n . We do, however, now g i v e t h e term 5' which i s n e c e s s a r y
i n order
t h a t t h e above c l a u s e s s h o u l d d e t e r m i n e what i t means t o f o r c e an atomic formula involving involve
5.
Namely,
5' = { < p , < n , m > > :n < l h ( p )
5, s o t h a t g e n e r a l l y pll-A
K
m = ( p ) n ] . Note t h a t
does n o t
5'
i s a formula w i t h o u t 5.
Remark: We have l o g i c w i t h no n e g a t i o n symbol, and i n s t e a d a falsum symbol i n The above d e f i n i t i o n shows t h a t p[l-lA
terms of which n e g a t i o n can be d e f i n e d . iff Ve1qIl-A.
which i s t h e u s u a l c l a u s e .
Since
v
and
+
a r e c l a s s i c a l l y super-
f l u o u s , i f we u s e c l a s s i c a l l o g i c o u r d e f i n i t i o n r e d u c e s t o t h e u s u a l n o t i o n of forcing. Our n e x t g o a l i s t o g i v e t h e m o d i f i c a t i o n o f t h e above i n t e r p r e t a t i o n t h a t w i l l s u f f i c e f o r t h e o r i e s w i t h t h e e x p o n e n t i a t i o n axiom. Notation:
C
P
q'&p
We i n t r o d u c e some
= { q E C : q-1
means q p
K
lh(q) = n+lh(p)
x ' / p ={ < q , u ' > E x ' :
;
x ' / p i s read "x' r e s t r i c t e d t o p".
Our f o r c i n g i n t e r p r e t a t i o n w i l l be d e f i n e d i n t h e f o l l o w i n g way:
we s h a l l f i r s t
a s s o c i a t e t o e v e r y formula A w i t h f r e e v a r i a b l e s x a n o t h e r formula R A ( p , x ' ) ; we t h e n w r i t e pll-A t o a b b r e v i a t e R ( p , x ' / p ) tion t h a t the variables x' v x ' means V x ' ( G ( x ' )
-+
.
I n what f o l l o w s , ke make t h e conven-
a r e r e s t r i c t e d t o s o - c a l l e d "gogd s e t s " ; t h a t i s ,
, and ] x u
means 3 x ' ( G ( x ' )
mula d e f i n i n g t h e good s e t s , a s f o l l o w s . With
- for
&,
where G ( x ' )
is a A
for-
e x t e n s i o n a l e q u a l i t y , G(y') i s
\ d p v e v w ( < p , w Y y ' + 3 v ( < q , v > E y ' & v- w / q ) ) . It i s not o b v i o u s t h a t any good s e t s e x i s t ; we s h a l l e n c o u n t e r o u r f i r s t o n e s i n Lemma 6 . 3 below. Now h e r e a r e t h e
M. BEESON
38
clauses defining the formulae R (p,x'): A
Now, using the abbreviation pi/-A for R (p.x'/p), the clauses for A implication and universal quantification can be rewritten in exactly the form we gave for the simpler version of forcing!
____ Lemma
6.1
If p1I-A and q y
then qll-A.
Proof: A straightforward induction on the complexity of
A,
using crucially that
the primed variables are restricted to good sets, which in fact is built into the definition just to make the atomic case of this lemma work. Lemma 6 . 2
If p,j 11-A and p,ml/-(A+B), then for some k, we have p,k/i-B.
Proof: Let
n
=
max(k,j); by Lemma 6.1,
ql/-(A+B). Hence, for each q< p, --n
p,n/(-Aand p,nl/-(A+B). So for each q '
3 i(q,ill-B).
p,
7
Now there are only finitely many
Let k be larger than any of .the values of i which work for these finitely q p . many q. Then q5p + q,k 11-B. Set k = n+k . Then p,k//-B. Lemma 6.3 p[l-ncw iff
new; more precisely, there is for each n a term n' such
that pl1-nEw implies n e w
, and n e w
implies pI/-nEo with n' substitituted for
the corresponding free variable of the formula pi(-n. Remark: This is the analogue of saying new
is "self-realizing". We might call
a formula with this property "self-forcing".
Proof: We first define a functmn n' of n for use in the lemma, by induction: 0' is
0, and (n+l)' is I
that for all integers n, n' is a good set.
We now define
to do in order to complete the definition of forcing)
as
w'
(which we promised
{
assertions of the lemma may now be proved by a straightforward induction on n. Lemma 6.4 Let A be an arithmetical predicate. Proof: By induction on the complexity of tions
A.
Then for x e w , p//-A(x)iff A(x).
The basis case consists of the rela-
x=y+z, y=x.z, and successor. These relations have their set-theoretical
definitions, so matters are technically complicated. Consider how to prove (pll-x=y+l iff y=y+l). induction on z .
This is done by induction on x, first proving pl1-0 6 z by
Then we proceed to
+
and
., lust as in the set-theoretical de-
velopment of arithmetic. (This can all be done in
B).
Next we do the induction step of the lemma, in which A is, for instance, VzEwB(x,z).
Suppose pl/-A(x);then for some n, we have p,nll-(z€ w
Let z € w be given; using z ' produced in Lemma 6.3, we have pll-z € & I ;
+
B[x,z).
hence for-
CONTINUITY IN INTUITIONISTIC SET THEORIES Hence B ( x , z ) , by t h e i n d u c t i o n h y p o t h e s i s .
some j , we have p , j l i - - B ( x , z ) .
z was a r b i t r a r y , we have A ( x ) .
t h a t i s , 011-Vzzrw B ( x , z ) .
39
C o n v e r s e l y , suppose A ( x ) .
Since
We w i l l show $?/I-A(x);
We c l a i m q1l-z E w i m p l i e s q l I - B ( x , z ) .
Indeed, i f q l l - z e w
then z e w , s o B ( x , z ) ; h r n c e , by i n d u c t i o n h y p o t h e s i s , $?ll-B(x,z). I f y ' i s s u b s t i t u t e d f o r x ' i n t h e formula p I ( - A ( x ) , t h e r e s u l t i s
Lemma 6 . 5
l o g i c a l l y e q u i v a l e n t t o pil-A(y).
R
I n o t h e r words R A ( x ) ( p , x ' / p )
i s equivalent t o
(P,Y'/P).
A(y) Proof: By i n d u c t i o n on t h e complexlty of A . We a r e now ready t o s t a t e t h e soundness theorem f o r f o r c i n g . auxiliary theory described i n 5 2 , with a constant
L e t Ta b e t h e
5 f o r an element of t h e compact
space X . Theorem 6.1
L e t T be dny of t h e n o n - e x t e n s i o n a l s e t t h e o r i e s d i s c u s s e d
i n t h i s p a p e r , e x c e p t B-ext. T 4 , Z , o r ZF-+RDC.
Proof:
Thus T can he
(non-extensjonal) T1,T T 2 ' 3'
Then T a t A i m p l i e s T i - 3 n ( @ , n l l - A ) .
By i n d u c t i o n on t h e l e n g t h of t h e proof of A.
axioms and r u l e s , t h e n t h e s e t - t h e o r e t i c a l
axioms.
We have t o check t h e l o g i c a l We h e g i n w i t h modus ponens.
Suppose @,n[l-A and @ , m / l - ( A + B ) ; t h e n by Lemma 6 . 2 , @,klI-B f o r some k. Lemma 6 . 2 was proved w i t h i n
p.)
t h e l i s t on page 3 o f [Tr]).
t i o n a l axioms and r u l e s ( u s i n g e . g . q u a n t i f i e r axioms and r u l e s .
Consider t h e axiom ( VxAx
Consider f i r s t t h e c a s e when t i s a v a r i a b l e . \dx'3nYpSnq RA(p,x'/p).
(Note t h a t
We l e a v e t h e r e a d e r t o check t h e o t h e r p r o p o s i -
+
We t u r n t o t h e
A t ) , f o r some term t .
Suppose q l t V x A x ; t h a t i s ,
S u b s t i t u t e t ' f o r x ' ; t h e n f o r some n we have
q , n ) k A ( t ) , u s i n g Lemma 6 . 5 .
The c a s e i n which t 1s a term o t h e r t h a n a v a r i a b l e
i s handled t h e same way, p r o v i d e d we have a t hand a c o r r e s p o n d i n g term t ' w i t h t h e propert-y o f Lemma 6 . 5 .
We s h a l l g i v e , i n t h e c o u r s e of v e r i f y i n g t h e s e t -
t h e o r e t i c a l axioms, such a term t' f o r each term t .
The o t h e r q u a n t i f i e r axioms
and r u l e s can he t r e a t e d s i m i l a r l y . We now t u r n t o t h e n o n - l o g i c a l axioms, b e g i n n i n g w i t h t h e axiom cCX,
which has
t h e f o l l o w i n g form when w r i t t e n o u t : v n t w g m € w ( < n , m > E a& m M n )
&
V n , m ( < n , m > d a& < n , r > c a + m = r )
t/x€a]n,mEw(x=
V n , m ( p (a ( n ) , a ( m ) )< l/(n+l)+l/(m+l))
where p i s some r e c u r s i v e f u n c t i o n and M F i r s t we show pIl-nEw+
i s s o m e r e c u r s i v e sequence ( s e e 8 2 ) .
$ ? l l - v n ~ w 3 m ~ w ( < n , m >&E m<M -
3 m m E w ( < n , m > E a &m<M
).
m<M
is t r u e
).
L e t n and @ be g i v e n ; we c l a i m
Suppose q 2 h a s q / / - n E w ; t h e n n E w
Lemma 6 . 3 ; we must show q , j l l - 3 m E w ( < n . m > 6 5 Then r \ \ - < n , m > E a .
&
A l s o by Lemma 6 . 4 ,
&
m<M
)
f o r some j .
,
by
Take j = M + l + n .
r j - m ~ M n , s i n c e by d e f i n i t i o n
of C ,
M. BEESON
40
Next consider the conjunct x € 5 + 3n,m ~w(x=
m=(p)
&
. Thus
pll-nEw
E
mEw
E
(x=
larly; this completes the verification of the axiom We now turn t.o the set-theoretical axioms.
3 x\dy(y e x
c-t
.
A(y) )
are all of this form. will be forced by
a€X.
Consider an axiom of the form
Pairing, union, separation, exponentiation, and power set If we can form x' = {
0, as
is easily checked.
(If we had not been so careful in our
definition, we would have to form {
;
which cannot be done for the
We now check the axioms of this form one by one.
Separation: Here we have to form x' = {
&
yEa)}; that is,
This can be formed using separation and abstrac-
{
6.4.
@/l-(xaa -+ 3n,mEw(x=
To check A -separation, we have to prove that pl(-B(y) is a A
formula if
is; this is a simple induction on the complexity of 8.
rhe definition of x' just given also determines the term t' corresponding to the It has to be checked that x' as
term t associated with this separation axiom. just defined is a good set.
Generally if x' is defined as {
then x' is good, since if q 2 and p is in x', then q also forces C, so is in x', by definition of x i ;but cq,y'/q> is exactly what
we
, which is
must prove is in x' in order to show x' is good.
All the terms t'which
we shall exhibit in verifying the set-theoretical axioms have this form, so we need not repeat the argument in each case.
Union: Here we have to form x'
=
{
:
pll-3z(y€z
&
zEa)}; that is,
we want x' to consist of all
for some z ' , we have
is equivalent to
Hence we want x' to consist of all
u
{a'/q,b'/q}. qcc
Thus Ex' iff qII-(y=aVy=b).
This is the most difficult axiom to verify.
show how to form x'={
&
Dom(y)=a
E
Here we have to
Rng(y)C_ b].
The problem
advance a set to which y'/p must belong, so that x' can be formed Suppose @li-(Fcn(y)
&
Dom(y)=a
&
Rng (y)C b) ; then where must y '
lie? (Remember, we do not have power set available.) Introduce an equivalence relation @,nl/-z=wfor some n.
Then let
LZ']
an Rng(b') by defining z'=w'
iff
be the equivalence class of 2' under this re-
lation; [ z ' ] can be formed using A -separation.
Let S2 be the set of all
[='I
CONTINUITY I N INTUITIONISTIC SET THEORIES
for z'
i n R n g ( b ' ) (which e x i s t s by a b s t r a c t i o n ) and l e t S
n i t e s u b s e t s of S 2
41
b e t h e s e t of a l l f i -
( u s i n g e x p o n e n t i a t i o n , u n i o n , and a b s t r a c t i o n ) .
. 0
s e t of a l l f u n c t i o n s from a ' t o S { < q , < w ', z ' > / q > : Ea'
1
be t h e
1
V x y [ z j c f ( q , w ' / q ) &[XI€ f ( q , w ' / q )
& V Z '
Let S
I f f i s in S . , l e t F ( f ) be
+ Z ' S X ' )
1.
(We can form t h i s s e t by t h e A - s e p a r a t i o n axiom, s i n c e t h e e q u i v a l e n c e r e l a t i o n i s d e f i n e d by a A
'25
formula.)
I t i s t e m p t i n g t o t h i n k t h a t i f @ ) \ F c n ( y ) & Dom(y)=a & Rng(y)Lb, a s we have
assumed, t h e n y ' must be ( e x t e n s i o n a l l y ) F ( f ) f o r some f i n S check t h a t y '
C F(f),
f ( < q , w ' / q > ) ={
where
12'1 :
z'(Rng(b')
&
3r(q
1
.
Now, we can
rll-<w,z>Eyl.
( T h i s s e t i s f i n i t e , b e c a u s e 0 , n ~ ~ - ~ ! z C - b ( < w , z > Cfyo)r some n; and it can b e formed u s i n g a b s t r a c t i o n and s e p a r a t i o n .
We n o t e t h a t " f i n i t e " means t o b e t h e
range of some f u n c t i o n d e f i n e d on some i n t e g e r , i n t h e i n t u i t i o n i s t i c c o n t e x t ; i.e.
t o be of bounded s i z e . )
-
However, p o s s i b l y some < q , < w ' , z ' > / q > E F ( f ) may
n o t have qlI-<w,z>€y, a l t h o u g h z i s unique such t h a t t h i s problem, l e t u s s a y y '
0
n,wYi
q,nl/-<w,z>Ey.
To solve
iff
\ d z ' € R n g ( b ' ) \ d ~ 0 ( q l l - < w , z >t~t qy/ l - < w , z > EY 1 1 .
So \ d w ' E : R n g ( a t ) 3 n ( y 'l Y n r W F ( f ) ) .
y;
=
i < q , < w ' , z ' > / q > :q k 0
f o r some f i n i t e s u b s e t S o f C . we can form ( b y
A -separation)
w'c- a '
u
n
E
Now, i f y ' N n , w y ; ,
then
/q>Ey;V q € s )
&
S i n c e t h e s e t of a l l f i n i t e s u b s e t s of C e x i s t s , {y;:
{ y ' : y'-. w
yi)
n,w
F(f)}
f o r each n,w',y;.
Thus
( u s i n g a b s t r a c t i o n and u n i o n ) .
Thus, i f we form by t h e c o l l e c t i o n axiom t h e s e t
011-
(Fcn(y)
&
Dom(y)=a & R n g ( y ) c b )
implies y ' t S j .
a s e t Sp such t h a t pII- ( F c n ( y ) & Dom(y)=a 3 ,:S and s e t collection again, s e t S = Pt c
u
x ' = t'-p,y'/p>CG C x S : p l l - ( F c n ( y )
&
&
S i m i l a r l y , we can c o n s t r u c t
Rng(y)C_b) i m p l i e s y ' / p E.'s
Dom(y)=a
&
3
Using
Rng(y)Cb]
T h i s completes t h e v e r i f i c a t i o n of t h e e x p o n e n t i a t i o n axiom.
There seems t o be
no hope of e l i m i n a t i n g t h e need f o r c o l l e c t i o n i n forming S j ;
a b s t r a c t i o n i s de-
s i g n e d f o r c o l l e c t i n g s e t s formed by s e p a r a t i o n , b u t h e r e we have t o c o l l e c t s e t s
M. BEESON
42
formed by union. It is worth remarking that replacement would suffice in place of collection. Infinity. We have already given the constant w its associated term w ' , and we have defined a function n
n', in the proof of Lemma 6 . 3 . , with whose aid the formula
n E w was proved to be "self-forcing". Let P ( z ) be the property, 0 E z .s Vx(x E z
-t
x+l E z ) , where x+l is the set-theoretic successor function.
Using n' and the definition of w ' ,
1-
is also forced, for if q
we easily see that 011- P(w). Moreover, P(z) +wcz
P ( z ) then one proves by induction that for every integer
n, some extension of q forces n E z . We note that only restricted induction is required. Foundation. We first note some facts about well-founded relations. We say (R,<) is well-founded if TI holds on (R,{)
for sets (not formulae): thus the foundation
axiom says that (W,E) is well-founded, for each transitive set W. Suppose (W,Q) is a well-founded relation, and (R,<) is a relation such that for some function F:R+W,
4
we have a
b
+
Q(F(a) ,F(b)). Then (R,()
is well-founded. Next, let (W,Q) be a
b iff 3x E W(a R x & x R b). n+l (as well as their union) is a well-founded relation on W. A special
well-founded relation, and defineRo=Q,Rn+l by a R Then each R
case of this is when Q is E; then, for example, if <x,y> E z y Rj Now consider the relation R on any subset of C
X
A,
2.
defined by iff
). Hence
+ F(
&
z +
y
z ' , the set z '
0
y E
2).
We must prove pi /-Vu(u E W
= {
Vu Vq 6 p(Eyb
+ +
+
p E 2 ' 1 . Then pII- y
E z ; ) ,
&
tz
can be written
which is equivalent to
E z;), which is equivalent to
VR
8.
u E z ) . Define for each set
+
y E z ) is equivalent to
V R
Since R is a well-
founded relation, and since R is defined by a do formula, we conclude Vcy' ,p> E W;(
+
y E
2)
, which
was what we had to prove. This completes the verification of the foundation axiom.
43
CONTINUITY IN INTUITIONISTIC SET THEORIES Strong C o l l e c t i o n .
v < q , x > Ea ' / p 3 y ' , n ( q , n l j - A ( x , y ) ) .
t a i n i n g a y ' such t h a t
3n(q,nll-A(x,y))
y ' i n W a r i s e s t h i s way. q,nIl-A(x,y))}.
That i s ,
Suppose pi[- V x G a 3 y A ( x , y ) .
Applying c o l l e c t i o n , w e g e t some W conf o r each < q , x ' > t a ' / p , and such t h a t each
Put z ' = { < q , y ' > : y'b W
Then pII-(\Jxc a g y c z A ( x , y )
&
&
3x'(
R n g ( y ' ) ( < q , x ' > La
&
V y e z 3 x 6 a A ( s , y ) ) , so
@ f o r c e s t h e s t r o n g c o l l e c t i o n axiom. Bounded Dependent Choice.
Suppose p ( ( -v x & a 3 y c a Q(x,y)
duce z ' such t h a t pll-(Fcn(y)
&
&
x E a.
We w i l l pro-
Oom(z)=w & z(O)=x & Vnrw Q ( z ( n ) , z ( n + l ) ) ) .
From t h e h y p o t h e s i s about p , we g e t & aQ ( x , y ) ) ; t h a t i s \dEa ' j y ' t R n g ( y ' ) 3 r ~ q ( r l l - y ~
Applying dependent c h o i c e , we g e t a
\ d < q , x ' > E a ' 3 < r , y ' > € a ' rl(-Q(x,y). sequence <'n 2'
=
, x ' > C a ' , w i t h q =p and x;, n
and q n + l / / - Q ( x n , x n + l ) . Now we d e f i n e
=XI,
< p , < n , x > ' / p > : p z q n } . Here n ' i s a s d e f i n e d i n Lemma 6 . 3 , and
i s a t e r m b u i l t from n' and x ' a s d i s c u s s e d i n t h e v e r i f i c a t i o n of t h e p a i r i n g
axiom. The r e s t of t h e argument
iS
routine.
Now only f o u r axioms remain t o be checked:
(numerical) i n d u c t i o n , power s e t ,
r e l a t i v i z e d dependent c h o i c e s , and t r a n s f i n i t e i n d u c t i o n . We omit t h e s e v e r i f i c a t i o n s , s i n c e t h e p r o o f s f o r RDC and T I are e x a c t l y l i k e t h e p r o o f s for DC and f o u n d a t i o n , and s i n c e t h e proof f o r numerical i n d u c t i o n i s completely s t r a i g h t forward.
A s r e g a r d s power s e t :
when c o n s i d e r i n g t h e o r i e s with power s e t , t h e r e
is no need t o use t h e complicated f o r c i n g i n t e r p r e t a t i o n given h e r e ; i n s t e a d , one
should r e t u r n t o t h e simpler d e f i n i t i o n f i r s t given.
With t h a t d e f i n i t i o n , t h e
v e r i f i c a t i o n of power set i s a l s o completely s t r a i g h t f o r w a r d . This completes t h e proof of Theorem 6.1. Lemma 6 . 6
L e t A be a r i t h m e t i c i n
5, b u t n o t c o n t a i n i n g
formula of second o r d e r a r i t h m e t i c ) .
tJf &x(f(lh(p))=p
-f
A(f)).
Then p)FA(a) A,
l i k e Lemma 6.4.
Let T be any of t h e s e t t h e o r i e s d i s c u s s e d i n
T may be e i t h e r e x t e n s i o n a l o r non-extensional.
f o r some m
( i n i t s formulation a s a
For each f i x e d A t h i s i s provable i n B.
Proof: By i n d u c t i o n on t h e complexity of Theorem 6 . 2
-+
i f and only i f
and k , Ta I - ( { e J " ( O ) >
&
h(m0)
t h i s paper except B ;
Suppose Ta I-{e)'(O)
ew
. Then,
determines { e j a ( 0 ) ) .
Remark: The p h r a s e i n t h e l a s t l i n e of t h e theorem means t h a t t h e Turing machine computing { e } B ( 0 ) h a l t s u s i n g only t h e f i r s t m < k. -
v a l u e s of a , and y i e l d s a value
This can be expressed i n a r i t h m e t i c using t h e T - p r e d i c a t e , without mention-
ing t h e c o n s t a n t
a,
which seems t o appear i n t h e formula.
Note t h a t , i n t h i s c a s e ,
M. BEESON
44
m will be a modulus of uniform continuity for {ela(O) regarded as a function of a.
Proof:
Suppose Ta I - { e > a ( 0 ) € w .
By the results of Section 3, if T is extensional
we can replace T by the non-extensional version, and still Ta will prove {e}d(0)c;w .
Hence, by Theorem 6.1, arguing in Ta, for some n , we have
0,n(l-3i,msw(i={e}a(0)
is determined by a ( m ) ) .
Since this statement does not
involve 5 , as discussed above, it is provable in T , not just Ta. Let pSnO.
Now argue in T:
Then (using Lemma 6.3), for some i and m, we have for some j,
p,j I(-(i={e}“(O)
is determined by a ( m ) ) .
If we choose j large enough, the same
j will work for all p< 0, so that by increasing n, we may assume that for p< @ n n we have some i and m, depending on p, such that p)I-(i={e}”(O) is determined
by a ( m ) ) .
be the largest of these values of m, over all p< 0 , and let k
Let in
be the largest of the values of i.
Then if p<
p,j l/-({e}5(0) is determined by a ( m )
&
-n
{e}”(O)L
0, we
n
have
k) , for some value of j depending
on p, since the formula in the last line is a consequence of the one forced by p. Taking the maximum of these values of j, we have 0,n+j
11-
is determined by i ( m
({e}%O)
)
& { e } a ( 0 ) l kl
.
By Lemma 6.6, this last formula is true, since it is forced by every condition of length n+j.
(Note that it can be expressed without using implication.)
Remembering that we have been arguing in T, we have just proved that T
1-3k~mo({e}a(0)
is determined by a ( m o )
bility for T, we get T I - ( { e > a [ O ) some numerals
6
and i;.
&
{ g } (0)Lk). Applying explicit defina-
is determined by
)
&
- a {e)-(O)>
This completes the proof of the theorem.
k)
for
CONTINUITY IN INTUITIONISTIC SET THEORIES 87.
45
The Main Theorems about Continuity
In the introduction, we have discussed the various derived rules related to continuity and local continuity, which form the focus of our wor':.
In this section,
we plan to establish results for intuitionistic set theories analogous to those obtained for Feferman's theories in LB1-J. Those results are of two kinds: rived rules, and consistency or independence results.
de-
In the preceding sections,
we have done all the necessary work to establish the general metamathematical properties which were shown in [Bl] continuity to hold.
to be sufficient for the derived rules of
For the consistency results, we have something yet to prove
in this section. Before we give our main results, we shall fulfill the promise made in the introduction to explain further the Principle of Local Uniform Continuity. In formulating such a principle, we wish to state something like the Principle of Local Continuity, except that we want 6
to depend only on
E.
The most curious thing
about the Principle of Local Uniform Continuity is that we cannot express exactly what we mean in the usual predicate calculus.
What we really mean is
That is, b depends only on a, and 6
.
only on
E
Of course, one can say some-
thing in the s u a l predicate calculus which is equivalent to this under some axiom of choice, but that is beside the point.
We have not pursued this matter further,
but instead have formulated a weaker version of Local Uniform Continuity, which nevertheless has all the interesting consequences mentioned in the introduction. When we refer to Local Uniform Continuity in the rest of this paper, what we mean is this version, called LUC(X,Y)
in 1.13.
We now state our main theorem
on derived rules. Theorem 7.1
-
Let T be T2 (similar to Myhill's CST), TJ,T4, Z, or ZF
+
RDC,
or the non-extensional version of any of these theories. Then T is closed under the rules of local continuity, continuous choice, local uniform continuity, Heine-Borel's rule, and all the rules discussed in [el]. Corollary
If the hypothesis of one of these rules is provable in
g ,
the
con-
clusion is provable in T2, i.e. requires at most some instances of induction and collection beyond
Proof:
E.
The necessary conditions laid out in [Bl]
set theories in Theorem
5.4, Lemma 5.2,
have been derived for these
and Theorem 6.2.
M. BEESON
46 Remark: In case
T is a non-extensional theory, we can allow an arbitrary formula
in place of a definable set in the rule of local continuity, local uniform continuity, and continuous choice. We now shall obtain some consistency results complementing these results on derived rules.
These results concern the principles corresponding to the derived
rules we have already studied; their statements are obtained from the rules in the obvious way, namely: if a rule says, from A infer B, then the corresponding principle 1 s A
+
B.
Foremost among the principles we study are the Principle of
Local Continuity, the Principle of Local Uniform Continuity, and the Principle of Continuous Choice.
Note that the Principle of Local Continuity implies the sim-
pler Principle of Continuity, that every function from a complete separable metric space to a separable metric space is continuous; and the Principle of Local Llniform Continuity implies that every function from a compact metric space to a separable metric space is uniformly continuous. jectures of [Fr 2 1
Thus our results include the con-
concerning functions from NN to N and from ZN to N.
The
Principle of Local Uniform Continuity also implies Heine-Borel's theorem. At the end of [MI],
there is a "postscript" announcing certain theorems of Fried-
man, which include special cases of some of the consistency results in this section; see also [B3,§3] in the last line of LMl],
for related results.
The theorem announced for Friedman
about the axiom of choice for the reals with the prefix
b x 3 y A(x,y) instead of v x 3 ! y A(x,y), is false; the axiom is refutable, as discussed in [Bl]
.
We are going to prove our consistency theorems using realizability.
The key
to these proofs lies in the construction in [Bl] of a so-called "weak BRFT" which all operations on NN are continuous. To explain what this means: Let
in S
be
a set, and suppose C is a class of partial functions from S to S, of several variables, including a pairing function and a binary "universal function" @(e,x) such that each unary function f in C is
Xx@(e,x) for some e.
some other, less important conditions spelled out in [El], "weak BRFT".
If (S,C) satisfies
then it is called a
The use of such structures is that the functions in C can be used
in place of recursive functions for realizability. As mentioned, in [Bl] a specific weak BRFT is constructed in which all operations from NN to N are continuous.
(Each weak BRFT contains a copy of the integers, calling some element
0
and using p(x,O) for successor, where p is the pairing function. Thus it makes sense to speak of operations from NN to N.)
As a matter of fact, two specific
weak BRFT's are constructed with this property: on ZN
in one of them, all operations
to N are uniformly continuous, and in the other, there is a continuous,
but not uniformly continuous function on 2N to N.
Call these weak BRFT's S
and
CONTINUITY IN INTUITIONISTIC SET THEORIES S1,
47
respectively.
Theorem 7.2
+ RDC
ZF-
is consistent with Church's thesis CT plus
"All
functions
from a complete separable space X to a separable space Y are continuous". Remark:
Continuity cannot be improved to uniform continuity without dropping
Church's thesis. Theorem 7.3
ZF-
+
RDC is consistent with the Principle of Local Continuity
plus "There is a non-uniformly continuous, continuous function from 2N to N." Theorem 7.4
ZF-
+
RDC is consistent with the Principle of Local Uniform Con-
tinuity.
-
Corollary: ZF
+ RDC is consistent with
"All
functions from a compact metric
space to a separable metric space are uniformly continuous." Proof: The idea of all the proofs is to use realizability to prove the theorem for ZF- + RDC - ext, and then use the ideas of Theorem 3.1 to prove it for ZF-
+
RDC.
We first show how to use realizahility to prove the theorems for
T-ext, where for simplicity we write T for ZF-
+
RDC.
Generally speaking, realizability interpretations can be either formal or informal; that is, e r predicate.
A
can be either a formula of the formal system, or an informal
For instance, Kleene's original interpretation for arithmetic can
be taken either way.
The q-realizability given earlier in this paper for set
theory is necessarily formal, however, since it is not clear how to interpret xi informally. Of course we can also do (formal) "1945-realizability" for set theory, which is analogous to Kleene's original "1945-realizability" (as it has come to he called) for arithmetic.
Here are the clauses defining this interpre-
tation: e r xEy
is
e r
(A & B)
is
(elo r
<e,x> €y
e r
(AVB)
is
((e)o=O + (e), r
A &
(e)l r
(elo#O + (ell r e r
(A
+
e r3 x
B)
A
e r d x ~
is
va(a r A
is 3 x e r
+
-
&
B))
iel(a) r
B)
A
is kjx e r A
We then have the soundness theorem, T-ext C A implies numeral e.
B
A)
T-ext.1
e
r
A
for some
The proof of the soundness theorem is so similar to the soundness
theorem for q-realizability that we do not write it out here. Now Kreisel-Lacombe-Schoenfield's theorem asserts that every effective operation
M. BEESON
48
from NN to N is continuous; and the same theorem is true for complete separable metric spaces in place of iVN and separable spaces in place of N. (Kreisel-LacombeSchoenfield's theorem is proved, for instance, in Rogers [R, p. 3621 ; the reader will have no difficulty making the extension mentioned.)
Moreover, if X and Y are
complete separable and separable metric spaces, respectively, with X in standard form, then KLS(X,Y) (in obvious notation) is 1945-realized, as is proved in [BZ]. Now, it is easy to see that (1) Church's thesis is realized using 1945-realizability, and (2) with Church's thesis, KLS(X,Y) is equivalent to the assertion that all functions from X to Y are continuous.
It follows that "all functions from X
to Y are continuous" is realized, and hence consistent with T-ext, by the soundness theorem for 1945-realizability. Thus Theorem 7.2 is proved with T-ext in place of T. Now we turn to the proofs of Theorems 7.3 and 7.4.
In the definition of
realizability given above, there is nothing particularly sacred about the recursive functions. If we have any weak BRFT which can be defined and proved to be a weak BRFT in T-ext, we can use it for (formal) 1945-realizability. That is, instead of using {e)(a)=y
we use @(e,a)=y, where @ is the universal function of
To be precise, instead of using
the weak BRFT.
the formula defining @(e,a)=y in T-ext.
3 n(T(e,a,n)
&
U(n)=y)
we use
A priori, it is possible that we might
have a weak BRFT which could be proved to be a weak BRFT without having a definable universal function, but that possibility doesn't occur here.
Also, one
Should add to the definition of e r ( A v B ) a proviso that (e) is an integer of the BRFT (each weak BRFT contains a copy of the integers). One can verify by reading the construction of S
and S1
iniBl] that their univer-
sal functions are definable, and that they can be proved in a very weak non-extensional set theory to be weak BRFT's. To verify that S
has the property that all
functions from 2N to N are uniformly continuous requirzs something like Konig's lemma, which goes beyond intuitionistic systems, but that doesn't affect the usefulness of S
0
for formal realizability, which only requires that we be able to
prove that it is a weak BRFT. It follows from the above discussion, and from a soundness proof inessentially different from the one give for q-realizability, that we can assign to each formula A another formula e r. A, for e realizes A in Si, and prove that T-ext
1-
I
A implies T-ext I-qe CSi(e ri A).
49
CONTINUITY IN INTUITIONISTIC SET THEORIES If A is
a
sentence, then I13e(e ro A) is true" can be regarded as an informal This interpretation can play the same role
realizability interpretation of A . that
''Npe(e
r A ) " does for Feferman's theories in
ments of 5 3 . 2 of
[Bl],
If one reads the argu-
making the substitution just mentioned, one finds that
Theorem 7.3 is proved, with T-ext in place of T. arguments of 5 3 . 3 of
1.11.
BI
, substituting *'Ie(e rl
Similarly, if one reads the for
A)"
'8
finds that Theorem 7.4 is proved, with T-ext in place of T.
&e(e
r A)", one
That is, the proofs
that Local Continuity and related properties are or are not realized are not dependent on the particular theory in detail, but only on the existence of a sound realizability interpretation in some weak BRFT with the properties of S c.r S1, respectively. Thus Theorems 7.2, 7.3, and 7.4 are proved for T-ext. improve this result to T instead of T-ext.
*
from T to T-ext given in Theorem 3.1.
Now we discuss how to
We have to consider the interpretation
Suppose for simplicity that we are
working with a two-sorted version of T, with variables for integers and variables for sets.'
Recall that two sets are
of the other, and a & b if a leaves numbers alone.
x
if each member of one is
are
%
some member
if and only if they have the same values.
is any "extensional" subset of &IN,
same values as a then b c X , we have a E X iff
i.e. whenever a E X and b has the a€X.
separable metric space in standard form (see 5 2 )
In particular, any complete
is-such an extensional subset
Similarly, if X and Y are complete separable spaces and P is an extension-
of N".
al subset of
X X
the following: T-ext
'I,
The interpretation for two-sorted T
It is then easy to check that two functions from N to N
(as sets of ordered pairs) Thus if
' I ,
some c e b .
I-
Y
in the sense of
1 2 , then X E P iff
xEP.
We shall now prove
Let A be an instance of the Principle of Local Continuity.
(A ++ A*).
Then
We consider the conjuncts of the hypothesis of Local Continu-
ity one-by-one. Making use of the "standard form" of X, we may suppose that all references to X in the Principle of Local Continuity are implicit: that is, " a E X " actually is "a€NN
&
Conv(a)", where Conv is a formula expressing the convergence conditions
on a, i.e. Conv(a)~vn,m€N(o(an,am) < (l/m)+(l/n)), recursive function.
where
CJ
.
is a certain
Hence, the hypothesis "X is a complete separable metric
space" no longer needs to actually occur.
Similarly for Y.
thesis "P is extensional", which says d(a,a')=O
E
d'(b,b')=O
where d and d' are the metrics of X and Y respectively.
Consider the hypoE
P(a,b) + P(a',b'),
We have seen already
that P is equivalent to P*; by a similar argument it follovis that (d(a,a')=O) is equivalent to (d(a,a')=O)*; hence, "P is extensional" is equivalent to its interpretation.
*
M. BEESON
50 Consider t h e h y p o t h e s i s ,
"VaEX {b: P(a,b)
&
b e y } is closed".
We do n o t have
t o check t h i s one, s i n c e t h e s p e c i a l c a s e Y=N i m p l i e s t h e g e n e r a l c a s e , provably i n a v e r y weak t h e o r y p l u s a s i m p l e axiom of c h o i c e AC shown i n
[Bl].
which i s r e a l i z e d , a s
N
is
However, t h e r e a d e r who w i s h e s can v e r i f y d i r e c t l y t h a t t h i s
*
hypothesis, too, i s equivalent t o i t s
interpretation.
F i n a l l y , c o n s i d e r v a E X 2 b EY P ( a , b ) . The i n t e r p r e t a t i o n o f t h i s i s
'd a E X
3 b Y~ P * ( a , b ) ; which we have s e e n is j u s t t h e o r i g i n a l formula a g a i n .
The c o n c l u s i o n of Local C o n t i n u i t y can b e d e a l t w i t h s i m i l a r l y .
Hence, each
i n s t a n c e A of t h e P r i n c i p l e o f Local C o n t i n u i t y i s p r o v a b l y e q u i v a l e n t t o A*. Now we prove t h e c o n s i s t e n c y o f T
+
LC(X,Y).
I f i t i s n o t c o n s i s t e n t , t h e n some
c o n j u n c t i o n o f i n s t a n c e s o f L C ( X , Y ) , s a y B , i m p l i e s 0=1 i n T. 3.1,
B*
i m p l i e s 0=1 i n T-ext.
Then, by Theorem
But B* i s p r o v a b l y e q u i v a l e n t t o B , i n T-ext.
Hence L C ( X , Y ) i s i n c o n s i s t e n t w i t h T - e x t , which p o s s i b i l i t y we have a l r e a d y r u l e d o u t by r e a l i z a b i l i t y .
The p r o o f s of Theorems 7 . 2 , 7 . 3 , and 7 . 4 can be completed by checking t h a t t h e o t h e r statements involved a r e a l s o e q u iv a le n t t o t h e i r
*
interpretations.
The
b a s i c r e a s o n why t h i s works seems t o be t h a t none of t h e s e s t a t e m e n t s mention The u s e of s t a n d a r d form f o r
o b j e c t s of type h i g h e r t h a n f u n c t i o n s from NN t o N.
complete s e p a r a b l e s p a c e s r e d u c e s e v e r y t h i n g t o low t y p e s . t h a t " A l l f u n c t i o n s from NN t o N a r e c o n t i n u o u s " tation.
Now "F: NN
-f
N"
is
Fcn(F)
&
l e n t t o Fcn(F).
But t h e n
*
i s equivalent t o its
\da,b(atNN
< a , b > tF
&
F c n ( F ) * s a y s t h a t i f < a , b > E F and < a , @ € F t h e n b'w. a 'wome a ' and b = F ( a ' ) .
We check, f o r i n s t a n c e ,
-f
interpre-
b E N ) . Now
But < a , b > e F s a y s
. E N N and so b = F ( a ) . ffence F c n ( F ) *
The argument shows a l s o t h a t < a , b > EF i f f
< a , b > € F.
is equiva-
Together
w i t h what we have a l r e a d y p r o v e d , t h i s s u f f i c e s t o complete t h e proof t h a t "F: NN + N "
is e q u i v a l e n t t o i t s i n t e r p r e t a t i o n .
Next, "m i s a modulus f o r F
a t y" i s ~ y E N N ( t f i i m ( z ( i ) = y ( i+) )F ( z ) = F ( y ) , which i s e q u i v a l e n t t o i t s own i n t e r p r e t a t i o n i n view of t h e f a c t t h a t
< a , b >E F i f f
Hence, "All f u n c t i o n s from NN t o N a r e c o n t i n u o u s " i n t e r p r e t a t i o n , a s claimed. 7.4 can be t r e a t e d s i m i l a r l y . Footnote 1 : ____
< a , b > E F , a s proved above.
i s e q u i v a l e n t t o i t s own
The r e s t of t h e s t a t e m e n t s i n Theorems 7.2, T h i s completes t h e p r o o f .
We have f o r m u l a t e d o u r s e t t h e o r i e s T w i t h a c o n s t a n t w
Neumann i n t e g e r s .
7 . 3 , and
f o r t h e von
A l t e r n a t e l y one may use a t w o - s o r t e d t h e o r y T L w i t h one s o r t o f
v a r i a b l e s f o r numbers and one s o r t f o r s e t s ( o r e q u i v a l e n t l y , one can u s e two unary predicates.)
A t f i r s t g l a n c e i t may seem t h a t T and T2 a r e t r i v i a l l y e q u i v a -
l e n t , b u t t h e problem i s more s u b t l e .
T 2 can be e a s i l y i n t e r p r e t e d i n T.
But
t h e c o n v e r s e i s more d i f f i c u l t , s i n c e T2 d o e s n o t n e c e s s a r i l y p r o v e t h e e x i s t e n c e of t h e von Neumann i n t e g e r s .
However, i f T c o n t a i n s c o l l e c t i o n , t h e n T 2 does
51
CONTINUITY IN INTUITIONISTIC SET THEORIES
prove the existence of the von Neumann integers, and T can easily be interpreted 2 in T In particular, the application we make of T2 in the consistency proofs of
.
97 fall under this remark. Errata: (1) In Lemma 0.2 of [Bl],page 260, the hypothesis should state that for i, j 5 k , we have p (a.,a,) i (1/4i) 1
3
+
(1/4j). When the lemma is applied on page 298,
we may assume that b satisfies this hypothesis, by replacing bn by b4n
.
(2) Theorem 2.4 of [El], p. 303, is correct, but something must be added to the proof at line 26, for as it stands, the proof applies only if X' is provably compact, which we could only assure in general if X is locally compact. (This is related to a defect of Bishop's definition of continuity pointed out by Hayashi: continuity on compact sets does not guarantee pointwise continuity unless the space is locally compact). T o correct the proof, we appeal at line 26 to the rule of local uniform continuity with a parameter X' for a compact subspace of X. Note that a compact subset X ' of X can be coded as a function from N to N, since it is given by a function assigning to each rational
E
> 0 a finite E-approximation to X', and
since X is in standard form, each member of X is a sequence of integers. Thus, in the notation of Section 2.8 of [BlI,Q(e) ++
e codes a compact subset of X is an
allowable choice of a set of parameters. In Section 2.8, the rule of local continuity with parameters is derived; the rule of local uniform continuity with parameter may be similarly treated.
References B1
M. Beeson, Principles of continuous choice and continuity of functions in formal systems for constructive mathematics. Annals of Math. Logic 12 (1977) 249-322.
B2
-,
B3
---,
B4
-,
B5
-,
Bi
E. Bishop, Foundations of constructive analysis, McGraw-Hill, 1967.
Fe
S. Feferman, A language and axioms for explicit mathematics, in Algebra and Logic, Lecture Notes in Mathematics No. 450, Springer-Verlag.
Frl
H. Friedpan, The consistency of classical set theory relative to a set theory with intu+tionistic logic. JSL 38 (1973) 315-319.
Fr2
-,
On the area of harmonic surfaces, Proc. AMS 69 (1978) 143-147.
Extensionality and choice in constructive mathematics, to appear in Pacific J. Math. Goodman's theorem and beyond, to appear in Pacific J. Math.
The non-continuous dependence of surfaces of least area on the boundary curve, Pacific J. Math 70 (1977) 11-17.
Set-theoretic foundations for constructive analysis, Annals of Math. 105 (1977) 1-28.
52 Fr3
M. BEESON __ , Some applications of Kleene's methods for intuitionistic systems, in Proceedings of the Summer Logic Conference at Cambridge, August 1971, Springer-Verlag.
M1
J. Myhill, Constructive set theory, JSL 40 (1975) 347-383.
M2
-, New axioms for constructive mathematics (preliminary report).
M3
-, Some properties of intuitionistic Zermelo-Frankel set theory, in Proceedings of the Summer Logic Conference at Cambridge, August 1971, 206-231.
T
F. Tomi, A perturbation theorem for surfaces of constant mean curvature, Math. Zeitschrift 141 (1975) 253-264.
Tr
A.S. Troelstra, Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics No. 344, Springer-Verlag, Berlin.
LOGIC COLLOQUIUM 78 M. Boffa, D . van Dalen, K . McAloon (ed8.I 0 North-HoZland Publishing Company, 1979
STABLE A L G E B R A I C THEORIES Grc g o r y C h e r l i n Ru to,e r s l l n i vc r s i t y
1
A b s t r a c t : I\" s u r v e y the c l n s s i f i c a t i o n o f s t n h l e t i i c o r i e s o f m o d u l e s , r i n q s , and g r o u p s . 2 I. Introduction I11 5 1 w e q i v e a n e l e m e n t a r y i n t r o d u c t i o n t o s t a b i l i t y theory.
E3 1 . The s t a h i l i t y s p e c t r u m t h e o r e m . 1.1
set:
Let
Stahle theories. T be a f i r s t o r d e r theory, l e t
A
b e a model o f
T
,
and
.
I)ef(A) = t h e Boolean a l g e b r a o f d e f i n a h l e s u h s e t s of A tlere we t a k e " d e f i n a b l e " t o mean: f i r s t o r d e r d e f i n a h l e u s i n g p a r a m e t e r s f r o m !I. S t a b i l i t y t h e o r y s t a r t s w i t h t h e q u e s t i o n : how c a n we measure t h e c o m p l e x i t y 01 t h e R o o l e a n a l g e b r a I k f ( A ) ? To s t u d y I ) ef ( A ) i t is convenient t o pass t o i t s Stone space: SA = t h e Stone s p a c e of lkf(A) T h u s S A i s a t o p o l o g i c a l s p a c e whose p o i n t s a r e t h e maximal f i l t e r s i n Def(A) , o r t o p u t t h e m a t t e r more e x p l i c i t l y : n p o i n t p o f SA i s a c o l l e c t i o n o f d e f i n a h l e s i ~ b s e t so f A p o s s e s s i n g t h e f i n i t e i n t e r s e c t i o n p r o p e r t y p a n d s u c h t h a t p i s maximal w i t h r e s p e c t t o t h i s property. A c o a r s e measure of t h e c o m p l e x i t y of Ikf(A) is c i v e n by t h e c a r d i n a l i t y o f i t s S t o n e s p a c e SA - and t h i s s u f f i c e s f o r s t a h i l i t y t h e o r y . C l e a r l y c a r d ( S h ) t c a r d (A) ( J u s t a s s i c n t o e a c h R E A t h e c o r r e s p o n d i n g p r i n c i p a l f i 1t e r p a = a l l d e f i n a b l e s e t s i\.hich c o n t a i n a . ) C a l l a m o d e l A o f T t a m e i f f card(S:A) = c a r d ( A ) Then our basic notion is the follnwing: Ilefinition: For a n y c a r d i n a l h we s a y t h a t t h e t h e o r y T i s A - s t a b l e i f a l l models A o f 1' w h i c h h a v e c a r d i n a l i t y i, a r e t a w . h'e a l s o s a y t h a t h i s a s t a b i l i t y c a r d i n a l f o r T
.
.
.
1.2
Ex am p l es .
There a r e e s s e n t i a l l y o n l y four r e l e v a n t examples: Let T he s i m p l y t h c t h e o r y of i n f i n i t e s e t s (eqiiipped lisakple 1 . wit t h e e q u a l i t y r e l a t i o n a n d n o f u r t h e r s t r u c t u r e ) . Then f o r an y
1
Preparation of t h i s survey :;SF G r a n t :ICS76-064911 ,401
2
l 3 i h l i o j i r a p h i c a l a n 3 h i s t o r i c a l n o t e s wi 11 h e found m i n g l c d a t t h e e n d o f t h e a r t i c l e , Oath s o r t s a r e t h o r o u g h l y i n c o m n l e t e .
WAS
.
53
s u p p o r t e d by
G. CHERLIN
54
model X o f T , n e f ( X ) c o n t a i n s j u s t t h e f i n i t e s e t s a n d t h e i r c o m p l e m e n t s , t h e c o f i n i t e s e t s . The S t o n e s p a c e SX h a s i n a d d i t i o n t o t h e p r i n c i p a l f i l t e r s j u s t o n e more f i l t e r , t h e f i l t e r o f a l l c o f i n i t e s e t s . Thus T i s a - s t a b l e f o r a l l h
.
Example 2 . with
Let
T
he t h e t h e o r y o f s e t s
independent suhsets
No
w h i c h a r e eq u i p p e- 1
I'
[ r e p r e s e n t e d i n t h e language of
P,
T
by u n a r y p r e d i c a t e s ) . The i n d e p e n d e n c e c o n d i t i o n means : (ind) G i v e n two d i s j o i n t f i n i t e s e t s o f i n d i c e s I , J , t h c n
0 pi
/7 ( p - 1 3 ~ )
A
Then t h e d e f i n a b l e s e t s i n a model B o o l e a n a l g e b r a ) by t h e s e t s dence c o n d i t i o n Pi
i s n o n em p t y .
j EJ
i E 1
P
of
will b e g e n e r a t z d ( a s a
T
and t h e f i n i t e s e t s . T h e i n d e p e n -
PI,
i s j u s t t h e f i n i t e i n t e r s e c t i o n of t h e s e t s
(ind)
and t h e c o m p l e m e n t s o f t h e r e s t o f t h e s e t s
f'i
.
This immediately
g i v e s us ZNo e x t r a e l e m e n t s o f t h e Stone s p a c e SP , r e y a r d l e s s of the c a r d i n a l i t y of P I t i s t h e n e a s y t o see t h a t T i s h - ~ t a l i l e For h 3 2"' , a n d A - u n s t a h l e f o r A < 2'
.
ExanipleL
T
Lct
.
be t h e t h e o r y o f s e t s
I'.
cqriipperl w i t h
No
equivalence r e l a t i o n s such t h a t : h a s i n f i n i t e l y many e q u i v a l e n c e c l a s s e s .
1.
El
2.
En+,
subdivides each equivalence class of
into
En
i n f i n i t e l y many e q u i v a l e n c e c l a s s e s . F or E a model o f T t h e B o o l e a n a l E e h r a l ) e f ( E ) i s g e n e r a t e < ! b y the e q u i v a l e n c c c l i s s c s o f t h e v a r i o u s En toSether with the f i n i t e s e t s . The m ai n way t o c o n s t r u c t c l c i n e n t s o f t h e S t o n e s p a c e SE i s t o c h o o s e a s e q u e n c e o f s e t s C , 2 C2 2 C 3 s o t h a t e a c h Ci i s
...
a n E i - e q u i v a l e n c e c l a s s , and t h e n t o e x t e n d { C i l p r i n c i p a l f i l t e r . \ow i f E h a s c a r d i n a l i t y h
t o a maximal n o n then the c a r d i n a l i t y
o f SE w i l l d ep en d somewhat on t h e s t r u c t u r e o f E , b u t i n t h e w o r s t c a s e i t w i l l be p o s s i h l e t o c h o o s e e a c h o f C,,Cz,C5, in h
...
different
ways, producing
d i f f e r e n t elements of
AH'
t h e n e a s y t o show t h a t t h e s t a h i l i t y c a r d i n a l s f o r those cardinals
h
satisfying: hH0 = h
T
SI1
.
I t is
are cxactly
. .
Example 4 . Let T he t h e t h e o r y o f d e n s c l i n e a r o r d e r i n q s 2 'Then f o r a n y mode1 Q o f T t h e 5 o o l e n n a g e h r a k f ( n ) i s g e n e r a t e d by t h e r a y s :
.
( - - , a ) and ( a , - ) w h e r e a E Q a n d -0 < Q < The e l e m e n t s o f S? a r e b a s i c a l l y k d e k i n t l c i i t s , an d t h e r e a r e inany o f them. I n f a c t T i s A - u n s t a h l e f o r a l l A , w h i ch t r a n s l a t e s i n t o
55
STABLE ALGEBRAIC THEORIES
a classical fact: For a l l X t h e r e i s a dense l i n e a r o r d e r i n g
Q of c a r d i n a l i t y X s i t t i n g as a d e n s e s r i h s e t o f a l i n e a r o r d e r i n g o f c a r d i n a l i t y g r e a t e r than A
.
I t r e m a i n s t o be s e e n i n w h a t r e s p e c t t h e s e e x a m p l e s a r e t y p i c a l . 1.3
'[he s t a h i l i t y s p e c t r u m t h e o r e m .
For a f i r s t o r d e r t h e o r y T i n a c o u n t n h l e l a n g u a w t h e s e t o f T m u s t h e a s i n on e o f t h e f o u r e x a m p l e s
stability cardinals for a hove : 2.
a l l cardinals. a l l c a r d i n a l s from
3.
j u s t c a r d i n a l s o f t h e form
1.
2'0
on
. X
= Xxo
.
no c a r d i n a l s . T h i s i s t h e c o n t e n t o f t h e s t a h i l i t y s p e c t r u m t h e o r e m , w h i c h may be 4.
r e f o r m u l a t e d t o a p p l y t o ~ l n c o r i n t a b l e la n g ! u aq es a s f o l l o w s . C a l l a t h e o r y T s t a b l e i f i t i s A - s t a b l e f o r some X , an d u n s t a b l e o t h e r wise. F o r for
T
.
T
stable, let
6(T)
he the smallest s t a h i l i t y cardinal
( S t a h i l i t y S p e c t r i m Theorem). I f T is s t a h l e then there i s a c a r d i n a l K(T) such t h a t t h e s t a h i l i t y c a r d i n a l s X f o r T a r e c h a r a c t e r i z e d by : 1. X 2 6(T)
Theorem 1
2.
Theorem 2
X
6(T) 3 ?card(T)
and
K(T) 5 c a r d [ T ) +
.
T h i s c o r r e s p o n d s n i c e l y t o t h e s i t u a t i o n i n Examples 2 , 3 ahove.
.
is t h e s t a b i l i t g exponent f o r T T is said T i s s t a h l e f o r a l l c a r d i n a l s ahoi-e b(T) , w hi c h i s e q u i v a l e n t t o t h e c o n d i t i o n t h a t T h a v e c o u n t a b l e s t a h j l i t y exponent. ' T r a d i t i o n d i c t a t e s t h a t we w r i t e " w - c t a h l e " f o r " S o - s t a h l e " I t is a basic fact t h a t W-stahility implies s t a h i l i t y in a l l cardin a l s ; t h i s c a n he r e a d o f f f r o m Theorem 1 U n f o r t u n a t e l y a s k e t c h o f t h e p r o o f o f t h i s t h e o r e m woirld l e a d u s t o o f a r f r o m o u r main t o p i c . 'The b a s i c i d e a o f t h e p r o o f may h e
Terminology:
K(T)
t o be s u p e r s t a b Z e i f
.
.
summarized a s f o l l o u s . Obviously t h e f i r s t s t e p i s t o qet a manageahle d e f i n i t i o n o f t h e i)nce t h i s h a s h e e n d o n e , o n e i s o h l i q c t l t o p r o v e two invariant K(T) theorem :
.
56
G. CHERLIN
The I n s t a h i l i t y C a r d i n a l T h e o r e m . thnt: ,IK > x then
is
T
, tlien
= A
K
<
K(T)
and
X
such
A-unstahle.
'The S t a h i l i t y C a r d i n a l 'Thcnrcm. ,I<'('~)
Civen
Roiigiilv
.r
is
speaking,
Given
,I >
h-stni>Ic. '[T) is Jcfinetl
RT-
siicli t h a t
6(T)
the largest cardinal for
which i t i s e a s y t o p r o v e t h e I n s t a b i l i t y C a r d i n a l Theorem, a f t e r w h i c h one d i s c o v e r s t h a t w i t h c o n s i d e r ~ h l ee f f o r t t h e S t a l l i l i t y C a i - d i n a l Theorem c a n h e p r o v c d . 'This c o n c l u c i e s o u r i n t r o d u c t i o n t o s t a h i l i t y t h e o r y . I t w o u l d b e i n t e r e s t i n g t o discriss apnlications of s t a h i l i t y theory t o s p e c i f i c p r o b l e m s , e . : . t h c s t r u c t i r r e o f d i f f e r e n t i a l l y c l o s e d f i c l t l s , h u t we will omit t h i s t o p i c , r c f c r r i n n t h e r c a d e r t o 1 2 , 2 0 1
.
The work o f
G a r a v a g l i a C101 i s i n t e r e s t i n g i n t h i s c o n n e c t i o n .
I 2. Stahlc alFchraic theories. IZe c o n s i d e r t h c f o l l o w i n g q i i c s t i o n :
Let
A
b e a modrile, n r i n g , o r a g r o u p . I f t h e t h e o r y o f
s t a b l e , t h e n w h a t can he s a i d
about t h e structirre o f
,I i s
A ?
I n t!ie p r e s e n t s e c t i o n u c s u m m a r i z e many o f t h e known r e s u l t s . The
folloir.in< convention i s useiril: i f a striictiire t n e o r y . we s a y more b r i e f l y t h a t
A
has A-stable
,2
i s A-stnhle.
2.1 Modules. The h n s i c f a c t i s :
'Theorem 1 .
All m o d u l e s a r e s t a h l e .
'This does n o t r e a l l y t r i v i a l i z c t h e h a s c q u e s t o n , s j n c e one a l s o w a n t s a c h n r a c t e r i z a t i o n o f s u p e r s t a b l e and w - s a h l e m o d u l e s . Such c h a r a c t c r i z a t i o n s have hecn foiind; t h e y i n v o l v e d e s c e n d i n g c h a i n c o n d i t i o n s , and w i l l h e p r e s e n t e d u n d e the next section.
t h a t headin!!
a t t h e end o f
2.2 I
matrix rings
'.ln(l))
o v e r s t a h l c d i v i s on r i n q s
n
.
S t a h l e d i v i s i o n r i n g s have n o t heen c l a s s i f i e d , s o w e must settle for less: A s u p e r s t a b l e d i v i s i o n r i n q i s f i n i t e dimension;il over Theorem 4 . i t s c e n t e r (which i s a s u n e r s t a h l e f i e l d ) .
Theorem 5,
An w - s t a b l e f i c l d i s a l g e h r a i c a l l y c l o s e d o r f i n i t e .
57
STABLE ALGEBRAIC THEORIES
'The d a y
b e f o r e t h i s t a l k was d e l i v e r e d S h e l a h completed t h e
p r o o f t h a t Theorem 5 a p p l i e s a l s o t o s u p e r s t a b l e f i e l d s , s o we c a n combine Theorems 3 . 4 , and t h e s t r e n n t h e n e d Theorem 5 t o Ret: :1 s e m i s i m p l e s i i p e r s t n h l e r i n g i s t h e d i r e c t sum o f a f i n i t e r i n g a n d f i n i t e l y many m a t r i x r i n g s Mn(F) o v e r a l g e b r a i c a l l y
Corollary
.
.
F
closed f i e l d s
"4 will p r o v e T h e o r e m 5 i n S 5 u s i n q t e c h n i q u e s d e v e l o p e d i n t h e
study o f w-stable groups; t h i s i s t h e argument t h a t g e n e r a l i z e s t o t h e s u p e r s t a b l e case. The c l a s s i f i c a t i o n o f s t a h l e f i e l d s a n d d i v i s i o n r i n g s i s v e r y much a n o p e n p r o h l e m . .r\ f e w n o n t r i v i a l s t a h l e f i e l d s a r e known, n a m e l y the separably closed f i e l d s . I t is certai:ily possible t h a t s t a b i l i t y alone implies sepnrahle closure.
.
The l a r g e s t p a p i n t h e r e s u l t s a h o v e l i e : i n Theorem 2
.
Problem What c a n h e s a i d a h o i i t a n i l n o t e n t r i n q i f i t i s known t o -~ be s t a h l e ( o r s u p e r s t a h l e . o r w - s t a h l e ) ? A s i m p l e c o n s t r u c t i o n shows t h a t t h i s prohlem i s a t l e a s t d i f f i c u l t . Let A l r A 2 . A 3 h e a h e l i a n g r o u n s a n d l e t R : A1 x Az
be a b i l i n e a r map.
Ih'e
-C
A3
associate to
R = A1
ci)
6
a rinq
defined by:
T:
A2 A A3
as an a d d i t i v e group, w i t h t h e f o l l o w i n g m u l t i p l i c a t i o n : ( a 1 , a ~ , a 3 ) ( b l , h 2 , h 3 ) = I Q , n 9 8 ( a 1 , h 2 ) + R(b1,aZ)) T h i s r i n g i s n i l p o t e n t : xyz = 0 f o r any x , y , z in
". .
commutative. Furthermore i f t h e s t r u c t u r e c o n s i s t i n c of e q u i p p e d w i t h t h e rrlnp t h i s 1 - s t 3 h i 1i t y
.
n
i s a A-stahle s t r t i c t u r e then
To a p p l y t h i s c o n s t r u c t i o n , t a k e ;Iny 1 - s t a h l e r i n g f o r t h e map
R
t h e m u l t i p l i c a t i o n map o n 12 : 11 : A x A + A , f < ( r , s ) = r s
. !I
i s even
41 ,.12 ,A3 will inherit
!:
A
and t a k e
.
Then t h e a b o v e c o n s t r u c t i o n p r o d u c e s a new r i n g q v l i o s e u n d e r l y i n : s e t i s !Id 17 d A T h i s s u g 2 e s t s t h a t tlic c l a s s i f i c a t i o n o f s t a b l e
.
n i l p o t e n t r i n q s o f cxnonent 3 i s a l r e a d y very d i f f i c u l t . Croups. rhe m e t h o d s n v a i l a l , l e f o r tlic c l a s s i f i c a t i on o f s t a h l e Groups will b e d c s c r i h e t l i n < : e t a i l he lo^. Thc f o l l o ~ ~ ~ i snpf ei c i a l r e s u l t s h a v e b e e n o b t a i n e d , amonq o t h e r s : I f G i s nn K , - c n t e g o r i c a l { J - s t a b l c g r o u p t h e n G i s 'Thcorem 6 . :ibe 1i nn -h y - f i n i t e 2.3
.
58
G.
CHERLIN
( A group i s a h e l i a n - h y - f i n i t e
f i n i t e index.) Theorem 7 .
If
G
.
n i l p o t e n t -by- f i n i t e Theorem 8 . If G
i s an
i f i t h a s an a h e l i a n s u h g r o i i p o f
Ho-categorical
s t a b l e croup then i t is
i s a s t a b l e l o c a l l y n i l p o t e n t croup then
solvable. The t e c h n i q u e s c l i s c u s s e i l h e l o w
\\ere
is
G
ilevclonec' i n t h e c o u r s e o
proving these three r e s u l t s . Some f u r t h e r r e s i i l t s o n qroiips o f *llorlcv r a n k a t mos
threc
11
iv
h e d i s c u s s e d i n $ 6 (we i n t r o d u c c ' l o r l e y r a n k i n S 3 . 1 ) . :$hat a r e t h c n r o s p e c t s f o r o l l t a i n i n f i sonic ::enera1 r e s u l t s ab o i i t s t a b l e (superstahlc, W - s t a h l e ) groups? This discussion i n 5 6 w i l l i n d i c a t e o n e d i r e c t i o n i n w h i c h such r e s i i l t s m i T I i t l i e . Q n t h e neq:it i v c s i d e , j u s t as i n thc case of r i n c s , i t i s a f a c t o f l i f e t h a t t h e r e a r e many e x a n p l e s o f s t a h l e g r o u p s , a n d h e n c e C e n e r a l r e s u l S c a n n o t he t r i v i a l . The m ai n s o u r c e o f s t a b l e g r o u p s i s t h e f o l l o w i n g .
Let R b e a
s t a b l e c o m m u t a t i v e r i n g w i t h i d e n t i t y . Thcn a n y o f t h e u s u a l m a t r groups taken with c o e f f i c i e n t s i n
.
R
X
will i n h e r i t t h c s t n h i l i t y
p r o p e r t i e s o f I< Q n c e x a m p l e , c x p l o i t e d h y \ l a l ' c c v , i s o b t a i n e d Iiy using upper t r i a n g u l a r iinipotent 7 x 7 matrices. i . e . :
For t h i s p a r t i c u l a r example i t i s n o t n e c e s s a r y t o havc commutative, o r even a s s o c i a t i v e .
R
I I. w-stable qroups.
!Ve w i 11 d e s c r i h e t h e s p e c i f i c mode 1 - t h e o r e t i c m a c h i n e r y applicahle t o the s t r u c t u r a l a n a l y s i s o f d - s t a h l e groups. I n 111. a p p l i c a t i o n s o f t h e s e t e c h n i q u e s a r e discrisscd.
5 3. Chain c o n d i t i o n s f o r s t a h l e qroups_ 3.1 'The w - s t a h l e n C C .
-D e f i n i t i o n .
'The qrorip
G
s a t i s f i e s t h e o ~ - s t a b Z e D C C ( = t l escen t l i n f i
chain condition) i f f every descendins chain of definahlc suhqroups o f
G
is finite.
Theorem 1 .
If
G
i s W-stable then
G
s a t i s f i e s t h e W - s t a b l e DCC
The p r o o f o f t h i s t h eo rcm r e s t s on m o d c l - t h e o r e t i c m achi ner y: :lorley rank: I f A i s any W - s t a h l c s t r u c t i i r e then i t i s p o s s i b l e t o a s s i g n
59
STABLE ALGEBRAIC THEORIES
t o each definable suhset a = rank(S)
of
S
c a l l e d t h e (Morley) rank o f
an o r d i n a l :
A
i n s u c h a way t h a t t h e f o l l o w i n g
S
a xi om s are s a t i s f i e d : (mor ) I f S c o n t a i n s i n f i n i t e l y many m u t u a l l y d i s j o i n t d e f i n a b l e
...
t h e n f o r some i r a n k ( S i ) < r a n k ( S ) * s u b s e t s S1 , S 2 . I f S i s d e f i n e d i n A by t h e f o r m u l a $ ( x ) , i f A ' is an
(ee 1
elementary extension of
A
,
and i f
S'
is the subset of
.
d e f i n e d b y t h e same f o r m u l a , t h e n r a n k ( S ) = r a n k ( S ' ) I n d e e d , i f we t r y t o a s s i g n t o S t h e l e a s t o r d i n a l c o m p a t i b l e A'
w i t h ( m o r ) an d ( c e ) t h e n a n i n d u c t i v e d e f i n i t i o n o f r a n k e m e r g e s . (In p a r t i c u l a r t h e rank
0
s e t s w i l l b e j u s t t h e f i n i t e s e t s . ) The
o n l y p r o b l e m i s t o show t h a t e v e r y s e t S e v e n t u a l l y d o e s g e t a s s i g n e d a r a n k , and t h i s i s proved u s i n g t h e w - s t a b i l i t y o f A
and
t h e f o l l o w i n g " s p l i t t i n g argument" : I f we suppose t h a t show t h a t subsets
S &
A
remains unranked, then it is easy t o
s p l i t s a s t h e d i s j o i n t u n i o n o f two u n r a n k e d d e f i n a b l e
S
Sl,S2
.
I f we s p l i t
i n t h e same way an d i t e r a t e t h i s
S1,Sz
c o n s t r u c t i o n i n d e f i n i t e l y , w e w i l l o b t a i n a h i n a r y b r a n c h i n g t r e e of unranked d e f i n a b l e sets:
w he r e e a c h s e t i n t h e t r e e i s t h e d i s j o i n t u n i o n o f t h e p a i r b e l o w .
i s a countahle elementary substructure o f A such t h a t , t h e n j u s t a s i n Example A, N 3,§1 t h e p a t h s through t h i s tree w i l l produce 2 0 d i s t i n c t elements o f t h e S t o n e s p a c e SAo , c o n t r a d i c t i n g t h e w - s t a b i l i t y o f A If
A,
e v e r y set i n t h i s t r e e i s d e f i n a b l e over
.
Supplement: Vorley d e g r e e . i f A is w - s t a b l e and S I A i s d e f i n a b l e a n d of t h e n i t i s e a s y t o show t h a t t h e r e i s a l a r g e s t i n t e g e r t h a t i n some e l e m e n t a r y e x t e n s i o n o f A : (deg)
S
c a n h e decomposed i n t o t h e d i s j o i n t u n i o n o f
.
rank a d such d
,
defin-.
a b l e suhsets o f rank a The T h i s m ax i m al i n t e g e r d i s c a l l e d t h e Y o r l e y d e g f e e o f S f o 11o w i n g i s s t r a i gh t f o r w a r d : (sum) I f S o f r a n k a i s t h e d i s j o i n t u n i o n o f S1 a n d S 2 d e f i n a b l e s u b s e t s of rank a , then : degree(S) = degree(Sl) + degree(S2)
.
.
,
G. CHERLIN
60
Let
he t l c f i n n h l e subgrorir,s o f t h e w - s t a l i l c C? t h c n : o r e l s e tile r a n k s a r e e q u a l e i t h e r rank(l;,) < raiik[C,) and then tleqrce(C1) < depree(G2)
Lcnnin 2 .
structiirc
.\
G1
.
If
C_
c1
I;?
+
r ' r o or: l'!ic r i g h t c o s e t s o f G 1 i n G 2 a l l h a v e t h e same r a n k arid d e r r e e ( t h e y c a n be i c l e n t i f i c d liy nienns o f d e f i n a b l e 1 - 1 c o r r e s p o n i l e i i c c s ) . Tlic lemma t h e r e f o r e f o l l o w s f r o m ( s u m ) a b o v e a n d tlie f a c t t h x t t h e r e a r e a t l e a s t two s u c h c o s e t s . P r o o f o r Theorem 1 Immctliate Crom 1,ciiimn 2
.
.
3 . 2 l'hc s u p e r s t ; i h l c DCC
Ilefinitiqn.
1 group
I ;
s a t i s f i e s t h e supez9stabte DCC i f f t h e r c i s
no i n f i n i t e . ~ . l e s c e n t l i nc~h a i n of
I ;
CGi
a r e .a11 i n f i n i t e . If
Theorem 3 . i)CC.
Gl
> I;?>
siic!i t h a t tlie i n d i c e s : : G.
1+1
of d e f i n a h l e suhqroups
1
is superstahle then
C
...
G
s a t i s f i e s the sunerstable
'This d c p e n l s on t h e f o l l o i ~ ~ i n m f io r l e l - t h c o r e t i c m a c h i n e r y :
Shelah rlegrcci I f A i s a n y s u p e r s t a h l e s t r u c t i ~ r ct h e n i t i s p o s s i b l e t o n s s i y n t o e a c h d e f i n a h l e s u h s e t S o f A and o r d i n a l : rr
= Deg(S)
called the [Shelah) dcnree o f
i n s u c h a way t h a t t h e f o l l o w i n g
S
axioms a r c s a t i s f i e J I j a r g o n t o he c x n l i c a t e J b e l o w ) : (she)
I f S c o n t a i n s v e r y many n - m u t u a l l y d i s j o i n t u n i f o r m l y d e f i n a b l e s u b s e t s S 1 ,S2,. t h e n f o r some l)eg(Si) < (r)eg(S1 ( e c1 as f o r ' l o r l e y rank 'Ye r m i no 1o a y :
.
1 . n-niutually d i s j o i n t :
.
..
t h e i n t e r s e c t i o n of any
n
i
o f them
i s emqty. 2 . i i i i i f o r m l ~ i _ d c f i n a b l e : t h e r e i s a s i n p l e formiila
d(x,y)
s u c h t h a t e a c h o f tlic s e t s S i i s d e f i n a h l e h y R f o r m u l a o f t h e form $(x,,:) for a s u i t a b l e choice of the elements
-
a
2 s i n ti
in
2 .
3 . 1 a s p l i t t i n g a r g u m e n t shows t h a t e v e r y d e f i n a h l e s e t
c a n bc a s s i g i i c d a S h e I a h d e p r e e . 'Then Theorem 3 f o l l o w s d i r e c t l y f r o m t h e f o l l o w i n g f a c t , whose p r o o f t h e r e a d e r s h o u l d s u p p l y :
61
STABLE ALGEBRAIC THEORIES
-Lemma -
If
4.
G,
a r e r l c f i n a h l e siihjirou: s o f t h e s i i p e r s t a h l e
G2
CG, : G 2 1 i,: i n f i n i t e t h e n t h r S h e l a h d e g r c c o f i s s m a l l e r t h a n t h a t o f G, .
structure G2
! I
anti
3 . 3 The s t a b l e c h a i n c o n d i t i o n s .
Definition.
1.
s a t i s f i e s t!ie o b v i o u s s t a b l e CC i f f a n y c h a i n
I;
o f u n i f o r m l y d c f i n a h l e s i i h ~ r o u p so f
is f i n i t r .
G
2. G s a t i s f i e s t h e ( f u l l ) s t a b l e C c i f f a n y c h a i n of i n t e r s e c t i o n s o f m i f o r n l y d e f i n a b l e s u h g r o u n s o f (; i s f i n i t e . Example. T h e c e n t r a l i z e r s C!?) of s i n z l c elerrcnts g i n I; a r e u n i f o r m l y d e f i n a b l e , and t h e i r i n t e r s e c t i o n s a r e t h e c e n t r a l i z e r s o f arliitrary suhsets of
G
.
'Thus t h c stable C C i n c l u d e s a c h a i n c o n d i -
t i o n on c e n t r a l i z e r s . Aiiothcr esamnlr nccurs i n 6 5
.
'I'lic r o l l o w i n g r e s i i l t i s n o n t r i v i a l c v c n f o r s i i n e r s t a h l e Z r o u p s . fheorcrri 5 . I f G i s s t a l i l c t h e n G s a t i s f i e s tlic s t a h l c (:C ilcmark.
I.ct
n
he a s t a h l e s t r l . i c t i i r c ,
,I
i l c f i n a b l c h i n a r y r e l a t i o n hctir.ccn n - t i i n l c s linearly
-P r o o f :
orklci- a n y i n f i n i t c s u ! > s c t o f
,
I?
an i n t c q e r , and
in
,ln
1
.
.
Then
i1
I<
:I
cannot
(sketch) :
,An e l e n c n t n r y arniinicnt shows t h a t
.1
.
w i l l sketch a n r o o f o f t h i s t ! i e o r e r i .
:lye
;in(l h e n c c
may t a l i c
KC
n = 1
I i n c a r I y o r d c r s n n i n f i n i t e srilrsct
a n e l e m e n t a r y e x t e n s i o n 1'
or
In
i n h ! r i t 5 tlic s t . i ! , i l i t y o f
~ i t l i n 1 1 tl o s s o f centrality. r h e n i f 1
of
Y
i t is
I
i n iiliich
CRSY
t o find
1inc:irly o r d e r s
I?
:I
d c n s c l i i i e a r o r d e r i n g o f a n y c!csirck! f o r v . 'l'!icn t h e n c i l c k i n J c u t s i n t h i s o r d c r i n g may be use.1 t o s o n s t r i i c t e l e m c n t s o f t h c
, c o n t r a t l i c t i i i ~ :t h e s t n h i l i t y o f
1 (cf.
iisaiiililc 4
/It t h i s y o i n t wc see t h n t t h c o l ) v i o l r s s t n h l c (:C o b v i o u s f o r s t a h l e !:rou?s. 1 y J c f i n : i l ) I c !:roiips
5 1 S
Trir!cc,l, a n i ! i T i n i t c c o i l c c t i o n o f i i n i forril-
1 inc:trly nr,icrc,l
nl)lc 1inc:ir orderin:
,
really
iiii(!cr
i n c l i i s i o n incliiccs a t i e f i n -
o f t h r n : i r a m t c r s iisc(l to , l c f i n e t h e !!rouijs,
c o n t r a d i c t i n g tlic a h o v e rcninrli. Tlic p o i n t i n t h e p r v o f o r t h ?
rccliiced t o t h c ol.,vioiis s t r i h l c CC I,crnn;a 6 .
I.ct
s t a b l e groap
h e n f a ~ i l vo f iiriiToi-vnly J c T i n n h l e sul~r:ro11!3so f a
F G
.
'I':icn t i i e r c i s a n i n t c ; : c r
scctiori o f r l c v c n t s o f of F. Corol 1 a r y
5
Vi
f u l l s t n i i l r CC i s t h a t i t c a n he
, nnmcly:
ti1
F,G
F
11
siich t h a t a n y i n t e r -
i s i n fact nn intcrscction o r a s ;ihovc l e t
t r n r y i n t e r s e c t i o n s o f !:rotips
ticlonoin:
n
clcacnts
F'
h c t 4 c f;imi l y o f n r h i -
to
1:
.
Thcn t h e q r o u p s i n
G. CHERLIN
62
F’
a r e uni formly d e l i n a h l c .
C l c a r l y I.emma h q i v c s t h e c o r o l l a r y , a n d t h e c o r o l l a r y r e d u c e s t h e s t a h l e C C t o t h e o h v i o r i s C C . T h e p o i n t t h e n i s t o n r o v e Lemma 6 . T h i s J c p e n d s on t h e f o l l o i i i n g , w h i c h c a n he p r o v c \ l d i r e c t l y on t h e h a s i s o f 7’ 1ki:iark. If -_
>
A
.
(Proof omitteil].
s n s t a h l e s t r u c t u r e ant1
F
i s a family o f inde-
p e n d e n t u n i f o r m l y d c f i n a h l e s l i h s c t s o f :I t h e n F ( I n d e p e n d e n c e was d e f i n c r l i n 1:xnmplc 2 , S I . ? .) P r o o f o f Lemma h
is finite.
skctch) :
we c a n f i n d n i n d e p e n d e n t g r o u p s i n F , t h e n i n a n e l e m e n t a r y e x t e n s i o n o f :I we c a n f i n d an i n f i n i t c s e t o f i n dependent u n i f o r m l y d c f i n a h l e groiips, c o n t r n d i c t i n q t h e p r e c e d i n g I f f o r each
n
r e n a r k . Thus € o r some
we c a n n o t f i n d
n
n+l
independent groups.
‘Ke c l a i m e v e r y i n t e r s e c t i o n o f e l e m e n t s o f t o an i n t e r s e c t i o n o f
n
elements of
o h t a i n an i n t e r s e c t i o n o f n + l qrouns n o t a n i n t e r s e c t i o n o f a n y n o f them
.
F
F can he reduced 4 s s u m i n g t h e c o n t r a r y , we
i n F which i s I t s u f f i c e s t o show t h a t
Gl,...,Cn+l
.
G i a r e i n d e p e n d e n t . I f one t h i n k s o f t h i s as a v a r i a n t o f t h e C h i n e s e R e m a in d e r Theorem w i t h t h e Gi p l a y i n g t h e r o l e o f maximal i t l e a l s t h e n t h e p r o o f i s e v i d e n t . t h i s implies t h a t the oroups
3 . 4 C l a s s i f i c a t i o n o f modules.
IVe w i l l Xive t h e c l a s s i f i c a t i o n o f s u p e r s t a h l e and W - s t a h l e Fo r
modu l es w i t h o u t p r o o f . C o n s i !er m o d u l e s o v e r a f i x e d r i n z R a ny s y s t e m E o f l i n e a r e q u n t i o n s i n v o l v i n c . p a r a x e t e r s From
.
R , an from ‘7 , and unknowns x l , . ,xk l e t ‘I(E) d e n o t e t h e s e t o f p a r a m e t e r s m i n ‘~1 f o r w h i c h t h e s y s t e m E becomes s o l v a h l e i n ‘1 upon s e t t i n g x = m Then Y ( E ) is a s u b g r o u p o f ’1 Let o = ot! d e n o t e t h e f a m i l y o f a l l s u c h s u b g r o u p s o f 21 Theorem 8 . The f o l l o w i n g a r e e q i i i v a l e n t : 1. ‘1 is W-stable. 2. ’7 s a t i s f i e s t h e d c s c c n d i n ~c h a i n c o n d i t i o n r e l a t i v e t o groups i n P T h e a n a l o g o f Theorem R f o r s i i n e r s t a h i l i t y i s a l s o c o r r e c t .
unspecified parameter
.
..
x
.
.
.
6 4 . C o n n ect ct l g r o i i n s . 4.1
Ilcfinition.
The i d e n t i t y c o m n o n e n t . l h c ?roup
G
i s c o n n e c t e d i f € i t c o n t a i n s no d e f i n a b l e
proper subqroun o f f i n i t e index.
63
STABLE ALGEBRAIC THEORIES
Definition.
If
f i n i t e i n d e x in
we may s e t :
11 =
contains
C;
C
, GO
cnnncctcd d e f i n a b l e suhgroup
R
t h c n i t i s e a s y t o see t h a t
II
of
II
i s unique, s o
.
We e m p h a s i z e t h a t Go may n o t e x i s t . !\'hen C" e x i s t s we s a y t h a t G i s c o n n e c t e d - b y - f i n i t e . Y o t c t h a t Co i s n o r m a l i n C; when i t exists. Thcorem 1 . I f C; i s w - s t a h l e t h e n C; i s c o n n e c t e d - h y - f i n i t e . P r o o f : ___ 'This i s a n a p p l i c a t i o n o f t h e w - s t a h l c !)CC.
If
G
i s not itself
c o n n e c t c d t h c n i t 113s a d i s c o n n e c t i n g s u b q r o u p i I ( i . c . a d e f i i i n h l c p r o n c r s u b g r o u p o f f i n i t c i n d c x ) . I F 11 i s n o t c o n n e c t e d i t h a s n d i s c o n n e c t i n ! : suhj:roiin 11' [ : o n t i n u i n ? i n t h i s manlier wc c o n s t r u c t a c h i n o f d c f i n : t l > l c s u h n r o u p s o f C, '\'here t h e c h a i n s t o p s we s i l l f i n d t h e i d e n t i t y c o m n o n e n t o f C; ( I n a s i m i l a r v e i n t h e st:il>lc C C c a n be iiscci t o n r o v e : I f C, i s s t a h l c a : i d H - c n t c q o r i c a l t h e n G i s c o n n e c Theorem 2 . ted-by-fini te. 'This i s a u s c f u l f i r s t s t e p i n t h e p r o o f o f Theorem 7 o f 5 2 . 3 ) T h c r c i s a n o t h e r v a r i a t i o n on t h i s theme w h i ch p l a y s an important role in B 5 : Theorem 3 . 1,et I: h e ;in i n f i n i t e s t a h l e f i c l d . 'Then t h e a d d i t i v e c r o u p o f I' i s c o n n e c t e d .
.
.
.
? roof : -
F
.
Suppose .I i s R r ! c F i n a h l e a d d i t i v e s i i l i ~ r o i io~f f i n i t c i n d e x i n ITc \ \ , i l l show t h a t I = I:
.
For a n y n o n z e r o r l c v e n t
a
of
thc scalar multiplc
1:
a o a i n an : i t l d i t i v e s u h y i - o u p o f f i n i t e i n i t c x i n i n t e r s e c t i o n of a l l groups i n
0
.
Sy
I:
.
t h e s t a h l e CC
Let A.
is
a:\
be t h e
A.
is
R
finite
i n t e r s e c t i o n o f siicii q r o u p s , and as s u c h i s o f f i n i t e i n d e x i n ilowevcr b y c o n s t r u c t i o n t h e n r ! r li ti v e
yroup
2,
I:
.
is closed under
m u l t i p l i c a t i o n b y e l e m e n t s of F , i . c . i s n n i t l c a l o f I: i s I: o r (0) , a n d s i n c e (0) i s n o t o f f i n i t e i n d e x i n h a v e :lo = F , Iiencc .1 = F , a s d e s i r e d .
. Th u s F
;lo
we
4.2 Consequences o f c o n n c c t e d n c s s . !Ye h a v e s e e n t h n t c o n n c c t e t l q r o u n s a r i s e i n c a s e s o f i n t e r s t . :Yc will now e s t a h l i s h some o f t h e i r p r o n e r t i c s .
Theorem
4
( S u r j c c t i v i t y T h c o r e n i) .
s t a b l e groun and l e t h : G + C
Let
G
he a connected super-
G. CHERLIN
64
be a d e f i n a b l e endomorphism o f
G
w i t h f i n i t e k e r n e l . Then
h
is
surjective. Proof: L e t G h a v e S h e l a h d e q r e e a and l e t I I b e t h e image o f h Viewed somewhat a b s t r a c t l y , H i s t h e r e s u l t o f c o l l a p s i n g G by a d e f i n a h l e e q u i v a l e n c e r e l a t i o n whose e q u i v a l e n c e c l a s s e s a r e o f f i x e d f i n i t e s i z e . Such a c o l l a p s e p r e s e r v e s S h e l a h d e g r e e , s o I1 a l s o h a s
.
S h e l a h d e c r e e a ( d e t a i l s o m i t t e d ) . Looking a t t h e way G b r e a k s up i n t o c o s e t s modulo I1 , we s e e t h a t t h e i n d e x C G : l { l must b e f i n i t e
i s connected, so fi = G , a s claimed. The f o l l o w i n g r e s u l t i l l u s t r a t e s a c e r t a i n p r o o f t e c h n i q u e . L e t G be a c o n n e c t e d g r o u p and l e t N be a f i n i t e Theorem 5 . (by ( s h e ) , $ 3 ) . Rut
normal s u b g r o u p o f
G
G
. Then
i s contained i n t h e c e n t e r of
\I
G
.
Proof:
Let t h e e l e m e n t s o f G a c t on t h e e l e m e n t s o f V v i a c o n j u g a t i o n , s o t h a t e a c h e l e m e n t g E G i n d u c e s a p e r m u t a t i o n IJ of R N I n t h i s way w e g e t a homomorphism:
.
a :
G
.+
Permutations of
N
.
is t h e c e n t r a l i z e r of Y Since G/K i s isomorphic w i t h a group o f p e r m u t a t i o n s of N , the index [ G : K l i s f i n i t e . S i n c e K i s d e f i n a b l e and G i s c o n n e c t e d we qet K = G , s o G c e n t r a l i z e s Y , a s c l a i m e d . The main r e s u l t on c o n n e c t e d g r o u p s i s t h e f o l l o w i n g : Theorem 6 ( I n d e c o m p o s a h i l i t y Theorem). Let G be a n W - s t a b l e g r o u p . Then t h e f o l l o w i n c a r e e q u i v a l e n t : 1. G i s connected. 2. G has Vorley decree 1 whose k e r n e l
K
.
The p r o o f c l o s e l y r e s e m b l e s t h e p r o o f o f Theorem 5 t e c h n i c a l d i f f i c u l t i e s which w i l l he g l o s s e d o v e r . -P r o o f ( s o - c a l l e d ) :
2 1 : entirely trivial. 1 + 2 : Let G h a v e Morley rank a and Y o r l e y d e g r e e We know t h a t t h e r e i s a d e c o m p o s i t i o n :
,
except f o r
+
G =
(dec) into
d
s1
u
s2 u
... u
d e f i n a b l e s e t s o f rank
.
Sd and d e c r e e
a
d
1
.
Furthermore i t i s
n o t h a r d t o show t h a t t h i s d e c o m p o s i t i o n i s e s s e n t i a l l y u n i q u e i n t h e s e n s e t h a t g i v e n a second such decomposition: G = Tl
U
T,
U
...
t h e r e w i l l he a u n i q u e p e r m u t a t i o n
U
Td o
o f t h e i n d i c e s such t h a t
65
STABLE ALGEBRAIC THEORIES
rank(SiATia)
( A = symmetric d i f f e r c n c e )
< a
.
I t i s convenient t o abhrciriate t h i s contlition hy: 5. = T. la
W e c a n u s e t h e ( l e c o n i p o s i t i o n ( d e c ) t o d c f i n e a n a c t i o n o f i; as a qroup o f permutations on 1 ..,d a s f o l l o w s . i
,.
...
which m u s t e s s e n t i a l l y c o i n c i d e w i t h ( d e c ) a s e x p l a i n e d a b o v e a f t e r
a permutation
a
6
of the indices
1,
.
...,d
.
Yore e x p l i c i t l y : S . q : S . Thiis we h a v e a p e r m u t a t i o n r c p r e 10 a o f I; , a n d we may(I c o n s i d e r i t s k e r n e l K hrgirin!: a s
.
sentation
i n t h e p r o o f o f Thcnrcm 5 wc c o n c l r i d e t h a t
K
=
G (however i t i s
J;O
l o n g e r o b v i o u s t h a t K i s r l e f i n a ! , l e , an:l t h i s ? r e s e n t s t h e main t e c h n i c a l c o r n n l i c a t i o i i i n t h e a r g u m e n t ) . ' l a k i n ? t h i s c x p l i c i t , we have t h a t f o r a l l c, i n C, : (fix) sic: : S i f o r i = 1 ,...,:i
.
low w i t h a c o n s i d e r a b l e sinorint o f "hand-wnvin!:"
.
wc a r e i n s i q ' i t
o f a c o n t r a d i c t i o n i f tl > 1 F o r fixcLl 7 i n S 1 and f o r most s 2 i n S ? t h e lcft-!iari,lc(l v e r s i o n o f ( f i x ) y i c l d s : $S2
E
5,
.
Ilcnce i f we c a n f i n t l a n c l e m e n t "gcncric"
s
in
r e l a t i v e t o t h e elcnicnts o f q14
5
c,?
T2
SIl
w h i c h i s i n some s e n s e we c o n c l u t l e :
,
Lthich c o r i t r n ' l i c t s ( f i x ) . T ! i i s o c n c r i c i t y nr:i.imcnt r i g o r o u s by T o i n ? t o a n e l e m e n t r y e x t e n s i o n
r;'
c a n c a s i l y hc r:-;idc of
r;
.
G. CHERLIN
66
q
1.
+
aq
2.
q
=
K
=
and
p E
i s a Kummer e x t e n e i o n ,
K
i.e.
K = F(a)
or p and K i s an A r t i n - S c k r e i e r F ( a ) where aP-a E F
.
extension, i.e.
T h i s w i l l he comhined w i t h t h e f o l l o w i n g r e s u l t : Lemma 1 . Let F h e a n i n f i n i t e w - s t a b l e f i e l d . Let t h e f o l l o w i n q maps: h ( a ) = an h ( a ) = aP-a
1. 2.
Then
h
with
,
F
i f
F
has c h a r a c t c r i s t i c
p
>
b e one o f
h
0
.
is surjective.
L e t u s f i r s t s e e how t o c o m p l e t e t h e p r o o f o f Theorem 5 o f 5 2 . 2
.
u s i n g Lemma 1
P r o o f o f Theorem 2 . 2 . 5 :
i s an i n f i n i t e w - s t a h l e f i e l d w h i c h i s n o t a l g e b r a i c a l l y c l o s e d . n y Lemma 1 w i t h n = p ( i f F h a s c h a r a c t c r i s t i c p > 0) F i s p e r f e c t , h e n c e h a s a C a l o i s e x t e n s i o n K o f f i n i t e d e E r e e . ,?mong a l l p a i r s (F,K) o f f i e l d s satisfying: i. F i s i n f i n i t e and w - s t a b l e . Suppose t o w a r d a c o n t r a d i c t i o n t h a t
F
i s 3 C a l o i s e x t e n s i o n o f F o f f i n i t c d e s r e e (1 , c h o o s e a p a i r f o r w h i c h t h e d e g r e e (1 i s m i n i m a l . I t i s t h e n e a s y t o v e r i f y t h a t we h a v e a r r i v e d a t t h e s i t u a t i o n d e s c r i b e d b y F a c t .I, n a m e l y t h a t q i s p r i m e and xq-1 splits in F ilence w e have a Kummer e x t e n s i o n o r a n A r t i n - S c h r e i e r e x t e n s i o n . Ilowcver i t i s nil i m m e d i a t e c o n s e q u e n c e o f I.emma 1 t h a t F h a s n o s u c h e x t e n s i o n s , a n d we h a v e t h e d e s i r e d c o n t r a d i c t i o n . ii.
K
.
P r o o f o f 1.emma 1 : Let
G1 , C 2
elements of
F
he r e s p e c t i v c l y t h e m u l t i p l i c a t i v c group o f nonzero and t h e a d d i t i v e c r o u p o i
1.'
.
I f t h e s e two g r o u p s
a r e c o n n e c t e d , t h e n t h e d e s i r e d s u r j e c t i v i t y r e s u l t s f o l l o w imniedia t e l y from t h e S u r j e c t i v i t y T h e o r e m (4 4 . 2 ) . Theorem 3 o f S 4 . 1 t e l l s 11s t h a t C 2 i s c o n n e c t c d . I t r c , . , ? i n s t o h e s e e n t h a t C,
is
connected. By t h e I n d e c o m n o s a h i 1 i t y Theorerr ( T h e o r e m 6 o f 5 4 . 2 ) following are equivalent: 1.
(F,+)
2.
I:
3.
(F',.)
is
il
c o n n e c t e ! l o,roiip
has hlorley degree 1
i s a connected group
llence t h e C o n n e c t i v i t y o f a n d we a r e f i n i s h e d .
the
C7
.
implies the connectivity of
C,
,
67
STABLE ALGEBRAIC THEORIES
S 6 . Gr o u p s o f s m a l l V o r l e y r a n k . 6.1
Results.
1Jsin.q t h e I n d e c o m p o s a h i l i t y Theorem o f $ 4 . 2 i t i s p o s s i b l e t o Civc a r a t h e r t h o r o u q h a n a l y s i s o f w - s t a h l e Rr o u p s o f r a n k s o n e and t w o, a s w e l l a s - t o a more l i m i t e d e x t e n t - r a n k t h r e e . S i n c e an a r b i t r a r y w - s t a h l e prouy, i s c o n n e c t e d - b y - f i n i t e fine oursclves t o the connected casc. Tlieorcm 1. Let G ____ 1. If n = 1
( $ 4.1)
w e may c o n -
bc a connected w-stable group o f \forley rank t h e n I; i s a h e l i a n .
2.
Tf
n
=
2
then
3.
If
n
=
3
and i f
is solvable.
C
G
c o n t a i n s a subgroup o f rank
e i t h e r I; i s s o l v a h l e o r e l s e G o f t h e form S I , ( ? , F ) or PSL(2,F) closed ficld
1:
.
2 then i s isomorphic t o a group f o r some a l g e b r a i c a l l y
(Yotation: CL(2,F) i s t h e g r o u p o f i n v e r t i b l e 2 x 2 matrices over F , SL(2,F) i s t h e subgroup o f m a t r i c e s o f determinant 1 a nd I ’ S L ( 2 , F ) is simple.) 6.2
n.
is the quotient of
SL(2,F)
,
hy i t s c e n t e r . PSL(2,F)
c r o u p s of ‘lorley rank one.
lVc h e g i n w i t h a s i m n l e c r o u p - t h e o r e t i c lemma:
Lemma 2 . --
Let
!I
h e a g r o u p i n w h i c h a l l e l e m e n t s d i f f e r e n t from 1
a r e c o n j u g a t e an d o f f i n i t e o r d e r . Then
II
h a s a t m o st two e l e m e n t s .
Proof: Assume I 1 i s n o n t r i v i a l and f i x a i n 11 o f p r i m e o r d e r p Then I 1 i s o f e x p o n e n t p , a n d i f p = 2 a s t a n d a r d e x e r c i s e y i e l d s t h a t I1 i s c o m m u t a t i v e , h e n c e e q u a l t o Z 2 I f p > 2 we o b t a i n a c o n t r a d i c t i o n a s f o l l o w s . Let g E G c o n j u g a t e a t o 3 - l . Then s i n c e p i s odd w e S e t : a = aPP = =
.
.
contradictinq
.
p > 2 We c a n co m b i n e t h i s lemma w i t h t h e I n d e c o m p o s a h i l i t y Theorem o f 5 4 . 2 to nhtain: I f I; i s a n i n f i n i t e w - s t a b l e g r o u p t h e n G c o n t a i n s Theorem 3 . an i n f i n i t e a h e l i a n d c f i n a h l c s u h y r o u p .
P-ro0 f : S u p p o s e t h a t I; i s a c o u n t e r e x a m p l e . We may t a k e G t o h a v e l e a s t p o s s i h l e \ l o r l e y r a n k and t h e n l e a s t p o s s i b l e V o r l e y d e g r e e . A11 p r o p e r d e f i n a b l e s u h q r o u p s 11 o f G a r e f i n i t e , s i n c e o t h e r -
wise t h e t h e o r r m would n n p l y t o
11
and h ence t o
G
.
In particular
G. CHEKLIN
68
i s c o n n e c t e d , a n d h c n c c liy t h e I n d e c o m p o s a b i l i t y Theorem h a s C 'lorley dcqrce 1
.
Let t h e r a n k o f
w i l l show t h a t i n
I;
C/Z
.
h e a and l e t t h e c e n t e r o f C h e Z !Ve a l l n o n i d c n t i t y elements a r e conjugate and
o f f i n i t e o r d e r , w h i c h i n c o n j u n c t i o n w i t h t h e a b o v e lemma y i e l d s a contradiction. t h e c c n t r a l i z e r C(a) of a i n G is a proFor a i n C-7 p e r J e f i n a h l e s u h p r o u n n f (; , h e n c e f i n i t e . 111 p a r t i c u l a r a i s o f f i n i t e o r d e r , s i n c , e a i s i n C(a) , and i t r e m a i n s t o c o n s i d e r t h e c o n j u j i a c y c l a s s o f a , wiiicii we c t e n o t e
.
T h e r e i s a n a t u r a l i d e n t i f i c a t i o n o f A(: !\.it11 c g s e t s o f G modulo C ( a ) , ant1 t h i i s ns i n t h e p r o o f o f t h e S i i r j e c t i v i t y T h e o r e m
of J 3 . 2 it follok's t h n t
aC,
has rank
. 'Thus
a
since
has Vorley
G
d e q r c e 1 i t f o l l o w s t h a t t h e r e i s room i n C f o r o n l y o n e s u c h eqiiivalence c l a s s , i . e . a l l n o n c c n t r a l clemcnts o i G are conjugate, and hence a l l n o n t r i v i a l e l e m e n t s o f
a r e c o n j u q a t e . Thus
G/Z
I.cmina 2 a p p l i e s t o c o m : > l c t c t h e a r g u m e n t . P a r t 1 o f Theorem 1 i s now a n i m m e t l i a t e c o n s e q u c n c c o f Thcorcm 3 . 6.7
.I1p c h r n i c j:roiins.
I t i s a n n r o p r i a t e a t t h i s p o i n t t o srimmarizc sonic b a s i c f a c t s
rrom t h c t h c n r y o f a l r . c h r n i c S r o u p s w h i c h m t i v a t c t h e a n a l y s i s u s e d t o c s t n h l i s h t h e s e c o n d nnd t h i r d n r l r t s o f T h c o r e n 1 , e v e n t h o u p h I w i l l n o t oive any o r t h a t a n a l y s i s i n d e t a i l . ConsiJer an nln.chraic m a t r i x groun a su1i::roiin
of
cicnts i n
F
I;
ovcr a f i e l d
F
.
G
is
i l c f i n c c ! as t h e s e t o f m a t r i c e s w i t h c o c f f i whose c n t r i c s s a t i s f y c e r t a i n a o l y n o m i a l e q u a t i o n s : I;T,[n,F)
n 1( a i j ) = ? , . . . , n k ( a i j )
= 0
.
T y p i c a l q r o u n s c!efinrtl i n t h i s illanner :ire: S l . ( n , F ) , tlcfinc,tl !>y d c t ( n . . ) - l = '7 IJ
( t ! i c i i p n c r t r i a n 1 ; i i l a r q r o u p ) , t l c f i n e ~ lb y t h e cc~untions ;I. . = n for j < i
T(ri,l')
.
1J
Ice t a k e t!ie h n s e f i c l d
I:
t o 1)c n l q e h r a i c a l l y c l o s c J . 'l'hcn a l l s u c h
croups nre w-stnlilc. I f ii t o p o l o x y i s y l a c c t l o n I; h y t n k i n q t h e z e r o - s e t s o f a r b i t r a r y Tystcnls o f n o l y n o m i n l s t o fie c l o s c t l t h e n onc p r o v c s t h a t t h e c o n n e c t e d c o n i n o n r n t o f t h e i r l e n t i t y i n C, i s a s u h q r o u p , d e n o t e d I;' , ant1 t I i : i t t h c i n t l c x o f ';I i n I; i s f i n i t e . The a s T i c c t o f t h e t h e o r y t h a t i s o f i n t c r e s t ;it t h i s p o i n t i s the
s t r i i c t u r c o f s i m p l c ; i l q c h r a i c m a t r i x q r o u n s , and s p c c i f i c a l l y t h e
69
STABLE ALGEBRAIC THEORIES
Ilruhnt d e c o m p o s i t i o n , which wc will h r i e f l y Clescrihc. .Is a D r e l i m i n a r y
we need t o d i s c u s s s o r e 1 s u b g r o a p s and WeyZ g r o u p s . 1. I j o r e l sril)p,roups. Let
h e a c o n n e c t e d a l p e h r a i c m a t r i x g r o t i n . 11ninximal s o l v a b l e H o f G i s c a l l e d a Bore2 s d g r o u p . I n t h i s c o n -
G
connected suhoroup
t e x t a gooil d e a l o f i n f o r m a t i o n is o h t a i n n h l e rcithorit f u r t h e r h y p o -
t h e s e s , sucli a s :
.
Fact R
I S i t h t h e nhnvc ' n y n o t h e s c s a n d n o t a t i o : i :
.
1.
R
i s i t s or\'n n o r m n l i z c r i n
2.
G
i s t h e iinion o f t h e c o n j u y a t e s o f
G
.
B
Vc can c o n s i d e r R n r c l sii!>o,rnuns o f w - s t a b l e o , r o u p s u s i n g t h c
same d e f i n i t i o n : maxim:;
s o l v n l ~ l cc o n n c c t e i l d e r i n n h l e s u b g r o u p s .
c o u r s e t h e T u n e r a 1 r c s i i l t s c o n c c r n i n c R o r e l sirhqroups o f a l g e b r a i c m a t r i x g r o u n s niny n o t t r n n s f c r t n t h i s cxtenricd c o n t e x t , and i n d e c i t i s n o t knol%.n i,!iethcr a n y o f t'iem rcmnin v a l i d . Thc I 3 r u ' l n t d e c o m p o s i t i o n o f t o t l o u h l c c o s c t s modulo
.
13
i s s i m p l y i t s tlccomposition i n -
G
I n t h e c o n t c x t o f a1j:chraic
grotips t h i s
i s t i c d up w i t h t h e '\Icy1 !i.roun, irrhich iic h r i c f l y d e s c r i h c n e x t . The Yeyl groirp.
2.
lixaniplc: Let sits insidc G actinn o f
Sll
i; =
.
Gl,[n,F)
The symmetric q r o u p
Sn
on
n
letters
a s t h e s c t o f p c r n u t n t i o n m a t r i c e s . Furtheriiiorc t h e a s a g r o u p o f pcrriiiitntions can he s e e n i n
G
by
a t t h e a c t i o n o f t h e p e r m i i t a t i o n w a t r i c c s on t h e STroup
lookin:
iliagonnl m a t r i c e s
3
of
fcnnjuqation hy a nermutation matrix induces n
permutation of t h e diagonal c n t r i c s ) . You i f i i e t a k e
.qs
trianyular natrices, o f 1; relativc to R C =
;1
Ilorel siih:roirp
=
T(n,F)
of
i ;
thc iiroup of u p p e r
, t h e n t h e tlnuhle c o s e t d c c o r n o s i t i o n
can he p i i t i n t h c f o r n : 30!3 ;
LJ
O
v a r i e s o v e r p c r m ~ i t a t i o nm a t r i c e s ,
o
o r i n o t h e r iv.nrrl.s t h c doiuhlc c o s c t s n r e n n t i r r a l l y p n r a m e t r i z e d b y t h e g r o u p S, , w!iic'i i s t!ic 'Ycyl g r o u p o f G l , ( n , I I ) . Thc n o t i o n o f a ',\'cyl !!rotin
d o c s n o t s c c n t o j i c n c r : i l i z c ii:uch
bcyond t h e c l a s s o f m a t r i x y r o u n s .
F i r s t onc nce
t o r u s , which r1i:iy hc d c f i n c c l ns n conncctci! diagonaZizab Ze s u b g r o u p D of
C
.
I f one has t h i s n o t i o n ,
x i t h i n a iiorel suhqroup
x r o u p o f aiitomorniiis.ns o f piiisr::~o f
I)
, sct
Then o n e S e t s :
I\'
1;'
( t h c 'Scyl y r o i i p ) hc t h e
ii!iich a r e indL!ccd !iy i n n e r nutonior-
I:
G , i.e. lettin:
noriiinlizcr o f
tlicri nnc c a n t a k e a inaximal t o r u s
, ant1 l c t
11
C
= Y/(:
Jic t \ c
.
~ c n t r a l i ~ co fr
1)
and
N
the
G. CHERLIN
70
.
Fact C
i s a f i n i t e group. I f C; i s e . g . s i m p l e a n d !!! i s a s y s t e m o f r e p r e s e n t a t i v e s f o r W i n G t h e n G = BlVR (This i s t h e Bruhat decomposition.) A l l o f t h i s becomes r a t h e r t r i v i a l when w e f o c u s O U T a t t e n t i o n on P S L ( 2 , F ) . Here t h e Weyl g r o u p i s c y c l i c o f o r d e r 2 ( a s i n GL(2,F)) and t h e g e n e r a t o r o f W i s r e p r e s e n t e d b y t h e m a t r i x : 1.
IV
2.
.
3 . Philosophy. The r e l e v a n c e o f t h i s m a t e r i a l t o g r o u p s o f s m a l l Morley r a n k l i e s i n t h e f o l l o w i n g . I n t h e a n a l y s i s o f g r o u p s o f r a n k 2 o r 3 one a t t e m p t s f i r s t t o g e t a "Bruhat decomposition" f o r G ( a double cos e t d e c o m p o s t i o n f o r G r e l a t i v e t o a B o r e l s u b g r o u p , made r e a s o n a b l y e x p l i c i t ) . Depending on t h e a s s u m p t i o n s made on G t h i s c a n l e a d e i t h e r t o a c o n t r a d i c t i o n O T t o g e n e r a t o r s and r e l a t i o n s f o r G . The c d n t e n t o f Theorem 3 i s t h a t a n y i n f i n i t e w - s t a b l e g r o u p h a s a n o n t r i v i a l B o r e l s u b g r o u p . T h i s i s a weak r e s u l t , and h e n c e w e
were f o r c e d i n t h e t h i r d p a r t o f Theorem 1 t o make a n a d d i t i o n a l a s s u m p t i o n which s i m p l y amounts t o t h e e x i s t e n c e o f a R o r e l s u b g r o u p of rank 2 ( t a k i n g i n t o a c c o u n t t h e r e s u l t o f p a r t 2 ) . Groups o f r a n k 2 i I n p r o v i n g p a r t two o f Theorem 1 , we c o n s i d e r a c o n n e c t e d w - s t a b l e g r o u p G o f r a n k 2 which i s n o t s o l v a b l e , make a d e t a i l e d s t r u c t u r a l a n a l y s i s o f G , and e v e n t u a l l y o b t a i n a c o n t r a d i c t i o n . The f i n a l c o n t r a d i c t i o n may be p u t i n v a r i o u s f o r m s . One s u c h i s a s follows. 6.4
By a p r e l i m i n a r y r e d u c t i o n one may assume t h a t G h a s n o n o n t r i v i a l p r o p e r d e f i n a b l e n o r m a l s u b g r o u p . Let I d e n o t e t h e s e t o f i n v o l u t i o n s i n G ( i . e . elements of o r d e r two). W e can derive t h e following information: 1. I i s nonempty. 2.
Thus
I
U
(11
The e l e m e n t s o f I commute w i t h e a c h o t h e r . i s a d e f i n a b l e commutative normal subgroup of
G
,
and we o b t a i n a c o n t r a d i c t i o n . F a c t s 1 , 2 a r e o b t a i n e d on t h e b a s i s of t h e f o l l o w i n g : 1.1
f o r some i n v o l u t i o n Rorel subgroup).
G = R
U
RwR
w
(here
B
is a
STABLE ALGEBRAIC THEORIES
2.1
N(R)
f o r any Bore1 subgroup
= R
71
is the normalizer in G ) .
(N(R)
of
R
2.2
B1 n B 2 = 1
f o r d i s t i n c t Rorel subgroups
2.3
The c e n t r a l i z e r o f a n y n o n t r i v i a l e l e m e n t o f Rorel s u b g r o u p .
.
I t remains t o be s e e n t h a t Clearly 1.1 y i e l d s 1 yield 2 T h i s goes a s f o l l o w s .
Let
.
i,j
be d i s t i n c t involutions i n
.
is a
G
2.1-3
an d l e t
G
.
B1,RZ
together
. Let
b = ij
be t h e c e n t r a l i z e r o f b i n G I f we c o n j u g a t e b by i a s i m p l e c a l c u l a t i o n shows t h a t t h e r e s u l t i s b-1 I t follows t h a t
R
.
and t h e c o n j u g a t e o f R by i h av e a n o n t r i v i a l i n t e r s e c t i o n , s o 2 . 2 an d 2 . 3 i m p l y t h a t i n o r m a l i z e s R Th en 2 . 1 i m p l i e s t h a t
R
.
.
i s in R , i.e. i centralizes i j I t follows immediately t h a t a n d j commute, as d e s i r e d . W e n e e d t o b r i n g o u t more c l e a r l y t h e r o l e o f t h e B r u h a t decomp o s i t i o n i n a l l of t h i s . We h a v e s e e n i t i n a m i n o r r o l e , a s t h e i i
c o n t e x t i n which a n o n t r i v i a l i n v o l u t i o n o c c u r s . I n f a c t t h e f i r s t s t e p i n t h e g r o u p - t h e o r e t i c a n a l y s i s , on which t h e o t h e r s depend which i s a l m o s t t r i v i a l ) , i s t h e f o l l o w i n g :
(except 2.2, 1.1.1
I f x E G does n o t normalize t h e Rorel subgroup t h e n G = N ( B ) t' BxR
B
.
,
T h i s i s a l r e a d y a v e r s i o n o f t h e Rruhat decomposition, and reduces t o t h e d e s i r e d d e c o m p o s i t i o n when w e s u b s e q u e n t l y show t h a t
X(B) = B .
The p r o o f o f 1 . 1 . 1 i s i n s t r u c t i v e . One shows t h a t N ( B ) r a n k 1 a n d RxB h a s r a n k 2 f o r x o u t s i d e N(R) Since G
.
has is
c o n n e c t e d , t h e I n d e c o m p o s a b i l i t y Theorem t e l l s us t h a t
G can have of rank 2 1.1.1 f o l l o w s . Now f r o m 1 . 1 . 1 w e c a n a l r e a d y g e t o u r i n v o l u t i o n w i t h t h e h e l p
o n l y one d o u b l e coset of 2.2
. Yamely,
1.1.1
x-1 = b l x b 2 Set
w = xh2
and
implies t h a t t h e r e i s an equation:
with b
.
RxR
=
b;'
bl,b2 b2
.
in
R
.
An e a s y c o m p u t a t i o n shows t h a t
w ' = b . Hence : b = bW SO
w'
= 1
E
,
R n Bw = 1
(by 2.2)
,
as desired.
T h i s i l h s t r a t e s t h e way i n w h i c h e v e n a p a r t i a l B r u h a t decompos i t i o n f a c i l i t a t e s an a n a l y s i s o f t h e s t r u c t u r e o f G
.
G. CHERLIN
72
6.5
Open q u e s t i o n s ,
Present techniques f o r analyzing simple w-stable groups w i l l p r o b a b l y n o t be a b l e t o h a n d l e t h e p r o b l e m o f p r o v i n g t h a t Bore1 s u b g r o u p s a r e r e a s o n a b l y l a r g e . The t h e o r y o f a l g e b r a i c g r o u p s s u g g e s t s t h a t i t would be more r e a s o n a b l e t o a t t e m p t
an a n a l y s i s o f s i m p l e l i n e a r W - s t a b l e groups, i . e . groups o f l i n e a r transformations a c t i n g on a v e c t o r s p a c e V i n s u c h a way t h a t t h e g r o u p t o g e t h e r w i t h i t s a c t i o n on V c o n s t i t u t e s an w - s t a b l e s t r u c t u r e . We c o n j e c t u r e accordingly: Conjecture A. Any s i m p l e l i n e a r w - s t a b l e g r o u p o f f i n i t e Morley rank i s an a l g e b r a i c group. A more s p e c i a l i z e d v a r i a n t would b e : Any l o c a l l y f i n i t e s i m p l e U - s t a b l e g r o u p o f f i n i t e Conjezture B. Morley r a n k i s an a l g e b r a i c g r o u p . A p r o o f o f t h i s would u s e t h e t e c h n i q u e s o f f i n i t e g r o u p t h e o r y blown up i n t h e manner o f [ 2 2 1
.
Notes.
5 1.
The s t a b i l i t y s p e c t r u m t h e o r e m . T h i s s e c t i o n i s b a s e d on [ 4 1
w - S t a b i l i t y i s found f i r s t i n 113
,
,
p r i m a r i l y Chapter I 1 1
.
a s a peripheral notion. S t a b i l i t y
becomes a c e n t r a l n o t i o n i n S h e l a h ' s p a p e r 5 , e . g .
131
.
A calm i n t r o d u c t i o n t o t h e s u b j e c t i s f o u n d i n 121 References:
.
1.
Y.Morley, 5 1 4-538.
" C a t e g o r i c i t y i n power",
2.
G.Sacks, 1972
S a t u r a t e d Yodel T h e o r v
3.
S.Shelah, " S t a b i l i t y , t h e f.c.p., Math. Logic 3 ( 1 9 7 1 ) , 217-362
4.
S.Shelah, C l a s s i f i c a t i o n T h e o r v and t h e Xiymher o f Nanisomotp h i c Y o d e l s , N o r t h - H o l l a n d , Amsterdam 1978 Stable algebraic theories.
5 2.
.
.
Trans.AMS 1 1 4 ( 1 9 6 5 ) ,
,
B e n j a m i n , R e a d i n g , 'lass.
and s u p e r s t a b i l i t y " , Annals
.
For a s u r v e y o f s t a b i l i t y , i n c l u d i n g m o d e l - t h e o r e t i c and a l g e b r a i c a p p l i c a t i o n s , s e e 1181
.
2.1
Yodules.
A s t r a i g h t f o r w a r d v e r s i o n o f t h e s t a h i l i t y o f modules i s g i v e n i n 1181 F i s h e r announced an e q u i v a l e n t t h e o r e m i n 1141 , and Baur found a p r o o f i n d e p e n d e n t l y C71
.
.
The c h a r a c t e r i z a t i o n s o f w - s t a b l e and s u p e r s t a b l e modules were d e v e l o p e d by G a r a v a g l i a and V a c i n t y r e , and G a r a v a g l i a a l o n e , r e s p e c t i v e l y 1161. G a r a v a g l i a h a s shown t h a t t h e r e i s s u b s t a n t i a l l y more
73
STABLE ALGEBRAIC THEORIES
t o t h e t h e o r y o f w - s t a b l e modules t h a n t h e c l a s s i f i c a t i o n p r o b l e m , i . e . t h e y o c c u r i n p r o f u s i o n , and t h e y h a v e n o n t r i v i a l p r o p e r t i e s [161 2.2 R i n g s . The c l a s s i f i c a t i o n o f s t a b l e s e m i s i m p l e r i n g s modulo s t a b l e d i v i s i o n r i n g s i s i n [5,121 S u p e r s t a b l e d i v i s i o n r i n g s are c l a s s i The s t i m u l u s t o t h i s work f i e d modulo s u p e r s t a b l e f i e l d s i n 1101 came from M a c i n t y r e ' s c l a s s i f i c a t i o n o f t h e w - s t a b l e f i e l d s i n 1 1 7 1
.
.
.
.
2.3 Groups. The t h e o r e m s on H , - c a t e g o r i c a l s t a b l e g r o u p s a r e i n [91 and [131. The s o l v a b i l i t y o f s t a b l e l o c a l l y n i l p o t e n t g r o u p s i s i n [61
.
References: J . B a l d w i n and B.Rose , " U , - c a t e g o r i c i t y and s t a b i l i t y o f r i n g s " , 5. J . A l g . 45 (19771, 1-16 J . R a l d w i n and J . S a x 1 , " L o g i c a l s t a b i l i t y i n g r o u p t h e o r y " , 6. J . A u s t r a l i a n Math. S O C . 21 (1976), 267-276 W.Baur, "On N o - c a t e g o r i c a l modules", JSL 40 (1975) , 213-220 7. 8. W.Baur, " E l i m i n a t i o n o f q u a n t i f i e r s f o r modules", t o a p p e a r . W.Baur, G . C h e r l i n , and A . M a c i n t y r e , "On t o t a l l y c a t e g o r i c a l 9. g r o u p s and r i n g s " , J . A l g . , t o a p p e a r . lo. G . C h e r l i n , " S u p e r s t a b l e d i v i s i o n r i n g s " , P r o c e e d i n g s ASL Europ e a n Summer M e e t i n g , Wroclaw, P o l a n d , 1977, t o a p p e a r . 11. G . C h e r l i n , J . R e i n e k e , " S t a b i l i t y and c a t e g o r i c i t y f o r commutat i v e r i n g s " , A n n a l s ?.lath. L o g i c lo (1976). 376-399. 12. U Fe 1gne r , "N k a t e go r i s che T h e o r i e n n i ch t -kommut a t i v e r Ringe" , Fund. Math. 8!!-(1975), 331-346. 13. U . F e l g n e r , " N o - c a t e g o r i c a l s t a b l e g r o u p s " , Math. 2 . 160 (1978), 27-49. 14. E . F i s h e r , "Powers o f s a t u r a t e d modules", JSL 37 (1972), 777 15. S . G a r a v a g l i a , " D i r e c t p r o d u c t d e c o m p o s i t i o n o f t h e o r i e s o f modules", JSL, t o a p p e a r . 16. S Gar a v a g 1i a , " De comp o s i t i on o f t o t a 11y t r a n s c e n de n t a 1 modules ' I , t o appear. 17. A . M a c i n t y r e , "On w1 - c a t e g o r i c a l t h e o r i e s o f f i e l d s " , Fund.Math. 71 (1971). 1-25 18. S She 1ah , "The l a z y mode 1- t h e o r e t i c i a n s g u i d e t o s t a b i l i t y ," i n S i x D a y s of Mode2 T h e o r y , e d . P . H e n r a r d , E d i t i o n s C a s t e l l a , Albeuve. S w i t z e r l a n d .
.
.
.
.
.
.
.
.
5 3. Chain c o n d i t i o n s . The given i n o f [61 g o i n g on
.
.
w - s t a b l e DCC i s d i s c u s s e d i n 1191 The ' s u p e r s t a b l e DCC i s C181 The s t a b l e CC i s i m p l i c i t i n t h e a r g u m e n t on p . 274 The s t u d y o f c h a i n c o n d i t i o n s i n g r o u p t h e o r y h a s been f o r some t i m e 1201
.
.
G. CHERLIN
74
Y o r l e y r a n k i s d e v e l o p e d a n d e x p l o i t e d i n C11. S h e l a h d e g r e e i s one of v a r i o u s m o d i f i c a t i o n o f t h i s n o t i o n m a n u f a c t u r e d see t h e f i r s t two c h a p t e r s o f [ 3 1 .
by S h e l a h ;
References : 19.
2 0.
A . M a c i n t y r e , "On 0 1 - c a t e g o r i c a l t h e o r i e s o f a b e l i a n g r o u p s " , Fund.Yath. 70 ( 1 9 7 1 ) , 2 5 3 - 2 7 0 I). Rob i n s o n , W t e n e s s -con d i t i on s and S e n e r a 1i z e d-So 1v a b l e Groups , S p r i n g e r , N e w York, 1 9 7 2
.
.
5 4 . Connected g r o u p s . The u s e o f a s u i t a b l e n o t i o n o f c o n n e c t e d n e s s i s s u g g e s t e d by [221
.
The I n d e c o m p o s a b i l i t y Theorem f o r w - s t a b l e g r o u p s i s p r o v e d i n [211 A weaker v e r s i o n i s i n [ 2 3 1
.
.
References : 21.
G . C h e r l i n , "Groups o f s m a l l Morley r a n k " , A n n a l s Y a t h . L o g i c , t o appear.
22.
O.Kege1 and R . W e h r f r i t z , L o c a l l y F i n i t e Groups, N o r t h - H o l l a n d , Amsterdam 1 9 7 3 B . Z i l ' b e r , "Groups and r i n g s w i t h c a t e g o r i c a l t h e o r i e s " , Fund. Yath. - 9 5 ( 1 9 7 7 ) , 1 7 3 - 1 8 8 ( R u s s i a n ) .
23.
§ 6 .
.
Groups o f s m a l l V o r l e y r a n k .
.
The d e t a i l s a r e i n ( 2 1 3 The r a n k 1 c a s e i s i n 1 2 5 1 ( g i v e n t h e I n d e c o m p o s a b i l i t y T h e o r e m ) . Rackground on a l g e b r a i c g r o u p s i s i n [241
.
Ileferences: 24.
J.Humphreys, 1975 .
25.
J . R e i n e k e , ' V i n i m a l e Gruppen" 357-359
.
L i n e a r A l g e b r a i c Groups
, Z.
,
S p r i n g e r , New York,
\ l a t h . Logik 21 ( 1 9 7 5 ) ,
L O G I C COLLOQUIUM 78 M. Boffa, D. van DaZen, K. McAloon ( e d s . ) 0 North-Holland Publishing Company, 1979
iln r 6 s u l t a t
de
ZOTI
contradiction r e l a t i v e au
s u j e t d e l a c o n r j e c t u r e d e SOLOVAY
R e n d DAVII) U n i v e r s i t d T o u l o u s e Le M i r a i l 1 0 9 rue V a u q u e l i n 31081 T o u l o u s e Cedex FRA N CC
SOLOVAY a v a i t
: si a c O n e s t
un e n s e m b l e q u i n e c o n s t r u i t
a l o r s a e s t g 6 n e r i q u e p o u r un e n s e m b l e d e c o n d i t i o n s d e L .
p a s 0*,
JENSEN
coniecturd
(cf.
J E ) a m o n t r 6 q u e c e c i p o u v a i t d t r e f a u x . Nous m o n t r o n s
ici,
en u t i l i s a n t
peut
G t r e n i d e p a r un r 6 e l a q u i e s t s i n g l e t o n
l a p r e u v e d e JENSEN,
q u e l a c o n j e c t u r e d e SOLOVAY
fli
dans L(a).
ddmontrer dans ZF: Cons
(ZF)
-+
Cons L
-+
t 3!xcp(x)
( Z F C t G C H t ,Oc
a cardinal)
Vx(cp(x)
t
+
+
(a c a r d i n a l dans
Va
V = L(x)
x
&
4
L 8 x n'est
p a s g 6 n C r i q u e s u r L p o u r un e n s e m b l e d e c o n d i t i o n s . ) ) On o b t i e n t
l e modele
o b t e n u p a r JENSEN (*)
1) Z F C
+
cherchg
(cf.
comme e x t e n s i o n g e n e r i q u e d ' u n m o d ' e l e
JE) qui v 6 r i f i e :
GCH t 3 A ( A c
H2
t
V
=
L(A) t A n'est
pas g6n6rique
sur L p o u r u n e n s e m b l e d e c o n d i t i o n s ) 2) v x (x c 3)
(A n'est K
K2 il
H1
-+
XEL)
Va ( a c a r d i n a l d a n s L
pas gdndrique
+
c1 c a r d i n a l
8 cf(n)L
=
cf(n))
sur L p a r c e q u e , d a n s L ( A ) , p o u r t o u t
e x i s t e B E P(K)
- L t e l que pour t o u t a
<
Notre preuve u t i l i s e a l o r s les a r b r e s de S o u s l i n e t 75
K,
B ll
a E
L.)
les ensembles
R. DAVID
76
r e e l qui est singleton TI
presque disjoints pour coder A p a r un (voir (JE-JO);
1
2'
(JE-SO); (H))
Dans la suite nous adoptons la terminologie suivante. Pour un ordre partiel P , "P a la condition de chaine K" signifie que toute K chaine d6croissante dans P a un minorant dans P ; "P a la condition d'antichaine
<
K" signifie que toute famille d'glgments de P deux 2
deux incompatibles est de cardinalit6 infgrieure 3 K ; s i G est M ggngrique sur P , V ( x ) est l'interpr6tation p a r G dans M(G) G
des
termes x E M du langage forcing pour P. Soit donc M
0
un mn??le
de ( * I ;
M dans la suite L designera L 0
0
LnMO.
Dans M0 soit Q l'ensemble de conditions qui permet de coder gengriquement A par une partie de H : 1
Q = { ( s , u ) / s c H1 ; u avec l'ordre:
05
(s,u)
C
<
A ;
[u(
IsI,
(s',u')
++
s'
s
3
E
va E
E u
3
s
u' E
n
s c st,
est une famille, dans L , de parties presque disjointes
de K 1 ( M o et L ayant les msmes cardinaux, la notion est absolue). Lemme 1 1) Q est inclus dans L et s i q. q' E Q , q G q'ssi Lk q G 9'.
2) Q a la condition d e chaine dgnombrable et la condition
d'antichaine
< H2
Soit G un M o ggngrique sur Q et M 1 = MO(G).
V
On a :
M I= 3 B c H 1
1
L(B).
Dgfinition Soit ((Tn)nEw,f)
la fainille d'arbres construite, dans L ,
par (JE-JO); soit P l'ensemble de conditions associ6:
P = {p / dom p = n E Vi avec l'ordre
: p
<
q
tf
dom p
3
<
n p
i E T i E f(pi+l) 2 pi}
dom q & V i E dom q
P n = {p E B / dom p = n).
qi c Pi
RESULTAT DE
UN
77
NON CONTRADICTION
Lemme 2 S o i t N un m o d a l e
d e ZF e t T u n
a r b r e de Souslin
C e s t un e n s e m b l e d e c o n d i t i o n s d e f o r c i n g a y a n t
dans N.
Si
l a c o n d i t i o n de
c h a i n e d c n o m b r a b l e e t si G e s t N g e n c r i q u e s u r C a l o r s N(G) satisfait preuve
e s t un a r b r e d e S o u s l i n .
: T
: voir
(DE-JO).
Lemrne 3
< H 1'
Dans MI P a l a c o n d i t i o n d ' a n t i c h a i n e
U Pn; i l s u f f i t donc de montrer que chaque P a l a n = Pn T ( b n ) o? d6signe c o n d i t i o n d ' a n t i c h a i n e < K1. O r P n+l P
preuve:
*
l'it6ration
d e s f o r c i n g s e t T(b
bn a j o u t e e d a n s T
n
par f
e s t l'image
)
.
l e forcing P
par
*
-1
de l a branche
T ( b n ) e s t un s o u s - a r b r e
normal
il s u f f i t donc de rnontrer que :
de T,+l;
1 ) P o e s t un a r b r e d e S o u s l i n
dans M
1'
2) Vn T ( b ) e s t u n a r b r e d e Souslin d a n s M l ( b
D a n s (JE-JO) i l e s t m o n t r 6 q u e
: Vn
T(b
1.
e s t un a r b r e d e S o u s l i n
)
dans L ( b n ) ;
*
To e s t S o u s l i n d a n s L , d o n c p a r par
*
soit n
l e s lemmes
E P
e t
5
l a formule
: p
H-
(oh
r
soit soit H
il l ' e s t
2
a u s s i d a n s I.?
q E Q et H1
M2 = M O ( H )
x
t e l q u e E.!
t M~
1
1'
pas Souslin
s a t i s f a i t Y'(p,x)
<
H 2 un M
2
=
I+Y C ; , : )
Q t e l que ( p , q ) E H e t
(X)
e s t une a n t i c h a i n e non
MO(H2)(H1).
satisfait
d 6 n o m b r a b l e d e T ( F ) . p E HI
r.)
x
c: g 6 n 6 r i q u e sur P
MO(HL)(H2)
: q
satisfait
est
02 W(p,z
sur P )
l e gcn6rique
t e l que
E M~
dans
N1.
e s t u n e a n t i c h a i n e non d g n o m b r a b l e d e T
x
q E H 2 d o n c MO(H
n'est
Pn a d o n c l a c o n d i t i o n d ' a n t i c h a i n e
e s t l e nom c a n o n i q u e p o u r
0
- 2) il l ' e s t a u s s i d a n s M o e t
le p r e m i e r e n t i e r t e l q u e T ( b Ell(bn).
soit p
1 et
(*
: p
K-
V
H2
-
donc M2 s a t i s f a i t
une a n t i c h a i n e non dsnombrahle
dans T(b'
).
:
(02
X' =
"H1
(V, 2
Cz))
est
b t n e s t l a branche
R. DAVID
78
dans Tn
5 H
associge
e s t uii a r h r e
T(b',)
1
(ce qui donnera l a contradiction
Montrons
0
(H 1 ) =
de T ,
b,lo(b'
e s t S o u s l i n d a n s L(S',);
)
A a d a n s I1
A
u n nom
P
d'aprss
satisfait:
2
le
qui e s t dans
une a n t i c h a i n e
L ( p a r *-2) d o n c A e s t d a n s
de'
0
il
la condition
d a n s M (H ) une
es!
O r si ( q l ) l E w
dgcroissante d'bl6ments
d a n s P I (b'n) 3
es!
2 d e v o i r q u e Q a , dans MO(H1),
lemme
de c h a i n e dgnombrable.
donc aus-,i dans
Pour n o n t r e r que T e s t Souslin dans M2
L ( b I n ) donc d s n o n b r a b l e ) . suffit,
; si A
HI
(car T c
)
que M
de S o u s l i n .
On s a i t q u e T = T(h' PI
cherchse)
chaine
1
2 , ( q l ) l E w e s t en f a i t d a n s
M0
donc
d u lemme b i e n
et
d'une
( l e n n e 1) a un m i n o r a n t . Cela r d s u l t e immgdiatement
connu s u i v a n t
induction triviale. Lemme 4 S i T e s t un
:ur
T et
si f : w
C e c i dckCve
orbre de Souslin !I
+
e s t da!,s !I(G)
l a p r e u v e d u lemme
d a n s N,
s i G e s t N ggngrique
a l o r s I e s t en
fait
dans N.
3.
D6f i n i t ion S i 5 c H1 c
E C(B) *
c
0
(Sa)n
ordonne d 6 f i n i
par
est,
8
Q
tf
danz
L,
Il
u~
3
une
R
va
famille
E
u 1
5
n ~ ( p c) ~ ( p ' )
de p a r t i e s presque
disjointes
1 d e w e t oh ~ ( p d) c s i g n e l ' e n s e n b l e d e s 6 1 6 m e r i t ;
de hauteur
{ p i / i E dom p }
).
Lemne
:
(p,u) p E P 8 u c B fini
(p,u) G (pl,u') (0;
s o i t C(3) L ' e r i s e s i h l e
-
pour
l e s d g t a i l s v o i r (H) -
0
sous
5
S o i t 9 c til
tel
10 c o ~ ~ d i t j o rd l' a n t i c t l s i n e
preuve: seulement
si o(p)
0
u(:>'),
si p et p'
:1
q u e !Il
<
K
L(5).
A l o r s dan:
!Il,
C(B)
a
1'
(p,u) e t
le sont;
V
(p',u')
s i (pa,
sont
compatibles
si et
u iy. ) n < K L e s t u n e a n t i c h a i n e d e
C(5) o n p e u t s u p p o ? c r ( p u i s q u c ~ ( p e) s t u n e p a r t i e f i n i e d e w ) q u e :
GSULTAT DE NON
UN Va,B
=
O(pa)
79
CONTRADICTION
e s t une a n t i c h a i n e de P,
o ( p )done ( p a ) a a q
P
ce q u i e s t
impossible.
Fin de l a p r e u v e du th6orSme S o i t G un M 1
in
a =
Ew / s(p,u)
alors M
1
en e f f e t
E
e6n6rique s u r C(B).
1
n E o(p)j
G
(G) s a t i s f a i t
Dans M ( G ) s o i t
: V
= L(a)
&
a e s t s i n g l e t o n Jl
1 2'
C(B) r 6 a l i s e e n mSme t e m p s l e s p r o p r i h t e s d e c o d a g e d e B
e t rend l e g6n6rique a s s o c i 6 a d 6 f i n i s s a b l e p a r une formule ( p o u r l e s d 6 t a i l s v o i r (H)). lql
n: .
La d g f i n i t i o n d e a e s t i n d 6 p e n d a n t e d e
e t c c l a ach6ve donc l a preuve du th6orbme.
Kkfhrences (DE-JO)
: K.
DCVLIN
&
H.
JOHIJSBRATEN
The S o u s l i n P r o b l e m ,
L e c t u r e Notes i n M a t h s , V o l 4 0 5 ,
1974, Springer-Verlag.
(JE)
: R.
JENSEN
Coding t h e u n i v e r s e by a r e a 2 , u n b u h l i s h e d N o t e s , 1 9 7 5 . (JC-JO)
: R.
JENSEN 8 H .
JOHNSBRATEN 1
A New C o n s t r u c t i o n oJ" a A 3 non c o n s t r u c t i b l e s u b s e t 0;
(JE-SO)
: R.
w,
Fund.
Math.
JEI\JSEN & R .
(81) 1 9 7 4 .
SOLOVAY
Some A p p l i c a t i o n s oJ" a l m o s t d i s j o i n t s e t s M a t h e m a t i c a l L o g i c and F o u n d a t i o n s o f s e t t h e o r y (Jerusalem,
(H)
: L.
1968) e d . B a r - H i l l e l .
HARRINGTON
Long Projective well-orderings Annals o f Math.
L o g i c (12), 1 9 7 7 .
LOGIC COLLOQUIUM 78
M. Boffa, D . van Dalen, K . McAloon (eds.)
0 North-Holland Publishing Company, 1979
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS 0. DEMUTH and A. KUEERA
Charles University, Prague The paper is devoted to some questions of constructive mathematics (CM) in the sense of the Markov school. Its aim is first a short outline of the fundamental principles of this trend in mathematics, and secondly a survey of basic results concerning constructive mathematical analysis, obtained within the Prague seminar. §
1.
Introduction
CM, in the sense of the Russian school (headed by A.A. Markov and N.A. Sanin), has been created on the basis of criticism of foundations of mathematics, as formulated consistently by L.E.J. Brouwer and H. Weyl. The aim was to find foundations for mathematics that would be as simple and secure as possible and would be free from too far reaching idealisations, foundations in which the concept of effectivity would be the principal. In this connection, the following facts are important: 1) intuitive natural numbers (NNs) are indispensable for mathematics, 2) from the historical point of view, the development of mathematics was substantially influenced by applications of mathematics where solutions of problems consisted, de facto, in transformation of particular information coded by words. We should notice that words in an alphabet can be taken for material representatives of intuitive NNs. Thus, the means necessary for algorithmic processing of words are indispensable for any sufficiently rich mathematical theory. The constructive program and its realization shows that these means necessary for processing of words are also sufficient for the development of wide mathematical theories, comparable to classical mathematics as to richness of results and methods. Let us c6aracterize the basic features of constructive mathematics : 1) in CM are studied so called constructive objects, i.e. the words 81
0 . DEMUTH and A.
82
KUEERA
in certain alphabets and the objects which can be coded in a finite way by words, 2) the abstraction of potential realizability and theory of algorithms are used, 3) a specific interpretation of mathematical propositions is used. To put it shortly, the main subject of CM is the study of possibilities of algorithmical transformation of coded information about mathematical objects. This fact delimitates the space in which CM can be developed and, at the same time, it distinguishes CM from the intuitionistic program as well from other constructive trends as, e.g.
Bishop's one. In this context we would like to mention that the
use of theory of algorithms, mentioned above, results in some similarity of constructive mathematical analysis and recursive analysis. The use of theory o f algorithms and the way how the system of constructive concepts is built require the use of specific interpretation of mathematical propositions. A s for our opinion the classical Aristotle logic is not in harmony with the needs of theory of algorithms. N.A. sanin suggested successfully the constructive interpretation of mathematical propositions congenial to ideas of A . N . Kolmogorov and S.C. Kleene. It is based on the intuitionistic interpretation of logical connectives and quactifiers, theory of algorithms and Markov's principle. The substance of constructive interpretation is the algorithmical, i.e. the effective interpretation of the existential quantifier and the disjunction. The basic concepts and results of constructive mathematical analysis can be acquainted with, for example, in 1 2 1 , [ 131, 141, 1161, 1171. A summary exposition and detailed bibliography up to 1971 can be found in [ 61. The present paper deals with the structure of the real line in CM, with the questions of differentiability and with the theory of integral. Corresponding results of classical mathematics can be found in [ 11 and. [ 2 1 . Note, that a list of necessary concepts and notations, used in the sequel, is introduced in short in 82. 42.
Basic concepts
The basic objects of constructive mathematical analysis studied in this paper are words in the alphabet E containing, among others, the letters: 0,I , - , / , 0 , 0 , A , V etc.. By t * we denote the The signs A , a , denote empty set of all words in the alphabet
:.
REMARKS ON C O N S T R U C T I V E MATHEMATICAL A N A L Y S I S
83
word, g r a p h i c a l e q u a l i t y a n d c o n d i t i o n a l ( g r a p h i c a l ) e q u a l i t y , resp e c t i v e l y . The s y m b o l s U, V p l a y t h e r o l e o f v a r i a b l e s f o r t h e e l e m e n t s o f E*. Markov a l g o r i t h m s , which w e d e a l w i t h , a r e t h e a l g o r i t h m s o v e r the alphabet
:.
The a p p l i c a b i l i t y o f a n a l g o r i t h m
t o a word P is
d e n o t e d by ! O r ( P ) . The t e r m " s e t " i s u n d e r s t o o d i n t h e same s e n s e a s i n [ 1 3 1 . W e
mn%
,
write
"2
t h e d i f f e r e n c e of
m\Z
I
,
and
f o r t h e u n i o n , t h e i n t e r s e c t i o n and
respectively.
By a s y s t e m o f words o f a c e r t a i n t y p e w e mean a n y l i s t o f words of t h e t y p e . S y s t e m s o f w o r d s w i l l b e d e n o t e d by {Vi}y=o.
By a f i -
n i t e s e t w e mean a s e t f o r w h i c h a l i s t o f a l l i t s e l e m e n t s c a n b e g i v e n , and by a n i n f i n i t e s e t w e mean a s e t d i f f e r e n t from any f i n i t e set.
1 By NNs w e mean t h e words 0, 0
0 1I
,... .
The s e t o f a l l NNs
w i l l b e d e n o t e d by N. The s y m b o l s i , j , k , m , n , p , q , s , t v a r i a b l e s f o r NNs. The r a t i o n a l numbers c e r t a i n words i n
:.
p l a y t h e role of
( R t N s ) are i n t r o d u c e d as
The s e t o f a l l R t N s w i l l b e d e n o t e d by Q. The
symbols a , b , c , d p l a y t h e r o l e o f v a r i a b l e s f o r R t N s . A s a b b r e v i a t i o n a l n o t a t i o n f o r NNs a n d RtNs w e s h a l l employ t h e s t a n d a r d n o t a t i o n of t h e form 2 , -3,
2 etc. 5
.
L e t us n o t e t h a t t h e a r i t h m e t i c a l opera-
t i o n s o v e r t h e s e numbers a r e , i n l i n e w i t h t h e r e q u i r e m e n t s o f CM, realized algorithmically. W e u s e t h e c o n s t r u c t i v e i n t e r p r e t a t i o n o f m a t h e m a t i c a l propo-
sitions (1131,
[ 1 4 ] ) . The s u b s t a n c e o f i t i s t h e a l g o r i t h m i c a l i n -
t e r p r e t a t i o n o f 3 and v . F o r e x a m p l e , t h e f o r m u l a VU3V A ( U , V ) h o l d s i f f t h e r e e x i s t s a Markov a l g o r i t h m VU( !&(U)
& A(U,
&(U)))
such t h a t
holds. A d i s j u n c t i v e formula holds i f f it
i s p o s s i b l e a l g o r i t h m i c a l l y d e t e r m i n e t h e member o f i t w h i c h h o l d s . L e t us r e c a l l t h e importance of s o - c a l l e d normal f o r m u l a s ( i . e . f o r mulas n o t c o n t a i n i n g 3 a n d v ) .
E.g.,
any normal f o r m u l a i s e q u i v a -
l e n t t o i t s own d o u b l e n e g a t i o n . By a normal s e t w e mean any s e t f o r w h i c h t h e r e l a t i o n of memb e r s h i p c a n b e g i v e n by a normal f o r m u l a . I t s h o u l d b e n o t e d t h a t
w e d e a l o n l y w i t h v a r i a b l e s f o r which t h e domains o f a d m i s s i b l e v a l u e s are normal s e t s . L e t u s f i x Markov a l g o r i t h m s c o r r e s p o n d e n c e b e t w e e n :* algorithm inverse t o
G
and
a n d N, w h e r e
G.With
e s t a b l i s h i n g one-to-one maps N o n t o E * and
is
t h e h e l p o f t h i s numbering w e c a r r y
o v e r c o n c e p t s i n t r o d u c e d i n i t i a l l y f o r s e t s o f NNs t o t h e sets o f
04
0 . DEMUTH and A.
KUEERA
words. Also, the well-known equivalence of Markov algorithms and partial recursive functions (PRFs) can be expressed in this way. The use of either Markov algorithms of PRFs depends on practical needs of the context. A s for the relative computability we use the constructive reformulation of its characterization, given in [lll. The only difference is caused by the constructive icterpretation of the existential quantifier. In fact, if B is a set of NNs and m , k and n are NNs B then by ( d Bwe mean the B-PRF with the index m, by (m) (k) 2 n we denote --3st((k,n,s,t) E Wp(m) & Ds E B & Dt E N\B) and by !(m)B(k) B p), (for notation see 111, 5 9 . 2 ) . For dewe denote ,73p((m) (k) IC.
tails and for the employment of relative computability in CM see [541. We consider the relativized PRFs as predicatively defined correspondences. Let us note, that these predicates are equivalent to normal formulas. @-PRFs are just the PFRs and therefore ( d is an indexing of PRFs. We usually write ( m ) instead of (m) It should be noted that we are interested, owing to the natural
'.
'
connection between concepts of constructive mathematical analysis and arithmetical predicates, only in the computability relative to jumps of empty set. It is known from the results of E.M. Gold and P. Putnam that the @(")-PRFs ( 1 Q n) can be represented on the basis of recursive functions by means of non-effective limits. Without leaving constructive program concerning effective processes we improve, by the use of relative computability, our ability to handle effective procedures. The advantage of the improvement consists in both substantial simplification and clearness of formulations. Relativized algorithms can be introduced on the basis of relativized PRFs. Let B be a set of NNs and let m be a NN. The correspong((dB(%(U))) is called a Bdence U(dBn defined by U(dBJ(U) algorithm with the index m. Let us note that @-algorithms are just the correspondences realizable by Markov algorithms. If B is a set of NNs and F is a B-algorithm then for any word P we denote by Fp a B-algorithm such that VV(Fp(V) F(PV)) 2
IC.
.
In such a manner as the constructive interpretation of 3 and v is expressed by means of Markov algorithms we can analogically define the relativized existential quantifier and the relativized disjunction on the basis of relativized algorithms. Let B be a set " ) for of NNs. We write aBUA1(U) ("B-exists a word U A2 for 3m(!j(dBj(0) & Al(i(dBj(0))), and we write A1
...
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
sBV( (V x A 3 A 1 )
&
(V x A 3 A2)), where A 1 , A
2
85
have no occurences
of variables m and V. In such a manner the interpretation of the quantifier is reduced to the interpretation of the quantifier 3 . Of course, 3 and '3 are equivalent. Further, 3 implies for any set B, and implies 713. Let B be a set of N N s and l e t m be a normal set. A B-algorithm F is said to be 1) a B-sequence of elements of if Vn( !F(n) & F(n) E , 2) a B-sequence of B-algorithms of a certain type if for any NN n Fno is a B-algorithm of this type. For any set% and any word P we denote b y m , the set {U : PU Em}. BY a sequences of sets of a certain type we mean a set 2 such that VU(U € % 3 3nV(U E noV)) holds and for any NN n
m
m)
4
atn, 5
is a set of this type. In the sequel we present B-sequences (of words or B-algorithms) and sequences of sets by their "members" using notations I . . . 1: and 1.. .In, respectively, Of course, e.9. "{P,}: is a B-sequence of words" means that there exists a B-algorithm F such that
Vn( !F(n) & F(n) E Pn) . In what follows we deal with various @(")-concepts. If n = 0 we, as a rule, omit the sign 6 in the corresponding notations. E.g., we speak of sequences instead of +sequences, continuity instead of +continuity etc.. The central concept of mathematical analysis is the concept of real number. There exist more constructive formulas mutually nonequivalent that characterize, from the classical point of view, the fundamentality i.e. cauchyness. Indeed, a sequence F of RtNs is said to be i) fundamental if Vn3mVk(m k 3 IF(k)-F(m) I < 2-"),
ii) pseudo-fundamental if vn~13mVk(m < k 3 IF(k)-F(m) I < 2-"). Let us note that in accordance with constructive interpretation of propositions the fundamentally of F means the existence of an algorithmical regulator of fundamentality of F, i.e. the existence of an algorithm transforming every NN n to the corresponding m. In the case of pseudo-fundamentality of F the formula is equivalent to a normal formula and its interpretation does not differ from the classical one, specifically the existence of algorithmical regulator is not required. As known, in the case of Specker sequence such algorithmical regulator does not exist, indeed. For sequences of RtNs we obtain the concept of d(")-fundamen-
0 . DEMUTH and A . KUEERA
86
tality so that we replace in i) 3 by 36 (n). It should be noted that a sequence of RtNs is pseudo-fundamental iff it is @'-fundamental. Let us mention that the similar situation, we have met in the case of fundamentality, is also found in the case of constructive formulations of other concepts such as convergence, continuity, uniform continuity etc.. In quite analogical way we receive concepts of #(n) -convergence and pseudo-continuity etc.. Let us-agree that the sign convergence.
-*
is used for denoting the convergence, i.e. @-
A constructive real number (CRN) is a RtN of a word of the form mOn, where m,n are NNs, [(m)l is a sequence of RtNs, [I( n)l is a sequence of NNs being a regulator of fundamentality of [(m)l. A pseudo-number (PN) is a RtN or a word of the form mQ, where m is a NN and [(m)l is a pseudo-fundamental sequence of RtNs. The set of all PNs is denoted by r. The symbols x,y,z,v,w play the role of variables for CRNs, the symbols c , q play the role of variables for PNs. For any RtN a we denote by a sequence of RtNs such that
Vn(&(n) P a) , for any NN m we denote by the algorithm [I( m)l. Let us note that on the basis of 6(")-fundarnental $(")-sequences of RtNs we can obtain "arithmetical real numbers". In this connection it is worthwhile to note that PNs are, de facto, real numbers constructive relatively to d ' (6'-CRNs). On the set of all CRNs and PNs the relations of equality and order are defined as predicates in the obvious way. Basic algebraic operations over these numbers are realized algorithmically. A s for the arithmetical complexity, the set of CRNs is TI -corn2
plete and the set of PNs (i.e. 71) is Il 3 -complete. It is known that the set of CRNs (with euclidean metric) forms a complete separable metric space (cf. I 1 4 1 1 By a segment (resp. interval) we mean any word of the form x A y (resp. x V y), where x,y are CRNs and x < y . If H is a segment (resp. interval) then Ep(H), Er(H) denote the left and right endpoints, respectively, IHI denotes the length of H, i.e. Er(H) - Ep(H), and (H)' denotes the interval Ep (H) V Er(H). A segment (resp. interval) the endpoints of which are RtNs is called rational segment (resp. rational interval). Relation of membership for CRNs and PNs to a segment (resp. interval) is defined in a natural way.
87
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
E
>
I t i s known (Lacombe, C e j t i n , Z a s l a v s k i j ) t h a t f o r any R t N
0 t h e r e e x i s t s a s e q u e n c e of r a t i o n a l i n t e r v a l s which e f f e c -
t i v e l y c o v e r s a l l C R N s and s u c h t h a t t h e sum of t h e l e n g t h s of an a r b i t r a r y f i n i t e s e t of t h e s e i n t e r v a l s i s l e s s t h a n
In the
E.
s e q u e l w e u s e t h e f o l l o w i n g t y p e of c o v e r i n g s ( [ 1 7 1 ) . A sequence @ of non-overlapping
r a t i o n a l segments ( r e s p . seg-
O A 1 i s s a i d t o be a c o v e r i n g ( r e s p . a r e a l
ments) c o n t a i n e d i n
c o v e r i n g ) i f E p ( O ( 0 ) ) = 0 , E r ( @ ( l ) ) = 1 and Vx(x E &
O A 1
3 3pq(Er(O(p)) = E Q ( @ ( q ) & ) E p ( @ ( p ) )Q x
x G E r ( @ ( q ) ) ) )h o l d .
A c o v e r i n g @ i s s a i d t o be
a)
regular i f the series
b)
singular i f
C l @ ( k )I c o n v e r g e s t o 1 , k
~ d w V p (Z i @ ( k )1 Q w
< 11.
k% L e t us n o t e t h a t t h e r e e x i s t s b o t h r e g u l a r and s i n g u l a r
coverings. The f o l l o w i n g lemma ( c f . [ 1 7 1 ) e n a b l e s u s t o g i v e a r e a s o n a b l e d e f i n i t i o n of t h e c o n s t r u c t i v e c o n c e p t of a l m o s t everywhere f o r CRNs
.
Lemma 2 . 1 .
L e t @ be a s e q u e n c e o f s e g m e n t s , x A y a segment,
and l e t t h e series Z I Q ( k )I c o n v e r g e s t o a CRN less t h a n Ix A yI k Then t h e r e e x i s t s a CRN w s u c h t h a t w E x V y & -3k(w 6 @ ( k ) ) .
.
C o r o l l a r y . For any c o v e r i n g 0 a)
i f t h e series Z l O ( k ) k
1
converges t h e n 0 is r e g u l a r ;
b)
i f the series ZIO(k) k
I
d o e s n o t converge t h e n $I i s s i n g u l a r and
t h i s series pseudo-converges
t o a P N n o t e q u a l t o any CRN.
A s e q u e n c e {Hnln of n o n - o v e r l a p p i n g
segments i s termed a n s
-
s e t and a CRN z i s termed a measure o f {HnIn i f t h e series Z n l H n [ c o n v e r g e s t o z . I f P i s a CRN o r a PN and {Hnln i s a n s -set U
t h e n w e w r i t e P E {Hnln f o r 1 1 3 n ( P E H n ) .
A p r o p e r t y A of C R N s i s s a i d t o h o l d f o r a l m o s t e v e r y ( a . e . 1 CRN x ( r e s p . f o r a . e .
CRN x from a segment H )
i f there exists a
sequence o f S o - s e t s { d n I n s u c h t h a t f o r any NN n t h e measure o f
d n i s less
t h a n 2-"
and f o r any CRN x ( r e s p . x E H ) ,
88
0. DEMUTH and A. KUEERA
i ( x E d n ) 3 A(x)
holds.
"A p r o p e r t y A o f PNs h o l d s f o r a . e .
e i t h e r by f u l l r e l a t i v i z a t i o n t o
8'
PN" c a n b e i n t r o d u c e d
of t h e d e f i n i t i o n j u s t given
( r e c a l l , t h a t PNs a r e , d e f a c t o , d ' - C R N s ) , t i v i z e d c o n c e p t s ( c f . [ 441 )
o r without using rela-
.
L e t us c o n s i d e r c o n s t r u c t i v e analogues o f t h e concept of a function of a real v a r i a b l e . An a l g o r i t h m f i s c a l l e d a c o n s t r u c t i v e f u n c t i o n of a r e a l v a r i a b l e (CFRV) i f t h e f o l l o w i n g c o n d i t i o n s are s a t i s f i e d : 1)
2)
! f ( x ) , t h e n f ( x ) i s a CRN,
f o r a n y CRN x , i f
t/xy(!f(x) & x = y 2 ! f ( y ) & f ( x ) = f ( y ) ) . An a l g o r i t h m f i s c a l l e d a v - o p e r a t o r
i f t h e following condi-
cq(!f(c) & f ( c ) E v & ( 5 = q 3 f ( c ) = f ( q ) ) ) .
tion is satisfied
One o f t h e well-known
r e s u l t s c o n c e r n i n g CFRVs i s t h e t h e o r e m
s t a t i n g t h e c o n t i n u i t y o f a n y CRFV a t e v e r y CRN i n i t s domain ( p r o v e d by K r e i s e l , Lacombe, S h o e n f i e l d , M o s c h o v a k i s , C e j t i n ) . On t h e o t h e r h a n d , e v e r y w h e r e d e f i n e d CFRVs ( t h o u g h c o n t i n u o u s ) need n o t b e e i t h e r u n i f o r m l y c o n t i n u o u s o r bounded on
O A 1 . As f o r do-
mains o f CFRVs l e t u s o n l y n o t e t h a t t h e y a re t h e s e t s o f t h e t y p e Gg
( i n t h e e f f e c t i v e sense)
( c f . Eernov
41
,
i n d e p e n d e n t l y i n [ 461 )
,
b u t t h e y n e e d n o t b e open ( F r i e d b e r g ) . F o r f u r t h e r r e s u l t s on domains see [ 461. A s f o r t h e a r i t h m e t i c a l complexity, t h e set of
CFRVs i s I13-complete
(indices of)
([ 641).
F o r b r e v i t y , e v e r y w h e r e d e f i n e d CFRVs, c o n s t a n t o n b o t h {x : x
<
O } and { x : 1
<
L e t F,G b e f u n c t i o n s . G,
By F*G w e d e n o t e a s u p e r p o s i t i o n o f F and
i . e . a f u n c t i o n s u c h t h a t VX(F*G(X) = F ( G ( x ) ) ) . I f F i s a n i n -
c r e a s i n g on O A l F-l
x}, are c a l l e d simply functions.
f u n c t i o n , F ( 0 ) = 0 & F ( 1 ) = 1 , t h e n w e d e n o t e by
t h e i n v e r s e f u n c t i o n o f F.
L e t {H a s e q u e n c e o f s e g m e n t s , xo A x1 a s e g m e n t , and F n n be a f u n c t i o n . Then, 1)
% ( { H n l n ) means:
IHnlnzO, 2)
{Hn}n i s a s e q u e n c e o f n o n - o v e r l a p p i n g
and 13n( 0 E (H,)
v 1 E (H,)
-
i f x ( { H n I n ) h o l d s , t h e n [F,{H,In]
on a n y H n t s a t i s f y i n g t l x ( i 3 n ( x E (H,) 3)
O)
0
,
aenotes a function,
1
linear
3 [ F , I H ~ )( ~ x )I = F ( x ) ) ,
F i X o A X " d e n o t e s a f u n c t i o n s u c h t h a t V z ( F I X o A X 1( lz ) =
F(max(min(z,xl) , x o ) ) ) .
segments,
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS L e t us n o t e t h a t f o r any c o v e r i n g 0 w e have
89
-
J((@).
F u r t h e r , f o r any f u n c t i o n F pseudo-uniformly
c o n t i n u o u s on
0 A 1 a n - o p e r a t o r d e n o t e d by Op[F] c a n b e c o n s t r u c t e d s u c h t h a t VxS(x =
5
2 F ( x ) = Op[F]
( 5 ) ) . L e t us n o t e t h a t a f u n c t i o n i s
pseudo-uniformly continuous i f f it i s @'-uniformly continuous. L e t F be a f u n c t i o n ,
a s e g m e n t , P (resp. R) a word b e i n g
xAy
a CRN o r a PN. Then 1)
A ( F , XAy ) d e n o t e s ( F ( y )
2)
by D ( R , F , P ) w e d e n o t e
-
F(x)),
( " R i s a d e r i v a t i v e of F a t P " ) ,
3 ) by Dcl(R,F,P) w e d e n o t e ( 1 ) w h e r e 3 i s r e p l a c e d by --3 ("R i s a p s e u d o - d e r i v a t i v e o f F a t P " ) ,
("F i s f i n i t e l y p s e u d o - d i f f e r e n t i a b l e 5) D
a t P")
,
w e d e f i n e DcQ ( + = - , F , P ) , Dcl( ( - - , F , P ) , and Dcn ( + - , F , P ) , (-m,F,P) ( u p p e r a n d lower p s e u d o - d e r i v a t e , r e s p e c t i v e l y ) , i n a
Q n a t u r a l way.
-C
L e t us n o t e t h a t i f a f u n c t i o n i s f i n i t e l y p s e u d o - d i f f e r e n t i a b l e a t a PN t h e n t h e v a l u e o f t h e c o r r e s p o n d i n g " p s e u d o - d e r i v a t i v e ' ' need n o t b e a PN b u t , i n g e n e r a l , a r e a l number c o n s t r u c t i v e r e l a t i v e l y t o @ ( 2 ) . The q u e s t i o n s of u p p e r a n d lower p s e u d o - d e r i v a t i v e s are s t i l l more c o m p l i c a t e d . F o r d e t a i l s see [ 511. By Theorem 5 . 4 o f [ 1 6 1 w e c a n c o n s t r u c t f o r any u n i f o r m l y continuous function F 1)
t r a n s f o r m i n g a n y segment H t o a CRN
a l g o r i t h m s ( S , F ) and ( 1 , F )
being a 1.u.b.
and g . l . b . ,
r e s p e c t i v e l y , o f v a l u e s assumed by F on
HI
2)
a l g o r i t h m s ( w , F ) and
( w , F ) (H)
N
( S , F ) (H)
-
(
(
0,F)
s u c h t h a t f o r a n y segment H
1 , F ) (H) a n d
( 0 , F ) (HI = ( 1 , F ) (H) A ( S , F ) (H). Lemma 2.2.
L e t F be a u n i f o r m l y c o n t i n u o u s f u n c t i o n . Then t h e r e
e x i s t s a sequencp o f CRNs { z k I k s u c h t h a t Vabcy(0 G b
<
c G 1 &
( F ( a ) = y v ( S , F ) ( a h b ) = y v ( 1 , F ) ( x A y ) = y) 2 3 k ( y = z k ) ) , and, c o n s e q u e n t l y , f o r any r a t i o n a l segment a A t
5
O A l
and f o r
0 . DEMUTH and A. KUCERA
90
any CRN y such that 13n(y = zn) we have 1)
(3x(x E aAb & F(x) = y) v d x ( x E aAb & F(x) = y)), if y E (0,F) (aAb), then C R N s x1,x2 can be constructed such that
2) a < x1 < x2 < b x 1 G X G X )2 .
&
F(xl) = F(x2) = y
&
Vx(x
E
aAb
&
F(x) = y
2
Let F be a function, H 5 O A l a segment, and z a CRN. By BVS(Z,F,H) we mean that variation sums of F on H are bounded by z . By Var(z,F,H) we denote BVS(z,F,H) & Vn -BVS(z-2-”,F,H). Further, F is called to be 1) a function of bounded variation on H if 3v Var(v,F,H), 2) a function of weakly (resp. quasi-weakly) bounded variation on H if there exists (resp. cannot fail to exist) a NN m such that BVS (m,F,H) If F is a function of bounded variation on OA1 then by 1 1 6 1 it is uniformly continuous and there exists a function V[F] such that Vw(0 < w < 1 3 Var(V[F] (w),F,OAw)). Further, any function of quasiweakly bounded variation on a segment is pseudo-uniformly continu-
.
o u s on the segment.
Obviously, the class of functions of weakly bounded variation is closed under the basic arithmetical operations but this important property does not hold in the case of functions of bounded variation. For any CRN z we denote by hZ a function such that Vx(hZ(x) = z.max min(x,l) , O ) ) . We introduce the condition c1 important in the sequel. A function F is said to fulfil the condition a (in symbols, cu(F)) if Va3.z Var(z,f - ha,OA1). Example 2.1. There exist non-decreasing functions F 1,F2 fulfilling the condition N such that the function F1-F2 is not a function of bounded variation on any segment contained in OAl. Lemma 2.3. If F is a function increasing on OAl such that F(0) = 0 & F(1) = 1 then a(F) iff a(F-’).
a
A function F is said to fulfil the condition (resp. QcPlin symbols, a(F) (resp. (F))- if for any NN m there exists (resp. there cannot fail to exist) a N N n such that for any system of non-overlapping rational segments {aiAbij:=o
a,,
91
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS S
L: laiAbil
i=O
<
Remark 2 . 1 .
2-”
S
3
z I
i=O
(F,aiAbi)
I <
2-m h o l d s .
L e t F be a f u n c t i o n .
I f a ( F ) t h e n F i s a u n i f o r m l y c o n t i n u o u s f u n c t i o n o f weakly
1)
bounded v a r i a t i o n o n O A l ( b u t , i n g e n e r a l , n o t b e i n g a f u n c t i o n o f bounded v a r i a t i o n o n O A l ) . 2)
a ( F ) i f f f o r a n y S o - s e t {Hn}n s u c h t h a t % ( { H n j n ) , t h e s e r i e s
Z ~ A ( F , H ~I ) c o n v e r g e s . n 3) I f ( F ) t h e n F i s a f u n c t i o n o f q u a s i - w e a k l y bounded v a r i a -
acP
t i o n on O A l . A f u n c t i o n F i s s a i d t o b e a b s o l u t e l y c o n t i n u o u s o n a segment
H,
H
5 OAl,
i f t h e r e e x i s t s a sequence o f polygonal f u n c t i o n s { G } n n h o l d s ; AC(F) means t h a t F i s abso-
s u c h t h a t Vn BVS(2-”,F-Gn,H) l u t e l y continuous on O A l .
L e t us n o t e t h a t a n y a b s o l u t e l y c o n t i n u o u s f u n c t i o n ( o n O A 1 ) i s
a f u n c t i o n o f bounded v a r i a t i o n o n O A l f u l f i l l i n g t h e c o n d i t i o n
a.
But a stronger r e s u l t , being of i n t e r e s t i n t h e sequel, holds.
Theorem 2 . 1 .
acQ( F )
F o r a n y f u n c t i o n F, AC(F) i f f
a(F). ( C f .
E
L e t X b e a class o f f u n c t i o n s
1)
z-closed
So-set
(F) & a ( F ) i f f
1331, [ 3 4 1 ) . K is called
i f f o r a n y f u n c t i o n s F1,FZ f r o m K ,
a n y C R N s v,w,
and any f u n c t i o n cp i n c r e a s i n g on O A 1 s u c h t h a t Q ( 0 ) = 0 & c p ( 1 ) & AC (cp)
IF~I
any
{Hn}n s u c h t h a t Z ( { H n I n ) , a n y segment H c o n t a i n e d i n O A 1 & AC
E K,
(cp-’)
( F ~ +2F) E K ,
( F 1 ) I H 1 E X , F1* (3m(lFll
>A)
= 1
t h e following conditions are s a t i s f i e d : ( v . F ~ + w )E K ,
( F ~ . F ~E) K ,
I F ~ , { H , } ~ IE K ,
E K and
31
F1
EK),
2 ) A - c l o s e d i f i t is Ti-closed and f o r a n y f u n c t i o n F and any i n c r e a s i n g s e q u e n c e { x } of C R N s from O V l , s u c h t h a t x 1, n n n n-tthe following holds OAxnl Vn ( F EK) 3 F E K ; _ +
3)
V-closed
i f f o r a n y s e q u e n c e o f f u n c t i o n s {Fn}n a n d a n y f u n c -
t i o n F w e h a v e Vn F
E X & BVS(2-n,Fn-F,0Al)) 3 F E K . n It t u r n s o u t t h a t t h e class o f f u n c t i o n s a b s o l u t e l y c o n t i n u o u s
on O A 1 i s b o t h A - c l o s e d
and V - c l o s e d .
0 . DEMUTH and A . KUEERA
92
53.
Some r e s u l t s c o n c e r n i n g C R N s a n d PNs
T h e r e i s a c l o s e c o n n e c t i o n b e t w e e n c o v e r i n g s and p r o p e r t i e s
sets. C e j t i n ( 1 3 1 ) s t u d i e d pseudocuts, i . e . r.e. sets A of
of r.e.
of R t N s such t h a t A = { a : 3b(b pseudocut A(A*p
E A &
a
<
B)}, a n d p r o v e d : a
A+Q) i s s t r o n g l y u n d e c i d a b l e i f f t h e r e e x i s t s a
&
s t r o n g l o w e r i n g a l g o r i t h m f o r A , i . e . a n a l g o r i t h m t r a n s f o r m i n g any CRN x t o a r a t i o n a l i n t e r v a l H c o n t a i n i n g x and s u c h t h a t f o r t h e
s e t B, B
=
{ a : a E HI, T ~ ( B _C A v B n A
131, [ 7 1 i t fol l o ws :
= @)
h o l d s . Note t h a t f r o m
i f A i s a p s e u d o c u t f o r which 0 E A
&
1
9
A
then A i s s t r o n g l y undecidable i f f t h e r e e x i s t s a covering 0 such
>
t h a t Vb( b
3n( OAb _C & Q ( i ) ) )F ) u r.t h e r , a pseudo-
0 3 (b E A
c u t A i s wtt-complete
(A+@ & AIQ)
i f f t h e r e e x i s t s an a l g o r i t h m
t r a n s f o r m i n g any i n d e x o f a r . e . s e t W of R t N s c o n t a i n e d i n Q \ A t o a RtN
E
dently
>
0 less t h a n t h e " d i s t a n c e " b e t w e e n W and A ( [ 5 1 , i n d e p e n -
631 1 .
These f a c t s have f u r t h e r a p p l i c a t i o n s i n c o n s t r u c t i v e to p o lo g y . We c a n c o n s t r u c t a t o p o l o g i c a l l i n e a r s p a c e o f p a i r s of C R N s w i t h an e f f e c t i v e l y s e p a r a b l e t o p o l o g y which i s n o t e u c l i d e a n .
It turns
o u t t h a t s p a c e s of t h i s t y p e are complete i f f t h e convergence i n them i s t h e e u c l i d e a n o n e . Note t h a t b o t h c o m p l e t e and n o n - c o m p l e t e s p a c e s of t h i s t y p e w e r e c o n s t r u c t e d
([ 611,
[ 621).
I t i s p o s s i b l e by means o f c o v e r i n g s t o g i v e e x a m p l e s o f f u n c -
t i o n s having p e c u l i a r p r o p e r t i e s . I n t h i s connection, t h e s o r t i n g of c o v e r i n g s i s of i n t e r e s t . For example, l e t us t a k e f u n c t i o n s i n c r e a s i n g on O A l and mapping O A 1 o n t o O A l . Any s u c h f u n c t i o n t r a n s f o r m s , i n a n a t u r a l way, any r e a l c o v e r i n g t o a r e a l c o v e r i n g . Two coverings are c a l l e d equivalent i f they can be transformed t o each o t h e r i n t h i s way. Lemma 3 . 1 .
1)
Any e q u i v a l e n c e c l a s s o f r e a l c o v e r i n g s c o n t a i n s a r e g u l a r
covering. 2)
There e x i s t s an e q u i v a l e n c e class of real c o v e r i n g s c o n t a i n i n g
only r e g u l a r real coverings. Example 3.1. g(0) = 0
&
T h e r e e x i s t a f u n c t i o n g i n c r e a s i n g on O A 1 ,
g ( 1 ) = 1 , and a s i n g u l a r c o v e r i n g Q and a r e g u l a r
covering Y such t h a t g
=
[ g , Y ] and g t r a n s f o r m s any Y(n) o n t o 0 ( n ) . -1
From Theorem 2 . 1 and Lemma 2 . 3 i t f o l l o w s t h a t AC(g
)
,
a ( 9 ),
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
93
~ q , ( g )and, hence, TAC(g).
Definition. A real covering 0 is called hereditarily regular if any equivalent real covering is regular. Let us note that hereditarily regular coverings are used in constructive theory of non-absolutely convergent integrals (see [ 481 )
.
Studying the pseudo-differentiability it is useful to divide PNs into two classes according to existence of a certain weak algorithmical regulator of fundamentality (see [431). Definition. A PN 5 is said to be a) a PN of the first class ('PN) if there exists a sequence of non-infinite recursive sets of NNs {Cmlm such that for any NNs m,q the Lebesque measure of the set x < max(L(n), {x : 773n( 0 < n < q & n 9 C & min(&(n) ,&(n+l)) m &(n+l)l) f is less than 2-m; b) c)
2
a PN of the second class ( PN) if 5 is not a 'PN; the set of all 'PNs (resp. 'PNs) is denoted by (resp. 'IT). As for the arithmetical complexity we have the following.
'IT
Theorem 3.1. ([ 6 4 1 ) The set of all 'PNs is I13-complete (and, hence, recursively isomorphic to the set of all PNs). The existence of "weak algorithmical regulator of fundamentality" for 'PNs causes the existence of coverings which, moreover, "weakly cover", i .e. pseudo-cover, 'n
.
Theorem 3.2. For any NN t there exists a sequence of rational segments { K ts f s such that
3) for any 'PN 5 there exists a non-infinite r.6. set of NN s C such that the segments R:, s E C, are non-overlapping and T-I3S(S
E
c
&
5
E Kk) ;
4) if {Dmlm is a sequence of r.e. sets (resp. non-infinite r.e. sets) of rational segments such that for any NN m the sum of the
0 . DEMUTH and A.
94
KUCERA
lengths of an arbitrary finite set of segments from Dn is less than 2-m, then there exist a NN mo and a r.e. set (resp. non-infinite r.e. set) of NNs C such that for any NN p, p E C, there exists a for which Kt c H holds, and any segment H from segment H from D mo P is covered by a finite number of segments Kt p E C.
Dm
P'
t iff Vt--3s(C E Ks)
Corollary 1. If 5 is a PN then 5 E 'TI
.
Corollary 2 . Almost every PN is a 2PN Definition. A covering 0 is called a 'TI-covering if
VC(5 E 'TI & 5 E OA1 2 1 4 k ( < 0(k))). From Theorem 3 . 2 we obtain the following. Theorem 3 . 3 . For any NN t there exists a 'Ir-covering m k that Vm( I Qt(k) I < 2-t) & Vs3k(K: n 0Al 5 U Ot(p)). k=O p=o
Qt such
Lemma 3 . 2 . There is no regular In-covering. For any lv-covering 0 the series Z I O(k) I 2) k pseudo-converges to a 2PN and, consequently, there exists a 2PN 5, such that 5 is a non-decreasing sequence of RtNs. 1)
L-9
Remark 3 . 1 . There exist singular coverings which are not '71coverings. It turns out that any equality class of PNs containing a 'PNconsists entirely of 'PNs. On the other hand, 'PNs are not closed under the basic arithmetical operations. Theorem 3 . 4 . 1) 2)
1 For any PN 5 , 5 E 'v iff - ~ 3 q ( q E TI & q = 5 ) . For any PN q there exist 'PNs C1,C2 such that q
=
5, + 5,.
Theorem 3 . 5 . Let 5 be a PN, { Q k j k a sequence of segments such that -vm3p(
P
Z /Qkl
k=O
Then 5 E TI'.
< m)
&
Vp-3q(p
<
q
&
5
E Qq)
.
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
95
Corollary. Let {E k Ik be a sequence of So-sets, 5 a 2PN, and for any NN k let the measure of Ek be less than 2-k. Then k --3k(c 9 E ) . Obviously, no (real) covering can pseudo-cover all PNs from OAl. Nevertheless, PNs can be "covered" in a more general way (see [ 81 and Theorem 3 from [ 431 ) Theorem 3 from [431 implies the existence of a sequence of non-infinite r.e. sets of non-overlapping rational segments, contained in 0 1, such that U J 5 U J , 2 IJI 2 21 , Vx3p(x U J ) and, for any J E at JER JERp JE
.
P+l
{nnIn
np
P
Remark 3.2. The behaviour of functions is "reasonable" in a neighbourhood of any 2PN (see 16). Therefore, the '=-coverings are of a special interest in constructive function theory.
54.
Lebesgue measurable and integrable objects
We introduce the sets S and L1 as constructive analogues of the classes of Lebesgue measurable a.e. finite on [O,lI functions and Lebesgue integrable on [O,l] functions. The members of these sets are indices (codes) of sequences of "step functions" having the corresponding properties. On this basis we introduce also the concept of Lebesgue measurability of sets of CRNs. The detailed study of the subject, including the n-dimensional case, can be found in I201 (see also [ 181, 211 - [ 231, (271, 1291, [301, (321, 1361 and [401). By rational step frames (s-frames) we mean the words of the
.
type aoyal.. .yan6blyb2.. bn, where n 2 1, Vi(0 &
0 = a .
< al... < an
= 1 & Vi(1
<
i
<
n 3 bi
<
i
<
n 3 ai E Q)
E Q).
-
There exists an algorithm 6 such that for any s-frame R the algorithm gR is a polygonal function which is an "indefinite integral of R on OAl". For any CRNs xo,xl,xo
<
xl, by
The operations absolute value, addition, subtraction and multi-
0. DEMUTH and A. KUEERA
96
plication (denoted by I , I A , ;, 6 ) over s-frames can be defined in a natural way. Further, operation over s-frames corresponding to X the CFRV 1+7x1 we denote by wo. Let us note that for any s-frame R we have V[gRI = & /Rlo By T we denote the set of all words of the form Bm, where m is a NN and [(m)n is a sequence of s-frames. Let R be an s-frame, R z aoyal...yan6blyb2...ybn, let Bm E - T and x,y CRNs, and let U (resp. V ) be a word being a CRN or a PN. Then 1) by PO(V,R,U) we denote (73i(O G i G n & U = ai) & ( (U < O v 1 < U < U < a . 3 V = bi)); 3 V = 0) & Vi(1 S i S n & a i- 1 2) by P(y,Bm,x) we mean that there exists a sequence of CRNs {ynjn z
-
.
such that VnPo(yn,U(m)l (n),x) and yn n,,y; 3)
by Pcp (V,Bm,U) we mean that there exists a sequence of PNs such that VnPo( cn,[( m) 1 (n),U) and {cnIn pseudo-converges to V. By S and L1 we denote the sets of elements of T such that for
any Ern E T Bm E s
Vn(
"1
1 0
wo([(m)J(n)
1
Bm E L~
Vn(OJ
0
l [ ( m ) n (n)
;U(m)](n+l)) ;[ ( m ) ]
lo
(n+l)
< 2-") and
<
2-").
Let us note that L1 _C S . There exists an algorithm zp'such that VV( !S(V) & V(V) E L1 & vx(x E ov1 3 P(V,LqV) ,XI)) . Algorithmical operations absolute value, addition, subtraction and multiplication, denoted by I I , +, -, ., and w over elements of T are introduced as natural extensions of the corresponding operations over s-frames to sequences of s-frames (cf. 1 2 7 1 ) . Let us note that 1)
for any Om E s I BP E s
1 Bml ,
Bm+Bp, Bm-Bp
(2)
and Bm.pp are elements of S and w(Bm) E L1, 2) for any $m E L1, gp E L1 and any CRN v, (2) and v(v). Bm are elements of L1. Let Om E T, Bp E T and let v be a CRN. Then 1) Bm = 0 (resp. 0 S Bm) means that Pcp (O,Bm,x) (resp. 3y(Pct (y,gm,x) & 0 d y ) ) holds for a.e. CRN x (from O A l ) ; 2) we write Bm = ~p (resp. Bm G Bp) for Bp-Bm = 0 (resp. 0 G Bp-Bm) ;
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS 3 ) we write Bm
(v).Bm
)
.
-
v
(resp. v.Bm
)
for B m
-
(v)
97
(resp.
Theorem 4.1.Let Brn E S. Then there exist a sequence of S o -sets and a sequence,of uniformly continuous functions { Q 1 n "-n such that for any N N n the measure of d n is less than 2 ,
gn+l 5 dn
and
Vx(~(xEdn)
3
P(rpn(x) ,Bm,x)e
VE (-(EE~")>P,~(oP[(P~I( 6 ) ,Bm,c)). Definitions. 1) A function F is said to be Lebesgue measurable (resp. Lebesgue integrable) on OAl, if there exists a BmES (resp. a BrnEL,) such that P(F(x),Brn,x) holds for a.e. CRN x from OAl. 2 ) An object BmES is called summable if there exists a BpELl such that Bm = Bp.
Theorem 4.2. Let BmES and BpEL1. Then 1) if IBmlGBp , then ern is summable and, consequently, 2)
Bm is summable iff lBm[ is summable.
Lemma 4.1. Any function of weakly bounded variation on OAl is Lebesgue integrable on OAl. There exist algorithms NNs
I,
11 11
and p s such that for any L1
mo and ml and any C R N s xo and xl, x 0 G 1'
1) if BmoEL1, then
a) J is applicable to the word xoLx10 n70 and transforms it into a CRh being the limit of the fundamental sequence of C R N s
2) if BmOES and BmlES, then
0. DEMUTH and A. KUEERA
98
Theorem4.3. 1) A function F is absolutely continuous on OA1 iff there exists a BmELl such that vx (0 < x
<
1 2 F(x)
-
X
F(0)
=
J, Bm 1 .
(3)
2) Let F be a function and let Bni€Ll such that (3) holds, then vxy ( 0
<
x
Y
2 var(Jx/Bml,F,x~y))
.
Theorem 4 . 4 . ( [ 361) Let Bm€L1. Then there exist a sequence. of "),,a sequence of uniformly continuous functions { Q } So-sets such that for any NN and an increasing sequence of N N s n the measure of is less than 2-" and n+l
vx (-I(x€")>P(tpn(x), Bm,x)
&
a < x < b & O < b - a < 2
-Pk
&
vkab (n
3 ~ a I B m - ~ n ( x )< l
<
k
&
2-k.laAbl)
);
consequently, a.e. CRN from OAl is a Lebesgue point. Corollary 1. Let F be a function and BmELl such that (3). Then 3y (P(y,gm,x)
&
D(y,F,x))
holds for a.e. CRN x.
Corollary 2. Let F be a function and let Bm€L1 such that (3.) Let H be a segment. Then 1) F is constant on H iff for a.e. CRN x from H holds P(O,Bm,x) ; 2) F is non-decreasing on H iff for a.e. CRN x from H 3y (P(y,Bm,x) & 0 < y) holds. The following theorem can be proved on the basis of preceding results. Theorem 4 . 5 . (L1, 11
1
([27]) 1 ) 11 11
is a norm for the set L1. L1 ) is a complete separable normed linear space.
2) p s is a metric for the set S . linear metric space.
(s,ps)
is a complete separable
Definition. Let {Fnln be a sequence of CPRVs, {Bmnjn a sequence of elements of the set T, and let BmcT. i s said to be almost uniformly fundamental 1) {FnIn (re~p.{Rrn~}~) k if for any NN k there exist an S - s e t d with measure less than 2-k and a sequence of N N s {ptIt zuch that for any CRN
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
99
x, xEOAl & 1 (x€dk!, Vn (!Fn(x)) holds (resp. there exists a sequence of CRNs {znIn such that Vn P(zn, Omn, x) holds) and {p& is a regulator of fundamentality of the sequence of CRNs {Fn(x) In (resp. {znIn). " {FnIn (resp.{BrnnIn almost uniformly converges to Bm" can be defined in an analogical way. 2)
Theorem 4.6. ([40]) Let {BmnIn be a sequence of elements of the set S. Then 1) if the series B ~ ~ ( B m ~ @ B r n converges, ~+~) then the sequence n {BmnIn is almost uniformly fundamental, 2)
if the sequence{Bmnln is almost uniformly fundamental, then a) it is fundamental in the space ( S , p s ) b) for any BmES, {Bm,},
and
almost uniformly converges to Bm
iff pS(6mn@m) n z O . Corollary. Let {FnIn be an a h o s t uniformly fundamental sequence of uniformly continuous functions. Then there exists a BmES such that {FnIn almost uniformly converges to gm. The following analogue of Levi's theorem can be proved with help of Theorems 4.5 and 4 . 6 . Theorem 4.7. Let {BmnIn be a sequence of elements of the set L1 1 converges. Then the sequence such that the series Z n o I Bmnl a is both fundamental in (L1, 11 IIL ) and almost { Z p= 0 Brn,}, 1
1
uniformly fundamental and, consequently, there exists a BmeLl l a 1 2 Bm - Bml -0 and the sequence such that 0 p=o p qa almost uniformly converges to Bm.
I
such Example 4.1. There are a function F and a a-operator that vxg([ = x 3 = F(x) &IF(x) I < 1) and F is not Lebesgue measurable. Consequently, we can construct a sequence of polygonal
a(6)
functions {FnIn such that Vn (IFnl < 1) and for any CRN x and any PN 5 the sequence of CRNs {Fn(x) In converges to F(x) and the
0. DEMUTH and A. KUEERA
100
.
sequence of P Ns {Op[ Fn] ( 5 )jn $'-converges to @ ( c ) Thus, this type of convergence is weaker than both almost uniform convergence in (S,pS). be called regular if Definitions. 1) A set of C R N s &!will 2 xEOAl & (x = y 2 y E n ) ) VXy ( ~ € 2 2 ) A regular set of C R N s a will be termed Lebesgue measurable and a CRN z will be called the measure of this set (in symbols, m ( z , x ) ) if there exists a BmELl such that P(l,Bm,x)) holds for a.e. CRN x (P(O,Bm,x) v P(l,Brn,x)) & (,Em and z = 1' Bm.
.
0
Remark 4.1. 1) For any Lebesgue measurable regular sets of C R N s
m, and m2and for any segment H the following sets m,U r n 2 , an, nm, , abt, \ a n 2 and n , n H,
where
mln H * {x : XE %?., & xEH} , are Lebesgue measurable regular sets of C R N s . 2) For any Lebesgue measurable regular set of C R N s there exists an absolutely continuous on O A l function F such that vxy (X < y 2 o G A(F,XAY) G I XAYI & ??2( A (F,XAY),?tn XAY)) holds and (D(O,F,x) v D(l,F,x)) & (D(l,F,x) xfEn ) & (D(O,F,x) l ( x E a ) ) holds for a.e. CRN x. Any CRN x such that D(1.F.x) (resp.D(O,F,x)) is said to be a point of density (resp. dispersion) for the set 3 ) For any S o - s e t 8 , d ~O A l , and any CRN z, which is the measure of 8 (in the sense of the definition from §2), the set {x : x E 8 } is a Lebesgue measurable regular set of C R N s and (z,{x : xE 8 3 ) holds. As a corollary of Levi's theorem we have :
n.
m
Theorem 4.8. "291) Let IRnlnbe a sequence of Lebesgue measurable regular sets of C R N s and let {znln be a sequence of C R N s such that Vn m ( z n , ( n.;l) U ( Xn+,\ 2 n)) holds and the series 2 z converges. Then the set n n { x :-I 7 3nVk(n < k 2 xE atk) } is a Lebesgue measurable regu.lar set of C R N s . The following statement reflects peculiar properties of singular coverings.
nn\
101
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
Theorem 4.9. ([49]) Let Q be a covering, letat be a Lebesgue measurable regular set of CRNs, let z be the measure of and
at
let
k
Vk ( 2: IO(i)l< 1 i=O
m(y,
xn O(q))
&
y
-
z ) hold. Then
<
2-’.
IO(q)l
Vp3qy (p < q
&
holds.
Example 4.2. Let 0 be a singular covering. Then {x : 3n ( Ep (O(n))< x < 5 . a) the regular set of CRNs ( E y (Q(n)) + Er(Q(n))))], is (in the effective sense) an open set which is not Lebesgue measurable ; b) there exists a uniformly continuous function F such that
n,
Vx ( 0
f
~
€
)
2holds (cf. Theorem 4.11).
Nevertheless, we can prove the following statement. Theorem 4.10. ( [ 2 9 ] ) Let& be a Lebesgue measurable regular set of CRNs and let z be the measure of this set. Then for any NN m there exists a Lebesgue measurable open regular set of C R N s
a
with measure less than z + 2-m such that 8 n O V 1 5 (thus, the set?? is “equivalent“ to a Lebesgue measurable set of the type G s ) A relation between Lebesgue measurability of objects (or functions) and that of regular sets of CRNs is described in the following
.
statements (cf. [ 401 )
.
Theorem 4.11. Let RmES. Then there exist a sequence of CRNs {ykIk
and an increasing sequence of NNs {p )
CRN y ,- 3k (Y = Y,) a) the s e t
{x : xE0A.l where
&
3z
9 9
,
(P(z,Bm,x)
&
such that for any
z*y)1,
(4)
.
* p < v * p < v * z = v * p e v * E > ,
(5) is a Lebesgue measurable regular set of CRNs, b) if we denote by w*” (for * satisfying ( 5 ) ) the measure of t h e set ( 4 ) , then
,=IY
= 0
&
=
w2Iy
z
1
- wGry =
1
-
w<”
0. DEMUTH and A. KUEERA
102
Theorem 4.12. Let B m m and let * be one of the signs > and > such that 1) 3 2 P(z,Bm,x) holds for a.e. CRN x from OAl. 2 ) there exist an everywhere dense sequence of C R N s {vkIk and a sequence of C R N s { w ~ }such ~ that a) for any NN k the set {x : xEOAl & 3 2 ( P ( z , Bm,x) & z * vk) } is a Lebesgue measurable regular set of C R N s and wk is the measure of this set, b) Vp3qVk ((q
< vk
3
wk < ) ' 2
&
(vk
< -q
wk
>
1
-
2-'))
holds. Then there exists a BtES such that f3t = Bm. Theorem 4.11 allows us to define for S the comcepts of both "convergence in measure" and "fundamentality in measure" in the way quite analogical to the classical one. Theorem 4.13. Let {Bmnjn be a sequence of elements of the set S and let BmES. Then 1) {Bmn}n is fundamental in measure iff it is fundamental in (S,PS)
2)
;
{Brnnln
is convergent in measure to Bm
ps(f3mn B Bm) n=O
iff
holds.
We can prove the following analogue of Lebesgue's theorem. Theorem 4.14. Let
Bq€Ll and let@nn}n be a sequence of
elements of the set S , fundamental in ( S , p S )
I
such that
Vn (IBmn[ < B q ) . Then there exist a sequence of elements of the set L1 {Bp,}, and a Bp€Ll for which 1 ) holds. Vn (Bmn = BP,) & ( J o l BP, -
PI
Example 4.3. There exists a sequence of polygonal functions {FnIn such that Vn ( 0 < Fn+l < Fn < 1) & Vx3n (Fn(x) = 0 ) holds and for any NN n the Lebesgue integral of Fn over O A l is greater than 4 . Let us note that the theorem on integration by parts and the first mean value theorem follow from preceding results. Let us consider the second mean value theorem "301).
103
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
Definition. A Bm€T will be termed non-decreasing, if for any NN n there exists an S -setdn with measure less than 2-" such that for any C R N s x and y o < x < y < 1 & - ( x 8") ~ & - (YE 3 3 vw (P(v,Bm,x) t~ P (w,Bm,y) & v < w) holds. Lemma 4.2. Let Bm be a non-decreasing element of the set T. Then there exists a BkES such that Bm = gk. Theorem 4.15. Let BpELl , let Bm be a non-decreasing element of the set T, and let yo and y1 be CRNs such that yo < Bm < yl. Then there exists a BqELl such that gq = f3rn.B~ and
Let us note that the double negation in ( 6 ) cannot be omitted.
§5.
Differentiability of functions at CRNs
Example 5.1. There exist functions F 1 and F2 such that 1) F1 is an increasing on Oal function which fulfils the Lipschitz
conaition on OA1 but which is not differentiable at any CRN from oAl (Cf. [ 261) ; 2) a(F2) holds but F2 is not pseudo-differentiable at any CRN (resp. at any 'PN) from OAl (cf. [ 441 ) . The example above shows that the direct analogues of classical results concerning the differentiability of functions of bounded variation does not hold in constructive mathematical analysis. In classical mathematics we have : any finite function differentiable almost everywhere on a segment is necessarily almost uniformly aifferentiable on it and the corresponding derivative is Lebesgue measurable. This fact leads us to the following definition. Definition. Let F be a function and BpES. A(F) we denote the following : there exist a sequence of So-sets {@In and a sequence of NNs {kmIm SUCH that for any 1) By
NN n the measure of d n is less than 2-" and vmxabcd (n < m & xEOAl & -I ( x E dn) & a < x < b A(F,cAd) max(l aAbl ,I cAdl ) < 2-km 3 A(F,aAb) laabl- I cAd I
&
c
<
x
<
2-m
),
<
d
&
0. DEMUTH arid A . KUEERA
104
2) By A(F,p,p) w e denote the following : there exist a sequence of S -sets {dnInand a sequence of NNs {kmIm such that for any NN n the measure of d n is less than 2-" and Vmxab (n G m
&
xEOAl
-
&
(x E d n )
A(F'aab) Jaml
3v (P(v,Bp,x)
- v
&
a
<x
&
lab1
< 2-km
3
<2-m)) holds.
Theorems 4.3 and 4.4 lead immediately to the following theorem. Theorem 5.1. Let F be an absolutely continuous on OA1 function. Then A(F) holds and for any amELl such that (3) we have R(F, Bm)
.
A ( F ) . Then Lemma 5.1. Let F be a function such that 1) F is Lebesgue measurable and there exists a 5pES such that J(F,BP), 2) for any NN n there exist an S -set {H 1 with measure less than P P 2-", a uniformly continuous function u, and a sequence of NNs {kmIm such that a) {Hpjp is a sequence of mutually disjoint segments, vp (HP
50
1)
3p (aE(Hp)O ) )
Er(H1) = 1
&
Va(0
<
a
<
1 3
,
b) for any PN 5 we have
Eg(H ) = 0 & 0
&
,
Vmab (a
for which C E O V 1
<5
&
0
&
< b-a <
3p ( C E ( H l o ) P 2-km 3
-I
holds,
D (Op [ Q l ( 5 ) , F,S).
In particular,
vx (xcOv1
73p (XE(H~)O) D (u,(x), F,x)) holds, is an absolutely continuous function (on OAl) which [F, i H p l p l fulfils the Lipschitz condition on OAl, and if further the function F is uniformly continuous, the series Z (w,F) (H ) converges. P P &
Corollary. Let F be a function such that 2 a PN. Then 7 -I 3rlD(rl,F,S) holds.
A(F)
and let 5
be
105
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
Theorem 5.2. ([48]) For every uniformly continuous function F we have : A(F) iff for any NN n there exists an So-set {H 1
P P
with measure less than 2-" such that ( {H 1 ) holds, [F,{H 1 1 P P P P is a function absolutely continuous on OAl, and the series Z (w,F) (H ) converges. P P Let us note that the class of all functions F such that holds is A- and V-closed.
on
R(F)
Theorem 5.3. (1361) Let F be a function of bounded variation OAl. Then A ( F ) E a ( F ) holas.
.
Then Corollary. Let F1 and F 2 be functions, a(F1) & a(F2) a ( F 1 + F2) holds iff (F1 + F ) is a function of bounded variation 2 on OAl. Theorem 5 . 4 . Let F be a function of bounded variation on OAl. Then a) we have a(F) E a(V[F]) and a(F) U(V[F]) and, consequently, AC(F) E AC(V[F]) , b) for any BmES such that A(F,Bm) we have A(V[FI,I Bml) The following theorem gives information concerning the connection between almost uniform differentiability and pseudodifferentiability almost everywhere.
.
Theorem 5.5. Let F be a uniformly continuous function. Then JCF) holds iff for any NN n there exist an S -set {H with P P measure less than 2-" and a uniformly continuou: function cp such that {H } P P
DCL
C
OAl, VX (XEOVl
&
73p (xEH
P
(rp(x),[ F, {Hp}p] ,x)) and the series
)
3
2 (w,F) (Hp)
P
converges.
On the basis of this theorem and theorem 2.1 we have the following.
OAl
([32]) A function F is absolutely continuous on Theorem 5.6. iff &(F) holas and there exists a BmES such that for
a.e. CRN x from OAl
3y (P(y,Bm,x)
&
DcL(y,F,x)) holds.
0. DEMUTH and A . KUEERA
106
Example 5.2.
There exist a uniformly continuous function F, 01 such that
a Lebesgue measurable function G and a a-operator Vx D(G(x),F,x)
&
VSDcp(
a(c), F , C )
&
7
A(F).
Theorem 5.7. ( 1 5 8 1 ) For every Bm€S there exists a uniformly continuous function F such that a(F,@m). Definition. A function F will be termed singular (on O A l ) if a(F) and for a.e. CRN x from OAl Dcp(O,F,x) holds.
,
Theorem 5.8. ( [ 3 5 ] ) Let F be a function and let v be a CRN such that Var(v,F,OAl). Then F is singular iff the following holds VZ Var(v + 121 ,F - h Z , OAl). Theorem 5.9. ( [ 3 5 I ) Each of the following two conditions (a) and (b) is necessary and sufficient for a function F to be expressible as the sum of an absolutely continuous function and a singular function. and there exists (a) F is a function of bounded variation on 0 ~ 1 a Bm€Ll such that fl(F,Bm) holds. (b) a(F) holds and the sequence of CRNs
-
hnl + V-[ F + hnl , OAl) In converges. (V+ and Vdenote positive and negative variation, respectively). {A(V+[F
Remark 5.1. 1 ) The function g from Example 3 . 1 is an increasing on 0111 function which fulfils the condition a but which is not the sum of an absolutely continuous function and a singular function. 2 ) Every uniformly continuous function F is expressible as !# * cp where cp is an increasing on O A 1 singular function and for A(!#,Bm)) holds. function J, 3m (BmES & Bm = 0 & We introduce the following concept as a constructive analogue of the concept of aproximate differentiability almost everywhere which is important for the Denjoy integral.
Definition. ( [ 5 0 ] ) Let F be a function and let @m€S. Then aap(F) ( r e s p . Aap(F,Bm)) means : for any NN n there exists an S -set {Hk}, with measure less than 2-" , ({Hklk), and such
107
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
Theorem 5.10. 1) For any function F and for any Bm€S we have a) a(F,Bm) II Aap(F,Bm) ; b) BaP(F) holds iff there exists a BqES such that holds.
aap(F,Bq)
2) If F is a function of bounded variation on OAl then A(F) aap(F) holds. 3) The class of all functions F such that V-closed.
Rap(F) holds is A- and
Example 5.3. There exist a uniformly continuous function F and an S -set {HnIn with measure less than 4 such that ( {HnIn ) &
x
&(F)
A ~ P ( F ) 4. AC([F, { H ~ ) ~& ~ 13XI (xEoAi
&
&
13n (XEH,)
&
DcP (F,x) ) holds. Example 5.4. There exist a uniformly continuous function F, a Lebesgue measurable function f and a covering 0 such that Vn AC(F[O(n)l) 56.
&
Vx D(f ( x ) ,F,x)
&
VEDcp ( F , O
&
1
aap(F).
Pseudo-differentiability of functions
By Example 5.1 there exist monotone functions satisfying the Lipschitz condition which are not differentiable at any CRN from OAl. At the same time, the importance of the concept of pseudodifferentiability is indicated by Theorems 5.6 and 5.5. Example 6.1. ([49]) There exists a non-decreasing function F such that for any Lebesque measurable regular set of C R N s a with measure less than 1 there exists a CRN x such that xEOVl & 7 (xE ) & DcQ ( + " ,F,x) holds. On the other hand, the following propositions hold. Lemma36.1. we have
7
([
441 ) For any function F and for any 2PN 5
(Dcp(- m,F,S) v D,,(+",F,E)).
0. DEMUTH and A . KUtERA
I08
Lemma 6.2. For any non-decreasing function F and for any Lebesgue measurable regular set of CRNs of positive measure we can realize a CRN x such that xEOAl & x E & ~ - Dcp(+m,F,x) holds.
a
Thus, for any non-decreasing function F we have : a) The inner measure of the set {x : DcP (+m,F,x)} is always equal to 0, but the outer measure of this set can be equal to 1 b) 1 DcP (+=,F,C) holds for any 2PN 5 (and, consequently, for a.e. PN 5 ) .
;
The described situation led to the following concept. Definition. A property A of CRNs is said to hold for w-almost every CRN x from OAl, if there exists a non-decreasing function G such that
VX (xEOV1
&
i
Dell
(+m
,G,x) 3 A(x))
holds.
The following constructive analogue of Vitali’s covering theorem holds. Theorem 6.1. ( [ 431 , [ 4 9 1 ) Let V be a property of rational segments such that for any rational segment aAb _C OAl and for any RtN r, 0
< r < I aAbl , we have (3cd (a < c < d < b
V(cAd) ) v 13cd (a < c
&
r
< I cad1
&
&
V(cAd)) )
r
< I cAdl
. Then
&
there
exist a sequence of rational segments {H } and a non-decreasing n n function G such that 3 ({Hn}n) & Vn (Hn 5 OAl 3 Hn 5 OV1 & V(Hn))
&
Vab ( 0
aAbnHn # $ ) I 0
a
&
<
<
a
Vc(E;EOVl &
2-m
&
1 & V(aAb) 3 3n ( f . 1
1
DcP ( + m,G,c)
V(aAb)) 3
1 1
&
Vm
aAbl
1
&
3ab (a < 5
&
3n (SE(Hn)O)).
It turned out, that many properties of rational segments, important for pseudo-differentiability of uniformly continuous functions, fulfil the presumption of this theorem. This fact and the following lemma enable us to prove Theorems 6.2, 6.4 and 6.5. Lemma 6.3. “ 4 4 1 ) Let F be a function and let w and z be CRNs such that w < z & Vabx ( 0 < a < b < 1 & (x = z v x = w) 3 1
(x =
A(Frapb) , aAb
) )
&
F(1)
-
F(0)
<
z holds.
Then there exists a
sequence of rational segments {HnIn .such that
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS (
{HnIn
)
Vn (Hn
&
5
OAl 3 w.IHnI
Vxy ( 0 Q x Gy 1 3 A([F,{H,},] the function ([F, {Hnln] - hZ
<
A(F,Hn)
)
109
&
, xAy) < z.lxAyl). Consequently, is decreasing on 0 ~ and, 1 thus
[F, {Hn}nl is a uniformly continuous function of weakly bounded variation on 0 ~ 1 . Theorem 6.2. ([49]) Let F be a uniformly continuous function. Then there exists a non-decreasing function G such that for any PN S , c E 001 & - DcP ( + m , G , c ) , we have
Consequently, ( 7 ) holds for any 'PN, and for w-almost every CRN x from OA1 we have 1 1 (gcp ( - - ,F,x) & Dcn( + m ,F,x) v 3qDcp (QrFrx) 1 . Corollary. Let F be a function of bounded variation on OA1 Then for w-almost every CTN x from 0 1 we have 3qDcp(q,F,x).
Corollary. Let F be a uniformly continuous function. Then for 2 any PN 5, Op[F] IT , we have -I
7
(Dcp( -
mtF,S)
DcP ( + =.,F,S)
&
Dcn (+m,FrC) v
DcP ( - m ,F,E;) v DcP (F.5)
v
).
The assumption of uniform continuity in Theorem 6.2 cannot be omitted
.
hxample 6.2. There exists a function F of weakly bounded variation on O A l such that 1) CLce(F) & VX ( 1 Dcn ( - m , F , ~ )& -I Dca(+OO,F,X,)) holds,
0. DEMUTH and A. KUEERA
110
2) for any non-decreasing function G there cannot fail to exist a CRN X such that XEOVl & - DcP (+-,G,x) & DcP (F,x). On the other hand, we can prove the following statements. Theorem 6.4. ( [ 441 ) For any function F of quasi-weakly bounded variation on OA1 and for any 2PN 5 we have DcP (F,5) It is useful to confront this result with Example 5.1. Theorem 6.5. ([451) Let F be a function. Then for a.e. PN 5 (7)
we have
and (DcP(F,S) 3 3nDcp(n,F,S) ) .
Example 6.3. 1) We can realize an increasing on O A ~function F and a 2PN So such that V5Dcp (FrS)
&
a c L ( F ) & vx3y D (y,F,x) &
q3nDcp(QtF,SO)
(Cf. 1441).
2) There exist a pseudo-uniformly continuous function F and 2PNs 5 1 and 5,
i -cp D
(-
such that
-rF,S2)
&
1
-
Ilea
DcP (+" ,F,S1) & (-,F,S1) & Dcg ( + - tF,S2) & 1 DcE (F,S2) (Cf.1 451). 7
Let us note, that for any pseudo-uniformly continuous functions F and G and. for any 2PN Dcg (F,S)
(OdF]
D,p(F,S)
D c (G,S) ~
w e have
DcP (O,F,5)) and, consequently,
~i
&
(Op[Fl ( 5 )
-
OP[G] (S))E1n
2
-
Dcg (OrF - G r S ) We conclude this 5 by giving two sufficient conditions for a function to be non-decreasing. Notation. For any function F and any CRN x we denote Vm
7
73nVab (a
<x
&
IaAbl
< 2-"
A(F,aAb) I aAbl
>
-2-m)
by gca[F1 (x) 2 0 . Theorem 6.6. (b511) Let F be a function and let {xnjn be a sequence of CRNs such that 1) for any CRN x such that 13n (x = x ) we have 7
pCp(--,F,x)
(resp.
7
i
(Dcp(+- ,F,x) v Dcp(--,F,x) v
Dcg (Ftx)) 1 , 2 ) I&[F1 (x) 2 0 holds for a.e. CRN x from OAl. Then F is non-decreasing.
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
111
Constructive analogues of the conditions (N) and (T1)
57.
The condition (N) , introduced by N. Lusin, is very important in the theory of the integral. As it is shown in 551 and in [ 561 the following condition (N)* is an appropriate constructive analogue of this condition. Definition. Part I : A function F , 0 < F < 1, is said to fulfil the condition (N)*, if it satisfies the following condition : (a) for any Lebesgue measurable regular set of CRNs at the set {y : -I -13x ( x € a & F ( x ) = y)} is Lebesgue measurable. With the help of Cejtin's theorem ( [ 2 1 ) we can prove the following statement. Theorem 7.1. A function F , 0 < F < 1, fulfils the condition (N)* iff F is uniformly continuous and there exists a sequence of NNs {k } such that for any NN p and any system of rational
P P
segments S U
i=0
, 1=0 ;. I aiAbil < 2-kp , the measure of
{aiAbi}?=,
( 0 , F ) (aiAbi)
is less than 2-'
.
By this theorem the definition just given can be extended as follows. Definition. Part I1 : We say that a function G fulfils the condition (N)* iff G i s uniformly continuous and the function F such that
F =
.
(G
-
min(( I,G
)
(OAl), O)), where
u = max(( S , G ) (OAl), ( w , G ) (OA1), 1) , fulfils the condition (a) from the part I of the definition. Let us give several results. Lemma 7.1. Every function F , for which a(F),fulfils the condition (N)*.
0 . DEMUTH and A. KUEERA
112
Example 7.1. ( [ 5 5 1 ) . We can realize a function F such that F < 1 and for any CRN y, 0 Q y 1, there exists a se-
a(F) & 0
quence of mutually non-equal C R N s {xnIn such that Vn( 0 < xn < 1 & F(xn) = y)
.
(N)*.
2)
Lemma 7.2. Let F and G be functions fulfilling the condition Then 1) F G fulfils the conditions (N)*, 1
V U E E n 3 O ~ [ F (I5 ) (-3m BVS(m,F,OAl) E
n)
(32 v a r ( z , ~ , 0 ~ 13)
ac,(F)) holds,
3 ) iff -m3rl(Dcp(?lF,x) & 0 G then F is non-decreasing.
n) holds for a.e.
R ( F ) &)
CRN x from OAl,
Theorem 7.2. A function F is absolutely continuous on OA1 iff F fulfils the condition ( N ) * and there exists a Dm E L1 such that 3y(P(y,Dm,x)
&
Dc,(y,F,x)) holds for a.e. CRN x from OAl.
On the basis of facts, mentioned in [ 551 and [ 4 2 1 , the following conditions turned out to be proper constructive analogues of the Banach condition (T1). Definition. We say that a function F fulfils the condition (T,)* (resp. (T1 ) " ) if there exist functions 10 and Y such that I is a function of bounded variation on OAl (resp. a ( $ ) holds) and A C ( w ) & F = cp*$ holds. Remark 7.1. 1) The condition (T1)A implies the condition (T,)*. 2) Any function of bounded variation on OAl fulfils the condition (T1)*. Theorem 7.3. Let a function F fulfil the condition (T,)*. Then 1 ) F is uniformly continuous, 2) if 0 Q F G 1 holds, then there exists a Em E S such that a) for a.e. CRN y from OAl there exists a NN k and an increasing for which P(k,Bm,y) & system of C R N s {x.)k 7 1=f Vx(F(x) = y E 3 j ( l < 7 < k & x = x.)) holds, 3
b) F is a function of bounded variation on OA1 iff Bm is summable. Theorem 7 . 4 .
A function F fulfils the condition a iff F is a
function of bounded variation on OA1 fulfilling the condition (TIP.
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
From Theorem 7.4 and Theorem 5.3 it follows that the function F 1 from Example 5.1 fulfils the conditions (Tl)* and but it does not fulfil the condition (Tl)'.
1 I3
a,
Theorem 7.5. Let F be a uniformly continuous function such that A(F). Then 1) F fulfils the condition (Tl)* then it fulfils the condition (T1)', too; 2) F fulfils both the conditions (N)* and (TIIA iff Vg(5 E1 71 2 Op[F] ( 5 ) E1 TI) holds. Lemma 7.3. Let F be a function fulfilling both the conditions (Tl)* (resp. (T1)A) and (N)* and let G be a function fulfilling the condition (T1)* (resp. (T1)' 1 . Let us note that any uniformly continuous function is the sum of two functions fulfilling the condition (T1)A. A s for differentiability of functions fulfilling the con-
dition (N)* (resp. (Tl)*), in CRNs, let us remember Example 5.1. On the other hand, for any function F which fulfils the condition (N)* there cannot fail to exist a NN p (resp. (Tl)*) and for any 2PN and an increasing system of P N s { ~ j } ~ , l such that Vc(Op[F] ( 5 ) = fl --3j(l < j < p & 5 = c . ) ) holds. By this fact, 7 Theorems 6.2 and 6.3 and by Lemma 7.2 the following statement holds. Theorem 7.6. For any uniformly continuous function F and for any P N 5 , Op[F] ( 5 ) E2 7 i , we have: 1) if F fulfils the condition (Tl)*, then
--(D CQ (-=,F,c) v DcP (F,C) v DcQ (+-,F,<)) holds; 2 ) if F fulfils the condition (N)* then DcQ(F,S) holds.
98.
Superpositions of absolutely continuous functions
Theorem 8.1. ( [ 4 2 1 ) . In order that a function be a superposition of two (resp. of n, 2 < n) absolutely continuous on OA1 functions, it is necessary and sufficient that the function fulfils both the conditions (Tl)* and (N)*.
114
0. DEMUTH and A. KUEERA
Corollary. A superposition of two absolutely continuous on OA1 functions is absolutely continuous on OAl iff it is a function of bounded variation on OAl. Theorem 8.2. ( [ 3 8 ] ) Let F be a uniformly continuous function 051, there exist such that for any rational segment aAb, aAb an S -set d whose measure is less than (aAbl and a uniformly U continuous function G for which Vx(xEaAb & 7 (xE d ) 2 D(G(x) ,F,x)) holds. Then F is the sum of two superpositions of absolutely continuous on OA1 functions. Theorem 6 . 3 . ( [ 3 9 ] ) Every uniformly continuous function is the sum of three superpositions of absolutely continuous on OA1 functions. Example 8.1. There exists an increasing on OAl function F which fulfils the Lipschitz condition on OAl and which, consequently, fulfils both the conditions ( N ) * and (T,)*, and such that a) Vx3y D(y,F,x) holds, b) the function F cannot be expressed as the sum cpl * cp2 + cp3 * (p4 + cp5 , where cp. is absolutely continuous on Oh1
for 1 < i < 5. On the other hand, we can prove the following theorem.
Theorem 8.4. ([41]) A function F fulfils both the conditions and (T,)* iff there exist an absolutely continuous on OAl function cp and a function Y of bounded variation on OAl such that CLW) & F = cp * Y holds. (N)*
Example 8.2. There exists a function F such that a) F fulfils the condition ( N ) * , b) for a.e. CRN y there exist a NN k and an increasing system of k 3j (1 < j < k & x = x.) ) , C R N s {xjlj=l such that Vx(F(x) = y 1
c) F is not expressible as a superposition of a finite number of functions fulfilling the condition act (cf. [411).
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS 59.
115
Generalized absolutely continuous functions and functions of generalized bounded variation ( [ S O ] )
The importance of these concepts in the theory of the Denjoy integral is well-known. As it is shown by Example 3 . 1 , to deal with constructive analogues of these concepts it is necessary to consider not only CRNs but also PNs. Definition. Let F be a function. Then a) Wa(F) means: F is a uniformly continuous function of quasiweakly bounaed variation on OD1 and A(F) holds: b) we write WAC(F) for Wa(F) & aCl(F). From results of 5 5 2 , 5 and 7 it follows: Lemma 9 . 1 . a) For any function F we have AC(F) (resp. a(F)) iff WAC(F) (resp. Wa(F)) holds and F is a function of bounded variation on OAl. b) For any function F we have WAC(F) (resp. AC(F)) iff Wcr(F) (resp. a(F)) and
5(5
E 'TI
2 Op[Fl
(5)
E In).
(8)
c) The class of all functions F for which Wu(F) (resp. WAC(F)) holds is both i-closed and V-closed.
Lemma 9 . 2 . Let VAW be a segment, {xkVyk}k a sequence of intervals and {{H:}n}m a sequence of sequences of segments such that VAW 5 o A l & ~ 1 3 5 5( E vhw & 4 k ( 5 E xkVyk)) & J({{H:}n}m) holds. Then there cannot fail to exist a rational segment aAb, a NN m and a PN n such that v < a < 0 < b < w & - 3 k ( q E xkVyk) & Vcn(< E aAb & 5 E (H:)O 3 - i 3 k ( c E xkVyk)) holds. Notations. Let F be a uniformly continuous function, { {H:}n}m a sequence of sequences of segments and let Y be a word, f o r which Y O A C v Y n a v Y n W A C v Y n W a . Then 1) YG(F,I{H:I;}~) means: J({IH:I~}~) & vmY( F,IH:}, 2)
YG, (F,{{H;I~},J
(resp. YG,(F,I{H:}~}~))
means:
(9) ):
0. DEMUTH and A . KUEERA
1 I6
YG(F,{{Hr}n}m), and for every NN rn the series C(o,F) (Hm) converges (resp. pseudo-converges)
.
Definition. Let Y and Z be words such that (9) and Z P G v Z
P
Go v Z
E
(10)
G,.
A function F is said to fulfil the condition YZ (in symbols, YZ(F)), if F is a uniformly continuous function and if there exists a sequence of S -sets {{H:}n}m such that YZ(F,{{Hr}n}rn). From lemmas 9.1 and 9.2 and results of § § 2 and 5 it follows: Theorem 9.1. Let Y be a word, F a function, xoAyo a segment and {Knln an So-set such that ( 9 ) and YG(F1 & xOAyO 5 OAl & TC({KnIn) --35(0 < 5 < 1 & -3n( 5 €(Kn)O)) hold. Then there cannot fail to
&
exist rational segments aOAbO and alAbl and a PN 0 such that < bl < 1 & -3n(rl E (Kn)O) holds and x < a. < bo < yo & 0 < al < 0
tne functions F[aoAboland [F,{Kn 1 n
fulfil the condition Y.
Theorem 9.2. Let F be a function and Y a word such that (9) holds. Then we have: 1)
(Y(F) 2 YG,(F))
2)
(YG1F) 2aap(F)f : if ctG,(F) holds, then F fulfils the condition (T1)A;
3) 4)
5)
&
(YG,(F) 2
(F)
&
YGo(F))
&
(YGo(F)
3
YG(F))
&
if WACG(F) holds then (8) and (AC(F) : 3 2 Var(z,F,OAl)) hold; if WACG(F) & (F) holds then F is a superposition of two absolutely continuous on OAl functions; if WaG(F) holds then F is the sum of two superpositions of absolutely continuous on OAl functions.
Let us note, that for each of the following conditions ACG,, WACGO, WaG,, WACG and WclG the class of all functions which fulfil the condition is A-closed, but is not V-closed. The class of all functions F such that WACGo(F) & holds is both A-closed and V-closed.
a(F)
Theorem 9.3. Let F be a uniformly continuous functions such that A(F). Then we have: 1)
a) if
+S(D~~(--,F,S)
then aGo(F) ;
&
DCl(+-,~,c))
(11)
REMARKS ON C O N S T R U C T I V E MATHEMATICAL ANALYSIS
117
b) if (11) and F fulfils the condition (T1)* then aG,(F) 2)
if
+S(D~,(--,F,S)
v
DCl(+m,~,~)
;
(12)
then ACG, (F). Theorem 9.4. Let F be a uniformly continuous functions such that pap(F) holds. Then we have: 1)
if (11) then crG(F);
2)
if (12) the ACG(F). Example 9.1. There exists a function F such that ACG(F), (12)
and -WACGo(F) hold. Theorem 9.5. Let F be a function such that aG,(F). exists an increasing on O A l function cp such that
Then there
Theorem 9.6. In order that a function F fulfil the condition ACG, it is necessary and sufficient that F is a uniformly continuous function for which A(F) holds and there exists an increasing on O A l function cp such that (131, AC(cp) and (14) hold. Theorem 9.7. Let F be a uniformly continuous function such that &F) holds and let cp be an increasing on OA1 function such that (13) and i3E(gc1(--,F*cp -1 ,c) & 6cl(+-,F*cp-1,E)) hold. Then we have
.
WcrGo (F)
Theorem 9.8. Let Bm E S and let F be a uniformly continuous function such that 0 < Bm & aap(F,Bm) and (8) hold. Then F is a nondecreasing absolutely continuous on OAl function and, consequently, a(F,Bm) holds and (3m is summable.
510.
Constructive Denjoy integrals ( [ 5 2 1 )
We base the study of the Denjoy integrals on their descriptive definition (using results of 5 9 ) . Definition. Let Bm E S.
0. DEMUTH and A. KUEERA
118
L e t Bm E S .
Definitions. 1)
A function F is c a l l e d
a ) i n d e f i n i t e Denjoy i n t e g r a l i n t h e r e s t r i c t e d s e n s e i n t e g r a l ) o f Bm o n O A l i f ACG,(F) b) indefinite
3'-integral
&
(3,-
,L((F,Bm) h o l d s ;
o f Bm o n O A l i f WACGo(F) & d ( F , B m )
holds ;
c ) i n d e f i n i t e Denjoy i n t e g r a l i n t h e w i d e s e n s e ( 8 - i n t e g r a l ) o f Bm on O A l i f WACG(F) & a a p ( F , B m ) h o l d s . 2) Bm i s c a l l e d ,-integrable (resp. 3 '-integrable, resp. i n t e g r a b l e ) on O A l i f t h e r e e x i s t s a f u n c t i o n b e i n g a n i n d e f i n i t e i n t e g r a l o f t h e t y p e o f Brn o n O A l . L e t us note t h a t 1)
f o r any f u n c t i o n F a n d a n y Em E S i f F i s a n i n d e f i n i t e Lebes-
gue i n t e g r a l ( r e s p . d ) , - i n t e g r a l ,
resp.
then F i s an i n d e f i n i t e a),-integral a)-integral)
2)
at-integral)
(resp.
o f Bm o n OAI
a'-integral,
o f Bm o n 0 ~ 1 ;
a n y summable o b j e c t ( f r o m S ) i s a , - i n t e g r a b l e
integrable (resp. &'-integrable)
8- i n t e g r a b l e )
.
Examples 5.4 a n d 9 . 1
object is
a n d any
3'-integrable
resp.
3,(resp.
show t h a t t h e r e e x i s t a m e a s u r a b l e f u n c t i o n
i n t e g r a b l e i n t h e s e n s e o f Newton, which i s n o t d ) - i n t e g r a b l e ,
a &-integrable object not being
and
a'-integrable.
Remark 1 0 . 1 . R e s u l t s o f 59 g i v e us i m m e d i a t e l y f o r any i n t e g r a l , introduced above, t h e following s t a t e m e n t s : theorems of monotonicity and o f d i s t r i b u t i v i t y o f t h e i n t e g r a l and t h e o r e m o n i n t e g r a t i o n by parts. Theorem 10.1.
A Bm E S i s summable i f f
Theorem 1 0 . 2 .
Let
(0
Ism] i s & - i n t e g r a b l e .
b e an i n c r e a s i n g on O A l f u n c t i o n such t h a t
AC(tp) & cp(0) = 0 & c p ( 1 ) = 1 & A C ( ( 0 - l )
a n d l e t Bm E S .
Then t h e r e e x i s t s a Bp E S s u c h t h a t 1)
3 v w ( P ( v r ~ m , c p ( x ) )& D(w,cp,x) & P ( v . w , p p , x ) )
x from O A l ;
h o l d s f o r a.e.
a,-
a f u n c t i o n F i s a n i n d e f i n i t e Lebesgue i n t e g r a l ( r e s p . i n t e g r a l , resp. ' - i n t e g r a l , r e s p . d)- i n t e g r a l ) o f Bm o n OAI 2)
a
CRN
iff
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
I19
t h e f u n c t i o n F*cp i s a n i n d e f i n i t e i n t e g r a l o f t h e t y p e o f Bp on O A 1 i f f t h e t y p e o f Bp o n O A l ;
Bm i s summable ( r e s p .
3)
resp.
a)-integrable)
Definition.
,-integrable,
resp.
8'-integrable,
o n O A l i f f s o i s Bp. b e a Lebesgue m e a s u r a b l e r e g u l a r set of CRNs
Let
and l e t Bp E L1 s u c h t h a t ( P ( O , B p , x ) v P ( l , B p , x ) ) h o l d s f o r a . e . CRN x f r o m O A 1 .
x
s a i d t o b e i n t e g r a b l e i n a g i v e n s e n s e o n t h e set i n t e g r a b l e i n t h e sense on O A l .
(P(l,Bp,x)
&
5
Then t h e o b j e c t Bm E S i s
at
i s B r n . 6 ~ is
F u r t h e r , any i n d e f i n i t e i n t e g r a l o f
t h e t y p e o f Bm.Bp o n O A l i s s a i d t o b e a n i n d e f i n i t e i n t e g r a l o f
a.
t h e t y p e o f Bm o n t h e s e t
L e t Bm E S , l e t F b e a f u n c t i o n a n d l e t { x , } ~ b e
Theorem 1 0 . 3 .
a n i n c r e a s i n g s e q u e n c e s o f C R N s f r o m O V 1 which c o n v e r g e s t o 1. I f f o r any NN n t h e f u n c t i o n F'OAxnl i s a n i n d e f i n i t e a , - i n t e g r a l (resp.
al-integral,
resp.
d)-integral)
o f e m o n OAxn, t h e n F i s
a n i n d e f i n i t e i n t e g r a l o f t h e t y p e o f Bm o n O A 1 . Theorem 1 0 . 4 . L e t Om E S a n d l e t {H } b e a n S -set s u c h t h a t n n % ( { H n l n ) h o l d s . L e t F b e a n i n d e f i n i t e d, , - i n t e g r a y o f gm o n t h e
set {x : x E O A l
&
f u n c t i o n s such t h a t
- ( x E {H } ) } and l e t { F n l n b e a s e q u e n c e o f n n
f o r any NN n
1)
5 OA1) then 5 O A l t h e n Fn
a) i f -(Hn
Fn = 0 ,
b ) i f Hn
is an i n d e f i n i t e a , - i n t e g r a l
of Bm on
Hn and F n ( 0 ) = 0 h o l d s ;
t h e series X(w,F n n ) (H,)
2)
converges.
Then t h e o b j e c t 8m i s a * - i n t e g r a b l e
o n Oh1 a n d t h e f u n c t i o n
m
(F
+
nZOFn) i s a n i n d e f i n i t e a , - i n t e g r a l
of
Bm o n
OAl.
L e t us note t h a t q u i t e a n a l o g i c a l theorem h o l d s f o r a l - i n t e g r a l ( r e s p . 8 - i n t e g r a l ) , t o o . F u r t h e r , i n t h e case of a - i n t e g r a l ,
w e r e p l a c e t h e a s s u m p t i o n 2 ) by 2')
t h e series g l A ( F n , H n )
I
c o n v e r g e s a n d ( w,F n) (Hn )
t h e n t h e theorem r emai n s v a l i d .
0,
if
I20
0. DEMUTH and A. KUCERA
011.
The constructive Perron integral ( [ 5 3 ] )
In classical mathematics the Perron integral is equivalent to the Denjoy integral in the restricted sense ( 8,-integral). As we shall see, in constructive mathematics $)*-integration includes the Perron integral and Perron's process of integration includes 'integral, but the Perron integral is not equivalent either to
a a*-
% -integral or to a'-integral. Let us note that this fact is. connected with the existence of a function which fulfils the condition ci but is not expressible as the sum of an absolutely
continuous function and a singular function (cf. Remark 5.1). Definition. Let Bm E S. A function G is termed 1)
major function of Bm on O A 1 if G is uniformly continuous,
G(0) = 0 &
1 3 ~ g c l ( - - , G , ~holds ) and there exists a Bp E S such that
& Bm BP; minor function of Bm on O A 1 if the function -G is a major function of -Bm on OAl.
J(G,BP) 2)
Remark 11.1. I) By Theorem 9.3 any major function fulfils the condition aGo. On the other hand, there exists a major function F such that ~ c i G ,(F) By results of 555 and 6 if G 1 and G2 are major functions of a 2) S and G 3 is a minor function of Brn on O A l then the function Bm
.
min(G1,G2) is a major function of Bm on O h 1 and the function (G1-G3) is non-decreasing.
Definition. A Bm E S is said to be integrable in the sense of Perron ( @-integrable) on OAl if for any NN t there exist a major function G of Bm on O A 1 and a minor function that A(G-G,OAl) < 2-t holds.
of Bm on O A 1 such
Let Bm E S be 6'-integrable on OAl. Then, by Remark 11.1 and there exists a uniformly continuous function f o being both the 1.u.b. of all major functions of Em on O A l and the g.1.b. of all minor functions of Bm on O A 1 and, consequently, F ( 0 ) = 0 & A(FO,Bm) 0 holds; a function F is said to be an indefinite Perron integral 55,
(p-integral) of Bm on O A l , if the function (F-F OAl.
0
is constant on
121
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
Theorem 11.1. A f u n c t i o n F i s a n i n d e f i n i t e P e r r o n i n t e g r a l o f
Bm
E S on O A l i f f F
i s uniformly continuous,
J I ( F , B ~h)o l d s
and
t h e r e e x i s t s a n i n c r e a s i n g o n O A 1 f u n c t i o n cp s u c h t h a t a(cp) & = 1 &
cp(0) = 0
VS(7Dcl(F,S)
2 Dcl(O,F
* v-',
Op[cp]
(5))).
By Theorem 11.1 and by 59 w e h a v e : 1)
Any i n d e f i n i t e
indefinite indefinite
@ - i n t e g r a l i s an i n d e f i n i t e a ' - i n t e g r a l ;
i s an i n d e f i n i t e
dl,-integral
any
@ - i n t e g r a l . T h u s , any
@ - i n t e g r a l i s t h e s u p e r p o s i t i o n o f t w o a b s o l u t e l y con-
tinuous functions.
2)
The c l a s s o f a l l i n d e f i n i t e
6'-integrals
i s A-closed,
b u t it i s
n o t V-closed. Remark 1 1 . 2 .
F o r t h e P e r r o n i n t e g r a l ( t h e a n a l o g u e s o f ) Remark
1 0 . 1 a n d Theorems 1 0 . 1 Theorem 1 1 . 2 .
-
10.4 hold.
L e t Bm E S and l e t F b e a u n i f o r m l y c o n t i n u o u s
f u n c t i o n such t h a t aap(F,Bm) & Vg(
5 E 'n
3 Op[ F]
( 5 ) E 'n).
Then 1)
F i s a n i n d e f i n i t e P e r r o n i n t e g r a l o f Bm o n 0111
i f f t h e r e e x i s t s a major f u n c t i o n o f Bm o n O A l ;
2)
F is an i n d e f i n i t e dD*-integral
o f Brn on OA1
i f f t h e r e e x i s t s a major f u n c t i o n o f f3m o n O A 1 f u l f i l l i n g t h e condition (N)*. Example 11.1. T h e r e e x i s t s a Bm E S ,
9'-integrable
on O A l ,
s u c h t h a t Bm h a s n o major f u n c t i o n on O A l . Example 1 1 . 2 .
T h e r e e x i s t s a Bm E S w h i c h i s
@ - i n t e g r a b l e on
O A l , b u t which i s n o t a * - i n t e g r a b l e o n O A l .
Remark 1 1 . 3 . L e t g be f u n c t i o n f r o m Example 3.1.
Then 9 i s a n
i n c r e a s i n g o n O A 1 f u n c t i o n a n d t h e r e e x i s t s a f3m E S s u c h t h a t A(g,Bm). T h u s , g i s a major f u n c t i o n o f Bm on O A l and h o i s a minor f u n c t i o n o f Bm o n O A l .
On t h e o t h e r h a n d , Bm i s n o t
on O A 1 a n d , c o n s e q u e n t l y , i t i s n o t
a -integrable
8 - i n t e g r a b l e on O A l ,
too.
0. DEMUTH and A . KUEERA
122
5 12.
7-integral
(I481 )
A generalization of the Lebesgue integral possessing many of the properties of the Perron integral can be introduced by specifically constructive means.
Definition. 1)
A function F is said to fulfil the condition
7:
(in symbols,
Y(F)), if F is uniformly continuous and for any increasing absolutely continuous on O A l function cp, cp(0) = 0 & ~ ( 1 )= 1 , w e have
R(F*V). 2)
L e t Bm E S. Then
a) a function on O A l , if b) Brn is said indefinite
F is said to be an indefinite y-integral of Brn Y ( F ) & a(F,Brn) holds; to be 7-integrable on O A l if there exists an ?-integral of Bm on O A l .
Remark 12.1. 1)
The class of all functions fulfilling the condition “;c is both
A-closed and V-closed. 2) For y-integral (the analogues of) Remark 1 0 . 1 and Theorems 1 0 . 1 - 10.3 hold and a certain analogue of Theorem 1 0 . 4 is valid. Theorem 1 2 . 1 . Let Brn E S and let F be an indefinite ?!-integral of Bm on O A l . Then 1) if 0 < Bm then F is absolutely continuous on O A l ; if brn is 8 -integrable on O h 1 then F is an indefinite 2)
a*
integral of Om on O A l ; if Bm has on O A 1 a major function then F is an indefinite 3) integral of
Brn on
3-
OAl.
On the other hand, there exists a function F , Y ( F ) , which does not fulfil the condition ( N ) * and, consequently, F is not an indefinite 8 -integral. Indeed, for any uniformly continuous function G and for any hereditarily regular covering 0 we have y ( [ G , Q ] ) inie can also realize a function F, ACG,(F) & -?(F).
.
Let us note that the theory of ?-integral is a l s o useful for the study of other non-absolutely convergent integrals, dealt by us Example 12.1. There exist a function F and a Om E S such that
REMARKS ON CONSTRUCTIVE MATHEEIATICAL ANALYSIS
123
1) Y ( F ) & WAC(F) & A(F,f3m) holds, and, consequently, Em is both ?-integrable and 8'-integrable on OA1, 2) there is no major function of f3m on OA1 and thus Em is not 6'-integrable on OAl.
513.
A constructive analogue of functions of the first Baire's class ( 1 4 7 1 )
Definition. An algorithm F is said to be an A @ -operator if Vxy(!F(x) & F(x) E n & ( x = y 3 F(x) = F(y)) & F(x) = F(min(max(x,O) , 1 ) 1 f holds. 2) We say that an A6-operator F belongs to the first Baire's class 1)
(in symbols, F E B1), if there exists a sequence of uniformly continuous functions {Gnln such that for any CRN x (from OA1) the sequence of CRNs {Gn(x)ln pseudo-converges to F(x). Remark 13.1. 1)
For any function G there exists an A@ -operator F such that
F E B1 & Vx(F(x) = G(x)). 2) If a function G is finitely pseudo-differentiable at any CRN from OV1 then there exists an A @ -operator F such that F E B1 & Vxfx E OV1 2 Dcl(F(x) ,G,x)f
.
Theorem 13.1. Let F be an A 0-operator, F E B1, let p be a NN and aAb a rational segment, aAb 5 OAl. Then there @'-exists a rational segment cad such that cad 5 aAb & Vxy(c < x < y < d 3 IF(X)
-
F(y) I
< 2-')
holds.
Example 13.1. 1) We can realize a function G which fulfils the Lipschitz condition on Oh1 and an A @ -operator F such that F E Bl & VxD (F(x),G,x) holds but F is not pseudo-continuous at any CRN cl from OVl. 2) There exists an A@ -operator F such that Vx-3m(F(x) = m) &
Vxy(0
<
x
<
1 3 -(F(x) = F(y))) and, consequently, -(F E B1).
Remark 13.2. Let F and G be A 4 -operators from B1. Then A 4 -
I24
0. DEMUTH and A. KUEERA
operators IF], (F+G), (F-G) belong to B1 and, further, if +x(G(x) = 0) holds then (F/G) E B1, too. We can prove the following analogue of Lebesgue's theorem. Theorem 13.2. An A@ -operator F belongs to the first Baire's a} class iff for any RtN a the sets {x : F(x) > a} and {x : F(x) are sets of the type G6 (in the effective sense). Theorem 13.3. An A@ -operator F belongs to the first Baire's class iff there exists an A @ -operator G such that Vx(F(x) = G(x)) & VxmTl3kVpy(k p & y = x IE(x), (PI - G(y),(P) I < 2-m) holds. Corollary. Let F be an A@ -operator. If there exists a sequence of functions {GnIn such that for any CRN x (from O A l ) the sequence of CRNs {Gn(x)In pseudo-converges to F(x), then F E Bl. Let us consider the question of sufficient conditions for belonging an A @ -operator to the first Baire's class. Definition. An A @ -operator F is termed strictly @'-continuous if for any NN k there exists a @'-sequence of rational intervals such that Vx-i3p(x E a ( p ) ) IF(x) - F(y) I < 2-k holds.
&
vpxy x E m(p)
y E &(p)
3
Theorem 13.4. Each of the following five conditions is sufficient for belonging an A @ -operator F to the first Baire's class: (a) F is @'-uniformly continuous. (b) F is strictly @'-continuous. (c) F is non-decreasing and pseudo-continuous at any CRN from O A l . (d) F is convex on O V 1 . (e) There exists a @'-sequence of CRNs 6? such that for any CRN x, 4m(x = @ (m)) , F is continuous at x. Example 13.2. There exist A @ -operators F1, F2 and F3 not belonging to the first Baire's class and such that F1 is @"-uniformly continuous, F 2 is @'-continuous at any CRN and F3 is non-decreasing.
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
125
Example 1 3 . 3 . There exist a CFRV F and an A'-operator G, which does not belong to the first Baire's class, such that Vx((!F(x) 2 0 < x < 1) & -.-.(G(x) = 0 v G(X) = 1) & (G(x) = 1 !F(x))) & Vx3qVy( Iy-xI < 2-' 3 G(y) G G(x)) and, consequently, G is upper semi-continuous at any CRN.
'
On the other hand, we can prove the following statement. Theorem 13.5. We can realize a sequence of A -operators {G 1 P P' belonging to the first Baire's class, such that for any CFRV F there exists a NN p for which Vx(0 G x G 1 & !F(x) 3 G (x) = F(x)) holds. ~~
P
0. DEMUTH and A.
I26
KUEERA
R e f e r e n c e s [l] Bary N. (1930) M6moire sur la repr6sentation finie des fonctions
continues, Math. Annalen 103, 185-248, 598-653.
[21 Cejtin G.S. (1962) Algorithmic operators in constructive metric spaces, Trudy Mat. Inst. Steklov. 67, 295-361; English transl., Amer. Math. SOC. Transl. (2) 64 (1967), 1-80. [31 Cejtin G . S . (1970) On upper bounds of recursively enumerable sets of constructive real numbers, Trudy Mat. Inst. Steklov. 113, 102-172; English transl., Amer. Math. SOC., Providence, R.1.1972. [41 Eernov V.P. (1974) On some properties of mappings of sheaf-spaces, Zap. nauc. sem. Leningrad. otd. Mat. Inst. Steklov. 40, 136-141, (Russian). [51 KanoviE 14.1. and Kusner B.A. (1976) Complexity of algorithms and Specker sequences, Issled. teor. alg. mat. log. 2, VC AN SSSR, 73-83, (Russian). [61 K u h e r B.A. (1973) Lectures on constructive mathematical analysis, Nauka, Moscow. (Russian). [71 K d n e r B.A. (19731 Coverings of separable spaces, Issled. teor. alg. mat. log. 1, VC AN SSSR, 235-246. (Russian). [81 K u h e r B.A. (1974) On a type of computable real functions, Dokl. Akad. Nauk SSSR 215, 259-262, (Russian). [91 Markov A.A. (1954) The theory of algorithms, Trudy Mat. Inst. Steklov. 42; English transl., Israel Program for Scient. Transl. Jerusalem, 1961. [lOIMarkov A.A. (1962) On constructive mathematics, Trudy Mat. Inst. Steklov. 67, 8-14; English transl., Amer. Math. SOC. Transl.(2) 98, (1971), 1-9. [llIRogers H., Jr. (1967) Theory of recursive functions and effective computability, McGraw-Hill, New York. [12]Saks S .
(1937) Theory of the Integral, New York, Hafner Publ.Co.
[13]Sanin N.A. (1953) On the constructive interpretation of mathematical judgements, Trudy Mat. Inst. Steklov. 52, 226-311; English transl., Amer. Math. SOC. Transl. (2) 23, 109-189, (1963). [14]Sanin N.A. (1962) Constructive real numbers and constructive function spaces, Trudy Mat. Inst. Steklov. 67, 15-294; English transl., Transl. Math. Monographs, 21, Amer. Math. SOC., Providence, R.I., 1968. [15];anin N.A. (1976) On the quantifier of limiting realizability, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.(LOMI) 60, 209-220, (Russian). [16]~slavskijI.D. (1962) Some properties of constructive real numbers and constructive functions, Trudy Mat. Inst. Steklov. 67, 385-457; English transl., Amer. Math. SOC. Transl. (2) 57, 1-84, (1966).
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
127
[17] Zaslavskij I.D. and Cejtin G.S. (1962) On singular coverings and related properties of constructive functions, Trudy Mat. Inst. Steklov. 67, 458-502; English transl., h e r . Math. SOC. Transl. (2) 98, (1971), 41-89.
Bibliography of Prague seminar on constructive mathematics Note that papers are in Russian, and CMUC = Comment. Math. Univ. Carolinae. [la] Demuth 0. (1965) On Lebesgue integration in constructive analysis, Dokl. Akad. Nauk SSSR, 160, 1239-1241. [19] Demuth. 0. (1967) The necessary and sufficient condition for the Riemann integrability of constructive functions, Dokl. Akad. Nauk SSSR, 176, 757-758. [20] Demuth 0. (1967) The Lebesgue integral and the concept of function measurability in constructive mathematics, docent thesis, Charles University, Prague. [21] Demuth 0. (1967) The Lebesgue integral in constructive analysis, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 4, 30-43. [22] Demuth 0. (1968) The Lebesgue integral and the concept of function measurability in constructive analysis, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 8, 21-28. [231 Demuth 0. (1968) A connection between Riemann and Lebesgue integrability of constructive functions, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 8, 29-31. [241 Demuth 0. (1968) Fubini's theorem for the Riemann integral in constructive mathematics, CMUC 9, 677-686. [251 Demuth 0. (1969) Note on the paper: Fubini's theorem for the Riemann integral in constructive mathematics, CMUC 10, 115-120. [26] Demuth 0. (1969) On the differentiability of constructive functions, CMUC 10, 167-175. [271 Demuth 0. (1969) The spaces Lr and S in constructive mathematics, CMUC 10, 261-284. [28] Demuth 0. (1969) Linear functionals in the constructive spaces Lr, CMUC 10, 357-390. [291 Demuth 0. (1969) The Lebesgue measurability of sets in constructive mathematics, CMUC 10, 463-492. [301 Demuth 0. (1970) Mean value theorems for the constructive Lebesgue integral, CMUC 11, 249-269. [311 Demuth 0. (1970) The representability of constructive functions of weakly bounded variation, CMUC 11, 421-434. [321 Demuth 0. (1970) The integrability of derivatives of constructive functions, CMUC 11, 667-691. [331 Demuth 0. (1970) Necessary and sufficient conditions for the absolute continuity of constructive functions, CMUC 11, 705-726
0. DEMUTH and A. KUCERA
128 [ 3 4 ] Demuth 0.
( 1 9 7 1 ) The superpositions of absolutely continuous constructive functions, CMUC 1 2 , 4 2 3 - 4 5 1 .
[ 3 5 ] Demuth 0.
( 1 9 7 1 ) A necessary and sufficient condition for the representability of constructive functions in the form of the sum of a singular and an absolutely continuous function, CMUC 1 2 , 5 8 7 - 6 1 0 .
[ 3 6 1 Demuth 0.
( 1 9 7 1 ) A certain condition for the differentiability of constructive functions of bounded variation, CMUC 12, 6 8 7 711.
[ 3 7 ] Demuth 0.
( 1 9 7 2 ) A necessary and sufficient condition for the representability of constructive functions in the form of the superposition of absolutely continuous functions, CMUC 13, 227-251.
( 1 9 7 2 ) A sufficient condition for the representability of constructive functions in the form of the sum of two superpositions of absolutely continuous functions, CMUC 1 3 , 2 6 5 - 2 8 2 .
1 3 8 1 Demuth 0.
[ 3 9 1 Demuth 0.
( 1 9 7 3 ) The representability of uniformly continuous constructive functions, CMUC 1 4 , 7 - 2 5 .
[ 4 0 1 Demuth 0.
( 1 9 7 3 ) Constructive analogue of the connection between the Lebesgue measurability of sets and of functions, CMUC 1 4 , 377-396.
[ 4 1 1 Demuth 0.
( 1 9 7 4 ) The representability of constructive functions possesing the properties ( S ) and (T1) in the form of superpositions, CMUC 1 5 , 4 9 - 6 4 .
[ 4 2 1 Demuth 0.
( 1 9 7 4 ) The connection between the representability of a constructive function in the form of a superposition of two absolutely continuous functions and the differentiability of this function, CMUC 1 5 , 1 9 5 - 2 1 0 .
[ 4 3 1 Demuth 0.
( 1 9 7 5 ) Constructive pseudonumbers, CMUC 1 6 ,
315-331.
[ 4 4 ] Demuth 0.
( 1 9 7 5 ) On differentiability of constructive functions of weakly bounded variation at pseudonumbers, CMUC 1 6 , 5 8 3 - 5 9 9 . ( 1 9 7 6 ) The constructive analogue of the Denjoy-Young theorem on derived numbers, CMUC 1 7 , 1 1 1 - 1 2 6 .
[ 4 5 1 Demuth 0.
( 1 9 7 6 ) On the domains of effective operators on recursive functions and of constructive functions of a real variable, CMUC 1 7 , 6 3 3 - 6 4 6 .
[ 4 6 ] Demuth 0.
( 1 9 7 7 ) On the constructive analogue of Baire classification of functions, CMUC 1 8 , 2 3 1 - 2 4 5 .
[ 4 7 ] Demuth 0. [ 4 8 ] Demuth 0.
( 1 9 7 7 ) On a generalization of the constructive Lebesgue integral, CMUC 1 8 , 499-514.
[ 4 9 ] Demuth 0.
( 1 9 7 8 ) On pseudodifferentiability of uniformly continuous constructive functions at constructive real numbers, CMUC 1 9 , 3 1 9 - 3 3 3 .
[SO]
Demuth 0. ( 1 9 7 8 ) The constructive analogues of generalized absolutely continuous functions and of functions of generalized
REMARKS ON CONSTRUCTIVE MATHEMATICAL ANALYSIS
129
bounded variation, CMUC 1 9 , 4 7 1 - 4 8 7 1 5 1 1 Demuth 0.
( 1 9 7 8 ) Some problems of the theory of constructive functions of a real variable, Acta Univ.Carolinae-Math.et Phys. 19,
61-96.
[ 5 2 ] Demuth 0. Constructive Denjoy integrals, (to appear). [ 5 3 ] Demuth 0. The constructive Perron integral, (to appear). [ 5 4 ] Demuth O., Kryl R.,KulSera A. ( 1 9 7 8 ) Application of theory of
functions partial recursive relative to number sets in constructive mathematics, Acta Univ.Carolinae-Math.et Phys. 1 9 , 15-60.
[ 5 5 ] Demuth O.,NgmeCkovZ L. ( 1 9 7 3 ) The constructive analogue of the property (T1), CMUC 1 4 , 4 2 1 - 4 3 9 . [ 5 6 1 Demuth O.,NemeEkovZ L. ( 1 9 7 3 ) The constructive analogues of the properties (N) and ( S ) , CMJC 14, 5 6 5 - 5 8 2 . [ 5 7 ] Filipec P. ( 1 9 7 6 ) On convergence of the Fourier series of a constructive function of weakly bounded variation, CMUC 1 7 , 755-769. [ 5 8 ] Kryl R. ( 1 9 7 4 ) On the constructive analogue of the Luzin's theorem, CMUC 1 5 , 4 6 5 - 4 8 0 . [ 5 9 1 KuEera A. ( 1 9 7 0 ) Weak convergence in constructive mathematics, CMUC 11, 2 8 5 - 3 0 8 . ( 1 9 7 1 ) Sufficient conditions of normability of linear operators in constructive mathematics, CMUC 1 2 , 3 7 7 - 3 9 9 .
[ 6 0 ] KuEera A.
( 2 9 7 4 ) Locally convex topologies of finite-dimensional constructive spaces, Teorija a1g.i mat.logika, VC AN SSSR,
1611 Kuzera A.
71-66.
[ 6 2 ] KuEera A. ( 1 9 7 4 ) Some types of recursively enumerable sets and
their topological characterizations in constructive functional analysis, Dissertation, Charles University, Prague.
[ 6 3 1 KuEera A .
( 1 9 7 7 ) Algorithmical non-approximability of least upper bounds of Dedekind cuts, CMUC 1 8 , 4 4 5 - 4 5 3 . ( 1 9 7 8 ) On the types of recursive isomorphism of some concepts of constructive analysis, CMUC 1 9 , 97-105.
[ 6 4 1 KuEera A.,KuBner B.A.
LOGIC CQLLOQUIUM 78
M. Boffa, D. van Dalen, K. McAloon (eds.) 0 North-Holland Publishing Coqany, 1979
THE DIOPHANTINE PROBLEM FOR POLYNOMIAL RINGS OF POSITIVE CHARACTERISTIC
J. DENEF* Institute f o r Advanced Study 1.
Introduction
.
1
The main r e s u l t s of t h e p r e s e n t p a p e r a r e T h e o r e m A and
T h e o r e m B below. THEOREM A. L e t R be any integral domain. R.
L e t p be the c h a r a c t e r i s t i c of
And let R[T] be t h e ring of polynomials over R in one variable T.
the diophantine problem f o r R[T] , with coefficients i n
z [TI, -
Then
is unsolvable PZ (this terminology is defined i n [8, S l ] ) , i. e. t h e r e is no algorithm to decide
whether o r not a polynomial equation (in s e v e r a l variables) with coefficients in
-[T] PZ
h a s a solution in R[T]. We proved t h i s i n [8,
5 21 for p
= 0,
by using a n idea of M. Davis and
H. Putnam [4]. However the method of [8] d o e s not work f o r p > 0.
In t h e
p r e s e n t paper we prove T h e o r e m A i n Section 2 f o r p > 2 and i n Section 3 f o r p = 2.
R. M. Robinson (141 proved that t h e e l e m e n t a r y theory of R[T] is
undecidable. The p r e s e n t paper is p a r t of a s e r i e s of papers: In [6], [lo] (in collaboration with Lipshitz) and [9] we proved that t h e diophantine problem is unsolvable f o r the r i n g of algebraic i n t e g e r s i n a totally real number field o r i n a quadratic extension of a totally r e a l number field, and we showed that e v e r y recursively enumerable relation is diophantine o v e r such a r i n g of algebraic integers.
(For
the definition of "diophantine relation" and related terminology, s e e [8, 511). In [8, § 31 we proved that the diophantine problem f o r t h e field of rational functions o v e r a formally real field is unsolvable.
And i n [7] we proved that e v e r y r e c u r -
sively enumerable relation i n Z [ T ] is diophantine o v e r Z [ T ] .
Of course, all
t h e s e r e s u l t s a r e based on t h e fact that t h e diophantine problem for Z is Supported i n p a r t by the Belgian "Nationaal Fonds voor Wetenschappelijk Qnderzoek. I ' 'We denote the s e t of natural n u m b e r s by IN, and t h e ring of integers by Z. 131
132
J. DENEF
unsolvable ( s e e e.g.
[3], [5]). The outstanding open problem is whether the
diophantine problem f o r t h e field of rationals is unsolvable. L e t m e now sketch the idea involved in the proof of T h e o r e m A.
When
R h a s positive c h a r a c t e r i s t i c , it m a k e s no s e n s e t o prove that Z is diophantine over R[T]. So we only can hope t o define a model of X i n R[T]. Using Pel1 sequences we can define a model of Z in R[T], which is diophantine over R[T], and f o r which addition and divisiljility are a l s o diophantine over
R[T] ( s e e Section 2). However t h i s does.not prove that the diophantine problem f o r R[T] is unsolvable, since A . Bel'tyukov [l] and L. Lipshitz [ll] proved that t h e existential theory of ( X ; t ,
1,
denoted by
thus a l b
--I -
1)
is decidable.
c :b
=
(The divisibility relation is
ac. ) On the other hand, Lipshitz [l2],
[13] proved that the positive existential theory of (cY;t,
1)
is undecidable i f
d
is a r i n g of a l g e b r a i c i n t e g e r s (of finite d e g r e e ) whose group of units is infinite and whose diophantine problem is unsolvable. finite, then the existential theory of (s';t,
1)
(If the group of units of
0 ' is
is decidable, s e e [ll]. ) In Section
4 of t h e p r e s e n t paper we prove the following theorem, by using t h e i d e a s of
Lipshitz [13]: THEOREM B. 1
a[--] by
In,
Let n be a fixed integer, and n > 1. Denote divisibility in thus f o r all x, y XIn
c
L -f Z:y=xqn
Y --Iq,f-
Then the positive existential theory of
Z;t,
no algorithm t o decide f o r m u l a s of the f o r m
In)
.
is undecidable, i. e. t h e r e is
S
where F . and G. a r e polynomials o v e r Z of d e g r e e one o r less, and w h e r e denotes a finite conjunction. COROLLARY.
ip
Let p be a fixed p r i m e number,
p > 1. Define t h e relation
by XI
P
y - - l f-r l N : y = t x p-.
f
133
THE DIOPHANTINE PROBLEM FOR POLYNOMIAL RINGS
1, 1’)
is undecidable.
Then the positive existential theory of (Z:+,
Indeed Theorem B implies the corollary since
Thus t o prove Theorem A, it is sufficient to show for some p that the
Ip
interpretation of
i n o u r model of Z, is diophantine over R[T].
is the characteristic of R.
out that t h i s can be done i f p
is that for a l l 1 , s
z
It t u r n s
The underlying idea
IN
when p is the c h a r a c t e r i s t i c of R. Polynomial rings of characteristic p f 0, 2.
2.
Let R be any integral domain of characteristic p f 2. Let a
R[T] and a
6
1 R.
Set
Q
fi.
Then a is not in the field of
=
fractions of R[T]. Indeed it is sufficient to prove t h i s when R is a field, and hence it suffices t o show that and a
cy
+a
1 R.
a r e in R,
6
Q
c
R[T].
Hence 2a
L
R,
Then
LY
-
a
which contradicts
4 R.
Let u(a) =
E.
Let
We define two sequences X,(a),
R[T], m c Z by
L e t R be an integral domain of characteristic p f 0, 2.
LEMMA 2.1. t
Suppose that
Let R be an integral domain of characteristic p f 2.
a c R[T] and a
a
1 R[T].
This enables u s to make the following definition:
DEFINITION. Y,(a)
(Y
since ( a - a ) ( a t a ) = -1.
R[T], a
1 R.
For all m, n
t
Let
Z we have:
1) X (a) (resp. Y ( a ) ) is equal to t h e polynomial obtained by substim m tuting a for T in Xm(T) ( r e s p . Y,(T)). The degree of the polynomial Xm(T) is m, T h e degree of the polynomial Y,(T) X-,(a)
=
Xm(a) and Y-,(a)
=
-Y
is m
m (a).
-
if m > 0. 1, i f m > 0.
J . DENEF
134 2 ) All solutions X, Y
6
X
2
R[T] of t h e Pel1 equation 2
- (a -1)Y
2
= 1
a r e given by X (a), Y (a) and -Xm(a), -Ym(a), with m m m
z.
2 = Xm(a)Xn(a) t (a -l)Ym(a)Yn(a),
3 ) X,+n(a)
Ymtn(a) 4) n l m
t
=
Xm(a)Yn(a) + Ym(a)Xn(a),
1
+-&
Y (a)IY ( a ) (At the left side denotes divisibility in Z , n m but a t the right side divisibility in R[T]. ) n n and X (a) = ap , if n > 0. ,(a) = (Xm(a))' 5) X mp Pn
6 ) X ( a t l ) = X,(a) m 7) Xm(a)
t 1
1 mod(a-1).
5
m
= t
pn for s o m e n e IN.
(Congruence in R[T]. )
PROOF.
1) F r o m (1) follows f o r m > 0 that
(3)
Xm(a) =
m
m
c
i even
m = (
-
C
i even
m-i
2
i/z
( i )a
(a -1)
(:))am
t t e r m s of lower d e g r e e i n a
...
2m-l m m- 1 a t (...)a i
T h e s a m e can b e done f o r Ym(a).
T h i s p r o v e s the a s s e r t i o n about the degree.
Applying the automorphism &(a)I-, -a(a) t o (l), we get X
m
m ( a ) - cY(a)Ym(a) = ( a - a ( a ) )
-m
= (ata(a))
Hence X
-m
(a)
=
Xm(a) and Y-,(a)
= -Y,(a).
2) We may suppose that R is a n algebraically closed field.
t h e notation we w r i t e
(Y
instead of @(a). L e t
T o simplify
135
THE DIOPHANTIN?? PROBLEM FOR POLYNOMIAL R I N G S
G
{X t a Y : X, Y c R[T] satisfy (2)).
=
Notice that G is a group with r e s p e c t t o multiplication.
We consider t h e field
K = R(T)(a), it is a field of algehraic functions i n one variable. terminology of [Z, Chapter I]. F i x X t (YYc G, the s e t of p l a c e s of K which a r e poles of T.
We use the
with X, Y c R[T].
L e t S be
Since K h a s d e g r e e two o v e r
R(T), S c o n s i s t s of a t most two p l a c e s ( s e e e.g. 12, Chapter IV thm. 1 p. 521). L e t P be a place of K not i n S. Obviously P is not a pole of X, Y o r T h u s P is not a pole of X t a Y or X - (YY.Since (XtaY)(X-aY) = 1. that P is not a z e r o of X t a Y . contained i n S. and a t
(Y
1 R.
T h u s all the z e r o e s and poles of X t a Y a r e
Now e v e r y element of K which is not in R h a s at l e a s t one T h u s S c o n s i s t s of at least two places, s i n c e a t a
z e r o and one pole. places of K.
(Y.
we see
c
G
Hence S = {L , Lz}, where L1 and L 2 a r e two different 1 T h u s the divisor of X t aY equals qLL - qL2, f o r s o m e
q c Z . T h i s gives u s a group homomorphism G
-Z
:X t
I t s kernel (YY G- q.
is {tl). Indeed i f X t (YY is i n t h e kernel, then X t (YYc R, and X = t 1. T h u s G/{+l} is an infinite cyclic group. L e t X t (YY, with X, Y
t
2 (XtaY)e,
c
R[T].
be a g e n e r a t o r f o r G/{+l}.
We have
For a good choice of t h e generator, we e e e e - i i i can suppose that e < IN. Hence a t (Y = t (XtaY) = 2 iFo(i)X (Y Y = (. . ) t (. . . ) Y a . T h u s Y divides 1 i n the ring R[T], hence Y c R. F r o m
a ta
=
for some e
hence Y = 0
Z.
.
(2) follows
(X-aY)(XtaY) = 1 - Y
If Y f f 1, then X since a
1 R.
-
2
L
a Y c R and X t aY c R.
R.
Hence 2 a Y c R,
But t h i s is i n contradiction with Y 11.
and Y = 0
Hence Y = t 1 and T h u s a t a is
X =t a. T h i s m e a n s that t at (Y is a g e n e r a t o r f o r G / {tl}. also a generator for G/{t1}. T h i s p r o v e s 2). 3 ) F r o m (1) follows
which p r o v e s 3 ) . 4) It is sufficient t o prove t h i s for n, m
m = nq,
with q
t
IN.
F r o m (1) follows
2 0.
First, suppose that
136
J. DENEF
Hence i- 1
Thus Yn(a) IY
nq
(a).
Conversely, suppose Yn(a)IYm(a). If n = 0, suppose n > 0.
Write m = nq
+
r,
with q, r
E
then m = 0,
N and 0 < r < n.
thus
From 3)
follows
Yn(a
Yn(a
Suppose r f 0,
then Y,(a) f 0.
Hence the d e g r e e of
Y ( a ) is l e s s o r equal
than the degree of Y ( a ) . F r o m 1) follows then that n < r, r < n. Thus r = 0, which proves 4). 5) F r o m (1) follows
which contradicts
THE DIOPHANTINE PROBLEM FOR POLYNOMIAL R I N G S
137
T h i s p r o v e s 5).
6 ) It is sufficient t o prove t h i s f o r m > 0. Then 5) i m p l i e s the left side of 6).
( a t l ) = X ( a ) + 1. Hence m f 0. m IN and q $ 0 mod p. F r o m 5 ) follows
Conversely suppose X with q, n
c
First, suppose m
m
pn
=
.
Write m = qpn ,
Hence X (atl) 9
=
X (a) t 1 9
F r o m 1) follows X (a) q
yaq t paq-' t t e r m s of lower d e g r e e in a
=
.. . = yaq t ( p t y 9 ) a q - l t ...
Xq(atl)
where y, q
p
c
0 mod p.
y(at1)q t p(at1)q-l t
=
R and Y { 0.
Thus if q > 2, n T h u s q = 1 and m = p
.
then p t yq
=
p, which contradicts
7 ) F r o m ( 3 ) (in the proof of 1)) follows
xm ( a ) 5 Moreover X ,(a)
a m :1 mod(a-1),
f o r m > 0.
= Xm(a).
Q. E. D.
-
From L e m m a 2.1 2) and 7) follows f o r all X, Y,a c R[T] with a (4)
( ZIm
(5)
X
2
6
-
z
:
x=
Xm(a) .A Y
=
Y,(a))
4
2 2 (a -1)Y = 1 A X: 1 mod(a-1).
Thus t h e relation ( 4 ) i n the v a r i a b l e s X. Y. a is diophantine over R[T]. that the relation ( 4 ) is only defined f o r a taking (5) as i t s definition (e. g. if a iff X = 1).
R that
=
I R,
but we can extend it f o r a
Notice L
R by
1 then the relation (4) is defined t o hold
J. DENEF
138
F r o m Lemma 2.1 1) follows
thus we can consider the set
a s a model of Z in R[T]. Z [TI. cients in
F~
This model is diophantine over R[T] with coeffi-
Moreover, by Lemma 2.1 3 ) and 4), addition and divisibility in that model
z
a r e diophantine over R[T]
with coefficients i n - [TI. PZ Thus by the corollary to Theorem B, we s e e that Theorem A, in the c a s e
of characteristic p quantifier
-
-[I, s -
LEMMA 2. 2. m, q
t
{
c Z
0, 2,
follows from Lemma 2. 2 below.
Indeed the existential
in Lemma 2. 2 can be replaced using (4)-
Let R be an integral domain of characteristic p
(5).
{ 0, 2. F o r all
Z we have
PROOF.
It is sufficient to prove the Lemma for m, q, I , s f N. F i r s t suppohe n mlPq, thus q = mp for some n t IN. If m = 0, then 9 = 0 and X,(T) =
X (T) = 1. Thus suppose rn { 0. Set a = X,(T) 9 2.1 6) implies (8). F r o m Lemma 2.1 5) follows
1 R.
Set
s = I = pn. Lemma
Thus also ( 7 ) is satisfied. Conversely, suppose (6), (7) and (81.
If m
=
0,
then a
=
1. Hence
X ( a ) = 1, by (5), and X ( T ) = 1 by (7). Thus q = 0. So, suppose m > 0, 1 q then a R. F r o m Lemma 2.1 1) follows s = P , since Xs(atl) and X,(a)
4
THE DIOPHANTINE PROBLEM FOR POLYNOMIAL RINGS
139
have the s a m e degree by (8). Hence (8) and Lemma 2.1 6 ) imply 1 some n
6
So, ( 7 ) gives X (T) = X q
mP
n n ap = ( X ~ ( T ) ) ’
n(T). Hence q
Polynomial rings of characteristic
,/z
mp
=
n
=
x
mP
1 R.
n
, for
n(~).
.
W e have
=
Q. E. D.
p = 2.
L e t R be a n integral domain of c h a r a c t e r i s t i c p a
p
IN. Thus by Lemma 2.1 5) and (6) we have
xP (a) =
3.
=
=
2.
Let a
E
R[T] and
a t 1 c R[T], and t h e formulas of Section 2 a r e no
However the method of Section 2 can be adapted by defining other
m o r e valid.
(a), Ym(a). Notice that the equation x2 t a x t 1 = 0 has no root m in the fraction field of R[T]. Indeed if x L R[T] satisfies this equation, then sequences X
x divides 1 in R[T],
hence x
L
R and a
c
R, which contradicts a
R.
This
enables u s to make the following definition: DEFINITION. a
t
Let R be an integral domain of c h a r a c t e r i s t i c p = 2.
R[T] and a
R.
define two sequences Xm(a). Ym(a) 6 R[T],
L E M M A 3.1.
a c R[T], a
m
c
F o r all m, n
t
We
Z by
Let R be an integral domain of characteristic p = 2.
1 R.
Let
Let cu(a) be a root of the equation x 2 t a x t 1 = 0.
Let
Z we have:
1) Xm(a) (resp. Y (a)) is equal to the polynomial obtained by substim tuting a for T in Xm(T) (resp. Ym(T)). The degree of the polynomial X (T) is m-2, m ( T ) is m-1, m
T h e degree of the polynomial Y
2) All solutions X, Y
c
R[T] of the equation
x2 t
(2)
a r e given by
aXYtY2=1
if m > 2. if m > 2.
140
J. DENEF
with m
6 ) (at1)Ym ( a t l ) = a Y m ( a ) t 1 <==> m = t 2 REMARK.
n
f o r some n
t
c
Z.
IN.
Many of the above f o r m u l a s r e m a i n t r u e f o r c h a r a c t e r i s t i c p
one r e p l a c e s t by
-
c h a r a c t e r i s t i c p f 2.
at s o m e places.
j
2,
if
But 5) and 6 ) do not g e n e r a l i s e to
T h u s t h e method of t h i s section only w o r k s in c h a r a c t e r -
i s t i c two. PROOF.
Notice that a ( a ) and (@(a))-’ a r e the r o o t s of t h e equation x2 t a x t 1
and that (cu(a))-’= a t a ( a ) . Taking t h i s into account, t h e proof of L e m m a
= 0,
3.1 is a l m o s t the s a m e a s t h e proof of L e m m a 2.1; except f o r 5) and the a s s e r t i o n about the d e g r e e in 1) which a r e proved by induction on n and m respectively.
Q. E. D.
From L e m m a 3.1 2), 3) and 4) follows that
can be used as a model f o r Z i n R[T], which is diophantine over R[T], and f o r which addition and divisibility is a l s o diophantine over R[T].
T h u s a s in
Section 2 we s e e that T h e o r e m A, in the c a s e of c h a r a c t e r i s t i c p
=
2,
follows
f r o m t h e c o r o l l a r y t o T h e o r e m B and L e m m a 3. 2 below. LEMMA 3. 2. m, q
c
L e t R be a n integral domain of c h a r a c t e r i s t i c p = 2.
Z we have:
For all
THE D I O P W T I N E PROBLEM FOR POLYNOMIAL R I N G S
TY ( T ) = aY,(a) q
141
A
(at1)Y ( a t l ) = aY,(a) t 1.
PROOF.
T h e proof is a l m o s t t h e s a m e as t h e proof of L e m m a 2.2,
now using
L e m m a 3.1 instead of L e m m a 2.1. 4.
Q. E. D.
The diophantine problem f o r addition and localised divisibility. In t h i s section we prove T h e o r e m B.
o v e r the integers.
F r o m now on, all v a r i a b l e s r u n
W e u s e t h e method of L. Lipshitz [13]. It is sufficient t o
show that t h e relation u
=
z 2 can be defined by
w h e r e Ki and L . a r e polynomials of d e g r e e one o r l e s s .
Indeed, t h i s implies
that a l s o t h e relation w = xy can be defined i n t h i s way, since w (2)
=
xy
-
c = a t b
LEMMA 4.1.
4w 0
=
2
( x t y ) - (X-y)
2
I n (atb-c).
L e t n > 1, and suppose x
In
1 and y
only if the following conditions (3). (4) and ( 5 ) hold: (3)
2 ~ n x t l l4 n y - 1 n
(5)
ny - kx
PROOF.
I n nx -
k,
1 n 1.
f o r all k satisfying
Then y = x
2 .
if and
lkl < n.
Obviously, if y = x2 then (3), (4) and (5) hold.
Conversely, suppose (3), (4) and (5) a r e satisfied.
Since n, 2nx t 1 and 2nx
a r e relatively p r i m e t o one another, ( 3 ) and (4) imply
-1
142
J. DENEF
2 Moreover 4n y - 1
4 0,
hence the above divisibility yields
I (2nxt1)(2nx-1)1< 14n2y-11,
whence
But x and y a r e integers, thus
Now we shall use (5) t o obtain a n inequality in t h e o t h e r direction. p r i m e number p,
For e v e r y
we define h(p) (depending on p, n, x and y) by
h(p)
=
0 if ny and x a r e divisible by the s a m e powers of p,
h(p) = 1 otherwise. If h
h(p) mod p,
then
i p lny - hx
(7)
-
i p lx,
f o r a l l positive i.
By the Chinese Remainder T h e o r e m t h e r e e x i s t s a n h (mod n) such that f o r every p r i m e p dividing n we have h Ihl < n and hx
choose h such that
ny
(8)
From Ihl < n and x
0,
Suppose y
=
-
Moreover, we can
T h u s f r o m ( 7 ) and (5) follows
hx I x(nx- h).
0 follows x(nx-h) > 0,
nlyI Since hx
-
0.
h(p) mod p.
hence (8) yields
lhxl 5 nx2 - hx.
2
we obtain Iyl 5 x ; and i n view of (6) we conclude y 2 x , then ( 3 ) yields
-
I
Hence 2nx t 1 - 2,
and one e a s i l y d e r i v e s a contradiction.
= t
x
2
.
Thus y = x
2
.
Q. E. D.
LEMMA 4. 2.
L e t n > 1.
Suppose the following conditions ( 9 ) , (lo), (11). and
143
THE DIOPHANTINE PROBLEM FOR POLYNOMIAL RINGS
2 2 11 n u - (nx-1) n
nz t nx
-
2nz t 1
In
-
1
2nz - 1
In nx -
1
nx
2 2n u s 1 I n n x - 1.
2 Then u = z PROOF. (9) by
1,
.
Since n and nz t nx - 1 a r e relatively prime we can replace and we easily obtain
1n
in
2 2 2 n z t n x - l l n u - n z .
2 Suppose u f z , then the above divisibility implies
Jnx-11
(13) Since n, 2nz t 1 and 2nz
-
-
nlzl 5 n
2
JuIt n 2 z 2 .
1 a r e relatively prime t o one another, (10) and (11)
imply
But nx
- 1f
0,
hence 2 2 4n z
(14)
-
1 5 Inx-11.
Analogously (12) yields (15)
2n
2
JuI-
1 5 Inx-11.
F r o m (13). (14) and (15) we obtain (n1z1)2 - n l z l
-
1 5 0.
If z f 0, then n l z I 2 and the above unequality cannot hold. Thus u = z 2 or z = 0. But when z = 0 one easily deduces from (13) and (15) that u = 0. Q.E.D.
144
J. DENEF
LEMMA 4. 3.
F o r any nonzero integer d t h e r e e x i s t s a n integer x satisfying
x l n l and d I n n x - 1. PROOF. n.
I
Write d a s d = dOdl where d O n 1 and dl is relatively p r i m e t o w h e r e cp denotes t h e E u l e r Function, then by t h e F e r m a t -
Set x = n'(dl)-l,
Euler T h e o r e m d 1
I
nx - 1. Hence d
PROOF O F THEOREM B.
In
1, y
nz t nx - 1
(16) =
z
2
.
nx
-
Q. E. D.
1.
We c l a i m that u = z 2 if and only if t h e r e exist
integers x and y satisfying x
Indeed suppose u
1
In
1, (3), (4), ( 5 ) , (lo), (ll), (12) and
I n n 2u -
2 n y t 2nx - 1.
2 By L e m m a 4.3 (with d =(Znztl)(Znz-l)(2n u t l ) ) t h e r e
is a n integer x satisfying (lo), (ll), (12) and x
In
1. Set y
=
2
x , then a l l the
conditions hold. Conversely, suppose t h e r e a r e x and y satisfying t h e above conditions. F r o m L e m m a 4.1 follows y
u
= z
2
=
x
2
.
Substituting t h i s in (16) yields (9).
, by L e m m a 4.2, and t h e claim is proved.
relation u
=
z
2
Hence
Thus we have shown that the
can be defined by a formula of t h e f o r m (1). T h i s completes
the proof of T h e o r e m B.
Q. E. D.
THE DIOPHANTINE PROBLEM FOR POLYNOMIAL RINGS
145
References 1. A. P. Bel'tyukov, Decidability of t h e u n i v e r s a l t h e o r y of n a t u r a l n u m b e r s with addition a n d d i v i s i b i l i t y ( R u s s i a n ) , Investigations i n c o n s t r u c t i v e m a t h e m a t i c s and m a t h e m a t i c a l l o g i c VII, S e m i n a r s of Steklov Math. Inst. ( L e n i n g r a d ) Vol. 60 (1976), 15-28. 2.
C. Chevalley, A l g e b r a i c functions of one v a r i a b l e , A m e r . Math. Soc., New York 1951.
3.
M. Davis, H i l b e r t ' s tenth p r o b l e m is unsolvable, A m e r . Math. Monthly,
. 4.
80 (1973), 233-269.
M. D a v i s and H. P u t n a m , Diophantine sets o v e r polynomial r i n g s , Illinois J. Math. 7 (1963), 251-256.
5.
M. Davis, Yu. M a t i j a s e v i c and J. Robinson, Diophantine equations: P o s i t i v e a s p e c t s of a n e g a t i v e solution, Proc. of S y m p o s i a in P u r e Math. 28 (1976), 323-378.
6.
J. Denef, H i l b e r t ' s t e n t h p r o b l e m f o r q u a d r a t i c r i n g s , P r o c . A m e r . Math.
7.
J. Denef, Diophantine sets o v e r I [ T ] , P r o c . A m e r . Math. SOC. 69 (1978),
8.
J. Denef, T h e diophantine p r o b l e m for polynomial r i n g s and f i e l d s of
9.
J. Denef, Diophantine sets o v e r a l g e b r a i c i n t e g e r r i n g s 11, T r a n s . A m e r .
SOC. 48 (1975), 214-220. 148-150. r a t i o n a l functions, T r a n s . A m e r . Math. SOC. 242 (19781, 391-399. Math. SOC. ( t o a p p e a r ) . 10. J. Denef a n d L . L i p s h i t z , Diophantine sets o v e r some r i n g s of a l g e b r a i c i n t e g e r s , J. London Math. SOC. (to a p p e a r ) . 11.
L . L i p s h i t z , T h e diophantine p r o b l e m f o r addition and divisibility, T r a n s . A m e r . Math. SOC. 235 (1978). 271-283.
12. L .
L i p s h i t z , Undecidable e x i s t e n t i a l p r o b l e m s f o r addition and divisibility i n a l g e b r a i c n u m b e r r i n g s , T r a n s . A m e r . Math. SOC. 241 (1978), 121128.
13. L . L i p s h i t z , Undecidable e x i s t e n t i a l p r o b l e m s f o r addition and divisibility i n a l g e b r a i c n u m b e r r i n g s 11, Proc. A m e r . Math. SOC. 64 (1977), 122128. 14.
R. M. Robinson, Undecidable r i n g s , T r a n s . A m e r . Math. SOC. 70 (1951), 137-159.
LOGIC COLLOQUIUM 78 M. Boffa, D . van Dalen, K . McAloon leda.1 0 North-Holland M l i s h i n g Company, 1979
Algorithms and Bounds for Polynomial Rings Lou van den Dries Mathematisch Instituut, R.U. Utrecht
IO.
Introduction
Let in the following X be a sequence of variables (X1,..,Xn). Consider polynomials f,fl,..,fk E K[X], all of degree GI, where K is a field. If f E d(fl ,.., fk), i.e. fe = Zhifi for some e ED? and certain hi E K[XI, then we can choose the exponent e and the degrees of the hi's all below a number H = H(n,d) GIN, which depends only on n,d (not on the polynomials or the field). Let me recall A. Robinson's compactness trick to prove this: Let c be the vector of coefficients of the polynomials, the number of coordinates of c being determined by (n,d,k). Then we have: K
b $(c)
iff
K b V{$~~(c)lrEm},
-
where $(c) expresses that f vanishes on all common zeros of fl,..,fk in the algebraic closure K of K, and $,(c) expresses that fe = Zhifi for some e d r and certain hi E K [ X ] of degree e. (Note that such an open formula $(y) exists by the quantifier elimination for algebraically closed fields, and that the equivalence above simply reformulates Hilbert's Nullstellensatz.)
So, with FL the elementary theory of fields, we have: FL b $(y)
+
vI$,(y) Ir GIN}.
By the compactness theorem there is H = H(n,d,k) G I N such that: FL k $(y) * V{$r(y) Ir
<
HI.
We can eliminate the dependence of H on k, because we need only consider k d = di%{g E K[X] Ideg(g) < d}. Note that-by Godel's Completeness Theorem we can even compute such an H from (n,d).. It may be interesting that we can apply the same trick to a weak version of Hilbert's Nullstellensatz
(dAn)
147
L. VAN DEN D R I E S
148
-
if fl,..,fk have a common zero in some extension field of
K , they have a common zero in an algebraic extension to give the following:
-
if fl,..,fk have a common zero in some extension field of K, they have a common zero in a finite extension L with [ L : K ] < HI, where H' can be computed from (n,d). In fact, some older proofs of Hilbert's Nullstellensatz, cf. [ 2 0 , 974-751, provide such bounds H and H', but in the (very elegant) approach which is fashionable today, cf. [ 9 , p. 255-2561, there is not the least indication for their existence. Therefore it seems to me that we may welcome such compactness arguments, which enable us to recover very easily a lot of information we seem to lose by an inconstructive approach. A few new results were actually proved in this way, related to Hilbert s 17th problem, [13, p. 2231, and desingularization, [ 61 , respect vely . N o w there remained a lot of polynomial rings over fields structive arguments, cf. [ 71 , I ways interested in to "explain"
.
of effective bounds in the theory obtained by very complicated con-
9 1 , [ 171, which A. Robinson was alby modeltheory, as is clear from But he made only a beginning with this
[ 131, [ 14, p. 5031, [ 161 in [ 151. One reason for such an approach is of course the very easy proofs it usually provides, another motive is that it might help in solving open problems in this area, for instance Ritt's problem in
differential algebra mentioned in [ 14, p. 5041. In [5, Chapter 41 I proved by model theory the existence of effective bounds in a number of cases where Robinson's trick doesn't work. (At the moment I can handle in this way all the positive results in Seidenberg's [ 171 .) To be precise I used the following compactness result: Basic Lemma Let T be an L-theory and (I$i)iEI,($j)jEJ families of L-sentences such that
. Then there are finite subsets I
of I, Jo of J with
149
ALGORITHMS AND BOUNDS FOR POLYNOMIAL R I N G S
Moreover, for such I . and Jo one has: A{$ili€Io} and
T
A{QiIi€I}
T
V{Jlj I jEJ1 * V{iClj I jEJo}.
++
If, in addition, L is recursive, T is r.e., I
=
J
= w,
and
are r.e., then such sets Io,Jo can be the sequences (@i)iEw, ( J , . ) . 3 1Ew computed effectively from given indices for T,I,J. (Robinson's compactness argument corresponds to the case that the set I is a singleton, i.e. there is only one $i.) Proof By hypothesis T U {$i} U { T I ) . ) is inconsistent. I Pick finite sets , I C J , Jo C J such that T U { ~ $ ~ l i € Iu~ }{ ~ J , ~ l j g J ~ } is inconsistent. Then T $i -t jZJoJ,j. 0 The converse implication is obvious. The computability of Io,Jo rn follows from GGdel's completeness theorem. In section 1 I will indicate how this lemma nicely combines with the wellknown techniques and results of commutative algebra: local-global principles ( [ 8 , p. 14-15]), Krull's intersection theorem, the primitive element theorem, etc., to obtain many of the bounds we are looking for. Also I mention some problems which seem to be open and should be investigated from this new point of view. In section 2 we will follow a quite different path, namely a non-standard approach, again initiated by A . Robinson, cf. [15]. It often favourably compares with the methods of 5 1 , for instance in treating associated primes and primary decomposition. Its main attraction however is that the results on bounds in their nonstandard formulation express very simple relations between two rings, a polynomial ring K[X] and a certain extension K[X]*, for instance: K [ X ] * is a faithfully flat K[X]-algebra, an ideal I of K[X] is prime, primary, radical, respectively, iff I.K[X]* has the same property as an ideal of K [ XI *. Section 3 concludes with remarks on still other aspects to this whole subject. My intention with this paper is not to give full and precise proofs,
L. VAN DEN DRIES
150
but to show the underlying ideas and some (hopefully) suggestive examples.
Compactness arguments
81.
We will keep the notations and conventions of the introduction. Let us also write (fl,..,fk) for the ideal generated by f1,..;f k -in K1 XI Then clearly:
.
f E (fl,..,fk) * K k VI$,(c) Ir (1.1) where Jlr(c) expresses in K that f = Zhifi for certain hi E K[X] of degree at most r. Krull's intersection theorem, cf. [ 9 , p. 1551,combined with a localglobal principle, implies: (fl,..,fk) = n {mm - + (fl,.. ,fk) Im €IN, m a maximal ideal of K[X] } . But K[ X] /( i (fl,. ,fk) f is a finite-dimensional K-algebra K[ XI , m + (fl,..,fk)), s o we get: x = X mod(!
.
mm
(1.2) f E (fl,..,fk) * K k A{$r(~)Ir €IN}, $,(c) expressing in K that f (x) = 0 for each ring K[ XI = f (x) = 0 and di?K[x] G r. such that fl(x) = k Combining (1.1) and (1.2) we see, that
...
FL t A{$,(Y)
.
= K[ xl,. ,xJ
Ir €IN * vI$~(Y)Ir GIN).
( 1 . 3 ) By the Basic Lemma of 90 this implies that one can compute a number A from (n,d) such that, if f E (fl,. ,fk), then f = Zhifi for certain hi E K[X] of degree at most A . If we apply this compactness argument to submodules of the free module K[XIP (instead of to ideals of K[X]) we obtain the following generalization:
.
(1.4) Given n,d,p E IN, we can compute A = A(n,d,p) €IN, such that for each field K and each system of linear equations fll Y1 +
...
+
flq yq = g1
. fpl Y1
+ ... +
fpq Yq = qp
(all fij ,qi E K[ XI of degree at most d)
there is a solution (hl,.. ,hq) E K[ X] with all h . of degree at 3 most A , if there is a solution in KIXIq at all, cf. [ 1 7 , p. 2771.
151
ALGORITHMS AND BOUNDS FOR POLYNOMIAL RINGS
Note that this result implies that the decision whether such a system has a solution can be reduced to the decision whether a certain systems of linear equations with coefficients in K has a solution. (1.5) In differential algebra a similar problem is still open, as 1 fas as I know (cf. 112, p. 1771): Let f,g E KIT} = K[T,dT,d*T be differential polynomials in one variable T over the differential field K. How to decide whether g E [fldSf(f,df,d2f,...) ? (1.2) suggests to consider first the following type of question:
,...
Does g E [f] hold, if g(t) = 0 for each differential ring extension S of K and each t E S, such that f(t) = 0 and S is (an an ordinary ring) a localization of a finitely generated K-algebra? (1.6) Problem Can we find a formula Max(y) (independent of K and the vector of
coefficients c of fl,..,fk) such that: K
+ Max(c)
* (fl,..,fk) is a maximal ideal?
By (1.3) we construct for each r ElN a formula K
k @,(c)
* 1 9 (fl,.., fk), and 1
(y) s.t.
@
,..,
(g,fl fk) for each g E K[ XI\ (fl,. ,fk) of degree at most r. Suppose now that (fl,. ,fk) is maximal ( so K[ XI / (fl,. . ,fk) is a field extension of finite degree over K) and assume for simplicity that char(K) = 0. Then by 9, p. 185, Th. 141 : (1.7)
.
(1.8)
K[ XI / (fl,.
. ,fk)
zK
.
E
K[ TI /p(T) , for some irreducible p E K[ TI
.
(1.9) Hence K V{$,(c) Ir €IN), where $,(c) expresses that for some irreducible p(T) of degree at most r an isomorphism as in in
(1.8) holds under which the residue class of T corresponds to the residue class of a polynomial in K[X] of degree at most r. (1.7) and (1.9) imply:
FL C At@,(y) Ir EIN1 * V{$,(y)
Ir E N ) .
Hence the basic lemma provides a formula Max(y) as required, and moreover we can compute a bound M = M(n,d) such that (1.10) (fl,..,fk) is maximal iff 1
9 (fl,.., fk) and 1
for each g E K[Xl\ (fl,-.,fr)of degree at most M.
E (g,fl,.
. ,q
L. VAN DEN DRIES
152
Along these lines I showed in [ 5 , Ch. 4 , 9 3 1 . (1.11) Given n,d €IN one can compute P = P(n,d) €IN, such that for each field K and each ideal I of K[X] generated by polynomials of degree at most d one has: I is prime * for all f,gEK[X] of degree at most P, if fg E I, then f f I or q f I. (1.12) Moreover, the proof also shows that, if the perfect field K is computable (in Rabin's sense, cf. [ 101 ) and there is an algorithm for factoring polynomials in one variable over K, then there is an algorithm such that, given fl,..,fk f K[X] , the algorithm determines the number of minimal primes of (fl,..,fk), and a finite set of generators for each of them.
(1.13) Modeltheoretically, Ritt's problem, cf. 112, p. 1781, which has origins in work of Lagrange and Laplace, can be formulated as follows: Is there an elementary, i.e. lSt order, condition on the coefficients of an irreducible differential polynomial f(T) f KIT) with f(0) = 0 and K a differential field of characteristic 0 (and where a bound on the order and degree of f is given), which expresses that g(0) = 0 for each g f KIT} vanishing on the generic zero of f over K? (These 9 form the unique prime differential ideal containing f but not its separant.) It is not difficult to express this by an infinite conjunction of formulas in the language of differential fields. S o we have 'only' to find an equivalent infinite disjunction. Useful hints might be provided by 131, where a necessary and sufficient condition is given in terms of the existence of rank 1 valuations with certain properties. Can this "existential" test be transformed in an infinite disjunction? (1.14) If n > 2, d > 1, k > 2 it does not seem to be known whether (independent of c and K there are formulas u n,d,k(Y) and n,d,k s.t.: (fl,.- ,fk) = (gl,. ,qk-l) for certain K b u~,~,~(c) +
g1 I
* .
rgk-1 E K[ XI
ALGORITHMS AND BOUNDS FOR POLYNOMIAL RINGS
153
In both cases there are of course infinite disjunctions available. would have as a consequence The existence of such formulas T 3,d,k the positive solution of the long standing problem whether every algebraic curve in C 3 is the intersection of 2 algebraic surfaces, cf. [ 8 , p. 2141. (This follows by modeltheory, because Cowsik & Nori, (41, have positively answered the analogue of the above settheoretic complete intersection problem for each prime characteristic.) (1.15) If n > m > 1, d > 1 it does not seem to be known whether there are formulas 71n,m,d,k(Y) "n,m,d,k (y) such that: K
b
* KIXl/(fl,..,fk
as K-algebras,
'
is prime and the fraction 'n,m,d,k (c) * (fl,.. ,fk)
field of KIX]/(fl ,.., fk) is K-isomorphic with K(Y1 ,.., Ym). For m = 1 and algebraically closed K, the right hand side of the second equivalence expresses that (fl,..,fk) is the ideal defined by a curve of genus 0 in affine n-space. So, with K restricted to exist. be algebraically closed, formulas p n,l,d,k
52.
A non-standard approach.
We keep the convention that
X
=
(Xl,..,Xn)
(2.1) Given n,d,k €IN there is a bound B = B(n,d,k) €IN such that for each field K and each homogeneous linear equation ( a ) flyl
+
..
+ fkYk
= 0
(fl,..,fk E K[X], all of degree at most d)
the solution subset of KIXIk is generated as K[X]-module by solutions g = (gl,..,gk) with deg(g)
(i.e. deg(gi)
<
B for
L. VAN DEN DRIES
154
i
=
l,..,k).
(2.2) Proof Let n,d,k be given and suppose such a bound B doesn't exist. So for each m €Dl there is a field Km and a system of type ( a ) over Km with k a solution in (Km[X]) which is not generated by solutions of degree Gn. Consider a structure containing all fields Km, polynomial rings
KmfX],lN, etc., and take an enlargement of this structure. By the saturatedness properties of enlargements there is an internal field K in this enlargement and an equation f l y l + + fkYk = 0 with fl,. ,fk E K[ XI * dgf the ring of internal polynomials in X over K, each fi of degree GI, such that at least one solution (gl,..,gk) E (K[X]*)k of the equation is not generated by solutions in KIXlk, where KIX] is naturally identified with the ring of finite degree
..
.
polynomials in K[ XI *. Note that fl, ,fk E K[ XI But this implies that K[ XI * is not a flat K[X]-module [ 8 , p.51. Hence ( 2 . 1 ) follows by contradiction from:
..
.
(2.3) K[X]* is a faithfully flat K[X]-module (where K[ X]* is the ring of internal polynomials in X over any internal field K in an enlargement)
.
.
An easy proof of ( 2 . 3 ) by induction on n is given in 5, p. 1 2 8 It is fairly obvious that ( 2 . 3 ) is in fact equivalent with the conjunction of ( 2 . 1 ) (flatness) and ( 1 . 3 ) ("faithfully"). (2.4)
In most cases (and certainly in the more difficult ones),
where bounds can be extracted from the proofs in [ 1 7 ] , they can be obtained much easier using the above approach. Of course we lose some information, even in comparison with 5 1 , namely the effectiveness of the bounds. Another difference with 51 is that the nontrivial proofs use induction on n. Let me mention some typical results, using the notation of ( 2 . 3 ) . Given an ideal I of K[X] the following hold: K(X)* ( sion of K(X) . (2.5)
(2.6)
def =
the fraction field of K[Xl*) is a regular exten-
I is prime iff I.K[X]* is prime in KIXl*.
155
ALGORITHMS AND BOUNDS FOR POLYNOMIAL RINGS
(2.7) fi.K[X]* = {f E KIX]*lf'
E I.K[X] for some w E * m ) =
d r n . (2.8)
If M is a K[ X] -module, then ASSK[
* (M@K[
K[ XI * )
=
Ip.K[ XI *
..
(2.9) If I = n n Im is a reduced primary decomposition of I, I being p -primary (k = l,..,m), then k k I.K[ X] * = I1.K[ X] *
..
fl
n Im.K[ X] * is a reduced primary
decomposition of
.
I .K[ X] * ,Ik.K[ XI * being p k .K[ XI *-primary (k = 1,. ,m)
.
(2.10) Comment (2.5) is an important lemma in proving (2.6). The reader will check easily that (2.6) is the non-standard version of (1.11). ( 2 . 7 ) is an easy consequence of (2.3) and (2.6), and it implies for example that the exponent e mentioned in the first lines of the introduction can be bounded by a function of n and the degrees of fl,..,fk only (and does not depend on the degree of f) (2.8) is an imme-
.
diate cons'equence of (2.3), (2.6) and 12, Th. 2, p. 1541. If one takes M = K[Xj/I it means that the associated primes of an ideal I of K[X] which is generated by polynomials of degree at most d, are themselves generated by polynomials of degree at most C = C(n,d), and that there are at most D = D(n,d) of them (K any field). (2.9) follows easily from (2.3), (2.61, (2.7) and (2.8). Its standard interpretation is that an ideal I of K[X] which is generated by polynomials of deqree at most d has a primary decomposltion I = I~ n n Im with m < E(n,d) and where each Ik is generated by polynomials of degree at most F(n,d) (K any field).
..
(2.11) Perhaps one should consider the open problems mentioned in 9 1 also in the context of this section. For instance, the first pro-
blem of (1.14) simply becomes: if I . K [ XI * is generated by m elements, is then I also generated by m elements?
L. VAN DEN D R I E S
156 53.
Concluding Remarks
(3.1)
In the preceding sections not any attempt was made to obtain
concrete, say exponential, expressions for the bounds. This is not possible with model theory alone. Still it seems interesting to give a model theoretic explanation why the bounds given in [ 1 7 ] tend to be (super) exponential in n and polynomial in d. (3.2) Except that the results of [ 1 7 ] are more precise - and l e s s easy to obtain - than in this paper, another contrast is the point
of view of [ 1 7 ] and [ 1 9 ] which is thoroughly constructivistic. This is also the case in 1 1 1 1 , [18], where some basic construction problems are solved for more general polynomial rings. (3.3) Still another approach to some of the material treated here, is discussed in [ l ] , where algorithms for reducing words to canonical forms are the leading theme.
References G.M. Bergman, The Diamond Lemma for Ring Theory, Advances in Math. 2 9 ( 1 9 7 8 ) , 1 7 8 - 2 1 8 . N. Bourbaki, Algsbre Commutative, Chapitres 3 et 4, Hermann, Paris ( 1 9 6 1 ) . R.M. Cohn, Solutions in the general solution, in: Contributions to Algebra, A collection of Papers Dedicated to Ellis Kolchin, Ed. H. Bass, P . J . Cassidy, J. Kovacic, Academic Press, New York ( 1 9 7 7 1 , 1 1 7 - 1 2 8 . R. Cowsik & M. Nori, Affine curves in characteristic p are set-theoretic complete intersections, Inventiones 4 5 ( 1 9 7 8 1 , 111-114.
L. van den Dries, Model Theory of Fields, Thesis, Utrecht, June 1 9 7 8 .
P.C. Eklof, Resolutions of singularities in prime characteristics for almost all primes, Trans. AMS 1 4 6 ( 1 9 6 9 1 , 429-438.
G. Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Annalen 9 5 ( 1 9 2 6 1 , 7 3 6 - 8 8 . T.Y. Lam, Serre's Conjecture, Lecture Notes in Math. 6 3 5 Springer-Verlag, Berlin ( 1 9 7 8 ) . S. Lang, Algebra, Addison-Wesley, Reading, Mass., ( 1 9 6 5 )
ALGORITHMS AND BOUNDS FOR POLYNOMIAL RINGS [ 101
R a b i n , Computable a l g e b r a :
M.
g e n e r a l t h e o r y a n d t h e o r y of
c o m p u t a b l e f i e l d s , T r a n s . AMS 9 5 ( 1 9 6 0 1 , [ 111
F.
341-360.
Richman, C o n s t r u c t i v e a s p e c t s o f n o e t h e r i a n r i n g s , P r o c .
AMS 4 4 ( 1 9 7 4 ) , I121
J.F.
I131
A.
436-441.
R i t t , D i f f e r e n t i a l A l g e b r a , American Math. SOC. C o l l .
P u b l i c a t i o n s , V o l . X X X I I I , N e w York,
(1950).
R o b i n s o n , I n t r o d u c t i o n t o Model T h e o r y a n d t o t h e
Metamathematics o f A l g e b r a , North-Holland Amsterdam ( 1 9 6 5 ) A.
157
.
P u b l . Comp.,
Robinson, Metamathematical problems, J.S.L.
38 (1973)
500-516. A.
R o b i n s o n , On bounds i n t h e t h e o r y o f p o l y n o m i a l i d e a l s ,
i n : S e l e c t e d Q u e s t i o n s of A l g e b r a a n d L o g i c : Mal'cev
Memorial Volume ( N o v o s i b i r s k 1 9 7 3 ) A.
,
245-252.
R o b i n s o n , A l g o r i t h m s i n A l g e b r a , i n : Model T h e o r y and
A l g e b r a , a Memorial T r i b u t e t o Abraham R o b i n s o n , e d : E.
S a r a c i n o and V.
Springer, Berlin, A.
(1977).
273-313.
S e i d e n b e r g , What i s N o e t h e r i a n ? , Rend. d e l Sem. M a t .
e F i s . d i Milano X L I V ( 1 9 7 4 ) , G.
498,
S e i d e n b e r g , C o n s t r u c t i o n s i n A l g e b r a , T r a n s . AMS 1 9 7
(19741, A.
W e i s p f e n n i n g , L e c t u r e Notes n r .
55-61.
S t o l z e n b e r g , C o n s t r u c t i v e n o r m a l i z a t i o n of a n a l g e b r a i c
v a r i e t y , B u l l . AMS 7 4 ( 1 9 6 8 ) , 5 9 5 - 5 9 9 . B.L.
v a n d e r Waerden, Moderne A l g e b r a 11, V e r l a g von J u l i u s
Springer, Berlin (1931).
LOGIC COLLOQUIUM 78
M. Boffa, D . van Dalen, K . McAloon leds.) 0 North-Holland Publishing Company, 1979
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES') SOLOMON FEFERMAN Dedicated to the memory of my friend and colleague, Karel de Leeuw Introduction and contents.
These lectures were designed to acquaint a general
logical audience with basic features of Bishop's approach to constructive mathematics (BCM) and with work on a certain formal system represented.
in which that can be To Several competing and rather different systems have been proposed
for the same purpose.
Thus, in addition t o the intrinsic interest of the subject
BCM provides an excellent case study for the process of formalization. The contents are divided into five parts, only the last of which assumes some prior background; in outline they are as follows. I.
--
Background and aims.
Part I gives an informal introduction to BCM which con-
trasts it both with everyday non-constructive mathematics as well as with the schools of constructivity previously established by Brouwer and Markov.
Towards
the end of this part we discuss general criteria of formalization, involving questions of adequacy and accord with the informal body of mathematics being represented. 11.
The theory To
.
In part I1 we present the language and axioms of
some natural subsystems and extensions. The adequacy of
To
To and
to BCM is sketched
and the question of its accord is discussed. Alternative formal systems proposed by Martin-L8f and Myhill are briefly compared in this connection. 111.
Models.
sented for To
A variety of models (in the classical sense of the word) are preand related theories.
One main purpose which these serve is t o
show how developments in BCM, when formalized in To
,
generalize corresponding
parts of classical mathematics and certain recursion-theoretic analogues.
They
are also used to obtain consistency and independenc.e results for some statements of mathematical interest. IV.
Realizability interpretations.
In contrast to models, the method of realiz-
ability (originating with Kleene) is distinctively associated with interpretations Text of lectures presented at Logic Colloquium 78 (University of Mons, Belgium, August 24-September 1, 1978). I am indebted to the organizers of this conference for proposing the lectures and for their helpful assistance in many ways. Research for and preparation of the text were supported in part by National Science Foundation grant MCS76-07163-A01. 159
160
S. FEFERMAN
of constructive theories.
It is here adapted to the formalism of
To
so
as to
obtain more delicate consistency (and conservation) results, in particular as concern axioms of choice and continuity principles.
V.
Relations with subsvstems of analysis. In this part one combines both proof-
theoretical and model-theoretical methods to obtain equivalence (in strength) of 1 various subsystems of the classical system S = (C2-AC) + (BI) with subsystems of
To.
For the full system To
question whether
S
one has an interpretation in
is equivalent to
To
S
,
but it is an open
a
We concentrate throughout on explanation and statement of results. Proofs are not given but some proof-ideas are indicated.
The basic source is Feferman
1975; this has been enriched considerably by the work of Beeson 1977.
The latter
gives both models and realizability interpretations which are used particularly for continuity principles; his work is described within Parts I11 and IV.
Im-
portant contributions to Part V have been made by Aczel, Buchholz, Friedman, Pohlers, and Sieg; detailed references are given in the text. Otherwise we draw principally on the unpublished notes Feferman 1976a. 1976b, and 1976c, which are now largely incorporated in the following. For the reader seeking a general introduction to the subject of constructivity and its formalizations (especially stemming from the schools of Brouwer and Markov) I would suggest the excellent survey article Troelstra 1977a; this contains an extensive bibliography.
I. 1. Ad hoc (local)
VS.
Background and aims
systematic (global) constructive mathematics.
At the local
level -one deals with particular questions of construction without regard to gen-
era1 principles or methods.
Frequently one knows an existential result guaran-
teeing the existence of a solution to a specific mathematical problem without knowing how it may be calculated, represented, or constructed. One then seeks to produce an explicit solution to the problem.
For examples familiar to logicians
we have: (i) decidability of p-adic fields (first existence by Ax and Kochen, followed by a concrete decision procedure by Cohen) and (ii) representability of positive definite real polynomials a s sums of squares of rational functions (existence by Artin, followed by recursive representations by Robinson and primitive recursive representations by Kreisel). At the global, systematic level one reconstructs whole portions of mathematics using entirely constructive notions and methods.
One of the main reasons
advanced for doing this is philosophical; it is based on a conception of mathematics which is opposed to the current underlying platonistic conception and has its source in human thought and constructions.
Such systematic redevelopment
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
161
according to constructive principles was initiated by Brouwer and carried on by Heyting and his students.
Subsequently another school of constructivity was de-
veloped in Russia by Markov and Shanin (cf. Troelstra 1977a for references on these two schools).
Finally the approach (here labeled BCM) was initiated in
Bishop 1967 and continued by him and his students, The main features of the first two schools will be described briefly below and those of BCM will be described at length.
2.
Constructivity in principle and constructivity in practice (feasibility).
No
matter how a constructive result is obtained (locally or globally) there is a question o f its actual computation or execution.
In this respect, even con-
structive existence results have a non-concrete character. A classical example is provided by Gauss' characterization of the regular polygons which are constructible by ruler and compass; the general theory had to be refined in order to give a feasible construction even of the 17-sided regular polygon.
For a (neg-
ative) example familiar to logicians, we may mention Tarski's primitive recursive decision procedure for the theory of reals.
It has been shown by Fischer and
Rabin that any decision procedure for the reals requires exponential time and so is unfeasible by present computational methods. 3.
General features of the platonistic conception. We describe these for a
point-by-point comparison with the constructivist conception in 54. 3.1.
Mathematical entities.
These are conceived to be external to u s and
independent of our thoughts and constructions. In its modern form, the most general mathematical entities are 3 . 3 below),
sets and
functions (which are interchangeable, cf.
Thus the Platonist conception is also called the Cantorian set-
theoretical conception of mathematics. 3.2.
Hence the & JC I. I
Mathematical statements are true or false.
employed
is the classical predicate calculus based on 2-valued semantics. The law of excluded middle prove 3 x $(x)
$V
1
$
leads u s to conclude 3 x q(x)V
it i s sufficient to prove i V x
Jlfx)
i
.
V x i $ ( x ) , Thus to This is the basis of the
use of the indirect method to obtain existential results: and draw a contradiction.
assume V x 7$(x)
Evidently there is no explicit solution provided by
such arguments. 3.3.
Interchangeabilitv of sets and functions.
Of course the former are
reduced to the latter via characteristic functions, Conversely, functions are regarded as many-one relations, which in t u r n are certain sets of ordered pairs. But the latter are'definable as sets, so functions are reduced to sets.
S . FEFERMAN
162
3.4.
A,B
Extensionality.
The principle V x (x E A
++
x E B)
-f
A = B
for sets
is justified by the consideration that sets exist independently of u s and of
any means of definition. 3.5.
Sets, then, may only be distinguished by their members.
Power set.
Since arbitrary subsets of a set are supposed to exist
independently and permanently, we may speak of their totality P ( A ) eration may be iterated, leading to the finite-type hierarchy. iteration the operation A H A U P(A)
.
This op-
For transfinite
gives a more convenient theory (the
z-
lative hierarchy). 3.6.
Subset formation. Any property
$(x)
of elements of A
determines
~
a subset B = {x E Al$(x)l
$I
(separation or comprehension urinciple).
may con-
tain quantified variables ranging over other sets, in particular over P(A) Such comprehension principles are impredicative: totality P(A) , which contains B 3.7.
B
.
is defined in terms of the
as an element.
The axiom of choice is usually agreed to be correct on the Cantorian
view, since there is no question as to
the choices are to be effected.
Then
one has the well-ordering theorem and the theory of finite and transfinite cardinals.
In consequence, such statements as the continuum hypothesis are taken to
have a definite truth-value, though undecided by all set-theoretical principles so far recognized to be correct (or even having some plausibility).
4. General features of the constructivist conception.2 ) 4.1.
Mathematical entities are only those which are understood directly
by humans or obtained from such by successive human constructions (e.g., by com-
...
bination into pairs (denoted as a ___ or sequences). The natural numbers 0,1,2, whole by N ) form basic entities which are generated by repeated adjunction of a single unit.
Both the processes of construction of mathematical entities and
of recognition of their properties are mental activities.
Such recognition is
the result either of direct intuition or of proofs based on principles inherent in the specific nature of the constructions used. induction for N 4.2.
For example, the principle of
directly follows the manner of its generation.
Mathematical statements do not communicate questions of truth or
falsity; they can only be assertions which communicate results of completed proofs.
The use of the logical particles is explained in terms of constructions
and proofs, roughly as follows: (i)
a proof of
($I
A
resp.; (ii) a proof of
$) ($
is given by a pair
(p,q)
of proofs of
v +) .consists of a proof of
$
$
and
$
,
or a proof of $
(together with the information as to which of these is proved); (iii) a proof of For further information and references on §4-18, cf. Troelstra 1977a.
CONSTRUCTIVE T H E O R I E S OF FUNCTIONS AND C L A S S E S
o f ((I any
163
I)) i s a c o n s t r u c t i v e o p e r a t i o n f o r which we recognize t h a t it w i l l convert
proof
q
b
of
s i s t s of a p a i r
4.
(iv)
i s a proof of
$(c);
i n t o a proof
(p,c)
where
p
( p q ) of
a proof of
& t ( x ) con-
( v ) a proof
p
of
Vx$(x)
i s a c o n s t r u c t i v e o p e r a t i o n f o r which we recognize t h a t it w i l l convert any object
'x' ) i n t o a proof ( p c ) of
c ( i n t h e intended range of t h e v a r i a b l e Taking
1.
$(c).
t o be an i d e n t i c a l l y f a l s e statement ( e . g . 0-1) which has no
proof, n e g a t i o n i s d e f i n e d by
(7@ )=
( @+ I ) ; t h u s proof o f a n e g a t i o n of a
statement (or of i t s a b s u r d i t y ) amounts t o c o n s t r u c t i v e r e c o g n i t i o n of t h e i m p o s s i b i l i t y of proof of t h a t s t a t e m e n t . one has a proof of
d,
A proof of
d,
QV-,
i s only given when
o r a proof of i t s a b s u r d i t y .
There i s a system of i n t u i t i o n i s t i c logic which i s recognized t o be c o r r e c t f o r t h i s i n t e r p r e t a t i o n of t h e l o g i c a l o p e r a t i o n s , b u t which does not y i e l d such ( a p p a r e n t l y ) unacceptable p r i n c i p l e s as t h e law of excluded middle
(LEM) o r i t s consequence
7 Vx -I
$ ( x ) + 3x b ( x ) .
Heyting has formulate3 t h i s
l o g i c i n such a way t h a t c l a s s i c a l l o g i c i s o b t a i n a b l e from i t simply by adj u n c t i o n of
LEM.
No f u r t h e r g e n e r a l l o g i c a l p r i n c i p l e s have been recognized
a s constructively evident.
(However, t h e r e i s no g e n e r a l l y recognized complete-
ness r e s u l t f o r i n t u i t i o n i s t i c l o g i c . )
4.5.
Functions a r e suppose3 t o be c o n s t r u c t i v e o p e r a t i o n s , t h e i 3 e a of
which was a l r e a d y contained i n
4.2 ( i i i ) , ( v ) . These a r e supposed t o be given by
a l g o r i t h m i c r u l e s of c o n s t r u c t i o n which can be e f f e c t e d by f i n i t e mechanical s t e p s of computation.
For r e l a t i o n s w i t h t h e r e c u r s i o n - t h e o r e t i c concept of
computable f u n c t i o n c f . 4 . 8 below.
4.4.
Sets a r e
,
only given by d e f i n i n g p r o p e r t i e s
f o r which we a r e
supposed t o know an3 understand t h e i r c o n d i t i o n f o r membership. t h e c o n d i t i o n for
x EN i s that
x
i s generated from
of a p p l i c a t i o n s of t h e successor o p e r a t i o n . c o n s i s t s o f a l l o r d e r e d pairs BA
, where
x EB*
such t h a t f o r each i s any s e t an3
B
=
iff
x :A
with
A,B
0
by a f i n i t e number
a r e s e t s then
x E A A y eB, i s a s e t .
B, which means t h a t
y c A , x(y)
x
x
E
B
C,
x
E
AA
O(X)
.
A X B, which
So also i s
i s a constructive operation
( i s d e f i n e d and) belongs t o
B.
d , ( x ) i s a w e l l - u n d e r s t o o d p r o p e r t y of members o f
ix e A ' @ ( x ) ) i n a s e t , with
4.5.
(x,y)
If
For example,
Finally, i f A
A
then
Mon-extensionality. Two r u l e s may have t h e same v a l u e s a t all
arguments (even provably s o ) , b u t t h e y a r e n o t i d e n t i f i e d u n l e s s t h e r u l e s a r e recognized t o be t h e same, a s r u l e s .
( T h i s allows f o r minor s y n t a c t i c v a r i a t i o n s
i n t h e p r e s e n t a t i o n of rules.) Two s e t s may have t h e same members, b u t they a r e not i d e n t i f i e d u n l e s s they a r e seen t o be &iven by t h e
properties.
t h e n o t i o n of i n t e n s i o n a l i d e n t i t y i m p l i c i t here, c f . 4 . 1 1 below.)
(For
164
FEFERMAN
S.
4.6. A
and
is 2 -
N a n - i n t e r c h a n g e a b i l i t y of s e t s an3 f u n c t i o n s .
is suchthat
f:A+[0,1)
c h a r a c t e r i s t i c -f u n c t i o n of
B
f
Those which do a r e c a l l e d d e c i d a b l e , o t h e r -
E.g. t h e s e t of exponents
i s t r u e i s ( p r e s e n t l y ) undecidable.
n
f o r which F e r m a t ' s l a s t theorem
I f every c o n s t r u c t i v e f u n c t i o n on
D
r e c u r s i v e t h e n every subset of
i s a subset of
Not every ( s u b ) s e t (of a g i v e n
(rel. to A).
s e t ) has a c h a r a c t e r i s t i c f u n c t i o n .
wise undecidable.
If B
VxeA[xcB<+f(x)=O] thenwe saythat
is
W
which i s r e c u r s i v e l y undecidable is un3eI n any case, s e t s a r e not r e d u c i b l e t o
c i d a b l e i n t h e c o n s t r u c t i v e sense. functions. If R
f : A - + B t h e n t h e graph of
has t h e p r o p e r t y
such
R
5AXB
i s a s e t , namely R = ( ( x , y ) e A X B l f ( x ) = y ) .
f
Vx E A 31 y ( x , y ) E R
determines a f u n c t i o n
.
The q u e s t i o n whether conversely, any i s a s p e c i a l c a s e of t h e following.
f:AIB
The conclusion i s t h a t f u n c t i o n s a r e not r e d u c i b l e t o s e t s .
4.7. The axiom of choice i s considered h e r e i n t h e schematic form
This l o o k s l i k e it ought t o be admitted u s i n g t h e i n t e r p r e t a t i o n of t h e connect i v e s i n 4.2.
However we have t o be c a r e f u l :
w r i t t e n out a s x
belongs t o
Vx[x E A + 3,y @ ( x , y ) ] g i v e s f o r each (i.e. that
A
pends on both
x
q, not of
an3
an3 x
* * *
1
and each proof
x
y ) where p* proves
1
p = p ( q , x ) , so t h a t
would be r e q u i r e 3 f o r
A'
q
that
A ) a proof
$ ( x , y ) . But
p*
p* de-
a l s o i s a f u n c t i o n of
y
Writing
(AC).
has t h e p r o p e r t y determining
3oes j u s t i f y a c c e p t i n g
of t h e hypothesis,
p
has t h e p r o p e r t y which d e f i n e s
q, i . e .
& as
x
'q i s a proof t h a t
(AC
x
3,y @ ( x , y ) which i s a p a i r p =(p,,
of
a proof
,
x
x
A for 9 t h i s i n f o r m a l argument E
t h e f o l l o w i n g modified p r i n c i p l e :
V X E A ~ ~ $ ' ( X , i~ )W V ~ V X E ~@A( X , f ( x , q ) ) .
We can d e r i v e choice of
(AC)
only f o r t h o s e s e t s
q, i . e . a f u n c t i o n
example of such a se-t; f o r
c
f o r which we have a c a n o n i c a l
A
]. Vx[x E A + x E ~ ( ~ ) A W
such t h a t
n E N , t h e b u i l d - u p of
c a t i o n t h a t we have a n a t u r a l number.
W
It may be noted t h a t t h e p r i n c i p l e
i s , f o r t h e same reasons, no more a s s u r e d i n g e n e r a l t h a n a r e d e a l t with f o r m a l l y i n t h e framework of
4.8. notation
Church's t h e s i s . Let
[e](n)
e,n,m,.
..
To
AC
DJ
.
(These p r i n c i p l e s
i n P a r t IV below.)
range over
a, and
f o r p a r t i a l recursive function application.
every ( t o t a l ) c o n s t r u c t i v e ~f u n c t i o n on
i s an
gives i t s e l f the v e r i f i -
take t h e usual
The t h e s i s t h a t
i s r e c u r s i v e i s r e f e r r e d t o as -
165
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
Church's T h e s i s i n t h e l i t e r a t u r e on i n t u i t i o n i s m (though it i s open t o argument whether Church himself had t h i s i n mind).
Formally we can express it ( i n a 2nd
o r d e r language) by
V f cNJ"3eVn[[e)(n)l A f ( n ) = ( e ) ( n ) ] .
(CT)
Note t h a t t h e converse t o Church's t h e s i s i s t h a t
T h i s f o l l o w s from ( A C ) m -which i s a c c e p t a b l e by
4.7.
There a r e some schemes re-
l a t e d t o (CT) which are e x p r e s s i b l e i n 1 s t order form and f o l l o w from (CT)
and
in particular:
(AC)m,
Vn % @ (n,m) +3e Vn[ ( e 1 ( n ) I A
( CTo)
Q (n, ( el ( n ) ) I
.
I t i s of l o g i c a l i n t e r e s t t h a t almost every known t h e o r y f o r m a l l y c o n s t r u c t i v e l y a c c e p t a b l e i s c o n s i s t e n t w i t h (CT ) .
T which i s i n -
However, t h e
As a n example
a c c e p t a b i l i t y of t h i s o r of ( C T ) i t s e l f i s a m a t t e r o f d i s p u t e .
of t h e k i n d of argument which can be made a g a i n s t it, c o n s i d e r t h e following.
Let
be a mathematician who works on deep problems of s e t t h e o r y and whose
J
mental behavior i s n o t d u p l i c a b l e by a machine.
Then t h e f u n c t i o n
f
d e f i n e d by
1 i f on t h e n o t h 3ay from now, J proves t h e nlth theorem of 0
ZF
otherwise
i s constructive but not recursive.
Perhaps a more convincing argument a g a i n s t
( C T ) i s t h a t u n s e r t h e c o n s t r u c t i v e i n t e r p r e t a t i o n of t h e l o g i c a l o p e r a t i o n s i f
it he13 we would have t o be a b l e t o p a s s c o n s t r u c t i v e l y from any (proof o f ) f cmrn
t o a Turing machine
e
which c a l c u l a t e s
f.
Thus even i f human mental
behavior i s b e l i e v e 3 t o be mechanical i n p r i n c i p l e , t h e r e i s no c o n s t r u c t i v e method o f a u p l i c a t i n g i t hy Turing machines.
A n argument
for
(CT)
on t h e o t h e r
hand, goes back t o what i s meant by c o n s t r u c t i v e o p e r a t i o n ; a t l e a s t i n t h e form e x p l a i n e 3 i n 4.3, t h i s would seem t o be j u s t i f i e d by Church's t h e s i s i n t h e u s u a l sense t h a t every f i n i t e a l g o r i t h m i c proce3ure can be c a r r i e d o u t by a Turing machine.
4.9.
Function s e t s and power s e t s .
INw
functions.
An argument can be made without
sets
If Church's t h e s i s i s accepted, t h e
i s p e r f e c t l y c l e a r : i t c o n s i s t s simply of t h e t o t a l r e c u r s i v e
meaning of
CT
t h a t we understand
BA
f o r any
A,B (whose c o n d i t i o n f o r membership was g i v e n i n 4 . 4 ) because our con-
c e p t i o n of c o n s t r u c t i v e o p e r a t i o n i s supposed t o be b a s i c ; t h i s does not mean t h a t "we know t h e t o t a l i t y of a l l c o n s t r u c t i v e o p e r a t i o n s from A
to
B".
The question
S . FEFERMAN
166
o f whether f o r each s e t
we have
A
even i f we a c c e p t
be d i f f e r e n t :
5
k(A) of a l l s u b s e t s of
standing o f what c o n s t i t u t e s an a r b i t r a r y p r o p e r t y o f elements of o f any
The c o n s t r u c t i v e s t a t u s o f
A.
A
seems t o
it i s n o t c l e a r t h a t we have a n under-
CT
I N , l e t alone
P(A) i s n o t s e t t l e d ; it i s of mathe-
m a t i c a l and l o g i c a l i n t e r e s t t o i n v e s t i g a t e t h e e f f e c t of assuming i t s e x i s t e n c e .
4.10.
Comprehension p r i n c i p l e s .
accepted) t h e n q u a n t i f i c a t i o n over and
a r e understood.
3x €A(...),
A,
I f a set
A
i s given (understood and
i . e . t h e l o g i c a l operations
Vx €A( . . . )
Hence any p r o p e r t y b u i l t using such o p e r a t i o n s
determines a s u b s e t of any given s e t .
If t h e e x i s t e n c e of power s e t s i s assumed
then t h i s l e a d s us t o i m p r e d i c a t i v e comprehension p r i n c i p l e s , i . e . e x i s t e n c e of
0
[x ~ A l o ( x ) ) where i n
we can q u a n t i f y over
P(A). Again, t h e c o n s t r u c t i v e
i s not s e t t l e d , w h i l e t h e i r r o l e and e f f e c t i s of
c h a r a c t e r of such p r i n c i p l e s interest.
4.11.
L i t e r a l , i n t e n s i o n a l and e x t e n s i o n a l i d e n t i t y .
In
4.6 we spoke of
f'unctions g i v e n by t h e same r u l e o r s e t s given by t h e same p r o p e r t y .
If we con-
c e n t r a t e on s y n t a c t i c r e p r e s e n t a t i o n of r u l e s , p r o p e r t i e s , e t c . , t h e n t h e riost obvious n o t i o n of sameness t o c o n s i d e r i s t h a t of 1 i t e r a . i i d e n t i t y , i . e . i d e n t i t y of s y n t a c t i c c o n f i g u r a t i o n s , symbol by symbol.
A l e s s d e f i n i t e b u t common i d e a
i s t h a t r u l e s , p r o p e r t i e s , e t c . a r e mental o b j e c t s which may have a v a r i e t y of For example, and most t r i v i a l l y , t h i s may be by a
syntactic representations. renaming
of bound v a r i a b l e s or o t h e r symbols.
More g e n e r a l l y , we may have r e -
p r e s e n t a t i o n s i n d i f f e r e n t , b u t i n t e r t r a n s l a t a b l e languages ( s o t h a t t h e s t r u c t u r e of t h e formal c o n f i g u r a t i o n s may a c t u a l l y change).
When two s y n t a c t i c ob-
j e c t s r e p r e s e n t t h e same mental o b j e c t t h e y a r e s a i d t o be i n t h e r e l a t i o n of intensional identity. Most f r e q u e n t l y i n mathematics we a r e concerned with v a r i o u s k i n d s o f def i n e d r e l a t i o n s of " e q u a l i t y " =A on a s e t relations. (nl,ml)=
x =
~
P lation
A,
which a r e simply equivalence
For example, when d e f i n i n g t h e i n t e g e r s
2
( n ,m ) w nl+m2
2
=
2
y &pI(x-y) f =Fg
,+ Vx
for E
x,y
A Vy
E
E
n, +ml
.
The s e t
Z .
A [x= y
A
3
Z as
When d e f i n i n g
f(x)
F =
Z
P
IN
X
M, we t a k e
we t a k e
-BA has d e f i n e d on it t h e r e -
Bf ( y ) 1.
All such e q u a l i t y r e l a t i o n s
a r e sometimes lumped t o g e t h e r ( p e r h a p s m i s l e a d i n g l y ) under t h e heading of tensional identity relations. (A,
to
=A)
(A/
ex-
I n C a n t o r i a n mathematics it i s common t o p a s s from
=A) so as t o r e p l a c e a l l e q u a l i t y r e l a t i o n s by l i t e r a l i d e n t i t y
u s i n g t h e axiom of e x t e n s i o n a l i t y f o r s e t s .
This p r a c t i c e i s n e i t h e r p o s s i b l e
(without e x t e n s i o n a l i t y ) nor n e c e s s a r y c o n s t r u c t i v e l y ; one simply makes c l e a r f o r each s e t Note. -
A
considered what e q u a l i t y r e l a t i o n i s being used i n a g i v e n c o n t e x t .
It i s common p r a c t i c e t o drop t h e s u b s c r i p t ' A ' from =A once t h a t i s
f i x e d i n any given d i s c u s s i o n .
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
5.
C o n s t r u c t i v e t h e o r y of r e a l numbers.
We s k e t c h h e r e how t h e preceding
p r i n c i p l e s a r e used t o s e t up a t h e o r y of r e a l numbers. successively +
,.
and a l l f u r t h e r p r i m i t i v e r e c u r s i v e o p e r a t i o n s .
.
way t o
7~
+
, . ,<
Then
.,<
4.11, and + ,
a r e d e f i n e d a s explained i n and
F i r s t of a l l , r e c u r s i o n
i s j u s t i f i e d d i r e c t l y by i t s manner of g e n e r a t i o n , so we can d e f i n e
W
on
167
Q i s t a k e n t o c o n s i s t of a l l ( x , y ) w i t h
a r e extenaed t o i t .
Z
and
=
z
a r e extended i n t h e s t a n d a r d x, y o 2
and
yf 0
QD c o n s i s t s of a l l sequences ( r ) n n
Next
Cauchy sequences of r a t i o n a l s a r e t h o s e f o r which t h e
of r a t i o n a l numbers.
Cauchy c o n d i t i o n i s c o n s t r u c t i v e l y s a t i s f i e d , i . e . f o r which we have a r a t e - o f convergence f u n c t i o n
(c)
Vk
By t h e s e t
IR
1-1 : IN
>
suhh t h a t
N
--f
1
3 p ( k ) [ / r n- rmI < r; 1 .
0 Vn, m
of r e a l numbers i s meant t h e s e t of all p a i r s x = ( ( r j , p ) with
satisfying ( C ) . Then we p u t x = * y for y = ( ( s n ) , v ) if (rn)E Q m ( r - s ) + 0. Real f u n c t i o n s ( o f k arguments) a r e of course t h o s e o p e r a t i o n s n
n
which p r e s e r v e -
f : IRk + IR
r e a l functions. with
-IR
. may
I n p a r t i c u l a r + and
.
F o r example, we may t a k e
be defined a s
((rn),pl)+((sn), p2) = ( ( r n + s n ) , w )
v ( k ) =max(pl(2k), p 2 ( 2 k ) ) .
order.
The f i r s t e s s e n t i a l d i f f e r e n c e i s met w i t h i n v e r s e and Given -1 -1 we seek x = ( ( r n ), v ) b u t t h e r e i s no obvious choice of
x=((rn), p )
x
u n l e s s we know a bound of Bn
2
(or y
1 m ( r n L r ; ) , and
< x)
if
(x-y)
away from 0.
x > 0 if
3m,k(x
> o and f i n a l l y x # y
cannot e s t a b l i s h c o n s t r u c t i v e l y t h a t all
(x, (m, k ) )
such t h a t
1x1
>
x
x
#
if
y V x =y
>
O(m, k ) i f
Then d e f i n e (x
.
>
y)
x > y
v (x < y).
We
I n v e r s e i s d e f i n e d for
T h i s i s n o t s t r i c t l y speaking a
0 (m,k).
IR, b u t only a subset by imbedcling; such s e t s a r e d e a l t with
s u b s e t of
systematically i n
BCM
a s w i l l be d e s c r i b e d i n
W e could o f c o u r s e d e f i n e t o take
Define
> O(m,k)).
v
x
2
0
t o be
Vk
e q u i v a l e n t d e f i n i t i o n s , nor i s Classically, t h e expression f(x) d e f i n e s a f u n c t i o n on
IR
x
z0
> 0 3m fin
={
x
2
2
by
x
m ( r
>
0 V x
n0
<
1 if
x>O
0
x < O
if
>
-
$14
below.
0 V x = O , b u t i t i s more u s e f u l 1 - ) ; t h e s e are n o t c o n s t r u c t i v e l y
P constructively justified.
which i s discontinuous, b u t t h i s does n o t make
c o n s t r u c t i v e sense as a d e f i n i t i o n .
Indeed t h e r e i s no e v i d e n t way t o o b t a i n
a S s c o n t i n u o u s f u n c t i o n ; t h e o r e t i c a l reasons for t h i s w i l l be produced l a t e r .
168
6.
S. FEFERMAN Brouwer's i n t u i t i o n i s m . ')
Brouwer b o t h explored g e n e r a l c o n s t r u c t i v e concepts
( e . g . c o n s t r u c t i v e o p e r a t i o n s , s e t s or "species"
,
o r d i n a l s , e t c . ) an3 c a r r i e d
out p a r t i c u l a r mathematical developments, e s p e c i a l l y i n a n a l y s i s . should be p o s s i b l e t o
prove
He thought it
c o n s t r u c t i v e l y t h a t every ( t o t a l ) r e a l f u n c t i o n i s
continuous and t h a t every r e a l f u n c t i o n on a c l o s e d bounde3 i n t e r v a l
i s uniformly continuous. choice sequence formation
.
F:[a,b]+B
For t h i s purpose he i n t r o d u c e d a new concept of
(f.c.s.) (r )
free
of which we know only a f i n i t e amount of i n -
( r o , . . , r k ) a t any g i v e n time, though we can proceed a s far o u t as
needed t o make a c a l c u l a t i o n .
The sequence may be pro3uced randomly, e . g . by
rolls of a d i e or o b s e r v a t i o n s of some random p h y s i c a l phenomena, r a t h e r t h a n by some mechanical l a w .
4,
I t makes sense t o o p e r a t e c o n s t r u c t i v e l y on such
sequences t o o b t a i n v a l u e s i n t h e operations
+ and
. are
3N or
Q
or new f . c . s . themselves.
e a s i l y defined f o r f . c. s.
Now if
For example,
f ( ( rn))
=
( sn)
s has been e s t a b l i s h e d , it can only have used a f i n i t e amount of m i n f o r m a t i o n about ( r n ); from t h i s p r i n c i p l e f o l l o w s t h e statement of c o n t i n u i t y and a v a l u e
of r e a l f u n c t i o n s where t h e r e a l s a r e understood i n t h e extended sense t o i n c l u d e a l l t h o s e g i v e n by f . c . s .
By some f u r t h e r ( l e s s immediately e v i d e n t )
p r i n c i p l e s Brouwer a l s o d e r i v e d t h e statement concerning uniform c o n t i n u i t y . Choice sequences need not be completely ' f r e e ' . w i t h or without r e s t r i c t i o n s on t h e i r v a l u e s .
sequences
( r n ) r e s t r i c t e s by
lrnl
5
Mn
They can be considered
For example, we can c o n s i d e r
where
(Mn)
i s g i v e n i n advance, or
i s i t s e l f produce3 by some r u l e depending on e a r l i e r v a l u e s of
( r n ) . Lawless A t the
sequences a r e t h o s e which a r e given without any r e s t r i c t i o n whatever.
o p p o s i t e end, l a w l i k e sequences a r e t h o s e which a r e completely determined i n advance by r u l e s .
The theory of r e a l s sketched i n §
5 may be i n t e r p r e t e d as
applying t o t h e l a t t e r kinds o f Cauchy sequences; f o r t h i s reason it i s sometimes c a l l e d l a w l i k e a n a l y s i s . Brouwer's a n a l y s i s based on f . c . s . has been s t u d i e d i n v a r i o u s l o g i c a l formalisms by Kleene, Vesley, K r e i s e l , T r o e l s t r a , van Dalen and o t h e r s (cf. Troelstra
l977a 19771, f o r r e f e r e n c e s ) .
Various p a r t s o f t h i s have t a k e n
s e t t l e d an3 coherent form (and have, i n c i d e n t a l l y , been shown c o n s i s t e n t ) .
B u t e f f o r t s t o t r e a t t h e most g e n e r a l concept o f f . c . s . have not y e t had a convincing outcome.
For mathematicians, Brouwer's t h e o r y has remained a c u r i o s i t y ;
~
1975
(collected
works,
5)
A p e r u s a l of Brouwer
v o l . 1 ) i s rewarding h e r e .
4)
Following a remark of T r o e l s t r a , H. J e r v e l l has t r a c e d back t h e i d e a o f f . c . s . t o papers of E. Bore1 i n 1912 which grew out of e a r l i e r d i s c u s s i o n s by t h e French mathematicians on t h e axiom o f choice. A t f i r s t Brouwer r e j e c t e d t h e i d e a b u t l a t e r (1917) accepted it and expanded it i n t o a theory.
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
169
it has largely been of interest to logicians. Moreover, the concepts are rather special to analysis and topology and seem to have little to do with other parts of mathematics. Historically, the actual development of intuitionistic mathematics got hung up around analysis because of the need to clarify Brouwer's ideas there. It should be remarked that the intuitionistic theory of f.c.s. is g consistent with classical mathematics, for we can prove
7
V(rn)[ 3m(rm=O)VVm(rm~O)1,
as is intuitively evident from the 'finite-information'principle. Relatedly, one can disprove Vx EIR(x
2
0
V
x
< 0), etc. This is in contrast to lawlike analysis,
which is a part of classical mathematics (if one does not assume
(CTo)).
7. The (Russian) school of Markov and Shanin. Here one accepts the scheme (CTo) an3 the laws of intuitionistic logic, but also the following non-intuitionistic law, called Markov's principle:
(m)
vn[O(n) v
7
Q (n)I A -,V n
7
@(n)
+
&Q(n)
(where 'n' ranges over W). The intuitive idea for ( M P ) is that under the hypothesis we can constructively find a solution n of the conclusion simply by performing a search through IN.
It may be shown that
(CTo) + (MP) is con-
is inconsistent with full classical
sistent over number theory though (CTo) logic there. (The consistency proof can be given by Kleene's recursive realizability, which w i l l be described in IV.)
Various parts of analysis can
be carried out under these assumptions, continuing the line sketched in $ 5 . For example, if
f
is continuous on
[a,b] then
exist. However, it cannot be prove3 that f mum) in [a,b].
inf f(x) a l x l b
takes on --
and
sup f(x) a l x z b
its minimum (resp. maxi-
The reason is provided by a well-known example due to Specker of
a recursively continuous function on [0,1]which has no recursive point at which f takes on its minimum. Various other basic results of classical analysis may also be contradicted by suitable recursion-theoretic examples, e.g. that if f is continuous on
[a,b] and f(a) < 0 then %(a
<x
A
f(x) =O).
(In the
Russian school it is admitted that there are some 'peculiarities' to their approach.)
8. Recursive analogues to classical mathematics. We have in mind here a series of studies concerning analogues to classical notions where one uses recursive functions ( o r functionals or sets) in place of arbitrary objects of the same type. To be mentiones in particular is the work of Dekker and Myhill. for set theory, Crossley for order theory, Malcev
and Rabin for algebra, and Specker an3 Lacombe
for analysis and topoloQi (cf. Feferman 1975 for references).
These have been
S. FEFERMAN
170
c a r r i e d out informally, w i t h no r e s t r i c t i o n on t h e l o g i c o r methods employed. I n e f f e c t , though, at l e a s t (CT) i s assumed (though not (CT ), which i s c l a s s i c a l l y i n c o n s i s t e n t ) , and indeed a corresponding btronger p r i n c i p l e i d e n t i f y i n g p a r t i a l f u n c t i o n s on
W with p a r t i a l recursive functions.
W to
types one
i n t h e D e k k e r - m h i l l t h e o r y of r e c u r s i v e equivalence (A-B)
-3f,g[f,gpartialrecursive~
~ ; . A : A -,BA
A ( g f ) r A = l AA (fg)l'B
A,B
where
= lB 1
g
For example,
defines p
~ +: A ~
,
BI. One p o s i t i v e r e s u l t which i s
may be a r b i t r a r y s u b s e t s of
proved f o r t h i s i s a form of t h e Cantor-Bernstein Theorem:
A
-
(BtC) A B
-
( A t D ) -,A
- B.
I n a n a l y s i s one c o n s i d e r s r e c u r s i v e r e a l numbers ( i . e . x = ( ( r ),+) r e c u r s i v e ) and r e c u r s i v e f u n c t i o n s of =(defined
(rn),p
way v i a r e c u r s i v e f u n c t i o n a l s on
BIN).
with b o t h
i n an a p p r o p r i a t e
A s with t h e Russian school, a number of
' p e c u l i a r i t i e s ' a r e met i n t h i s v e r s i o n of a n a l y s i s . While t h e s e p u r s u i t s a r e not c o n s t r u c t i v e t h e y can be r e l e v a n t t o cons t r u c t i v e approaches i n t h e f o l l o w i n g ways.
Where a r e c u r s i o n - t h e o r e t i c analogue
g i v e s a p o s i t i v e r e s u l t , i . e . where a c l a s s i c a l theorem c a r r i e s over, one can o f t e n prove t h e same theorem c o n s t r u c t i v e l y .
On t h e o t h e r hand, when a n e g a t i v e
r e s u l t i s o b t a i n e d by s u i t a b l e counter example, it i s u s u a l l y p o s s i b l e t o use such t o g e t u n d e r i v a b i l i t y of t h e c l a s s i c a l theorem i n a c o n s t r u c t i v e system. However, n e i t h e r of t h e s e i s automatic.
For example, t h e l e a s t number p r i n c i p l e
which i s f r e q u e n t l y a p p l i e d i n r e c u r s i o n - t h e o r e t i c arguments i s n o t c o n s t r u c t i v e l y d e r i v a b l e except f o r d e c i d a b l e
9.
Bishop's approach.
In
@.
1967 Bishop
p u b l i s h e d h i s Foundations
constructive
a n a l y s i s i n which he c a r r i e d o u t a n informal development of c o n s t r u c t i v e a n a l y s i s which looked much more l i k e modern a n a l y s i s t h a n anything aone p r e v i o u s l y by c o n s t r u c t i v i s t s and which went s u b s t a n t i a l l y f u r t h e r mathematically.
Bishop
works w i t h g e n e r a l n o t i o n s of f u n c t i o n and s e t regardej. i n i n f o r m a l c o n c t r u c t i v e terms.
He r e j e c t s t h e n o t i o n of f . c . s . as being obscure and unnecessary.
s t e a d of t r y i n g t o
prove
In-
t h a t a l l f u n c t i o n s of r e a l s a r e continuous, h i s view
i s : t h e r e i s n o t much of i n t e r e s t we can say about a r b i t r a r y f u n c t i o n s from SR
to
SR
o r from [ a , b ] t o
IR. Define C ( [ a , b ] , B) t o be t h e uniformly ?R and C ( S R , IR) t o be t h e f u n c t i o n s which
continuous f u n c t i o n s on [ a , b ] t o
171
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
are uniformly continuous on each compact interval.
(These definitions w i l l be ex-
plainej. in more detail below.) These are classes of central mathematical interest. In a sense, Bishop is working in lawlike analysis and the notions and principles he uses are contained either 3irectly or implicitly in Brouwer's intuitionism,
hut simply without f.c.s.
What is novel about Bishop's work is its spirit and
execution, which is much more like everyday modern mathematics than anything previously done in a systematic constructive way. unprepare3) mathematician could pick up Bishop
Indeed, a (philosophically
1967 and
read it as a straight piece
of classical Cantorian mathematics. what would be puzzling to him is the more involved choice of certain notions and proofs, unless he also saw in what sense these were dictated by constructive requirements. It is this which is least successfully explained by Bishop. One of the main aims of the logical study of BCM
is
to elicit its underlying principles and to show how they may be interpreted constructively, as well as classically. One is led to consider constructive theories of functions, sets and classes which relate to BCM as theories like ZermeloFraenkel relate to Cantorian mathematics.
Such systems could have been developed
years ago, before Bishop, but it must be acknowle&ged that the work itself provide3 both the stimulus an3 a test for the adequacy af proposed theories. 5 ) 10. Note on personal viewpoints. Bishop is a confirmed constructivist, as was
Brouwer. Just as with Brouwer, he places the doing of constructive mathematics ahead of its logical study, regarding the latter as inessential. I am constructivist ( n o r a Platonist- it is harder to say
I an.)
not
a
My main in-
terests are logical and as a logician I am particularly interested in various forms of explicit mathematics (constructive, recursive, predicative, hyperarithmetic, inductive,Borelian, etc.)
Of course this kind of position lends
itself to greater objectivity, but there is also the possibility of insensitivity to, or neglect of, what are considered by a given school to be essential points.
11. Criteria of formalization. How well does a formal theory T represent an informal body of mathematics M ?
We judge this in terms of its adequacy and
accordance. (i) T
is an adequate formalization of M
if every concept, argument and
result of M may be represented by a (basic or defined) concept, proof and theorem, resp. of (ii) T T
T.
is in accordance with (or faithful to) M if every basic concept of
correspgnds to a basic concept of M
and every axiom and rule of T
sponds to or is implicit in the assumptions and reasoning followed in M T
3oes not go beyond M
corre(i.e.
conceptually o r in principle).
5) Actually an informal constructive theory of functions and sets was outlined about the same time as Bishop's work in Tait 1968.
172
FEFERMAN
S.
Remark.
Formalisms always go s y n t a c t i c a l l y beyond what i s of o r d i n a r y i n t e r e s t ,
4A 4 A 4
e.g. i n p r a c t i c e we never assert
l a t ed .
4
or
--f
where
ji
4, Jr
a r e unre-
We may r e f i n e ( i ) , ( i i ) by c o n s i d e r i n g whether t h e r e p r e s e n t a t i o n i s d i r e c t
o r i n d i r e c t . The i d e a of being ( i ) ' d i r e c t l y adequate, r e s p . ( i i ) ' d i r e c t l y i n accordance
with
M
seems c l e a r .
We would say t h a t
i s i n d i r e c t l y adequate
(i)" T
M
adequate t o
5
M
i f t h e r e i s a theory
which can be t r a n s l a t e d i n t o
T*
directly
T (or otherwise reduced t o
T
in
an elementary way). i s indirectly
(ii)" T
reduced t o a t h e o r y
T
+
A goOj. f o r m a l i z a t i o n o f accordance with 12.
M
.
accordance
with
M
if
T
can be t r a n s l a t e d or
which i s d i r e c t l y i n accordance w i t h
M.
M i s one which i s b o t h d i r e c t l y adequate t o and i n
I l l u s t r a t i o n s of t h e s e c r i t e r i a . 12.1. M
Z
=
Z1 ( + , 2
Z
=
elementary number t h e o r y ( n o n - a n a l y t i c an3 n o n - a l g e b r a i c ) .
1 2 = P e a n o ' s a r i t h m e t i c w i t h a l l p r i m i t i v e r e c u r s i v e f u n c t i o n symbols.
=
a
)
= Peano's a r i t h m e t i c w i t h j u s t
f,.
.
2nd o r d e r a r i t h m e t i c with f u l l comprehension.
'2 i s d i r e c t l y adequate t o and d i r e c t l y i n accordance with M . 1 Z ( +,-) i s d i r e c t l y i n accordance with M b u t only i n d i r e c t l y adequate
1
t o it (by t r a n s l a t i o n of Z2 i s 3 i r e c t l y adequate t o
concept of
Z ).
M
b u t n o t i n accordance w i t h
M
since the
P ( W ) a s a completed t o t a l i t y i s i m p l i c i t l y assumed i n t h e
comprehension scheme of
Z2 .
12.2. M = c l a s s i c a l a n a l y s i s .
zW =
arithmetic i n a l l f i n i t e types.
UJ.
Z i s d i r e c t l y adequate t o M . Z2 i s i n d i r e c t l y adequate t o
M
by r e d u c t i o n o f t h e concepts t h a t a c t u a l l y
occur i n p r a c t i c e t o second o r d e r terms. Zw i s not
i n accordance with
t o t a l i t y 1R, but n o t 2Rm
M;
f o r i n c l a s s i c a l a n a l y s i s we assume t h e
as a t o t a l i t y .
t o t a l i t y i n t h e c a l c u l u s of v a r i a t i o n s .
[C(IR, I R ) i s assumed as a Functionals i n
C ( I R , TR+B
a r e t r e a t e d i n modern a n a l y s i s , b u t n o t higher t y p e o b j e c t s i n any e s s e n t i a l way.]
12.3. M
=
Cantorian set t h e o r y .
ZF = Z e r m e l o - F r a e n k e l s e t t h e o r y .
ZF i s d i r e c t l y adequate t o M; i s it i n accordance w i t h M? The q u e s t i o n i s r a i s e d s i n c e t h e i d e a of t h e cumulative h i e r a r c h y does n o t seem e s s e n t i a l t o M.
CONSTRUCTIVE THEORIES O F F'JNCTIONS
AND CLASSES
173
It i s e v i d e n t from t h e s e examples t h a t t h e a p p l i c a t i o n of t h e c r i t e r i a of f o r m a l i z a t i o n a r e rcasonably o b j e c t i v e , though t h e r e a r e c a s e s o f u n c e r t a i n t y .
13. Formal systems which have been proposed f o r E M . 13.1. HAu) + AC
Bishop 1970, Goodman-Myhill
where
1972, both considered f o r m a l i z a t i o n i n
i s i n t u i t i o n i s t i c a r i t h m e t i c extended t o f i n i t e t y p e s
HAu)
(HA = H e y t i n g ' s i n t u i t i o n i s t i c a r i t h m e t i c ) .
BCM.
HAw i s d i r e c t l y i n accordance with
The q u e s t i o n of B i s h o p ' s views on, and use of, AC
HAu) + AC
w i l l be t a k e n up below. 13.2.
Martin-Lbf
1975
i s more d e l i c a t e and
i s inadequate t o B i s h o p ' s t h e o r y
( t r a n s f i n i t e type theory).
Of
Sets.
This i s d i r e c t l y i n
accor3ance with BCM and adequate t o e v e r y t h i n g b u t Bishop's t h e o r y of i n d u c t i v e l y d e f i n e d c l a s s e s ( o r a i n a l s , Bore1 s e t s , e t c . ) ; it may also be n a t u r a l l y supplemented f o r t h e l a t t e r .
It t h u s c o n s t i t u t e s a goo3 f o r m a l i z a t i o n of
BCM.
However,
it i s s y n t a c t i c a l l y complicated, an3 n o t as f l e x i b l e t o work with a s o t h e r t h e o r i e s t o be d i s c u s s e d .
T h i s will be e x p l a i n e d i n more d e t a i l l a t e r .
t h a t S c o t t 1970 a n t i c i p a t e d Martin-Lbf
13.3.
Myhill
It should be added
1975 i n v a r i o u s ways.
1975 CST ( C o n s t r u c t i v e s e t t h e o r y ) , Friedman 1977.
CST i s a sub-
t h e o r y of IZFC/ZFC, i n t u i t i o n i s t i c ZFC,which l i k e ZFC assumes e x t e n s i o n a l i t y and i d e n t i f i e s f u n c t i o n s w i t h many one r e l a t i o n s . w i t h c o n s t r u c t i v e views, l e t a l o n e BCM.
w i l l be explained l a t e r .
Friedman
It
1977 h a s
Thus it i s n o t d i r e c t l y i n accord i n a i r e c t l y a3equate t o BCM, as
c o n s i d e r e 3 a number of such t h e o r i e s
and c h a r a c t e r i z e d t h e i r s t r e n g t h ; he has a l s o sketched i n t e r p r e t a t i o n i n t o cons t r u c t i v e l y j u s t i f i e d t h e o r i e s , t h u s i n d i r e c t l y i n accord with BCM.
13.4.
Peferman
1975(To). T h i s w i l l be d e s c r i b e d i n d e t a i l i n P a r t I1 below.
It i s d i r e c t l y adequate t o all of BCM. I s h a l l argue t h a t it
Accordance however i s a m a t t e r of d i s p u t e ;
To
i n accor3, a t l e a s t i n f i r e c t l y .
i s a type-free
t h e o r y which i s v e r y amenable t o metamathematical study and a p p l i c a t i o n s .
14. Some g e n e r a l f e a t u r e s of BCM.
A s already s a i d these incorporate t h e general
f e a t u r e s of c o n s t r u c t i v i s t mathematics o u t l i n e d i n
54, § 5
:
the logic i s in-
t u i t i o n i s t i c , f u n c t i o n s a r e given by r u l e s , s e t s by d e f i n i n g p r o p e r t i e s ; t h e s e a r e n o t i n t e r c h a n g e a b l e , and e x t e n s i o n a l i t y i s not assumed. f o l l o w i n g s l i g h t v a r i a n t s from $-§5 of subsets i n
We d e t a i l i n t h e
above; novel p o i n t s come w i t h t h e treatment
14.6.
14.1. Mathematical e n t i t i e s . The o n l y o b j e c t s which appear t o be considered by Bishop a r e n a t u r a l numbers, o p e r a t i o n s and from t h e s e by p a i r i n g .
sets, and
such as a r e generated
Each such i s considered t o b e p r e s e n t e d by a f i n i t e
symbolic e x p r e s s i o n .
1 4 . 2 . I d e n t i t y and e q u a l i t y . Two symbolic e x p r e s s i o n s a r e i d e n t i c a l i f t h e y a r e p r e s e n t e d i n t h e same way- a s i n 4.11. We may t a k e t h i s t o be l i t e r a l i d e n t i t y
174
S.
Each s e t considered has a t t a c h e d t o it one or more r e -
o r intensional identity. l a t i o n s of ' e q u a l i t y ' . identity,
FEFERMAN
Notation: Bishop w r i t e s
=
= f o r a n e q u a l i t y r e l a t i o n on a s e t .
for l i t e r a l or intensional
We s h a l l w r i t e i n s t e a d = f o r t h e
f i r s t an3 = f o r t h e second ( b u t when t h e r e i s no ambiguity we drop t h e subA script).
14.3. Operations and f u n c t i o n s : Given s e t s ~
A
t i o n from f(a)
in
B
to
B.
i s a f u n c t i o n from A
f
14.4. Function s e t s . functions
F(A,B).
to
B
if
a
Bishop says t h a t f o r each
O f course, i f we c o n s i d e r
a
applied
to
A
to o p e r a t i o n s ,
B.
t h e r e i s t h e s e t of a l l
A,B
en5owed w i t h t h e
etc.
F (or 0) s t a r t i n g with
Iterating
IR
those (xn)nzl
such t h a t
fies
Ix
1
%>k me an
n
N . We w r i t e
BA
I n t e g e r s , R a t i o n a l s , Reals. T h i s f o l l o w s
Bishop d e f i n e s
-XI
1
<-) - n
; for
E =
.
$5
for
allows us t o
N
O(A,B).
f o r N, 2
and
Q. However,
t o c o n s i s t o f t h e f o l l o w i n g s p e c i a l c l a s s of Cauchy sequences: 1 1 Ix -x <+ - f o r a l l n, m, ( s o t h e l i m i t x s a t i s n in - n m
IR'
1
c o n s i s t s of p a i r s
1
(5-k)
x
t h i s means
2
(x,k) E
> 0.
where
x
=
(x )
i s a r e a l and
( x , k ) ~ ~ + ( y , i s) 3 e f i n e d t o
=my*
14.6.
lR+ i s not a s u b s e t of R i n t h e u s u a l sense, b u t i s one i n
Subsets.
t h e following sense of Bishop. operation
i : A -tB
j e c t i o n of
A
such t h a t
into
Thus i f we t a k e vals (a,b), [a,b]
B).
For x
By a s u b s e t al=Aa2 in
i ( x , k ; = x we have etc.
in
B
(A,i)
B
is meant a s e t
A
and
i(a ) = i(a ) ( i . e . i i s an i n 1 B 2 we say x E A i f x = i ( y ) w i t h y E A . cf
( B + , i ) a s u b s e t of R. S i m i l a r l y , i n t e r -
R a r e given a s s u b s e t s i n t h i s g e n e r a l i z e d sense.
But we can a l s o c o n s i d e r s u b s e t s i n t h e u s u a l sense where on
of a l l
O(A,B)
Since we can form t h e s e s e t s we can c o n s i d e r o p e r a t i o n s
o b t a i n t h e f i n i t e t y p e hierarchy over
14.5.
opera-
a n element
A
= a- + f ( a ) = f ( a ) . 1 A 2 I B 2
as s e t s
A,B
in
l i t e r a l i d e n t i t y r e l a t i o n t h i s implies t h a t t h e r e i s t h e s e t o p e r a t i o n s from
i s an
(A, = A ) , (B, = B ) , f
i f it i s a r u l e which a s s i g n s t o each
i = i d e n t i t y operation
A.
14.7.
S e p a r a t i o n . Bishop r e c o g n i z e s t h a t each p r o p e r t y
elements of a s e t
S
determines t h e subset
A
i s a subset i n t h e u s u a l sense; e q u a l i t y i s d e f i n e d t o be
14.8.
Operations on s u b s e t s of a s e t .
subset from
14.6, t h e o p e r a t i o n s of
( c a t e g o r i c a l ) forms.
=p
.
applicable t o (Implicitly, t h i s
restricted t o
A.)
Using t h e more g e n e r a l concept of
union and
Suppose g i v e n s u b s e t s
P
= ~ ( X SE A P ( x ) ]
i n t e r s e c t i o n t a k e on more g e n e r a l (Ao, io) and
(Al,il)
of
B:
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
i ) is d e f i n e d by Bishop t o c o n s i s t of a l l p a i r s ( k , a ) where
(AoUAl,
or
a €Ao
k = l
a
and
E
A1;
of a l l
(ao,al)
(j(ao,al)
14.9.
=
aoe A.
with
al € A l
an3
i o ( a o ) . Note t h a t
T, is meant an o p e r a t i o n
f ( t ) = ( A t , i t ) of
an3
f
i ( a ) = il(al); 0
0
t h e n one t a k e s
Aox A
By a family of s u b s e t s of a s e t
which a s s o c i a t e s with each
t
E
T
1'
B, i n -
a sub-
B, i n such a way t h a t e q u a l s e t s a r e a s s o c i a t e d with
The family i s i n d i c a t e d by
equal in3ices.
=
i n t h i s sense i s c o n t a i n e d i n
AonAL
F a m i l i e s of s u b s e t s of a set.
dexed by
k = O and
i ( a ) for k = 0 , 1 . Note t h i s i s k (Aon A1, j ) i s f u r t h e r d e f i n e d t o c o n s i s t
further, i ( k , a )
e s s e n t i a l l y a form of d i s j o i n t union.
set
175
{ ( A t , i ))
or ( A t ) t f l
t tcT
o r just (At).
For s i m p l i c i t y , we s h a l l only c o n s i d e r i n t h e f o l l o w i n g t h e c a s e s t h a t t h e index set
T i s simply s u p p l i e d w i t h l i t e r a l i d e n t i t y as i t s e q u a l i t y r e l a t i o n .
14.10. Operations on f a m i l i e s . ( U At,i)
one d e f i n e s
u
and (
t ET
t ET
n At,j)
t ET
A ~ (=( x , t ) i x E A ~ ) and
A s an e x t e n s i o n of t h e o p e r a t i o n s i n 14.8
B
as s u b s e t s of
by:
i(x,t) =it(x);
The union i s a g a i n a form of d i s j o i n t sum t h a t we c a l l t h e b e denoted below by
1A t . t ET
join of {At ) ; it will
I n e f f e c t , t h e i n t e r s e c t i o n i s formed by s e p a r a t i o n
Il At = ( g l V t e T ( g ( t ) € A t ) ] , on which e q u a l i t y t E T V t c T ( g l ( t ) = g ( t ) ), i . e . V t c T ( it ( g1( t ) )= B i t ( g 2 ( t ) ) ) .
from t h e C a r t e s i a n product
i s d e f i n e d by
gl =g2
14.11.
At
An a l t e r n a t i v e d e f i n i t i o n of f a m i l y which
Pre-joined f a m i l i e s .
Bishop says could be considered i s a subset any such
we can d e f i n e
A
(extensionally).
B
of
T.
X
Then
Thus w e c a l l such a family pre-joined,
a l r e a d y g u a r a n t e e s e x i s t e n c e of i t s j o i n . axiom which -
A
f ( t ) = A t =(xi(x,t) C A I .
t e l l s us t h a t i f
f
C e r t a i n l y , given
U A t t e T i.e. i t s prescription A
=
I n g e n e r a l though we need a
i s a family
(At)tET
then t h e j o i n
join J(T,f)
exists.
14.12.
Bore1 s e t s .
These a r e i n d u c t i v e l y generated
in a
t o p o l o g i c a l space
from c e r t a i n b a s i c s e t s by t h e o p e r a t i o n s of union and i n t e r s e c t i o n a p p l i e d t o countable f a m i l i e s . A b s t r a c t l y t h i s has t h e form: (i)
(ii)
Bo
5B
(f:W + B )
implies (J(N, f ) E B A I ( A , f )
8
B).
( i i i ) i f a p r o p e r t y h o l d s of all elements of B an3 holds of J ( R , f ) an3 I ( l N , f ) whenever it h o l d s o f f ( n ) f o r eacg n, t h e n it holds of all elements of B.
176
S.
FEFERMAN
14.13 Principles in the general theory of integration. The Borel sets in 1967,
a measure space are used in the development of integration theory in Bishop
Ch.7. The theory of measure and integration was redeveloped by Bishop-Cheng
1972 without the use of Borel sets. This is an abstract theory, i.e. one starts with an arbitrary 'integration space' X completion L(X).
and associates with it a certain
It was pointed out by Friedman that the basic definitions in
the latter approach make prima-facie use of the power-set operation which, as we have seen, is constructively problematic. However, this is only necessary if one wants L(X)
again to be a set. The notion of being a member of L(X) fioes not require the power-set axiom and in that sense one can carry out abstract integration theory without this principle or the generalized inductive principles behind Borel s e t s .
F o r more detailed examination of the issue here cf. Feferman
1978 64. In any case, the potential (albeit marginal) mathematical utility of both generalized in3uctive and power-set principles in BCM makes them of interest f o r logical study. The former are incorporated directly in T o , since they have
constructive character.
15.
General features of BCM, c o n t d : Existential definitions and witnessing in-
formation. Notions which are defined classically using existential information are frequently replaced in BCM (as well as in other schools of constructive mathematics) by corresponding notions in which witnessing information is explicitlj shown. This is required to carry out constructive operations on the objects satisfying the given notion.
15.1. Examples.
.
(i) I R '
We have already explained its definition in $5 (as needed to
make the operation of inverse constructive). (ii) Limits of sequences of reals. By a convergent sequence of reals is meant a triple and m
((x,)
,
xo, m )
where x
and each xn(n
2
1) belongs to
is a function of positive integers such that (for all k
f o r all
n
>_ m(k); m is called
2
IR
1 o I< - k-
1)Ix -X
n a modulus-of-convergence function for the
sequence. (iii) Continuous functions. By a (uniformly) continuous function f compact interval I = [a,b] is meant a pair
+
preserving =B)and w:IR +IR w
+
(f,ui)
on a
with fcF(I,?R)(i.e. f:I + IR
such that /f(x)-f(y)/ 5
E
whenever lx-yl _ < W ( E ) ;
is called a modulus-of-uniform-continuity function for f .
The set of all
continuous functions from I to IR is denoted by C(1, IR); it is a subset of F(1,IR)by
i (f,w) =f.
(This function
ui
is neeaed for example when performing
the operation of integration of f over I.) 15.2.
The logical pattern. In each case we are spelling out a property
P(x) involving existential quantifiers. In the above examples (i)-(iii) P(x) is,
177
CONSTRUCTIVE T H E O R I E S OF F U N C T I O N S AND C L A S S E S
r e spect i v e l y :
2
Xk
(i)
( ii )
1(xk
> L, k
2 13m Vn 2
3xo Vk
m ( 1 xn-xo 1
1
5 r; )
> 0 VxVy(Ix-y1 < 6
(iii) V s > 0 3 6
<
+ lf(x)-f(y)l
With each such p r o p e r t y i s a s s o c i a t e d another
P*(x,w)
nessing information t h a t r e a l i z e s o r v e r i f i e s P(x)
.
6).
where
w
i s some w i t -
These p r o p e r t i e s a r e r e -
l a t e d by (1)
~ w ~ * ( x , w+ ) ~ ( x )and
(2)
if
(AC)
i s assumed,
We s h a l l a l s o c a l l c o n s i s t s of
xo
15.3.
~ ( x -), 3 w P * ( x , w ) .
a s p e l l e a - o u t form of
P*
m
t o g e t h e r with
He speaks of
coupled w i t h some s i d e i n f o r m a t i o n Then one t r e a t s o p e r a t i o n s on
A*
w
of a l l ( f , m )
A*
when it i s r e a l l y only
A*
as i f t h e y were o p e r a t i o n s on I
to
J:
b
f(x)*. a There i s a c e r t a i n c a s u a l n e s s i n Bishop
n e s s i n g information a s one goes along.
x
C ( 1 , I R ) +IR
1967
A.
F o r example,
i s o f f i c i a l l y de-
IR
s a t i s f y i n g t h e conBition of
i s not e x p l i c i t l y reveale3 i n t h e operation
J
A
which can be considered t o be a member.
of (uniformly) continuous f u n c t i o n s on
A
f i n e d as t h e s e t form
Often Bishop d e f i n e s a s e t
A*= ( ( x , w ) l P ( x , w ) ) ( t h e o f f i c i a l d e f i -
being a member of
x
k.
( x l P ( x ) ) ( t h e u n o f f i c i a l d e f i n i t i o n ) but then
says t h a t he i s r e a l l y d e f i n i n g t h e s e t
the set
For example, i n c a s e ( i i ) , w
Having your cake and e a t i n g it t o o .
i n what appears t o be t h e form nition).6)
P.
as a f u n c t i o n of
1 5 . 1 ( i i i ) . But
tu
when w r i t t e n i n t h e
about mentioning t h e w i t -
Constructivity i n theory requires t h a t
it be mentioned, b u t one i s l o o s e r i n p r a c t i c e i n o r d e r t o keep t h a t from g e t t i n g P r a c t i c e t h e n looks v e r y much l i k e everyday a n a l y s i s and it i s hard
t o o heavy.
t o see what t h e a i f f e r e n c e i s u n l e s s one t a k e s t h e o f f i c i a l d e f i n i t i o n s s e r i o u s l y .
15.4.
A concrete i l l u s t r a t i o n .
The preceding i s i l l u s t r a t e d by a r e l a -
t i v e l y simple example o f a proof from Bishop out s t e p s a r e a l r e a d y i m p l i c i t l y involved.
1967, b u t i n
which s e v e r a l s p e l l i n g -
T h i s i s f o r t h e theorem t h a t
continuous f u n c t i o n on a compact i n t e r v a l has a 1 . u . b .
every
The r e a d e r should compare
t h e f o l l o w i n g with t h e o r i g i n a l as i n d i c a t e d by t h e page r e f e r e n c e s . Definition (p.34). (i) x
5
c-x
with
for all
c
<
Spclled out, (E(E)
6)
in
A
c
i s c a l l e d a 1 . u . b . of xinA
and
5 IR and c 6 I R ) if > 0 there exists x i n A
A (for A
( i i ) f o r each
6
E
( i i ) r e q u i r e s t h a t we p r o v i d e a f u n c t i o n
and
c- g ( ~< )
g
such t h a t V € > O
E).
Nith r e f e r e n c e t o l > . l ( i ) - ( i itih)e r e a d e r should compare Bishop p p . 18,19,p p . 2 6 , 27 an3 p.34, r e s p .
1967
178
S.
FEFERMAN
Theorem ( p . 3 4 ) . Suppose
A
exist
such t h a t f o r each
yl, . . . , y n
numbers
lx-yll,
1.u.b.
in
A
i s a subset of
..., lx-ynI
is
5
such t h a t f o r each
1R x
in
E
>
there
0
a t l e a s t one of t h e
A
(Such a s e t i s c a l l e d t o t a l l y bounded).
E.
Then
exists.
A
S p e l l e d out, t h e d e f i n i t i o n of being t o t a l l y bounded ( c o n t a i n e d i n t h e statement of t h i s theorem) r e q u i r e s t h a t we have two f u n c t i o n s
e > 0, h(E)
each
(so
n = d h ( h ( e ) ) ) and f o r each
x
h
and
(n,(yl, . . . , y n ) )
i s a f i n i t e sequence in
in A
y.
e > 0,
and
A
such t h a t f o r
j
w i t h each
With t h i s understanding, t h e proof of t h e above theorem proceeds a s f o l l o w s . Given any
k
lx-yjI f e j ( x , l/k).
.
2
For, each each
y;
since that
rn
i s in Q
A
in
. .,yn)
such t h a t f o r each i s g i v e n by
h(l/k)
.
1 1 and l y l - y l l < - + (fromp.15). P 9 - P 9
y"' > yL P - P
.
.
.
for
.
=
id,. .,n.
&.
q
Then f o r
2
Take
k,
9 -
1
x
0
x
.
-1.) k
We c l a i m t h a t
i s a 1.u.b.
i (y )
PPL1'
i.e.
and f i n d
m
for
q >p
and t h e n
for
(5) is a (%)
Cauchy
converges
A.
.
. , y n as above; t h e n x-xk = ( x - y . ) + ( y . - y ) J J m(k) Hence (x-x ) = l i m (x-x ) < l i m ( 2 / k ) - 0 ; t h u s x f x f o r k k + x k+@ The f u n c t i o n g r e q u i r e d by t h e d e f i n i t i o n of 1 . u . b . i s provided x in
A,
choose yl,.
=%
g( E)
where k i s chosen so t h a t
from t h e i n f o r m a t i o n which p r e s e n t s Corollary (p.35). a
.
when j = j ( x , l / k ) .
a l l x in A . by
some
j by
r e q u i r e d f o r (1). Note t h a t rn i n ( 1 ) i s found as a f u n c t i o n
(2)
For, given any
=
p=4k
say r n = m ( k ) .
and we can f i n d i t s l i m i t
< 2 k
A,
in
p
ym > (yi
It f o l l o w s t h a t
y.
It i s e a s i l y shown t h a t Let x - y k - m(k) ' sequence of r e a l s . Using a r e s u l t from p.27, it i s shown t h a t
of
x
an3 of
I< - rn -< n we have y m -> rnax(yl, . . . , y ,) - k - l
with
- yllP -< P 1 y > y. - - as m1 k ly:
(yl,.
i s g i v e n a s a Cauchy sequence of r a t i o n a l s
y.
such t h a t
yl, . . . , y n
Next it i s shown t h a t
f o r some
(1)
1, choose
The choice of
If
f : [a,b]+B
E
k
<
E
,
such k being c a l c u l a b l e
as a member o f IR.
i s continuous t h e n
[ f ( x ) l x c [ a , b ] ] has
1.u.b.
Proof. f any
E
>
i s i m p l i c i t l y provided with a modulus-of-continuity f u n c t i o n 0 choose a - a
<
al
< .. . < a
=
b
such t h a t (aicl - a i )
< w ( E)
0 .
Given
f o r each
179
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES i=O,
..., n-1.
Let
a
5
x
5
b.
It i s claimed t h a t we can f i n d j such t h a t
To do t h i s we c o n s i d e r x = ( x ) and a. = ( a 1 ) each p r e s e n t e d as Ix-a.1 < w ( E ) . J P 1 P a Cauchy sequence of r a t i o n a l s . Every x can be compared w i t h each of P i a; ap ,..., a;. F o r p s u f f i c i e n t l y l a r g e (determined by W ( E ) ) , t h e r e q u i r e d
,...,
j
i s found from t h i s comparison.
T h i s shows t h a t A = ( f ( x ) l x
E
[a,b]) i s totally
boun3ed ( a s r e q u i r e d by t h e o f f i c i a l d e f i n i t i o n ) , and we can apply t h e preceding theorem. The r e a d e r may wish t o r e c o n s t i t u t e one o r two o t h e r p r o o f s from Bishop
1967 i n
t h e same manner. (Another i n s t r u c t i v e example i s provided by t h e proof of
t h e B a i r e c a t e g o r y theorem, p . 8 7 ) .
*
1 5 . 5 A t h e o r e t i c a l s e t t i n g f o r 15.3. I n P a r t IV we s h a l l p r e s e n t a t h e o r y extending relation
To
x e W A an3 i n which
t o t h e l o g i c a l p a t t e r n of 15.2, ' u n o f f i c i a l l y ' 3efined
TZ
TO
i n which t h e b a s i c membership r e l a t i o n i s r e f i n e d t o a 3-placed
A
To
can be reduced t o
x
E
A
i s d e f i n e d as
Xw(x c W A ) . With r e f e r e n c e
15.3 one has A*= [ ( x , W ) l x E
~ A ) ,so t h a t t h e
a c t u a l l y aetermines t h e ' o f f i c i a l l y ' d e f i n e d
A*
.
so t h a t i n t h i s t h e o r y we can have our ( c o n s t r u c t i v e )
cake and e a t it t o o .
15.6 Remark on w i t n e s s i n g d a t a i n c l a s s i c a l mathematics.
The p r a c t i c e of
suppressing o f f i c i a l p a r t s of t h e d e f i n i n g d a t a i s a l s o f r e q u e n t i n c l a s s i c a l mathematics, e . g . a l g e b r a i c o r t o p o l o g i c a l s t r u c t u r e s a r e simply r e f e r r e d t o by However t h e p r a c t i c e i s more wholesale i n BCM.
t h e i r un3erlying s e t s .
To.
11. The t h e o r y
A s p r e s e n t e 3 h e r e t h i s t h e o r y i s a minor m o d i f i c a t i o n of t h a t i n t r o d u c e d i n Feferman
1975; t h e
d i f f e r e n c e s a r e explained below.
For t h e r e a d e r ' s con-
venience a goo3 d e a l of t h e m a t e r i a l from s e c s . 2-3 l o c . c i t . i s i n c o r p o r a t e d i n t h e following.
There a r e a l s o some novel p o i n t s .
1. The language of
To ; s y n t a c t i c a l n o t i o n s .
1.1 V a r i a b l e s an3 c o n s t a n t s . The language
S(T0)
i s two-sorte3.
In3ividual ( o r general) variables
a,b,c, . . . , x ,
Class v a r i a b l e s
A,B,C
Individual constants
k, s,p,p1,p2, d,o, sw , p m cn, i, j
w
Class constants We use
,
t,tl,t2, ... t o range over v a r i a b l e s o r c o n s t a n t s of e i t h e r s o r t .
1 . 2 Atomic formulas a r e a l l t h o s e of t h e form tlEt2.
YIZ
,..., X,Y,Z
Is a d d i t i o n t h e r e i s a n atomic formula
interpreted as f a l s i t y .
1.3 Logical operations:
A , V,
+
,
V, B
t l = t 2 , App(t1, t2, t 3) i
and
w i t h no f r e e v a r i a b l e s ,
S. FEFERMAN
180
1 . 4 Formulas a r e g e n e r a t e d from atomic formulas by applying
A , V,
+,
,a, vx .
3x, vx
0,$, 8 range over A ) 0, ( 3 ~A ) Q
Notation:
B! x 0,
(VX E
E
a r b i t r a r y formulas.
(7
@(z, X ) is z , ? . I n such
0) = def
( @ + 1)
5,g, 4
a r e d e f i n e 3 as u s u a l .
. (0 +, t ) ,
r e p r e s e n t sequences
of v a r i a b l e s o r t e r m s ,
w r i t t e n f o r a formula a l l of whose f r e e
v a r i a b l e s a r e among
expressions a s
that 2.
i s not i n t h e l i s t 5 and s i m i l a r l y f o r
y
Informal i n t e r p r e t a t i o n of t h e language.
t h e f u l l u n i v e r s e of d i s c o u r s e of
Yy$(z,y,&)
i t i s assumed
3Y$(z,Ij,Y).
The i n d i v i d u a l v a r i a b l e s range over
hence a r e a l s o c a l l e d g e n e r a l v a r i a b l e s .
To, These a r e t o be thought of as mental o b j e c t s ( i n c l u d i n g
as symbolic r e p r e s e n t a t i o n s of such o b j e c t s .
rules
an3 p r o p e r t i e s ) o r
Then = i s i n t e r p r e t e d as i n t e n s i o n a l
i d e n t i t y o r , i n t h e l a t t e r view as l i t e r a l i d e n t i t y of s y n t a c t i c o b j e c t s .
i s understood t o ho13 when
relation
App(tl,t2,tj)
which has
value tS at
12
t2. S i n c e it i s n o t assumed t h a t every tlt2 = t3 f o r App(t , t , t ) . We w r i t e
1 2 3 ( T h i s n o t a t i o n i s expanded below.)
,z).
2
class v a r i a b l e s
The
as: t
Bz App ( t , t 1
range over one-placed p r o p e r t i e s . t
has t h e p r o p e r t y ( g i v e n b y ) X .
t
vidual
part
X i s interpreted
E
(We may a l s o t h i n k of ' c l a s s ' as s h o r t f o r
' c l a s s i f i c a t i o n ' ; c l a s s e s a r e n o t conceived e x t e n s i o n a l l y . ) only range over a
The
i s a ( r u l e o r ) operation
t h e argument
o p e r a t i o n i s t o t a l we s h a l l w r i t e
t t 1 for
tl
of t h e u n i v e r s e of d i s c o u r s e .
Class v a r i a b l e s
To express t h a t an i n d i -
happens t o be a c l a s s we simply u s e t h e formula C d ( t ) =def Y x ( t = X ) .
The c o n s t a n t s
a r e c e r t a i n b a s i c combinatory o p e r a t i o n s which p e r m i t
k,s
one t o form t h e c o n s t a n t o p e r a t i o n s and c a r r y o u t t h e p r o c e s s o f s u b s t i t u t i o n , r e s p . ; p,p1,p2
a r e o p e r a t i o n s of p a i r i n g an3 p r o j e c t i o n .
m,
Ti
denotes t h e c l a s s of n a t u r a l
which g i v e s d e f i n i t i o n - b y - c a s e s
on
numbers, with l e a s t element
an3 o p e r a t o r s of s u c c e s s o r
pm
.
The c o n s t a n t s
cn
,
0
i, j
where
i s a n operation
d
r e p r e s e n t c e r t a i n *-formation
sm
and predecessor operations,
corresponding r e s p e c t i v e l y t o comprehension, i n d u c t i v e g e n e r a t i o n and
3.
A p p l i c a t i o n terms. The language
operator
a ( - , -)
S(To)
i s f o r m a l l y extended by a b i n a r y
which i s i n t e r p r e t e d a s t h e o p e r a t i o n of a p p l i c a t i o n .
... t o range over t h e terms of a p p l i c a t i o n terms ( a . t . ' s ) . We w r i t e
T,T~,T~,
We u s e
t h i s extended language, which a r e c a l l e d T
~
Tf o r~
a r e generated from t h e v a r i a b l e s an3 c o n s t a n t s of Since
join.
C U ( T ~ T ~) .
S(To)
Thus t h e
a.t.'s
by c l o s u r e under
a.
a . t . ' s may not have d e f i n e d v a l u e s , r e l a t i o n s between t h e s e a r e explained
as formulas i n
S(To)
i n t h e following way:
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
[
(Tl
(7
1
( t = x ) when
: T2)
=3ef
)
=def 3 X
We w r i t e
&(T
1 72 ) . . .
)
( T ~ , ( T ~ ,
any
n
T2
=
t
L(To)
of
XI
.
X)
(
T
T
.
~
is
X )
... T~
~
=
~ T
~
) when
1. F i n a l l y ,
zx
&(T
i s known o r assumed.
( ~ ~ A1 ( T) 1) 2
A x E
X).
a r e supposed t o be a s s o c i a t e d t o t h e l e f t a s
We w r i t e
..., T ~ ) ) .
for
4.
T
T
C,
i s a term
x A Nx)).
(T E
Parentheses i n
2
=
(T
=
In particular,
(...(T
Z X
( T ~ = T ~ f) o r
=def
K T )
VX [T1
T
181
for
(7)
( T ~ , T ~ f) o r
T,
,...,
This explains t h e notation T'
=df sw
T
p~ T 1 2
Z ( T ~
.
T
and
) or
..., 'n)
( T ~ , T ~ ,
z ( ~ )f o r
Further syntactical notions.
4 . 1 S t r a t i f i e d formulas a r e t h o s e ( i n S(T0)) which c o n t a i n c l a s s v a r i a b l e s or constants
on t h e r i g h t - h a n d s i d e of
c o n t e x t s of t h e form
t E X or
t
E
N
atomic formulas, i . e . only i n
E
, where
t
i s an i n d i v i d u a l v a r i a b l e o r
Thus a l l t h e o t h e r atomic formulas of a s t r a t i f i e d formula a r e r e -
constant.
Formally, s t r a t i -
l a t i o n s of e q u a l i t y an3 a p p l i c a t i o n s between i n d i v i d u a l terms.
f i e d formulas may be thought of a s 2nd o r d e r formulas w i t h t h e s o r t of i n d i v i d u a l s specifying t h e 1st order l e v e l .
4.2
Elementary formulas a r e t h o s e s t r a t i f i e d formulas which c o n t a i n "0
bound c l a s s v a r i a b l e s . --
These a r e a l s o sometimes c a l l e d p r e d i c a t i v e formulas, I n an elementary formula
t h e o t h e r s being impredicative.
@ ( s5) ,
classes are
not r e f e r r e d t o i n any g e n e r a l way; we only r e q u i r e understanding membership i n t h e given
X.
.
Elementary formulas may a l s o be considered as t h e 1 s t o r d e r
( s t r a t i f i e d ) formulas.
4.3 Comprehension n o t a t i o n . Let n be t h e Gbdel number '@(x, @ with a s p e c i f i e 3 i n c l u s i v e l i s t of v a r i a b l e s x, y ,2 . We p u t [XI N.,y
f
2 )1
=def
Cn(Y,
y,
z7
of
2).
T h i s shows t h e p r o c e s s of c l a s s formation by comprehension a s a uniform f u n c t i o n of t h e parameters
y ,2
o f t h e d e f i n i n g formula
hension a r e not a l l e q u a l l y e v i d e n t ; t h e mc elementary
0, an3 only
1
@
.
The i n s t a n c e s of compre-
e v i d e n t a r e t h o s e corresponding t o
t h o s e a r e immediately accepted i n
To
.
S. FEFEFiMAN
182
5.
Axioms and l o g i c of
To T
5 . 1 The l o g i c of
i s t h a t of t h e i n t u i t i o n i s t i c two-sorted p r e d i c a t e
There i s i n a 3 3 i t i o n a
c a l c u l u s with e q u a l i t y . t h e s o r t s , namely
.
basic
o n t o l o g i c a l axiom r e l a t i n g
Note t h a t t h i s j u s t i f i e s t h e formalism i n 4.3
PXX(X=x).
above where one a p p l i e s o p e r a t i o n s t o c l a s s e s .
5.2
Non-logical axioms.
These f a l l i n t o s e v e r a l groups.
A P P(App1icative axioms) x y
(unicity) (constants)
Z A X
^.
y =
= w
W+Z
kxy L A kxy = x
( s u b s t i t u t i o n ) sxy I A s x y z z x z ( y z ) ( P a i r i n g , p r o j e c t i o n s ) pxyl A p z l A p z l A p ( p x y ) = x A p,(pxy)
2)
( d e f i n i t i o n by c a s e s on
(s successor, ,
The remaining axioms a r e cn(yl,. . . , y m ,
z1,.. .,zP)
1
2
=
1
y
x , y ~W + (x-y 43xyab = a ) A ( x f y +dxyab = b ) .
preaecessor) x , y
E N ~
X
class e x i s t e n c e axioms. 2 ,2 .
'
I
Ay ~l ~ p( X ' ) = X A X ' { O
M
A
( x ' = y ' +x=y).
Note from I ( i i i ) t h a t
I for a l l
E C A (FLementary comprehension).
[ rxl$(x,&
For each elementary
2)I = x A
w x
E
x
$ ( x , y , ~ ) we t a k e :
2 111
@(.,y,
c,
IN (Natural numbers) ( i )( c l o s u r e )
0 E I N A A x ( x E N- , x ' c N )
Q(o) A VX(@(X) = $ ( x , . . . ) i s an a r b i t r a r y
(ii) ( i n s u c t i o n ) where
J
@(x)
+ vx
--f $ ( X I ) )
E
formula of
N ~ ( x )
S(To).
(Join) Vx cABY(fx z Y ) + XX[j(A,f) = X A V z [ z E X <--f ExZy(z = ( x , y ) A x E A A E~ ~ x ) ] .)
I G (Inductive generation) ( i )( c l o s u r e ) E I I ( i ( A , R ) = I A V Y ( ( Y , X ) E R
11
iyc1) + x a I
(i i) ( i n d u c t i o n ) Vx ~ A ( A y ( y , x )E R + b ( y ) ) + @ ( x ) )+ (Vx c i ( A , R ) ) $ ( x ) @ ( x ) = @ ( x. ., . )
where
of axioms:
6. (a)
R e l a t i o n of
.
This completes t h e l i s t
as g i v e n h e r e w i t h t h a t o f Feferman
T A,B,C,
...,X , Y , Z
We d e f i n e 3 x ' = (x,O) and
axiom (c)
T
1975.
P r e v i o u s l y we took a one-sorted language with an a d d i t i o n a l p r e d i c a t e
the variables (b)
i s a n a r b i t r a r y formula of
To =APP+ECA+B + J + I G .
were i n t r o d u c e d by convention t o range over
p
=
pl
an3 took t h e axiom
(x,y)
#
A
=
W was d e f i n e d a s i ( A , R ) [ X / X = Ov x z ( p x)')
R = [Zl?k,y
B
(Z=
for
and
(y,~A ) x=y'))
CJ?
.
0 ; then t h e
AFP( v ) a s g i v e n h e r e w a s d e r i v e d .
The c o n s t a n t
CJ?(x);
(predecessor r e l a t i o n ) .
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
183
The reason f o r l i s t i n g W a s a s p e c i a l axiom here i s so a s t o consider t h e s t r e n g t h of
To-IG
while s t i l l including W.
Previously we used d e f i n i t i o n by cases on t h e universe i n s t e a d of j u s t on
(3)
W , a s w i l l be explained next.
7.
Some v a r i a n t s of t h e axioms which w i l l be considered. This i s t h e following i n place of
( D e f i n i t i o n by cases on t h e u n i v e r s e ) .
Dv
mp(v): ( x = y V x f y ) A LxyabIA ( x = y - t d x y a b = a ) A ( x f y + m a b = b ) .
Then
To+DV
Note.
i s equivalent t o t h e system
The clause (x,y C W + x = y V x f y )
of Feferman 1 9 5 .
To
was not needed i n
APP(v)
since it i s
derivable from t h e other axioms. CA
( F i r s t order comprehension).
CA
(Second order comprehension). The same scheme a s f o r
1
2
This i s another denotation of
---
ECA.
except t h a t
CAl
4
may now be any s t r a t i f i e d formula i n
[xl@(x,r,z)). CA1 r e s t r i c t e d t o formulas of t h e form
SEP
( F i r s t order s e p a r a t i o n ) . This i s
SEP
[xlx E A A Ji(x,y,z)l. (Second order s e p a r a t i o n ) . The same a s
WP
( R e s t r i c t e d induction 0"
1 -x c A A Ji(x,y,z) (with parameters
2
--
y,Z,A).
( ~ C A JiI (x,&Z)]
SEPl
i s written for
w i t h any s t r a t i f i e d
$.
N ) . Here one replaces t h e induction schema N ( i i )
by t h e s p e c i f i c instance OEXAYX(XEX+X'EX)+W_CX
(where ( X _C Y ) =def Vx(x E X + x
E
Note t h a t
Y )).
WP can be used t o derive
any instance + VX Q(x) i s known t o e x i s t a s a c l a s s .
4x0) A W N X ) + N X ' ) )
for which
(xI$(x,. .. ) )
I G r ( R e s t r i c t e d induction
for
Analogously replaces
IG).
I G ( i i ) by
V X C A ( B ~ ( ( Y , X ) E R + ~ E X +) x s X ] + i ( A , R ) C X . Remark.
We could formulate a g e n e r d i z e d inductive d e f i n i t i o n
pressing f o r any elementary
$ ( x , X ) ( =Ji(x,X, . . . ) )
i n which
X
axiom
GID
ex-
(lccurs only
p o s i t i v e l y , t h e existence of a c l a s s I which i s t h e l e a s t p r e d i c a t e s a t i s f y i n g
Vx[ $(x,X) + x EX]. This stronger axiom i s not evidently constructive (GIDP can be derived i f one accepts t h e impredicative comprehension p r i n c i p l e Note. Then
I G i t s e l f becomes s t i l l more evident i f one w r i t e s
i(A,R)
the
XR
3 -accessible
i s t h e c l a s s of
p i c t u r e d a s t h e elements of
A
elements of
A,
CA2).
for ( y , x ) c R .
which may be
s i t t i n g atop well-founded t r e e s branching by
relation.
MIGr ( R e s t r i c t e d monotone inductive d e f i n i t i o n ) VX3Y[ fX 1 Y 1 A VXl,X2[X1
y
5 X2
+ fX1
R I ( f I 5 I A VX[fX
5X
fX21 + +I
5 XI]
I84
S. FEFERMAN
By adjunction of a suitable constant, we could also express I uniformly a s a function of f .
Here
f
represents a monotone operation on classes t o classes.
G I D r ( i n t h e presence of
T h i s i s stronger than
ECA),
since p o s i t i v i t y i s
weaker than monotonicity.
8. P
Proauct and power c l a s s axioms. (Product axiom) Vxc A3Y(fx=.Y) +BXVz[zsX -Vxo
A(zxcfx)
1.
We s h a l l prove (11.2) t h a t P follows from To ; however, it does r o t i f ' C A l i s replaced by SEQ . POW' (Strong power c l a s s axiom). VA'ABVx[x s B c, 3 X ( X 5 AAx = X)]. POW (Weak power c l a s s axiom). VABBVX[XeB + Z X ( X z where X
3
A
A X=X) A
U(X
5A
- 1 3 Y (X'YAY
As remarked previously, t h e constructive s t a t u s of Note. -
sB))
1
i s t h e r e l a t i o n of extensional equality between classes,
Y
POW'
( o r even
POW) i s unclear.
Each of these can be expressed uniformly by adjunction of suitable con-
stants.
9.
Theories r e l a t e d t o
9.1 EMo
m0i
Remark.
T
which are t o be considered here.
= APP + ECA +
= APP + ECA
.
+ mr
The notation 'EM' comes from 'Explicit mathematics'
constructive theory i n t h e present framework.
EM
r
i s a minimal
Dv ,J, I G r an3 I G t o these theories.
adding,, variously,
9.2
.
We s h a l l consider t h e e f f e c t of
A theory of s e t s
so * So = APP + SEP
1
+
IN + J + P + I G .
The c l a s s variables here can be interpreted a s s e t s ("small" c l a s s e s ) . (The i s a3ded here according t o t h e remark made i n § 8 . )
product axiom P
9.3 A theory of s e t s and classes T o ( S ) . by adjoining a constant
This may be obtained from
A theory of t h i s character was
conditions are expressed by additional axioms. presented i n Feferman taken t o be p a i r s
10.
1978, but
(A,E)
Consequences of
EM$
considered semi-formally. Notation.
where
of greater generality, since s e t s t h e r e a r e E
i s an equality r e l a t i o n on
. Throughout
We s h a l l often use l e t t e r s l i k e ' f '
10.1
T*
mop .
, 'g' , 'h'
f o r individual variables
The l e t t e r s ' k ' , ' n ' , ' m ' , ' p '
a r e now re-
IN.
Abstraction.
f i n d a term
A.
t h e following a l l statements a r e t o be
They a r e a l l provable i n
being t r e a t e d i n operator situations. served t o range over
TO
S f o r t h e c l a s s of all sets, f o r which suitable closure
For each application term
w i t h variables
5
var
(7)-
(x)
T
and variable
such t h a t
x
we can
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
AX(T) o r
a r u l e , whether o r not
on
(i) If
is
T
(Intuitively,
T [ x ] ~ a t any
x, we take skk f o r
x.)
T
(iii) If
T=T1~2
x
x
T
,
we want
so we can take
1 2 applied h e r e ) . 10.2
Ax(x)
i s an i n d i v i d u a l term
(ii) I f
ST
7
hx(T[x]).
because it always denotes
T*k
The proof i s c a r r i e d out by induction
(just as i n t o t a l combinatory t h e o r i e s ) :
t
*
= I.
A VX [ T*X
T*$
i s denoted
T*
185
* T
=
rf
ST
* * 1
T
i n t h i s case.
2
We can f i n d a f i x e d
for
hx(t)
hyl
serves as rf.
g=hh
r
*
since k t x = t .
*
(T1x)(T2x)
2
(Only A P P ( i ) , ( i i ) a r e
such t h a t f o r a l l
f:
g = r f , Vx(gx =. f g x ) .
and f o r
h = hy hxf(yy)x, so
h h = A x ( f ( h h ) x ) . Thus
skkx=. kx(kx) = x. kt
~ * x ~ ~ 1 [ x ~ ~ 2But [ x ] .T1[x]T2[x]
The recursion theorem.
For t h e proof t a k e
since
t fx, we t a k e
f o r a l l y.
I n particular
hhl
an3
We can t a k e
r =h f ( ( h y ~ ( y y ) X ) ( ~ h x f ( y y ) X ) ) . 10.3 Recursion on W. For any
a,f
1-a
Namely,
we can f i n d
if
g
satisfying:
x = 0
i s found by t h e recursion theorem s o as t o make
g
gx
= SC o a ( f ( x , g ( p m X ) ) ) .
It follows t h a t go = a
(i)
(ii) gn'
1 f(n,gn)
Then f o r any
a,A,f
f o r any
n.
we have:
a € A A VnVy[y E A + f ( n , y )
(iii)
E
A
1
-+Vn(gneA),
with t h e conclusion being proved from t h e hypothesis by r e s t r i c t e d induction, since
( x / g xE A )
i s a class.
10.4 Arithmetic i n EMop
.
It follows from t h e preceding t h a t a l l primitive
recursive functions can be defined i n
EMoP. Furthermore, every a r i t h m e t i c a l
formula i s equivalent i n t h i s theory t o an elementary formula, hence defines a class.
Thus t h e scheme of induction f o r a r i t h m e t i c a l formulas holds i n
EMo' 10.5
n
m l n
-
Bounded and unbounded minima.
fm
sothat
EM$.
of (Heyting's) a r i t h m e t i c i s contained i n
Hence t h e i n t u i o n i s t i c system HA
Using recursion on
E J
we can define
~m~n(fmoN c,) n f m a ~and n f m = o ~ & <-n ( f m I o ) . m z n m l n
Then f u r t h e r we can obtain ( p m
_<
n ) ( f m = 0)
which i s defined un3er t h e same
S . FEFERMAN
I86 conditions.
By t h e r e c u r s i o n theorem one f i n d s (pm
-,
g ( f , n ) =.
Let
n ) ( f m - 0)
if
g(f,n')
'
pf=g(f,O).
i n case
2
g
&r~
5
n(fm
= 0)
otherwise.
I t i s seen t h a t
E n ( f n = O A Vm
such t h a t
i s d e f i n e d and e q u a l t o q ( f n
pf
.
< n(fm E N ) )
10.6 P a r t i a l r e c u r s i v e f u n c t i o n s ; forms of Church's Thesis. of p a r t i a l r e c u r s i v e f u n c t i o n s
EMor
[kl(n)
with
(k)
for
z U(pnTl(k,n,rn)).
Just
0)
I
The enumeration
k EW can now be d e f i n e d as u s u a l i n
We now i n t r o d u c e t h e function-mapping no-
tation: (f:A+B)
=def V x c A ( f x ~ B ) .
The t h r e e forms of Church's t h e s i s d e s c r i b e d i n
1.4.8 can
be formulated i n
EM p
as f o l l o w s . CTo
i s t h e scheme
CT1
is
Vf[(f:W +IN) + % V n ( f n = ( k ] ( n ) ) ]
CT2
is
V f [ V n ( f n l 4 f n ~ I N )+ % V n ( f n =
(CT2
was suggested by Beeson; i t e x p r e s s e s t h a t every p a r t i a l f u n c t i o n on
Vn%'$(n,m) + % v n [ ( k ] ( n ) I A ' $ ( n , ( k ] ( n ) ) ] (k](n))].
c o i n c i d e s with a p a r t i a l r e c u r s i v e f h n c t i o n t h e r e . )
as we remarked i n 4.8, CT + A C w + C T o 1 be t a k e n up i n P a r t s I11 and I V .
10.7
.
Obviously
The c o n s i s t e n c y of t h e s e w i t h
Elementary o p e r a t i o n s on c l a s s e s .
We now t u r n t o u s e s o f
IN
, an3
CT2 +CT1
To
ECA.
will
The
o p e r a t i o n s having c l a s s v a l u e s as f u n c t i o n s of t h e i n d i v i d u a l
following g i v e
and c l a s s parameters shown.
v
=
(xlx=xJ
A
=
(~111
(a,b] = [x/x=aVx=b]
-A = (Xlx
4 A]
A U B = ( X ~ X E A V X E B ]A, n B = [ x l x ~ A A x c B ] A
X
B = ( z l ? k c A S y ~ BZ = ( X , y ) l
BA
=
Df
=
f[A]
( f If:A
+ B] ( a l s o denoted
A + B)
(xlfx I) =
[ y l ' h EA(fx z Y)).
Evidently a l l t h e s e have t h e form
[xl'$(x,.
..)I
with
@ elementary.
10.8 The f i n i t e t y p e h i e r a r c h y and H A w , The f i n i t e t y p e symbols ( f . t . s . ) a r e g e n e r a t e d by t h e following elementary i n d u c t i v e d e f i n i t i o n : 0 and if
p , ~a r e f . t . s . t h e n so a l s o a r e
ukv = ( l , u , v ) and (u G v ) = ( 2 , u , v ) .
pku
and
p
L o , where
i s a f.t.S.
0 = (O,O),
The f . t . s . a r e enumerated by a f u n c t i o n on
CONSTRUCTIVE THEORIES OF FUNCTIONS AND C L A S S E S R
whose range t h u s forms a c l a s s t h a t we denote by FTS.
we can d e f i n e a f u n c t i o n gb =IN where
Nu
,
g
g(&)
gu.
N.
=
Using r e c u r s i o n on
IN
satisfying: = g u xgv
and
g ( u 4 v ) = (gu + 0 )
10.7.
x, + a r e t h e o p e r a t i o n s d e f i n e d i n
t h e value
187
u
For each
E
FTS we denote by
Thus
I N , Npku
For each p a r t i c u l a r
= Np X N
i n FTS
u
and
Np ju
=(Np + N )
we can prove i n EM
r
.
N NP
=
u
u
that
However, t h e statement
Vu
FTS [ CE ( N u )
6
r e q u i r e s a proof by i n d u c t i o n u s i n g t h e i m p r e d i c a t i v e p r o p e r t y
EMo b u t not
can be c a r r i e d o u t i n
.
% ( g u = X ) . This
mop
GBdel's n o t i o n of p r i m i t i v e r e c u r s i v e f u n c t i o n a l of f i n i t e can be i n t e r p r e t e d i n t o p a s s from
fo
EMo:
simply by u s i n g r e c u r s i o n on
N P I, u
f l"N(pk0ku I, u) g ( x , o ) =. f o ( x )
and
which i s o b t a i n e d u s i n g 10.3 uniformly i n d u c t i o n on
IN
x EN + g(x,n) €Nu P
that
The t h e o r y
HA"
v a r i a b l e s of each t y p e
functional^.^ )
g(x,n')
x.
,
to a
R
g
=
.
tm
(GBdel 1958)
The b a s i c scheme i s
satisfying fl(x,n,g(x,n)),
Now we can prove by r e s t r i c t e d i n -
hence
g cNpkO
>
as r e q u i r e d .
of i n t u i t i o n i s t i c a r i t h m e t i c i n a l l f i n i t e t y p e s has
u
E
FTS
and c o n s t a n t s f o r t h e p r i m i t i v e r e c u r s i v e
The axioms a r e t h o s e of
HA
t o g e t h e r w i t h t h e d e f i n i n g schemata
f o r a l l t h e s e f u n c t i o n a l s and, f i n a l l y , t h e i n d u c t i o n scheme f o r a l l formulae of t h e language. formula of stants
N
I n t e r p r e t i n g t h e v a r i a b l e s of t y p e
ul,...,
N
am
.
Hence i t d e f i n e s a c l a s s un3er
HA" _C EMo! Remark.
(5
t o range over
Nu
, each
i s e q u i v a l e n t t o an elementary formula with f i n i t e l y many con-
HA"
An i n t e n s i o n a l form of
HAm
ECA.
It foli.ows t h a t
.
, denoted
I-HA"
i s o b t a i n e d by a d j o i n i n g a
f u n c t i o n a l a t each t y p e l e v e l which d e c i d e s e q u a l i t y between o b j e c t s of t h a t type.
It i s e a s i l y seen t h a t
10.9 The e x t e n s i o n a l f i n i t e t y p e h i e r a r c h y . The system E - 3 " obtained HA" by a d j o i n i n g e x t e n s i o n a l i t y axioms i n a l l f i n i t e t y p e s can a l s o be i n t e r p r e t e d i n EM_* However, h e r e we i n t e r p r e t t h e v a r i a b l e s of t y p e u t o ' 7 ) C f . T r o e l s t r a 1973,Part 1 8 6 f o r a p r e c i s e d e s c r i p t i o n of HA" and t h e systems from
- .
I-HAw, E-HAW below.
S . FEFERMAN
f 88
range over
M
defined together with an e q u a l i t y r e l a t i o n
U'
follows :
M. = I N
Mpku
n = m
and
= Mp X
and
Mu
ct
by induction as
=
n=m
x =p;ca y tt plx = p y~ p2x =u p2y P
1
Mp L u = ( f \ f c M p - + M u A V x , y E M ( x = ~ + f x = ~ f y )and ] P
f
=(p A u)g
&
VX
E
M ( f x = gx) P
O
Equality between o b j e c t s of type
0
E-HALO _c EM^^ . 10.10
E
5A XA
P
.
i s interpreted as
Classes w i t h e q u a l i t y r e l a t i o n s .
and
E
These a r e simply p a i r s (A,E) where
i s an equivalence r e l a t i o n on
Then we w r i t e
A.
E (though t h i s n o t a t i o n i s ambiguous, since
(x,y)
E
with
A).
so a s t o obtain
=
E
While we can operate on classes-with-equality i n
we proceed more generally than i n Bishop
1967 and
1 0 . 1 1 Integers, r a t i o n a l s and r e a l s i n
x= y A
for
i s not uniquely associated
T
( o r i t s subtheories)
work w i t h c l a s s e s p e r se.
EMor
.
O u r d e f i n i t i o n s here
I, 14.5 ( i . e . e s s e n t i a l l y Bishop 1967 Ch.2)
follow
IN ; (n,m)
8 = INX
+ i s defined on on 1 . N
= (p,q)
2
2
c, n + q = m + p .
by ( n , m ) + z ( p , q ) = ( n + p , m + q ) , and so on f o r
Q = ( ( X , ~ ) ~ X E Z A Y E ~ fAoY]
; (X,y)=
t-rx.v=y*U
e (U,V)
. ,<
(x,y) +Q(u,v) = (xv+yu, u v ) , and so on f o r
i s embedaed i n
2
. ,<
i s embedded i n 2 , and s u b s c r i p t s a r e dropped.
a;+ = ( n i n e Z A n >
Q
, an3
01.
on
4.
s u b s c r i p t s a r e dropped.
We w r i t e
x
for
xn
when
x
Ez+
+A
IR = j x j x E ~ ++ Q A vn,mc z + ( / x -x < -1+ - )1I n m - n m x < -1) . e V k 2~' 3 m e Z'Vn > m ( n - k
1
Ix~-Y I
x
,
= A U ( X ~ ~ + Y ~ ~ )
+
x
=
kx
AU(X 2 k u . yZku) where
k=max(k , k ) X
i s t h e l e a s t i n t e g e r g r e a t e r than
IR+
= ((x,n)/xeIRAxn>
Q
i s embedded i n
1
Y
and f o r each
Ixll
x,
+ 2.
and I R 0 + = ( ( x \ x c I R A V n e Z + ( x n L -
n1 ) I .
IR, and t h e s u b s c r i p t s a r e dropped.
The elementary p r o p e r t i e s of t h e s e number systems can be developed i n d i r e c t l y following Bishop
1967.
Then t h e complex numbers
a s u s u a l and t h e i r p r o p e r t i e s derived i n t h e same way.
C
Mor
can be introduced
I89
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES Continuous functions and c l a s s i c a l a n a l y s i s .
10.12
A, B
with e q u a l i t y r e l a t i o n s =A , =B
Given any two c l a s s e s
respectively, t n e function c l a s s
F(A,B)
i s defined by F ( A , B ) = ( ~: A/ ~- + B A v x , y ~ ~ ( x = ~ y ~ f ( x ) = ~ f ( y ) ) ) .
I n p a r t i c u l a r i n a n a l y s i s one i s i n t e r e s t e d i n a,b E IR and { a , b ] = ( x SIR la _< x 5 b ) . N e x t f o r t h e (uniformly) continuous f u n c t i o n s on [ a , b ] , following 1.15.1, one takes This i s a subclass of (A-+B).
F(IR, IR) and F ( [ a , b ] , IR) where
C([a,b],lR) = ( ( f , w ) l f c F ( [ a , b I , IR) A w oF(IR+, IR+) Ve OIR+VX,Y
A
~ [ a , b l ( I x - y l5 w(e) + I f ( x )
- f(y)l <
e)
1,
and one t a k e s C(IR, ?R) = ( ( f , u ) ( f c F ( I R , B ) A Pa, b c I R ( a < b
( f , u ( a , b ) ) oC(fa,bI,
-t
i . e . t h e f u n c t i o n s continuous on each compact i n t e r v a l [ a , b ] .
mop
as b a s i s , c l a s s i c a l r e a l a n a l y s i s i s pursued i n Ch.2.
IR)]
S t a r t i n g with t h i s
j u s t as i n Bishop 1967,
The only p o i n t which r e q u i r e s c a r e f u l checking i s t h a t only r e s t r i c t e d
i s applied throughout.
induction on N
We s h a l l r e t u r n t o t h i s observation i n
$14 below. 11. Consequences of t h e j o i n axiom.
within
Here, u n l e s s otherwise specified, we work
EMor + J.
11.1 Families and j o i n s . By a family of c l a s s e s (B,), operation
f
such t h a t
Vx
E
j o i n axiom guarantees t h e existence of a c l a s s PertY Z C
>Bx
i s meant an
and where we w r i t e Bx f o r f x .
A m ( f x IY)
5 B XEA
The
with t h e defining pro-
< ~ Z k i Y A ' & [ Z = ( X , y )A y a B x l .
x cA
From t h i s we can define t h e union operation on classes-with-equality as explained By a pre-joined family on
by Bishop (1.14.10)above. B _C A x V . (yI ( x , y ) c B
A
i s meant a c l a s s
Associated with such i s a family i n t h e preceding sense by f x ( = B ) =
1; extensionally t h i s makes
B
B
2
XEA
11.2 Products.
Suppose given a family
f =(
.
B ~ ) ~ Let~ J~ = . XCA
Then we can d e f i n e
TI
xc A since
(x,gx)
E
J
H
B~
.
B x = ( g l V X ~ A ( ( x , g x ) o J 1)
gx E Bx
for
x
E
A.
From t h i s we can t r e a t i n t e r s e c t i o n
of a family of classes-with-equality (1.14.10). Remark.
There i s no evident way t o d e r i v e t h e product axiom
axioms i n t h e theory
So
11.3 The passage t o t r a n s f i n i t e types. that
P
from t h e remaining
of s e t s i n 9.2. Using t h e operations o f 10.7 we know
S. FEFERMAN
190
.
C L ( a ) A CL(b) - , C l ( a x b ) ACL(a+b)
Then by enumerability of FTS i n 10.8 we can c a r r y out a n induction t o prove Vu
FTSICL(Nu)I.
E
we can then form
ueFTS types, at type l e v e l u).
n
and
N
Nu
u sFTS
.
i n t h e theory EMo
However t h i s r e q u i r e s u n r e s t r i c t e d induction on
, which
Using J
i s t h e f i r s t move t o t r a n s f i n i t e
Then by successively applying t h e operations x and +
again we can move up t o l e v e l
u.2, t h e n
...,
w.3,
2
u)
,
etc.
A general p u r s u i t of
t h i s would be based on a theory of o r d i n a l s , which a r e t r e a t e d i n terms of wellfounded t r e e s
( " t r e e o r d i n a l s " ) i n c o n s t r u c t i v e mathematics and based on I G i n
t h e framework of
This i s taken up next.
To.
I n any case we s e e t h a t t h e Precise limits
passage t o lower t r a n s f i n i t e types can be e f f e c t e d i n f o r t h i s are provided by t h e proof theory of 12.
EMo
f
Consequences of t h e i n d u c t i v e generation axiom.
12.1 T r e e o r d i n a l s . are distinct.
Note
Define
supnf
2
= (O,O),
O E Q1' x E @ l + x + E Then
%=i(A,R)
f o r s u i t a b l e A, R .
Here we move t o f u l l
x + = (l,x),
sup f = ( 2 , a , f ) .
T
.
These
i s inductively generated as t h e l e a s t
= (2,a,f).
c l a s s such t h a t
EMo + J . J (Part V).
4,an3
(f:lN+Ej)
-+supmf€ Q 1 .
C l a s s i c a l l y t h e members of
B1
represent
countable o r f i n a l s with
(Note t h a t
x+
suprn hn(x). )
can be 3ropped i n f a v o r of
We can p i c t u r e members
~
of
as well-founded t r e e s :
D e f i n i t i o n by recursion on
+ i s a consequence of the recursion theorem.
B1
Q2 i s
inductively generated as t h e l e a s t c l a s s such t h a t
-0
E
@*,X € 0 2 +x+EQ2, and
( f : N - 8 2) +SUPmf€Q2
( f : 81 -+B2) .+sup f @l
The l a s t i s p i c t u r e d by
branching over
s2,'
sup
/.
.
E @ ~
f
9
,,
;ha''
Then we can o b t a i n analogously t h e existence of a c l a s s
@n f o r each
Using j o i n we can c a r r y t h i s on t o define t r a n s f i n i t e t r e e c l a s s e s : a
E
@l, and more generally f o r
a
E
Ob
w i t h any given b.
on t h i s m a t e r i a l c f . Feferman 1975 pp.99-100,
a l s o Feferman
n
G
N.
@a f o r
(For more d e t a i l s
1978 5 5 .)
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
191
12.2 IG and Bore1 sets. For simplicity we indicate the treatment in Baire
space INW
.
Let
be an enumeration of all finite sequences in W and
sn(n G N )
G n = (g cDJ* Ig(Jh(sn))
=
sn); these are the basic clopen sets. We take
n
and if f : W -63 then fn G da and n EW In more general spaces we can follow Bishop's treatment via comple-
the smallest class such that each GncO
u fn n GW
E
63.
63 to be
mente3 sets. In any case IG
suffices for this.
It would be appropriate at this point to take up the questions of adequacy and accord of T
and its subtheories with
BCM. However we first complete our
discussion of the consequences of various axioms with a look at Dv
and POW
(which are only marginally related to actual BCM). 13. Non-extensionality as a consequence of
DV
*
13.1 Non-extensionality of operations. The following is proved in Feferman
1975 $3.4 as a consequence of APP + DV 1
Bf,g[Bx(fx
:
z gx) + f =gl
The idea of the proof is first (using D ) to associate with each f an f*
*
V
that Df-Df* and f x z 0 whenever x we have f
total (i.e. Vx[fx I])
E
Df.
iff f*=k(O)=O*.
by cases DV we obtain a total operation e
such
Then, if extensionality is assumed Again using full definition
such that Tot (f) ,+ef
=
Ox
.
Diagonalization produces a contradiction.
13.2 Consistency of extensionality of operations in T the statement Vf,g[Bx(fx =. gx)
+
f:g].
.
Denote by M T O p
Using extensional term models for APP
(due to Barendregt) it will be shown in Part I11 that T o + M T O p is consistent. Hence the use of Dv
in 13.1 is essential.
(It will also be shown that
T +DV is consistent.)
13.3 Non-extensionality of classes. Denote by M T C e the statement B A , B It is also proved in Feferman 1975 $3.4 that = B].
[ Bx ( XG A + x E B ) + A
M T C e holds under the assumptions APP+ECA+DV with each f the class cf :(xlfx 1 ) ; c we would have Tot(f)
&if
=
.
The idea is to associate
itself is total. Then if M T C a held
V, from which one can proceed as in 13.1. With
reference to 13.2 we have the following. Question: Is T o + M T C e consistent? We can of course ask similar questions for the addition of EXTCe to sub-theories like EM, , So etc., for all of which the answer is not known.
13.4 Discussion. Dv is a perfectly reasonable axiom if we regard the
entities of our universe as being syntactic objects and
=
as literal identity.
It is less evident if the entities are viewed as mental objects and = is interpreted as intensional identity; however, it appears from writings of Kreisel an3 of Troelstra (cf. Troelstra 1975) that here also Dv
is to be accepted. Then,
S. FEFERMAN
192
f a r from being d i s t u r b i n g , t h e r e s u l t s of 13.1 and
13.3 add support t o t h e b a s i c 1.4.5, 4.11.
non-ext,ensional viewpoint of c o n s t r u c t i v e mathematics as p r e s e n t e d i n
14.
S t a t u s of t h e power-class axioms.
1 4 . 1 I n c o n s i s t e n c y of
POW
with J o i n .
To be more p r e c i s e i t i s shown t h a t
APP+ECA+J provesiPOW, t h e weak power-class axiom of t h a t t h e r e i s a weak power-class and
x
C
of
V , so
Vx[x
A = ( x ~ x ~ C A \ x , x ) , d B Then ).
E
58
above.
Let
B= 5
and
a e a 6 a e A o a c C A ( a , a ) / B ~ j a s C A ( a / a ), a k a ,
X E
Indeed, suppose
C + C e ( x ) ] AVXXY(YeC A X q y ! . f o r some
A'a
a
in. C ,
whichgivesa
contradiction. Consistency of
14.2
POW
It may a l s o be shown t h a t s e t s described i n
EM
with
. This
w i l l be proved i n
i s c o n s i s t e n t where
S +POW
9.2. Even though So c o n t a i n s
contradiction as i n
J
P a r t 111.
i s t h e t h e o r y of
So
we cannot ,derive a
14.1, s i n c e we d o n ' t have a u n i v e r s a l c l a s s V i n
The c o n s i s t e n c y o f f u r t h e r axioms i n t r o d u c e d i n
SO
5 7 above (such as 2nd o r d e r
comprehension ) will a l s o be t a k e n up i n P a r t 111.
15. Adequacy of ( s u b t h e o r i e s o f ) T 15.1 Adequacy of
To.
t o BCM. o---
The development o u t l i n e d i n §§10-12 p r o v i d e s a b a s i s
Moreover, t h i s i s accomplished by To following t h e i n f o r m a l mathematics a s explained i n 1 . 1 4 and 1.15. The o f f i c i a l
f o r t h e f o r m a l i z a t i o n of BCM i n
.
intended d e f i n i t i o n s come t o t h e f o r e f r o n t i n t h e p r o c e s s of f o r m a l i z a t i o n and
must always be k e p t i n mind.
When i n f o r m a l concepts and p r o o f s a r e s p e l l e d o u t
accordingly, one i s i n a p o s i t i o n from which f o r m a l i z a t i o n i n (This was i l l u s t r a t e d i n
directly.
d i r e c t l x --adequate t o BCM
1.15.4).
(as exemplified
T
can proceed
One may t h u s conclude t h a t
i n Bishop
1967).
To
2
It i s o f l o g i c a l i n -
t e r e s t t o s e e next how much o f BCM can be c a r r i e d o u t i n t h e o r i e s weaker t h a n 15.2
The r o l e of I G .
s e t s i n Bishop
1967, which
IG
Obviously
To.
i s used only f o r t h e t h e o r y of B o r e l
i n t u r n f i g u r e d i n t h e t h e o r y of measure and i n t e -
1.14.15, t h i s was supersede$ by a t r e a t m e n t w i t h 1972. The l a t t e r makes prima f a c i e use of t h e axiom POW, b u t j u s t t o form ( a complete i n t e g r a t i o n space) L(X) a class
gration.
A s was explained i n
o u t B o r e l s e t s i n Bishop-Cheng from any i n t e g r a t i o n space n o t i o n of of POW.
8,
f
X ; however i n t e g r a t i o n t h e o r y only r e q u i r e s t h e
b e i n g a member of
L(X), which i s d e f i n a b l e without t h e assumption
The conclusion i s t h a t
IG
i s unnecessary f o r t h e development of
a b s t r a c t i n t e g r a t i o n t h e o r y i n t h i s sense.
8
The r o l e of axioms tjike POW i n a b s t r a c t c o n s t r u c t i v e i n t e g r a t i o n t h e o r y i s s i u d i e d i n Feferrnan 197 $4.3. A modified form of POW f o r t h l s purpose can be 3 e r i v e d i n t h e t h e o r y of s e t s and c l a s s e s T ( S ) ( c f . 9 . 3 above); t h i s y i e l d s t h e c l a s s of s u b s e t s of any given c l a s s .
193
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
15.3
D i s p e n s a b i l i t y of t h e j o i n axiom.
We have seen i n 11.3 t h a t t h e axiom
lqu and I! L I D ) * UCFTS However i n a c t u a l a n a l y s i s one never consiciers f a m i l i e s of varying t y p e b u t only J
i s needed t o e f f e c t t h e passage t o t r a n s f i n i t e t y p e s ( e . g . t o
f a m i l i e s of s u b s e t s of a given s e t .
uc FTS
In t h e s e c a s e s one can t r y t o e l i m i n a t e
J by
r e p l a c i n g t h e n o t i o n of f a m i l y by t h a t of p r e - j o i n e d f a n i l y (11.1). It may be v e r i f i e d t h a t , except f o r t h e theory of B o r e l s e t s , t h i s replacement does indeeci l e a v e t h e t r e a t m e n t of a n a l y s i s
irl
BCM u n a f f e c t e d .
1 5 . 4 Adequacy of r e s t r i c t e d i n d u c t i o n 2 M . An exaniple where u n r e s t r i c t e d M was used i n an e s s e n t i a l way was g i v e n i n 11.3, namely t o prove
i n d u c t i o n on that for a l l
u EFTS, N
i s a class.
S i m i l a r l y , t h e p r i n c i p l e of u n r e s t r i c t e d
i n d u c t i o n i n I G i s used only t o show t h a t t h e o b j e c t s i n t h e B o r e l h i e r a r c h y actually are classes.
But f o r BCM without t r a n s f i n i t e t y p e s and without Borel
t h e o r y it appears t h a t only r e s t r i c t e d i n d u c t i o n
011
i s needed.
N
This has been
v e r i f i e d i n d e t a i l by Friedman (unpublished, b u t c f . l d . 2 below).
1 5 . 5 Adequacy of EMor
.
P o t t i n g 1 5 . 1 - 1 5 . 4 t o g e t h e r we conclude t h a t
i s adequate t o a l l of Bishop l$7 Borel s e t s
and t o a l l of Bishop-Cheng
o p e r a t i o n from c l a s s e s t o c l a s s e s .
1972 except
The q u e s t i o n of accordance of
16.1 Sets vs. classes.
T
V,
g r e a t e r d i r e c t accord. as
To
.
L(-)
Mop2
c o n s e r v a t i v e over
HA.
( o r i t s s u b t h e o r i e s ) with BCM.
Bishop does n o t speak of c l a s s e s and it i s questionable
whether he would countenance a u n i v e r s a l c l a s s e x p l i c i t l y i n accordance
for treating
T h i s i s of l o g i c a l (and e p i s t e m o l o g i c a l ) i n -
t e r e s t because, a s w i l l be shown i n P a r t
16.
EPIor
except f o r t h a t p a r t i n v o l v i n g t h e theory of
with
BCM.
Incidentally,
V.
The t h e o r y of
I n t h i s respect,
sets
To i s not
(9.2) i s here i n
S
So i s adequate t o t h e same p a r t of p r a c t i c e
16.2 The q u e s t i o n of o p e r a t i o n s with unbounded domains. There i s no e x p l i c i t d i s c u s s i o n by Bishop of o p e r a t i o n s with unbounded domains l i k e sulting
d=hx(x), e = h x b ( x y ) , etc.
k , s and t h e r e However, t h e i d e a o f such does seem t o be
i m p l i c i t i n h i s view of o p e r a t i o n s simply as r u l e s ; it i s f u r t h e r i m p l i c i t i n h i s u s e of o p e r a t i o n s such as C a r t e s i a n product and power on s e t s , s i n c e no c l a s s of a l l s e t s i s assumed a s an o b j e c t .
It i s my conclusion from t h e s e arguments t h a t
the use of o p e r a t i o n s with unbounded -
domains i s i m p l i c i t l y
&
accordance with BCM.
However, t h i s i s c l e a r l y s u b j e c t t o debate, e s p e c i a l l y s i n c e it l e a d s us t o t a l k about combinations l i k e Remark.
(xx) which appear f o r e i g n t o p r a c t i c e .
There i s a simple formal d e v i c e which p e r m i t s us t o r e p l a c e unbounded
combinatory operatZons by corresponding bounded ones an3 s t i l l achieve much t h e same mathematical e f f e c t s .
Namely, one i n t r o d u c e s formal " e x t e r n a l " o p e r a t i o n
symbols on ( v a r i a b l e ) c l a s s e s
(A,B, . . . )
e . g . kA,B
t h e arguments as s u b s c r i p t s ) w i t h axioms i i k e :
,
idA, e
A, B
etc. (writing
S. FEFERMAN
I94 kA,B 0 (A
+
Px E APY
( B +A)),
idA E (A + A ) ,
Px
E
A( i d (x) zx),
+((A -,B)
e E (A A, B
B ( kA,Bx~ = x ),
E
A
+B)),
etc.
What i s l o s t h e r e i s t h e p o s s i b i l i t y of reducing r e c u r s i o n ( o n m, o r any i ( A , R ) ) t o t h e combinators, s i n c e t h o s e r e a u c t i o n s make e s s e n t i a l u s e of t h e p o s s i b i l i t y of s e l f - a p p l i c a t i o n
(xx). Thus i n such a s t e p one must supplement t h e Ti, r e s p .
I G axionis, by s u i t a b l e axioms f o r r e c u r s i o n o p e r a t o r s .
16.3 The o t h e r p r i n c i p l e s . inductive generation.
These a r e comprehension, n a t u r a l numbers, j o i n and
If we a r e t o ju3ge t h e axioms f o r t h e s e s e p a r a t e l y from t h e
16.1, 16.2, we must n a t u r a l l y c o n s i d e r them i n weaker forms t h a t apply as
issues i n
well t o s e t s particular
and a r e given by e x t e r n a l r a t h e r t h a n i n t e r n a l o p e r a t i o n s .
CAI
i s t o be r e p l a c e d by
should be c l e a r from I . l 4 , 1 5
In
With such m o d i f i c a t i o n s i n mind, it
SEPl.
an3 10-l2 above t h a t t h e s e p r i n c i p l e s a r e c a l l e d
f o r i n BCM. Remark.
Beeson has a l s o r a i s e d a q u e s t i o n ( i n c o n v e r s a t i o n ) about t h e construc-
t i v i t y of t h e j o i n axoim, as formulated uniformly u s i n g j. H i s p o i n t i s t h a t t h e re-
A
s u l t should depend n o t only on
16.4 Conclusion.
f
and
b u t a l s o on a proof o f
The i s s u e s i n d i s p u t e a r e t h o s e i n
l i e v e a c a s e can be made
-
and s e t s by p r o p e r t i e s -
t h a t t h e u s e of both o p e r a t i o n s
Vx
E
A[CJ(fx)].
1 6 . 1 and 16.2. I be-
based on B i s h o p ’ s views of o p e r a t i o n s g i v e n by r u l e s
___-
_-_-
and of c l a s s e s (as w e l l as s e t s ) makes To
implicitly
with unbounded domains 2 accordance w i t h BCM.
However, t h e r e i s l i t t l e support f o r e x p l i c i t , A i r e c t accordance. Remark.
Since
mot
i s c o n s e r v a t i v e over
HA
an3 t h e l a t t e r i s c e r t a i n l y i n
d i r e c t accordance with BCM, t h e former i s consequently i n i n d i r e c t accordance with it.
l’7.
Comparison w i t h Martin-LBf 1975.
17.1
C h a r a c t e r of Martin-LBf’s system. T h i s i s a k i n 3 of l o g i c - f r e e t r a n s -
f i n i t e t y p e theory which i s Aenoted and terms f o r t y p e s
a unique t y p e .
...
A,B,C,
.
a
i s read:
E k
system ( c f . Prawitz form
x. E A . 1
1
.
. . .,
The b a s i c p r o p o s i t i o n s a r e of t h e form ao.4
where
There a r e terms f o r o b j e c t s a,b,c,
TT. ’)
The i n f o r m a l i d e a i s t h a t each o b j e c t i s of
a
and
is _ of -t y p e _
1971) f o r
a = b ,
A.
TT
i s based on a n a t u r a l de3uction
d e r i v i n g such p r o p o s i t i o n s from hypotheses of t h e
For example, .,uppose one has i n f e r r e 3 b [ x ]
E
B[x]
from x E A . 10)
9 ) A s s t a t e d by Martin-LBf, a s i g n i f i c a n t e a r l i e r attempt t o f o r m u l a t e such a t h e o r y was maje i n S c o t t
1970.
1 0 ) i.le simplify h e r e t h e form of assumptions a c t u a l l y g i v e n by Martin-LBf f o r TT.
CONSTRUCTIVE T H E O R I E S O F F U N C T I O N S AND C L A S S E S
I95
Then we have terms f o r a p p l i c a t i o n , a b s t r a c t i o n and C a r t e s i a n product r e l a t e d by the rules
S i m i l a r l y t h e r e a r e r u l e s f o r p a i r i n g , p r o j e c t i o n and join ( E x e A ) B [ x ] . S p e c i a l and A XB. There a r e r u l e s f o r t h e n a t u r a l c a s e s of product and j o i n a r e N. ( F i n i t e i n i t i a l segments numbers €4 u s i n g 0 , s ( s u c c e s s o r ) and r e c u r s i o n
7 -
N
of
N aye p r o v i d e 3 f o r t o o . )
k relation
I on
on
F u r t h e r , with each
as a f u n c t i o n of
A
(x,y)
E
AXA
.
A
i s associate3 t h e identity
Finally, there i s a
Vo
which
i s supposed t o be t h e t y p e o f a l l s m a l l types, and i s c l o s e d under t h e i n t r o d u c t i o n
Vn
r u l e s f o r t y p e s ; moreover, t h e r e i s f o r each that
i s a t y p e i n t h e system one proves
A
t h e predicate calculus i s represented i n between formulas and t y p e s . (a EA)
i s thought of as ' a
When a t y p e
A
E
a corresponding Vn+l Vn
f o r some
.
To prove
n. The syntax of
TT v i a t h e (Curry-Howard) correspondence
i s thought of a s a p r o p o s i t i o n t h e n
A
i s a proof of t h e p r o p o s i t i o n
A'
.
From t h i s , t h e
i n t u i t i o n i s t i c p r e 3 i c a t e cal.culus i s d e r i v e d u s i n g t h e (Brouwer-Heyting) exp l a n a t i o n of t h e l o g i c a l o p e r a t o r s i n terms of p r o o f s (1.4.2 above) Remark.
Logic i s assumed i n f o r m a l l y i n t h e e x p l a n a t i o n of t h e r u l e s .
17.2 Comparison of t h e system with
To.
TT does
provide f o r i n d u c t i v e l y
g e n e r a t e d t y p e s i n g e n e r a l , b u t r u l e s f o r them can b e a d j o i n e d along t h e sane l i n e s , f o l l o w i n g Martin-Lbf
1971.
preted i n
T
Vn
involving
Vo,...,Vn-l).
mar.11)
(each
With o r without such r u l e s , t h e system can be i n t e r -
i s i n $ u c t i v e l y g e n e r a t e d by c e r t a i n c l o s u r e c o n d i t i o n s
Furthermore, t h e system with no u n i v e r s e s c o n t a i n s
17.3 Adequacy o r
TT t o BCM.
By t h e preceding,
L B f ) i s adequate t o t h e same p o r t i o n o f BCM
=
M ?
TT ( a s given by Martin-
; when supplemented by i n -
d u c t i v e l y g e n e r a t e d t y p e s a s suggested i n 17.2 it i s a l s o adequate t o t h e same p o r t i o n of
BCM
a s a l l of
To
a c t u a l l y serves t o formalize.
17.4 Accordance with BCM. The t y p e s of TT can he i n t e r p r e t e d as s e t s i n Following 1.14 - 1 . 1 5 above it should be g r a n t e d t h a t TT
Gs
B i s h o p ' s sense.
d i r e c t accordance with BCM, a t l e a s t i n s o f a r a s ccncerns b a s i c concepts and principles.
The one r e s e r v a t i o n has t o do with i t s h e a v i l y s y n t a c t i c formulation
f o r t h e c o n d i t i o n s t o i n t r o d u c e an3 u s e t h e v a r i o u s k i n d s of terms.
This i s i n
t u r n n e c e s s i t a t e d by t h e requirement t h a t each o b j e c t i s assigned a t y p e . we cannot have an ' i n t e r n a l ' f u n c t i o n
To
( a s done i n which
B[x]
E
f
by Yx EALCL! ( f x ) ] ) h u t must u s e ' e x t e r n a l ' o b j e c t s
Vn
i s prove3 ( f o r some
Thus
of which i t i s proved YxeA[fx i s a t y p e ]
n ) under t h e hypothesis
B[x]
x EA.
of
There a r e
no i n 3 i c a t i o n s i n B i s h o p ' s w r i t i n g s t h a t would l e a d one n e c e s s a r i l y t o t a k e such a formal approach.
I n t h i s r e s p e c t , t h e l o o s e n e s s which
t y p e - f r e e c h a r a c t e r seems more i n a c c o r 3 with
BCM.
11) The exact r e l a t i o n s h i p s a r e not known t o me.
T
enjoys owink; t o i t s
196
S. FEFERMAh'
Remark.
To.
TT
The syntax of
i s e v i d e n t l y somewhat more complicated t h a n t h a t of
Some s i m p l i f i c a t i o n could presumably be made by assuming a l l of i n t u i t i o n i s t i c
logic at t h e outset.
I n any case, it i s much e a s i e r t o form a v a r i e t y of models
an3 i n t e r p r e t a t i o n s of
18. Comparison
a s we s h a l l s e e i n P a r t s 111, I V .
To,
w i t h M y h i l l ' s and Friedman's e x t e n s i o n a l systems.
18.1 The c h a r a c t e r of t h e s e systems. The system CST
1975 i s a subsystem of
introduced i n m h i l l
by which i s meant Zermelo-Fraenkel set t h e o r y
IZF(N)+DC
over t h e n a t u r a l fiumbers ( a s urelements)
with t h e l o g i c r e s t r i c t e d t o be i n t u i t i o -
n i s t i c and w i t h t h e axiom scheme of dependent c h o i c e s added. The n o t i o n s of and of f u n c t i o n a r e b o t h d e f i n e d h e r e j u s t as u s u a l i n
pair
ZF : i n o t h e r words
f u n c t i o n s a r e iAentifie3. with graphs of many-one r e l a t i o n s .
One t a k e s ( o v e r t h e
N ) t h e axioms of e x t e n s i o n a l i t y , unordered p a i r , union,A -sepa-
u s u a l axioms of
r a t i o n , 3omain an3 ranges of f u n c t i o n s , t h e s e t
of a l l f u n c t i o n s f : A + B
(A + B )
for given s e t s A,B (which i s t a k e n i n p l a c e of t h e power s e t axiom) and t h e replacement scheme
(Vx eA)Z!y$ ( x , y ) Finally, t h e pri-nciple
--t
3z[Fun(z)ADom ( 2 ) = A A Vx e A$(x, z ( x ) )
DC
( V x ~ A 3 y i A $ ( x , y )+ V X E A E I Z [ Z : N
i s taken, b u t n o t
AC,
system. U )
A,
IZF/PI).
i s restricted,
-x
A Vn c N @ ( z ( n ) ,~ ( n ' ) ) ]
s i n c e t h a t i s shown t o c o n t r a d i c t Church's t h e s i s i n t h e
$,
formulas
CST
The weakest of t h e s e i s denoted
replacement i s t a k e n only t o form and
considere$ a r e $?noted however
i A A z ( 0 )
1977 c o n s i d e r s a number of subsystems and e x t e n s i o n s of CST ( a l l
Friehan contained i n N
1.
DC
i s a l s o t a k e n only for such
T ,T_,T
1 2 3
i s equivalent t o
an3 T 4 .
T2
.
B.
[ (x E
0.
AI
In
- , induction
T4
.
( i i ) Friedman
rervative for sentences.)
18.2
Il:
1977 shows t h a t
BCM
for
The o t h e r systems
i t s e l f i s reducible t o
HA
They a r e deand i s con-
I
Adequacy of t h e s e systems t o BCM. The system
same p o r t i o n of
A)
(Beeson 1979 shows t h a t i t i s c o n s e r v a t i v e f o r a l l B + c l a s s i c a l l o g i c i s e q u i v a l e n t t o Zermelo s e t t h e o r y .
sentences.
By c o n t r a s t ,
-
B
E
We s h a l l n o t d e s c r i b e them h e r e ; 13)
Remarks. ( i ) Axioms of i n d u c t i v e g e n e r a t i o n are n o t t a k e n i n CST. rivable i n
on
B
@ ( x , y )J 1y
a s t h e system
i s adequate t o t h e
EMor - ( c f . t h e d i s c u s s i o n i n Frieciman 1977 p . 7 ) . does not seem t o have any f u r t h e r power
Though formally s t r o n g e r , t h e system
CST
f o r t h e a c t u a l mathematics involved.
The adequacy i n b o t h c a s e s i s i n 3 i r e c t .
The d e f i n i t i o n s of concepts do n o t f o l l o w B i s h o p ' s o f f i c i a l s p e l l e d - o u t d e f i n i t i o n s , b u t r a t h e r t h c correspon3ing c l a s s i c a l ones which use e x t e n s i o n a l i t y .
For example,
12) The reason wh AC b u t n o t DC i s problematic i n t h e framework of T w i l l be exulained i n $art IV below. 13) ThG r e a d e r may f i n d it useful t o r e a d my review of F r i e h a n 1977 which appeared i n Math. Reviews 55(1978) ~0.7748.
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
197
IR a r e t a k e n t o be equivalence c l a s s e s of Cauchy sequences. IR’ a r e t h o s e x E B such t h a t (&I cm+c)(x> 1 ) .
t h e r e a l numbers
The p o s i t i v e r e a l numbers
Thus a l l t h e d i s t i n c t i o n s and u s e of w i t n e s s i n g d a t a r e q u i r e d by Bishop i n o r d e r t o c a r r y o u t c o n s t r u c t i v e o p e r a t i o n s a r e e s s e n t i a l l y ignored.
18.3
The q u e s t i o n of accordance. These systems d i f f e r i n two e s s e n t i a l r e -
s p e c t s from t h e c o n s t r u c t i v e p o i n t of view which i s b a s i c t o BCM, at l e a s t as described i n
1.4.
Namely, e x t e n s i o n a l i t y i s accepted, i n v i o l a t i o n of
f u n c t i o n s a r e d e f i n e d i n terms of s e t s ( o f o r d e r e d p a i r s ) ,
1.4.6.
1.4.5, and
i n v i o l a t i o n of
It i s p l a i n t h e n t h a t any s e t t h e o r y which c o n t a i n s t h e e x t e n s i o n a l i t y
axiom and d e f i n e s t h e n o t i o n of f u n c t i o n i n t h i s way-and i n p a r t i c u l a r CST and
-B
- i s n o t i n d i r e c t accordance with BCM. By t h e r e d u c t i o n of
above,
-
B
B
t o HA
due t o Friedman (and Beeson) r e f e r r e d t o
i s c e r t a i n l y i n d y r e c t l y i n accordance with
As t o
BCM.14)
CST,
Myhill 1975 g i v e s a c o n s t r u c t i v e r e d u c t i o n v i a a r e a l i z a b i l i t y i n t e r p r e t a t i o n . More sharply, F r i e h a n 1977 o b t a i n s r e d u c t i o n of t h e e q u i v a l e n t
<
n i s t i c ramified a n a l y s i s i n l e v e l s
To minus
IG, cf.Part
V below).
T2
t o intuitio-
c0 (which i n t u r n i s i n t e r p r e t a b l e i n our
The system
i s also reducedloc. c i t .
T:,
t o an i n t u i t i o n i s t i c t h e o r y of one i n 3 u c t i v e l y d e f i n e d s e t , which i s c e r t a i n l y j u s t i f i e d by BCbl an3 i s contained i n our
To
.
Finally, t h e theory
T4
i s re-
d u c i b l e t o t h e f u l l 2nd o r d e r t h e o r y of s p e c i e s which i s contained i n EM + C A 0
b u t t h e accordance of t h e l a t t e r with BCM i s open t o 3 i s p u t e .
.
2 ’
I t should be mentioned t h a t i n Beeson‘s c o n t r i b u t i o n t o t h i s volume he shows f o r a number of i n t u i t i o n i s t i c e x t e n s i o n a l t h e o r i e s of s e t s how t o i n t e r p r e t them i n t h e i r s u b t h e o r i e s without e x t e n s i o n a l i t y . Beeson
1979 by
T h i s i s followed i n
c e r t a i n r e a l i z a b i l i t y i n t e r p r e t a t i o n s t o reduce t h e l a t t e r t h e o r i e s
t o s u b - t h e o r i e s of
To
, in
p a r t i c u l a r of
B
HA ( c o n s e r v a t i v e l y ) .
to
111. Models Throughout t h i s p a r t models w i l l be understood i n t h e usual s e t - t h e o r e t i c e l sense and t h u s w i l l s a t i s f y c l a s s i c a l l o g i c .
T h i s does not hold f o r t h e i n t e r -
p r e t a t i o n s t o be d e a l t w i t h i n P a r t I V .
1. A model of Let
91
=
EW
where
o
1975 s e c . 4 . 1 . )
( V , App, k,s,d,p,pl’P~,O,sN,pN)
b e any model of t h e axioms APP of T x
APP ( p r e s e n t e d i n Feferimn
To o v e r any model of
( i n 11.5).
i s t h e l e a s t subset of
V
Here
x
containing
0
E
TV
i s i n t e r p r e t e d as
and c l o s e d under
+ x’ = s x . (The i d e n t i f i c a t i o n of ?N a s a member o f V w i l l be explained N i n a moment.) P b b r e v i a t i o n s f o r a p p l i c a t i o n terms, p a i r i n g , comprehension a r e
x
1 4 ) F r i e h a n a l s o g i v e s an i n f o r m a l argument f o r t h e c o n s t r u c t i v e j u s t i f i c a t i o n
of B (and s t r o n g e r t h e o r i e s ) by i n t e r p r e t a t i o n i n a t h e o r y of s p e c i e s of f i n i t e type:
S . FEFERMAN
198 t a k e n j u s t a s i n 11.3,4.
Now we t a k e
in
f o r t h e c l a s s c o n s t a n t s and
V
o p e r a t i o n s , e . g . as follows: c z
31 = ( O , o ) ,
Next
CQa
( l , n , z ) , j ( a , f ) = ( z , a , f ) and i ( a , r ) = ( 3 , a , r ) .
=
by t a k i n g
' E '
CCQa+l a-
Ce
and x e a f l a
for each elementary and f o r any
yl,.
. .,y,
f o r each
a eCQa
f o r each
a+lc
--lxeaa, for
O(x,yl
I= Z
r
an3
aEclcy;
al, . . . , a
P and
cJcy-bl
C E
such t h a t ~
+
~
-,C 3
we have
Cia
Vx(xcaa+ f x
E
we have
Cia),
x , y ( x ~ ~ a A y ~ ~ f x ) ;
V IF V [ Vu[ u E aa A Vw( (w,u) car
->
,
O(x,yl,...,ym,al,...,ap);
f E V
and and
a sCLa
I n this
Z1,..., Z ) and n = r$(x,y,zj? P E CQa we have
,..., y,
V and
E
xcCXilc - ( I I , c Q ~ , E ~ )
c=j(a,f)e
.
1 5)
. . , Y m , al,. . .,a p)
= Cn(Yl,.
CLcy
i n i t s o r d i n a r y s e t - t h e o r e t i c e x t e n s i o n a l sense;
t h e c o n t e x t s e r v e s t o avoid ambiguity.
E
are interpreted
and l e t t i n g t h e c l a s s v a r i a b l e s range over
'E'
d e f i n i t i o n we s h a l l a l s o u s e
x
S(T )
i n which t h e formulas of
ca)
(%,Cia,
for
c
a ; a t s t a g e a one
a r e d e f i n e d by t r a n s f i n i t e r e c u r s i o n on
and
has a s t r u c t u r e
c = i ( a , r ) E CQ
-
and
cyfl
WE
I ) -u
E
I1 +x
E
I
;
has only t h o s e elements obtained by ( i ) - ( i v ) .
A,
For l i m i t
CQk =
u
CQa
and
For t h e f i n a l model of
(4)
C!
=
u CQ2 a
To we t a k e and
x c a e x c a a The axioms f o r way.
.
"A = &Ea
cr
E =
.
u ea , a
so t h a t f o r
To a r e v e r i f i e d t o ho13 i n
a
E
4=(1(, Ce,
In p a r t i c u l a r , by (1)we have f u l l i n d u c t i o n on
p r o p e r t i e s ) and by (2)(iv) we have full i n d u c t i o n on
Cla
x,
i n a straightforward
E )
?N
and any
( w i t h r e s p e c t t o any
i(A,R)
f o r any
A,R. It
i s f u r t h e r t o be noted t h a t i n checking elementary comprehension f o r
c
=
(x;O(x,y1,
.. . , y m ,
a1, ...,ap))
we need only know t h e meaning of x
This i s where t h e p r e d i c a t i v e c h a r a c t e r o f
0is
E
a .(lLizp).
used i n an e s s e n t i a l way.
A new
i 3 e a i s need& i f one wishes t o s a t i s f y s t r o n g e r comprehension schemes; t h a t i s explained i n t h e next s e c t i o n . Remarks. ( i ) V, d e f i n e d by ( x ( x = x ) , i s i n t e r p r e t e d as a c e r t a i n code c
in the
domain V of 9J; t h e n X E V has t h e same meaning i n t h e model as e x t e n s i o n a l l y .
1 5 ) I n Feferman 1975 we u s e d t h e symbol ' q ' i n p l a c e of d i s t i n g u i s h t h e two u s e s .
'E'
i n S ( T ) i n order t o
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES ( i i ) If t h e c o n s t a n t
d
of
satisfies
U
(x=y + dxyab = a ) A ( x f y + 3xyab
and
=
b)
Vx,y,a,b[jxyab 1 ] then t h e f u l l d e f i n i t i o n
DV i s of course s a t i s f i e d so we have a model of
axiom
199
T
- by - c a s e s
i n t h i s case.
+ DV
( i i i ) There i s an obvious m o 3 i f i c a t i o n of t h e c o n s t r u c t i o n above t o g e t a model
of
To
s t a r t i n g w i t h any
5
as long as c a r d (C") 2.
and
Cto
card (V)
such tha.t
E
Modification t o o b t a i n a model of
T +CA2.
(Feferman
The argument h e r e w i l l be much l e s s c o n s t r u c t i v e . of
APP
of
V
an3
E
Jr(x,y,z,X)
(3X Jr
Given any s t r a t i f i e d
9-
-
f o r each
c
=
c
Ji
P(V)
where
* -
TR
+
and F
4
and so
a
t h e n d e f i n e 3 as i n
in
, ancl
CJ
!lX* = (U,Ce*,
let
g1(4),
(= G ( y , A ) )
@
-
that
NX,Y >
From t h i s it f o l l o w s t h a t The axioms f o r
*
E)
t)
Ai=(x1xc a . ) f o r each a 1 Id,. . ,p.
..
let
ax
be t h e e x t e n s i o n Then C l *
i s an elementary s u b s t r u c t u r e of
6)
I= @(X,y1,...,y m , A1, ...,Ap)
order definitions i n
n* I=
NX,Y ->
CJ*.
F i n a l l y it i s prove3 by
T + CA 0 2 .
IN and I G c o u l d b e subsumed under V
and
:W
It
we have
CA
App(x,y,z)
Taking 0 and x
s i n c e t h e i r 2nd
2
9 a r e absolute.
The r e c u r s i o n - t h e o r e t i c model. Take
i s closed Q.
E*).
i s a model of
(U, C J , E )
.-,Ap
A1,.
in
Q-
i n t h e sense of o r d i n a r y r e c u r s i o n t h e o r y .
c,
WX'
[XI
( y ) =. z
t o be standard,
so as t o o b t a i n I s a t i s f y i n g k,s,d,p,pl,p2,SN,PN DV i s a u t o m a t i c a l l y s a t i s f i d . I n a d d i t i o n t h e n t o
we can e a s i l y choose c o n s t a n t s t h e axioms
APP. Note t h a t
( o r T + C A i f we f o l l o w $ 2 ) we a l s o have Church's t h e s i s f o r p a r t i a l To 0 2 functions vf z e V n ( f n =. ( e ) ( n ) )
(CT2)
t r u e i n ($ CI J, , E) functions.
:
we have
CJ*=[a*laeCJ].
rxl%*
ce, E ) I=
t
*
i n d u c t i o n on s t r a t i f i e d
3.
which has t h e property:
*(X,y,
@ and any sets
Remarks.
To each s t r a t i f i e d formula
F (x, y , g ) = X
f o l l o w s t h a t f o r each s t r a t i f i e d
(VJ,
I
i s t h e s e t of a l l s u b s e t s
..., y , ~V and a1 ,..., a E CJ,: P an3 . . - > Y, > alj . . ,ap) E O X E F (x,yl,. . ., y m , A1, .. . , A P), where
[ x I x ~ a ) c V f o r each F
)
x, yl,
W i t h the resulting (9J,CJ,c)
under t h e
, z)
= fn(x,Y13
x
,
1975, Addenam).
5,F 4 ( x ,-Y , -Z ) ) ) . t a k e $(x, y , z,X ) = Yx[xsX ct @ ( x ,y , z ) 1 , )= @ ( x , y ,3). We choose codes f n f o r t h e -
2,X )
[x,Vl
EW
4 and i n p l a c e of ( 2 ) ( i i ) i n $ 1 t a k e , for n = $(x,y,z,X)l
f o r each s t r a t i f i e d (2)(ii)'
-
$(x, y
G (y, 2)
So
(X,Y,
xeoIN t r x
S t a r t i n g w i t h any model
i s t h e s t a n d a r d membership r e l a t i o n .
I= ,
E
i s a s s i g n e d a Skolem f u n c t i o n
-
G$ = F 4
TJl = (91, P(V),
a s i n § 1, l e t
and
IN E CJo
( s o t h a t t h e r e i s room f o r a l l t h e c o d e s ) .
simply by t a k i n g
Hence CT1 i s a l s o t r u e .
e=f.
INm i s j u s t t h e c l a s s of r e c u r s i v e
On t h e o t h e r hand C T o ,
being c l a s s i c a l l y
S . FEFERMAN
200
false, i s not satisfied.
i s interWith r e f e r e n c e t o 1110.8-10.9,it may be seen t h a t ( N ) u UEFTS p r e t e d i n t h i s model as t h e h i e r a r c h y HRO of h e r e d i t a r i l y r e c u r s i v e o p e r a t i o n s and (Mu)asTs
as t h e h i e r a r c h y HE0 of h e r e d i t a r i l y e f f e c t i v e o p e r a t i o n s
1973, 124-127).
Troelstra
(cf.
Going on t o 10.11-10.12one s e e s t h a t t h e r e a l s
a r e i n t e r p r e t e d as i n r e c u r s i v e a n a l y s i s , and so on f o r F i n a l l y , with r e f e r e n c e t o 12.1, i t i s seen t h a t
C([a,b],
01,Q2,
IR
IR), e t c .
..., Q a , ...
are inter-
p r e t e d as forms of t h e Church-Kleene c o n s t r u c t i v e o r d i n a l n o t a t i o n c l a s s e s . Remark.
Any "enumerative" g e n e r a l i z a t i o n of r e c u r s i o n t h e o r y g i v e s r i s e t o a
model of
APP which, when extended t o a model of
i n t e r e s t i n g i n t e r p r e t a t i o n s of i t s concepts.
T
as i n 9 1 yields other
For f u r t h e r examples of such and
005-6 below c f . Feferman 1978, 3.2 -3.4.
4.
Independence r e s u l t s from ECM.
in
To, any model 5! =(!?!, C A ,
results for
4
Since a l l of
can be s a f e l y f o r m a l i z e d
ECM
To a u t o m a t i c a l l y p r o v i d e s independence
of
E )
which a r e c l a s s i c a l l y t r u e b u t f o r which
we t a k e t h e r e c u r s i o n - t h e o r e t i c model
03
of
N
!5f
0.
For example, i f
t o begin w i t h t h e n t h e example
due t o Specker of a r w u r s i v e l y ( u n i f o r m l y ) continuous f u n c t i o n on [ 0 , 1 ] which 3oes not t a k e on a r e c u r s i v e minimum shows t h a t t h e theorem of t h e minimum i s underivable i n BCM. not derivable -
&I
T
Indeed, t o be more p r e c i s e and even s t r o n g e r , by $2
it
+ Dv + C A2P -w i t h c l a s s i c a l logic. S i m i l a r l y f o r t h e o t h e r o
examples g i v i n g ' p e c u l i a r i t i e s ' of r e c u r s i v e a n a l y s i s and of t h e Russian school
of c o n s t r u c t i v e a n a l y s i s ( c f .
f o r which ( c l a s s i c a l ) T p r e t a t i o n or 'analogue'
P : V2
.
1b
b
then
i s a mathematical statement
has a r e c u r s i o n - t h e o r e t i c i n t e r -
Y of APP from t h e f o l l o w i n g information:
'-5 V and p r o j e c t i o n s
( i i ) an embedding of
so t h a t
(w,O,')
f o r which c a r d (3) 5
F :V 2 V
for all
F(x) of
FB3
x.
To obtain
tations.)
...
P.: V + V in V,
and
pxy
and
=
xi
,
3 of p a r t i a l
Then we can d e f i n e c o n s t a n t s f o r
=
P ( x , Y ) , pix
there exists
f
E
V
=
pi(x),
N
which r e p r e s e n t s
(By n o n - e x t e n s i o n a l i t y , each
F
I1
s n = n ' >(pNn')=n F, i . e .
w i l l have many represen-
2l we simply u s e p a i r i n g t o b u i l d codes f o r t h e c o n s t a n t s
N as w e l l a s f o r each sx
P. ( P ( x , , x 2 ) )
( i i i ) any c o l l e c t i o n
F
E
5.
i n d u c t i v e c l o s u r e c o n d i t i o n s on t h e r e l a t i o n t o it t h a t
V, we can g e n e r a t e a
( i )a p a i r i n g o p e r a t i o n
f o r which
card ( V ) .
A P P + D ~ i s s a t i s f i e d and
and such t h a t f o r each
k,s,
+ Dv + C A 2 o
Generating models of APP+DV . Given any i n f i n i t e s e t
model
fx,
0
The obverse of t h e p o i n t h e r e i s t h a t i f
Remark.
5.
1.7-8).
sxy
Then we r e g a r d t h e axioms of App(x,y,z).
a r e always d e f i n e d i n a simple way) one wants
X Z = ~ A ~ Z Z W A U W + ~ V (SXy)ZzV.
APP a s
In p a r t i c u l a r ( s e e i n g
CONSTRUCTIVE T H E O R I E S OF FUNCTIONS AND CLASSES
20 1
6. F u l l set-theoretic models of APP+DV . In particular, let V = Rh (the set of Define sets in the cumulative hierarchy of rank < A ) for some limit A > w . 0,' and predecessor on w an3 pairing and projections as usual. Let 3 be the class of all functions which (as sets) are members of model 9J of A ? ? + D V
in which every set-theoretic function is represented.
Proceed to build a model %
=
has the same elements as w functions from w we have
By $ 5 we obtain a
Rh.
to w .
(9I,Ce,
and
of T o + D V + C A 2
E)
('IN
+IN)
For the type symbols 1 =(0 >
(M1/ =1) : ( w + u )
an3
over 8 . In d , IN
consists of representatives of all
(%/ =e) I ( ( w
0 ),
+w),
+w)
2 = (1>O),
etc.
etc. Further
(IR/ = R ) is isomorphic to the reals in the set-theoretical sense, and the class of all functions from B to lR which preserve =IR is isomorphic (modulo the define3 equality between such functions) with the set of all real functions in the set-theoretic sense. Now C([a,b], B) consists of representatives of all uniformly continuous functions.
A is inaccessible. Each element a of Q1 has a naturally associated ordinal la/ < wl and w1 = la1 : a E Q ~ ) . More generally for any Suppose
w = { /bl : b ) . The Borel hierarchy in I N w as explained la1 in 11.l2.2 consists of representatives of the full Borel hierarchy in Baire space
Qa, we have
in the set-theoretic sense.
7. Generalizing classical, recursive and constructive mathematics. 7.1 It follows from $ 3 an3 $6 that any mathematical theorem T +Dv + CA2
4
of
with classical logic automatically generalizes a theorem of re-
cursive mathematics and of classical set-theoretic mathematics.
7.2 It also follows that for any sub-theory T of T o t Dv + CAP which is recognize3 as being constructively vali3 (so, the logic may be restricted) any mathematical theorem
$ of T generalizes one from classical, recursive and
constructive mathematics. In particular, this applies to T = T (if 11.15.4 is accepted). Remarks. (i) In a certain sense Martin-Lsf's TT
can also be considered to have
both set-theoretic an3 recursion-theoretic models, so
7.2 would also apply to it.
(ii) Myhill's CST (and relate3 theories) has immediate set-theoretic models, but no Airect recursion-theoretic model and, as we have seen in 11.17, its constructive interpretation is in dispute.
8. Term models. In the framework of To these have been given by Beeson 1977 (1.3) which is followed here; however, the i3eas are familiar from combinatory calculi (cf. Barendregt 1971, 1977). As w i l l be explaine3 in 8.2, the method works to eive a model X
of APP but not of A P ? + D V
8.1 Reduction of terms. Let ~
...
range over application terms as ex-
T,T~,T~,
plainc3 in 11.1.3. A reduction relation
.
T
>
1-
T
2
is defined inductively by the
202
FEFERMAN
S.
(i)
T
(ii)
T
(iii)
2
T
A T~
>
T*
A r2
1- 2 1-
1
kTITS
2
T1
(v)
ST1T2
73
2
(vii)
2
PN(sNT)2
7
T1
3 : : ~ T > 1 2 -
r1
(ix)
2
W e s h a l l use
T
2 2
TITj
P1(PT1T2)
(viii)
9
for O
w :
T
>
(iv)
(vi)
I,
n
following c l a u s e s , where we w r i t e
T
T~
+ rl
7;
+
2
7
T~
* 7,*
>
1 2 -
1 2
( T2 75 )
,
2
P2(PT1T2)
n f m
1'
4
T3
3 ; l i ~T > 1 2 -
T
2
o n l y as r e q u i r e d by (i)- ( v i i i ) .
T~
>
for l i t e r a l i d e n t i t y of terms.
T
1- 2
form (op i r r e d u c i b l e )
T ~ 'r2L
i f whenever
i n normal form i s denote3 by NF.
we have
i s sai3 t o be i n normal
T~
.
= T~
T
The s e t of terms
Note every term reduces t o a term i n NF.
operty) f o r 2 i s Church-Rosser theorem - ( o r 0 p r~ CR. If 2 T1 an3 2 T2 t h e n for some -*,
The
prove3 by s t a n d a r d methods:
and
TILT*
> -*
T,
2 -
A.-.
.
1\, ,' 2
'.'*
T
A s a c o r o l l a r y one has u n i c i t y of normal form i n t h e sense t h a t Tl,T2
8.2 Beeson
E
NFA
T
L
The mo3el of normal terms.
-
A
T1
2
T
T
V
The domain
of t h e model
1977 i s NF (which i s a l s o used h e r e t o 3enote
lation is: App(T1,T2,T5)
T1~T2jT3
NF
E
2 .
2 j T 1=
'u t a k e n by
?I). The a p p l i c a t i o n r e -
A (T1T2
2
r3
.
The c o n s t a n t s , a l l of which a r e i n NF, 3enote themselves i n t h i s model.
: are
because a l l
a model of APP
easy lemmas a r e proved by Beeson, where
axiom 0
E
D A Vx (x E IN + x'
(a) If
T
(b)
If
(c)
If
T
(3)
If
T
E
T1,r2
NF
then
E
APPw
IN ) .
i s c l o s e d and
A P P ~TI)
i s any sub-theory of
n
for
denotes
n
f m.
is
NF
The following
APP
p l u s t h e TN-closure
APPm
t (Tll
APPW F ( T 1 ) .
a r e c l o s e d terms an3
t h e n for some
m
f
i n NF and
E
w,
T
T
>
then
T
1- 2 then
ZT*
T + C A with 0 2
ii).
+(T-
E
NF(T
APP
C
IN-
2 T
+T
2
L
AT
-7,
1-,
T").
and i f
T
F ( TEIN
When couple3 with r e a l i z a b i l i t y metho33 i n P a r t I V t h e lemma ( 3 ) allows one t o o b t a i n t h e numerical i n s t a n t i a t i o n p r o p e r t y and t h e d i s j u n c t i o n p r o p e r t y f o r such T. Remark.
An e s s e n t i a l d i f f e r e n c e of
lJv
from t h e o t h e r d p p l i c a t i v e axioms appears
h e r e . I n t h e proof o f Lemma ( a ) we u s e t h a t i f
n f m then
APPmk
5 f
m.
To
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
t r y t o o b t a i n a corresponding r e s u l t f o r and
‘r1fT2
and
k
f
then
0
APPW+DVt(T1f
T r ) .
Dv
203
we would want t h a t i f
NF
T ~ , T6 ~
But t h a t i s n ’ t s o - f o r example,
k,OcNF
i s n o t provable ( s i n c e we can c o n s t r u c t a n a p p l i c a t i v e
but ( k f O )
model G. i n which k = 0).For t h e same r e a s o n w e c a n ’ t prove t h e d i s j u n c t i o n p r o p e r t y f o r T + D v . 16)
8.3
g (Barendregt 1971). I n s t e a d o f t a k i n g
An e x t e n s i o n a l term m
8.2, one t a k e s V t o be t h e s e t of a l l equivalence c l a s s e s l e a s t equivalence r e l a t i o n
3
such t h a t
[ T ~ ] ,[ ‘r3 1 ) tt T ~ c o n t a i n s some T sNF, t h a t
App( [ T1],
T
T~
T
.
+
T2
‘rl
as i n
V
[T]
of terms for
.
Then we t a k e
T2
the
By t h e CR p r o p e r t y i f an equivalence c l a s s
~
i s unique.
Then one s e e s t h a t
[ii] f [~i] whenever
S i n c e every a p p l i c a t i o n t e r m
n fm
so t h a t we o b t a i n a model of
APP.
[ TiIn
t h e moael, it s a t i s f i e s
i. e . every o p e r a t i o n h e r e i s
(TJ),
T
total.
denotes In
a d d i t i o n t h e model may be shown t o s a t i s f y t h e axiom of e x t e n s i o n a l i t y f o r terms. t o be e s s e n t i a l l y r e q u i r e d f o r t h e n o n - e x t e n s i o n a l i t y r e s u l t of
DV
This shows
11.13.1.
9.
Continuous f u n c t i o n models.
once more without
9.1
There a r e a g a i n models due t o Beeson
1977 (1.2),
D
V’
Continuous p a r t i a l f u n c t i o n a p p l i c a t i o n . The i d e a h e r e i s t o form a model
of APP which i s a k i n d of untyped v e r s i o n of t h e c l a s s of countable f u n c t i o n a l s of f i n i t e t y p e (which a r e h e r e d i t a r i l y continuous i n a c e r t a i n s e n s e ) due t o Kleene
1959
(and K r e i s e l ,
functions
f
members of
same volume).
from V
w
to w .
as f o l l o w s .
V
One t a k e s
A p p ( f , g , h ) ( o r f g = h) i s d e f i n e 3 f o r
A relation
F o r each
t o be t h e c l a s s of all p a r t i a l
n, t h e v a l u e of
3epend on only a f i n i t e amount of i n f o r m a t i o n about with (x,y) a p r i m i t i v e recursive p a i r i n g function. t a i n e d , when defined, by =
at
n
More p r e c i s e l y ,
a c t i n g continuously on
k+l t h e n k i s unique an3 we p u t
n n Beeson t h a t t h e n a t u r a l numbers can be embedded i n
(g),
h(n) = k . V
an3 t h e c o n s t a n t s i n t e r -
APPtDV
V i s not continuous.)
because d e f i n i t i o n by c a s e s on
63 of ( c l a s s i c a l ) T o + C A 2 9.2
from
mn(m),and i f
It i s shown by
G
APP.
h ( n ) i s ob-
so t h a t i f
p r e t e d i n such a way a s t o form a model
of
i s supposed t o
Let ( f ) , = A x . f ( x , n )
no i n f o r m a t i o n i s g i v e n by t h e i n i t i a l segment
(f)n((< )(m)) = 0
(f) ( ( r ) ( m ) )
(f)
h g.
(We c a n ’ t do t h e same f o r Now form a model
APP by s 2 .
Consistency of c o n t i n u i t y p r o p e r t i e s .
I t can be shown t h a t t h e model 63
s a t i s f i e s t h e f o l l o w i n g s t a t e m e n t s of i n t e r e s t : ( i ) Any o p e r a t i o n
f :lN -+IN
i s continuous ( i n t h e pro3uct t o p o l o g y ) .
( i i ) Any f u n c t i o n from a complete s e p a r a b l e m e t r i c space
h e t r i c space Y 16) Another e x p l a n a t i o n t h e r e i s no Church-Rosser as f o l l o w s t o correspond dTIT
2 3 4 -> 47 ’ 7
7
X
t o a separable
i s continuous. of t h e d i f f i c u l t y i s due t o Klop 1977, who has shown t h a t theorem f o r t h e c a l c u l u s w i t h t h e 2 r e l a t i o n augmented axiom: ~ T T TT > T- and T ~ , TE N~F T ~ + t o the
D~
3 4 -
3
AT^^
204
FEFERMAN
S.
I t follows t h a t t h e s e c o n t i n u i t y p r o p e r t i e s a r e c o n s i s t e n t w i t h T
+
CAz,
even
Moreover, by modifying t h e model so as t o t a k e V t o
allowing c l a s s i c a l l o g i c .
be t h e c l a s s of a l l p a r t i a l r e c u r s i v e f u n c t i o n s w e can a l s o s a t i s f y Church's thesis
for p a r t i a l functions.
CT2
matics i s contained i n existence
of
analysis.
Hence, t o t h e e x t e n t t h a t c o n s t r u c t i v e mathe-
,
T o + C f i 2 + CT2
we cannot prove c o n s t r u c t i v e l y
the
discontinuous f l m c t i o n s on t h e spaces of i n t e r e s t t o us i n o r d i n a r y
The main r e s u l t s of Beeson
1977
are i n c e r t a i n respects stronger
p o s i t i v e r e s u l t s f o r a v a r i e t y of i n t u i t i o n i s t i c t h e o r i e s T , t o t h e e f f e c t t h a t
then
term can be proved i n T to d e f i n e a f u n c t i o n (between s u i t a b l e s p a c e s ) i f 2 ---
it _ can proved
T h i s w i l l be explained more p r e c i s e l y i n PartIV
t o be continuous.
( c f . a l s o Beeson's corresponding r e s u l t s for i n t u i t i o n i s t i c t h e o r i e s of s e t s i n t h i s volume).
11.4
Topological models. (The m a t e r i a l of t h i s s e c t i o n and i t s a p p l i c a t i o n i n
10.
was developed i n c o l l a b o r a t i o n with my s t u d e n t Jan S t o n e . ) Let S be a t o p o l o g i c a l space and c a r d (5)
5
5
a f a m i l y of p a r t i a l continuous f u n c t i o n s from
c a r d (S).
d i s c r e t e topology. l o g i c a l spaces. fine
Sa
a
for
E
+,
We u s e
Let J
have
w _C V.
indicated i n
V
=
S to
S
with
and i s considered
its
W i t h
f o r t h e o p e r a t i o n s of d i s j o i n t sum of topo-
C
J =( O ) *
be t h e c l o s u r e of
(0) under p a i r i n g .
Then de-
by
S =w+S
Finally, take
s
i s assumed d i s j o i n t from
w
-
Sa.
acJ A mo3el 91
an3
S
( a , b ) = 'a
b'
'
Thus p a i r i n g and p r o j e c t i o n make sense on (V,
=
= , k, s,p,pl,p2,d,0,
V
and we
sN,pN)of APP i s g e n e r a t e d as
$ 5 , only now d e f i n i n g d x y u v j u s t f o r
x,y
E LO.
The choice of
codes can be arranged i n such a way t h a t f o r each f , --
the p a r t i a l function
We i l l u s t r a t e t h e argument f o r s x = (Z,x), s x y = ( 2 , x , y ) and g e n e r a t i o n of t h e r e l a t i o n
For example, i f
f
k,s
sxyz fipp
Ax(fx)
where we t a k e f*
-t
i s ( Z , x , y ) = ( Z , ( x , y ) ) and
s u i t a b l e neighborhood of
f
, we
have
Hence by i n d u c t i o n it f o l l o w s t h a t not work t o g i v e t h e axiom V
f*
=
V
k-1, kx = ( l , x ) , k x y = x , s=2,
It i s proved by i n d u c t i o n on t h e
=. x z ( y z ) .
that i f
continuous
f
and
(2,x*
x*z*(y*z*)
x* + x t h e n
f* + f
, y*
and
) where
xz(yz).
i
f'x* + f x
.
z* + z t h e n i n a x* + x, y* + y
.
Again t h e model does
because f u l l 3 e f i n i t i o n by c a s e s i s not continuous.
ll. A p p l i c a t i o n s t o independence of Cantor-Bernstein s t a t e m e n t s . 11.1 C a r d i n a l i t y r e l a t i o n s . t h e language of (i)
(x-Y)
These r e l a t i o n s between c l a s s e s a r e 3 e f i n e d i n
To a s f o l l o w s : -,=,g[f:x
(ii) ( X S ~ Y ) +Zf[f:X
+YAg:Y + X A V X c x ( g ( f x ) = x ) A V Y E Y ( f ( g y ) = y ) ]
+Y
A 'dx , x
1
2
~ x ( f x ~ = - ft xx -~x 1- , ) I
(iii) ( X < 2 Y ) + Z g [ g : Y - t X A VxcX ?IyceY(gy=x)] (iv)
( x < ~ Y )+ Z Z ( Y -
X+Z).
CONSTRUCTIVE THEORIES OF FUNCTIONS A N D CLASSES
205
.
Xi ) The statement of ____ Cantor-Bernstein i E (0,ll c a n be g i v e n i n one of t h r e e forms corresponding t o ( i i ) - ( i v ) :
Xo + X I
(The o p e r a t i o n
is
(CBli
Y
X li
A
Y
- Y.
+X
I t w i l l be shown t h a t each of t h e s e s t a t e -
The converse i n each c a s e i s t r i v i a l .
ments i s c o n s t r u c t i v e l y unprovable, by s u i t a b l e independence arguments. such r e s u l t s were o b t a i n e d by van Dalen
1968 i n
The f i r s t
t h e i n f o r m a l framework of Brouwer's
t h e o r y of f r e e c h o i c e sequences where maps between s u i t a b l e t o p o l o g i c a l spaces a r e n e c e s s a r i l y continuous.
We g i v e d i f f e r e n t arguments h e r e f o r t h e framework of
Independence of CB
11.2
c u r s i o n - t h e o r e t i c analogue of Let
X =IN
and
Y
5
W
Let
onto
Y ,
63 = ( 8 , CJ,
Y
can be chosen
otherwise
Y
E
be a model of
)
8 of ordinary r e c u r s i o n t h e o r y i n $ 3 .
BJ
any member of Ce with
enumerable ( e . g . such from
v
.
CBl
b u i l t from t h e S t r u c t u r e
T o + D V + CA2
This i s by f a i l u r e of t h e r e -
from T + D + C A 2 .
1-0
T
Y
but
Y
not recursively
There i s no map i n t h e mo3el
co-r.e.).
would be r . e . ; t h u s
IN
%
Y .
11.3 Independence of CB2 f r o m T o + C A 2 . Here we u s e an example from van 1968 b u t apply §lo i n s t e a d t o g e t t h e independence r e s u l t . Let X be 2 N considered as a t o p o l o g i c a l space, and Y = X + E where E c o n s i s t s of a s i n g l e p o i n t , t h u s i s o l a t e d i n Y . There a r e continuous maps F : X + Y and G: Y + X .
Dalen
Let
S=X+Y
F,G
represented i n
V=CaSa.
an3
onto
5
Thus i f
=
(F,G). By $11 we form a model U3= (8, CJ,
onto of To+CA2 w i t h
k x ( f x ) i n 'M being p a r t i a l continuous on
an3 every
Ql, X-Y
6)
i n t h i s model we would have
X
X+E,
homeomorphic t o
which i s f a l s e . Question: I s CB independent from T + D V +CA ? T h a t would of course f o l l o w i f 2 2 t h e r e c u r s i o n - t h e o r e t i c analogue of CB2 i s f a l s e .
11.4
Independence of
Hanf ( c f . Hahios but
X+X+Z,
y=X+Z
that
where X-Y+Z
we form a model
-
1963)
63 of
CBJ
from
V.
X-X+Z+Z
It follows f o r
f a i l s f o r t h i s topological interpretation. over
X
+Y
S=X+Z+Z
Now
by §lo, i n which t h e maps
a r e i n c l u d e d and every map i s continuous
i n t h e sense of c a r d i n a l equivalence i n t h i s
1968 can a l s o
The r e c u r s i o n - t h e o r e t i c analogue of
I n t h e i r argument t h e f a c t t h a t t h e u n i v e r s e way.
with
so CB3
(Another example of van Dalen
Remark.
X, Z
To+C$
By H a n f ' s r e s u l t ,
model.
Here we u s e an example due t o
i s t h e r e l a t i o n o f being homeomorphic.
g i v i n g t h e homeomorphism X - X + Z + Z on
.
To+CA2
of a p a i r of t o p o l o g i c a l spaces
CB3 V
be adapted t o t h i s p u r p o s e . )
2 true is
IN
by Dekker-IQhill 1960.
i s used i n an e s s e n t i a l
T h i s i s an example of a p o s i t i v e r e c u r s i v e analogue of a c l a s s i c a l s e t -
t h e o r e t i c r e s u l t which i s not subsumed Question.
Is
CB
3
independent from
under a theorem of
T o + D V + CAP ?
To
o r even
To+CA2.
206
S.
FEFERMAN
l2. A model of t h e weak power-class axiom.
I n t h e preceding we mostly chose
d i f f e r e n t models of APP t o g e t v a r i o u s c o n s i s t e n c y and independence r e s u l t s ; t h e only exception was i n s2. Here we modify t h e c o n s t r u c t i o n of CJ? and E so as t o g e t a model of POW.
This w i l l a l s o s a t i s f y
( R e c a l l i n c o n s i s t e n c y of POW
J.
Let code
with
EM + D +IG
o
2l be t h e r e c u r s i o n - t h e o r e t i c model of f o r t h e “ c l a s s of a l l classes‘’
ff
C&
simultaneously we f i r s t d e f i n e
.
and t h e n
u Ce
Put
APP+DV.
u
=
n<w
ten
, and
Cdn
.
For a
X €
m
u X E W ,
4 ten x
and f o r
a,r ECJ
€
Cda,
( T h i s procedure would not be
E .
d,
with
CAn
E
E ~ + <+ ~ x c E
c
We i n t r o d u c e a new
Take
an3
C a o = {l”,E]
elementary and
al,
..., a m €C e n )
I.
( i ( a , r ) l a , r ECJ?
For c E C ~ , x aic
‘@(x,r,zf
ClnU(c ( y , a ) / k = k--
=
b u t not t h e j o i n axiom
Now i n s t e a d o f d e f i n i n g
p o s s i b l e i f c l o s u r e under j o i n were r e q u i r e d . ) Centl
v
from 1 1 . 1 4 . 1 ) .
J
.
c
we d e f i n e
x Ena
X E 0
c
For
c = c (y,a), k k--
as follows:
‘ 3 X E C l . =
‘$(x, y, 2)’
, with
the
we p u t
ntl c
and
<+
(21, CJ?
c=i(a,r)
,E n ) I = N x , r with
c
4
>
a)
Cln
i
we p u t
‘ n + l c ~ V I ~ w ~ [ V u ( u ~ ~ a A t l w ( (+ ww ,s uI )) ~- > ~ ur E I ] + x E I ] .
It may be seen t h a t t h e r e s u l t i n g model s a t i s f i e s
§ 3 ).
E M o + D V + I G ( p l u s CT:,
as i n
Furthermore i t s a t i s f i e s VX[XE
Given any
A
a:
<-tYX(X=X)l.
we can f o r m a weak power c l a s s
P(A)
of
A
by t a k i n g
P(A) = [fAblb E Q 1 where fAE3 = A P B . Remark. One can a l s o a r r a n g e t o s a t i s f y CA2 by u s i n g t h e method of s 2 . c a n ’ t be d e r i v e 3 from
CA2
IV 1.
Background.
CA
1
+ P O W without j o i n .
Realizability interpretations
The d i s t i n c t i v e e f f e c t of r e s t r i c t i o n of t h e l o g i c Lo be i n -
t u i t i o n i s t i c i s of course not shown by s t a n d a r d models of t h e kind considered i n t h e p r e v i o u s p a r t 111.
The following a r e some s p e c i a l p r o p e r t i e s which a r e
t y p i c a l l y enjoyed t o some e x t e n t or o t h e r by v a r i o u s i n t u i t i o n i s t i c t h e o r i e s (ED),
some term
T
T
property (iv)
T
T,
we have
(ED)m
T
k @(T);
holds, i . r . i f
and f o r T
ksI.d,(n)
k d,
k ( @ V $ ) then
The d i s j u n c t i o n p r o p e r t y (DP), i . e . i f ( i i )the e x i s t e n t i a l d e f i n a b i l i t y p r o p e r t y (i)
i . e . if
’ I
T
1E X @ ( X )
c l o s e d under Church’s Rule
CR,,
i.e. i f
T
k$;
T:
t h e n for
c o n t a i n i n g a r i t h m e t i c : (iii) t h e t h e n for some ( s p e c i f i c ) n, T
i s c o n s i s t e n t w i t h t h e schematic form of Church‘s t h e s i s T
or
1Vn am$(n,m)
b@(n);
CTo ; ( v ) T
is
t h e n f o r some ( s p e c i f i c )
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
e, T k V n @ ( n , [ e ) ( n ) ) ;and f i n a l l y f o r
T
207
containing f u n c t i o n v a r i a b l e s : ( v i )
i s c o n s i s t e n t with various forms of t h e axiom of choice
AC,
T
and i s closed under
While, as remarked by K r e i s e l and T r o e l s t r a , t h e s e
corresponding choice r u l e s .
p r o p e r t i e s a r e n e i t h e r necessary nor s u f f i c i e n t f o r
T t o be constructive, much
of t h e metamathematics of c o n s t r u c t i v e t h e o r i e s revolves around t h e i r v e r i f i c a t i o n . The b a s i c methods t o o b t a i n such r e s u l t s a r e by r e a l i z a b i l i t y i n t e r p r e t a t i o n ~ . ' ~ ) These were introduced by Kleene i n 1945 with h i s notion of recursive r e a l i z a b i l i t y . Many extensions an3 v a r i a n t s have s i n c e been applied, due t o Kreisel, T r o e l s t r a , Be Jongh, J . R . Moschovakis
, Friedman,
Beeson and others. A
r a t h e r complete survey can be found i n T r o e l s t r a 1 9 3 Ch.111 or T r o e l s t r a l 9 7 a
54; it may be h e l p f u l f o r t h e r e a d e r t o look at t h e s e references i n connection with t h i s p a r t .
It i s u s e f u l t o d i s t i n g u i s h formal or i n t e r n a l r e a l i z a b i l i t y from informal
or e x t e r n a l r e a l i z a b i l i t y i n t e r p r e t a t i o n s , though very o f t e n t h e s e are coupled.
e(T) a new formula @ with
I n t h e former one a s s o c i a t e s with each formula @ of one a d d i t i o n a l f r e e v a r i a b l e
f, written
r e l a t i o n between mathematical o b j e c t s
f r
r e c u r s i v e r e a l i z a b i l i t y was of t h i s type: f
EW
and formulas of a r i t h m e t i c . )
@. I n
t h e l a t t e r one defines a
0. (Kleene's
of some s o r t and formulas
f
he defined a r e l a t i o n between numbers
External r e a l i z a b i l i t y i n t e r p r e t a t i o n s can
o f t e n be regarded as t h e reading of a formal
i s t h e approach we s h a l l take here.
f r @ i n a s p e c i f i c model M; t h a t I n any case t h e i d e a of f r @ i s t h a t f
packages t h e c o n s t r u c t i v e information (witnesses, p r o o f s ) which v e r i f i e s
@
; the
d e f i n i t i o n s a r e thus c l o s e l y r e l a t e d t o t h e informal i n t e r p r e t a t i o n of t h e l o g i c a l connectives i n 1 . 4 . 2 . By a r e a l i z a b i l i t y i n t e r p r e t a t i o n
@ w f r @ with each formula @ t h e language of
T')
s o r t of v a r i a b l e of
g
2.
T
such t h a t
Formal r e a l i z a b i l i t y of
1975;18)
E(T') T)
i s meant an a s s o c i a t i o n of a formula
having a t most one a d d i t i o n a l f r e e v a r i a b l e
S(T)
f.
f r @(of (Thus every
P ( T ' ) . ) This i f f o r each theorem @ of T we
must a l s o be included among those of
i n t e r p r e t a t i o n i s s a i d t o be sound f o r T have a term
&I
S(T)
(of t h e language of
T' k
( T
&
T'
r@).
X(To)
in itself.
This w a s introduced i n Feferman
v a r i a n t s from Feferman 1976b and Beeson 1977 will be explained below.
When @ i s w r i t t e n
@(x,5 )
we w r i t e
a d i s t i n g u i s h e d v a r i a b l e as i n
f r
@(x,X)
Ex @ we may w r i t e
for
f r @; when concentrating on
f r ( B @(x))
.
The i n t e r p r e -
t a t i o n i s deflned i n d u c t i v e l y as follows:
17) Another'method t o o b t a i n some of t h e s e p r o p e r t i e s i s due t o Kripke; c f . Smorynski's chapter on Kripke models i n T r o e l s t r a 1973. These models will be a p p l i e 3 a t one p o i n t i n P a r t V below.
18) It w a s pointed out by Beeson t h a t t h e c l a u s e t h e r e f o r d i s j u n c t i o n needed correction, as given i n (iii)below.
,
S. FEFERMAN
208
When it is necessary to distinguish this from other realizability interpretations as r l .
to be defined later, we shall subscript this r equivalent to trz
(2.9).
7
Note that fr(7 $)
is
3.
Essentially (V,S)-free formulas. This class of formulas are such as can be realized in a canonical way (if at all) and for that reason play a distinguished
role. We call $ essentially (V,S)-free if it is built up from formulas of the form ( T I ) , C.~?(T), ( T E X ) and ( ~ ~ - 7by ~ A) , -+ and V applied to either sort of variable. Note that the existential information in the first three formulas, written as & ( T = x ) , ~X(T=X) and application term (1) For each (2)
9,
ZX(T=-XA
x EX) can be represented by the
itself. The following lemmas are easily established for r = r 1' the formula (fr@) essentially (V,B)-free.
T
With each essentialu (V,B)-free $ g associated a term T with free variables @-contained in those of $ such that APPN /- [ $ + ( T $ r $) 1.
(3) If @
essentially (V,S)-free then APPm
b [ (fr@)
+
91.
Here (2) and (3) are proved by a simultaneous induction in order to take care of the case of implication, where we put Because of ( 2 ) we call T(@*$) = k(T*). T+ the canonical realizer of $ for ess. (V,B)-free @. Remark. Formulas of the kind that we call essentially (V,S)-free are often called
almost negative in the intuitionistic literature. -~
4. The scheme 'To assert is to realize' . This scheme consists of all formulas of the following form:
(A-r) 0 t, (3f)(fr@) which expresses equivalence of the assertion of $ with its realizability. By (2), (3) of 53 each instance of ( A - r ) in which 4 is ess.(V,S)-free is derivable in APPN . The scheme as a whole is itself realizable: (1) for any formula $ we can find a
A P P ~l - T r [ @
f-)
T
Such that
~(fr$)I.
19) The clause for (@-++) does not completely mirror the requirements for a constructive proof as expressed in 11.4.2, which calls for constructive recognition of Vz[ zr$-t(fz)r @ ] when zro is read ' z is a proof of $' ; similar remarks a p r i to the universal generalization cases.
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
For t h e proof one d e f i n e s
which r e a l i z e each implication.
T1,T2
t h a t ( z r 9 ) i s e s s . (V,Z)-free and so has a canonical r e a l i z e r .
Remark.
Thus
T~
i s t o be
Vz[zr@ + ('rlz)r (3f ( f r @ ) ) ] . This makes use of t h e f a c t from $ 3
chosen so t h a t construction
209
The converse
i s equally easy, and uses t h e f a c t ( 3 ) from $ 3 .
T~
( A - r ) i s suggested by t h e b a s i c t e n e t of c o n s t r u c t i v e reasoning I . k . 2 ,
t h a t a statement i s t o be a s s e r t e d only i f it i s proved.
Note.
It may be necessary t o d i s t i n g u i s h ( A - r ) f o r d i f f e r e n t r e a l i z a b i l i t y i n t e r -
For example we w r i t e (A-rl)
pretations.
5.
Axioms of choice.
f o r t h a t of $2.
The most general scheme considered here f o r t h e axiom of
choice t a k e s t h e form: v.sX3.y@(x,y)
(AC)
f o r v a r i a b l e X.
S p e c i a l cases of t h i s can be formulated f o r each term A\
denotes a c l a s s , e.g. t h e scheme t o
+wvxEX@(X,fX)
N , Tl jI,e t c .
We w r i t e ( A C ) A
There i s a c l o s e connection between t h e schemes
X = A.
which
f o r t h e r e s t r i c t i o n of (AC) an3
(A-r); we have: (A-rl)
(1) Vx[x E X +
For suppose
E
1,
X
(AC).
(A-r) f o r
@ ( x , y ) l . By
Vx,z[zr(x E X ) + ( g x z ) r S y $ ( x , y ) ] . ( x E X ) + or ( x
implies
pl(gxO ) r @(x,p,(gxO))
so
r = r l we can f i n d
g
such t h a t
But by c l a u s e ( i )of t h e d e f i n i t i o n of
.
But then
r1
,
@(x,p2 ( g x 0 ) ) holas,
f = Xx(p2 ( g x 0 ) ) i s a choice f i n c t i o n . (Note t h a t t h e proof j u s t u s e s t h e
so
ApP axioms.)
It may be of i n t e r e s t t o t h e reader t o s e e which i n s t a n c e s of (A-rl)
a r e implied by ( A C ) ; t h e y form a wide c l a s s . While full (AC) w i l l thus be r e a l i z e d i n t h e r l - i n t e r p r e t a t i o n ,
ri
not hold f o r o t h e r
t o be considered.
this w i l l
A s p e c i a l consequence of (AC) which
w i l l be r e a l i z e d even when (AC) i s not, i s t h e axiom scheme of depenaent choices:
a
(DC)
V ~ E X
Using t h e axioms standard way.
X @ ( X , Y ) +vx0
E
we can d e r i v e (DC) from (AC) i n e s s e n t i a l l y t h e
APP+W
Namely, given
g
such t h a t
by p r i m i t i v e recursion t o s a t i s f y f u l l in3.uction on The theory
of
To
CAl(i.e.
ECA).
n = r@(x,y,
27
such t h a t
have
that
fO
Vn[fn L A
= x0 ' @
Vx
E
X [gx EX
(fn, f n ' )
@(x,gx)1 we define
A
f n ' =. g ( f n ) .
1.
f
Then it i s proved by
TA-) . We do not have a soundness theorem f o r rl - r e a l i z a b i l i t y i n i t s e l f . The problem a r i s e s with t h e elementary comprehension scheme
6.
w
I
wEXm[fO = x o h ~ n ~ ( f n , f n ' ) ~ .
EX
realize
TO
%lcn(y,z) =
x
A VX[XE
we have t o show how t o convert any
wr(x
E
wr(x E X ) + x
X) E
X
.
x
u
(j
b(x,n, z)]) f o r
with
,z) i n t o a
ur @(x,y
x = c (y,Z) and conversely. But f o r r = r n 1 we Thus we would have t o o b t a i n @ .$ Zu(m i @ ), which i s
where
only g e n e r a l l y t r u e f o r e s s e n t i a l l y
(V,
9 ) - f r e e formulas.
However, t h i s d i f f i c u l t y
210
S. FEFERMAN
suggests an obvious modification of only f o r e s s e n t i a l l y APP+CA!-)
+ IN, and by
We claim t h a t
as
0.
(V,3)-free
EML-)P
--Serves t o
EM(-)p 0
mop ( I I . l o ) ,
similarly
t o a scheme
CAl
By
, the
,
CA!-)
where
i s taken
CA1
we mean t h e axiom system
EM:-)
same theory with induction on
restricted.
W
o b t a i n t h e same mathematical consequences
for
m . i F p G ,mL-) + J i n p l a c e of
EMo + J, resp. (11.11). The reason i s very simple:
BCM
m0r + J,
i n t h e f o r m a l i z a t i o n of BCM
by following Bishop’s o f f i c i a l d e f i n i t i o n s , we never make e s s e n t i a l use of
V
or
3 i n d e f i n i n g p r o p e r t i e s of s e t s - s i n c e t h e witnessing information i s always required t o accompany t h e p r e s e n t a t i o n of t h e elements of t h o s e s e t s ( r e c a l l 1.15). Hence
CAi-)
always s u f f i c e s i n p l a c e of
The only d i f f e r e n c e appears
CAI.
when we e n t e r t h e theory of o r d i n a l s and Bore1 s e t s (11.12).
Here one must make
a s l i g h t modification i n t h e I G axiom t o achieve t h e same r e s u l t s .
For example,
where A = { x / x = O V x = ( px ) + V x = s u p ( p 2 x ) ) , an8 W 2 x = s u p ( p2 x ) A X n ( y = p2*x n ) ) . Thus R i s defined N 2 2 3 i n a n e s s e n t i a l way. W e now mojify I G t o IG(-) by t a k i n g i(A,S) = I
previously we took Q l = i ( A , R ) R=((y,x)/(x=(p2x)+Ay=p2x)V using
t o satisfy instead V X C A A ( V Y , Z [ ( ~ , Y , X +) CYSC
II+
x E I )
as t h e c l o s u r e axiom and t h e n t a k i n g a corresponding i n d u c t i o n p r i n c i p l e . has t h e same e f f e c t a s t h e previous I G with TO
=
.
+ J +IG(-)
EM:-’
(y,x)
It is t h u s seen t h a t
&
mathematical consequences
as
BCM
E
R &3Z[(Z,y,x)
S
E
1.
This Let
serves t o o b t a i n t h e same
T 0( - )
X-Tas
To (11.10 - 11.12). (The theory
intro8uced i n Feferman 1 9 5 , where t h e soundness r e s u l t of t h e next s e c t i o n w a s outlined.)
7.
rl
Soundness theorem f o r
- realizability
of
It i s u s u a l l y
in itself.
TL-)
a r o u t i n e matter t o v e r i f y s o u n b e s s of t h e axioms and r u l e s of i n t u i t i o n i s t i c
1973
l o g i c f o r any reasonable r e a l i z a b i l i t y i n t e r p r e t a t i o n ( c f . T r o e l s t r a For t h e p r e s e n t
rl
interpretation,soundness of t h e l o g i c a l p a r t of
easily verified using the
TL-)
Ch.111). is
APP axioms t o provide t h e r e q u i s i t e c o n s t r u c t i o n s .
Going on t o t h e non-logical axioms, it i s s t r a i g h t f o r w a r d i n each case t o v e r i f y s o u n b e s s of each axiom o r scheme on t h e b a s i s of t h e corresponding p r i n c i p l e s themselves.
The reason i n t h e c a s e of
by u s e o f t h e p r o p e r t i e s of e s s e n t i a l l y case of t h e induction scheme on @ ( 0 )AVx(@(x)+ @ ( X I ) ) zl
with
z o r @ ( 0 ) and uO=z
induction on
and
i.
has already been explained i n § 6 ,
(V,3)-free formulas from $3.
we are r e q u i r e d t o g i v e a
LLX’
i s defined by r e c u r s i o n (uniformly from
t h i s way we conclude t h a t t h e following- holds f o r
Q
zo,zl)
to
z1~ ( u x ) and t h e r e q u i r e d conclusion i s proved by
I
One proceeds s i m i l a r l y f o r t h e r e a l i z a t i o n of
(1) with each theorem
I n the
which r e a l i z e s
T
+ V X ( X E ~ 4 @ ( x ) ) . ’This must show how t o convert any z0 ’ Yx,w[wr$(x) (z,xw)r$(x’) 1 i n t o some u = ‘ r z O z l w i t h
VX[XGIN + ( w ) r ( p ( x ) I ; u satisfy
7N
CAi-)
r
=
IG(-)
.
In
r-:
f TL-) can be a s s o c i a t e d a t e h
‘r
such t h a t TL-)b(‘rr@).
CONSTRUCTIVE THEORIES O F FUNCTIONS AND CLASSES
211
EM:-+,
Corresponding r e s u l t s can be o b t a i n e d f o r t h e v a r i o u s s u b t h e o r i e s
, ..I:-’+ J
EM:-)
I n a d d i t i o n , i f we t a k e
which have been considered.
b e t h e r e s t r i c t i o n of
CA(-) t o 2 ( V , S ) - f r e e s t r a t i f i e d formul.as, we
t o essentially
CA2
get:
r
(2)
1
- realizability
Now as we have seen i n § 4, t h e scheme
(A-r)
t r i v i a l way; moreover t o realize
I
T L - ) +,A:-)
i s also s o u 3 for
(AC)
r -realized i n a
i s itself
(A-r)
implies
i n itself.
by
i n a c o n s i s t e n t theory, s i n c e
F i n a l l y , it i s impossible
$5.
<+ I .
(fr,i)
W e may t h u s conclude
that:
r - r e a l i z a b i l i t y i s sound f o r Tb-’ + ( A - r ) & Tb-) and t h e same holds f o r 1 TL-) + CAL-) + (A-r) T i - ’ + ,A$-) ; hence TL-) + CA(-) + ( A - r ) an3, (there-
(3)
2
-fore also)
+ ,A:-)
TL-)
consistent.
+ AC
Coupled with t h e f o l l o w i n g we s e e now t h e metamathematical power o f r e a l i z a b i l i t y i n t e r p r e t a t i o n s a p p l i e d t o t h e o r i e s d e a l t with i n t h i s paper emerging.
8.
Consistency of Church’s T h e s i s
T
Then
CTo
consistent with
z a b i l i t y i n a model of
(CTb).
, by
i s r e a l i z e d i n t h i s model by s u i t a b l e
which d e s c r i p t i o n
f
=
Tk-) + C A i - ) + ( A - r ) We r e a d
for
r=r
r -x‘eali1
f.
$
T h i s i s t o convert any
g
realizing
We t a k e
(e](n)
=
p 2 ( g n ) f o r a l l n , from
i s o b t a i n e d very simply.
Closure p r o p e r t i e s of
T ( - ) and r e l a t e d t h e o r i e s .
9 . 1 r2 - r e a l i z a b i l i t y . I n o r d e r t o o b t a i n t h e d i s j u n c t i o n and e x i s t e n c e p r o p e r t i e s f o r TL-) we modify r1 i n a manner due t o Kleene, ( c a n e d q - s zability,
c f . Troelstra
1973
p.
189) ;
t h i s i s h e r e denoted by
r2.
Only t h e
following clauses a r e varied: ( i i i ) ’ [ f r ( ~ V J , ) 1 = [ p l f ~ F l A ( P l f = O - , 9 ) A ( ~ 2 f ) r $ ) 1A [Plf
$)I
(iv)’
[ f r ( @+
(vi)’
[fr 3 x K x ) I = [ N p 2 f )
( v i i i ) ’ [ f r RX
$(X),I
= VZ[$ A
=
(2.Q)
+
f 0 - q
A (p2f)r+1
(fz)r$I
A (plf)rQ(p2f)1
.
Cd(p2f) A $ ( p 2 f ) A ( p l f ) r @ ( p 2 f )
I t i s e a s i l y checked t h a t
the soundness -)
each he_ s u b -_ t h e o r_ ies of To -of -tr e a l i z e d j u s t as b e f o r e .
1
t h a t Yn3m@(n,m)+SeVn$(n,(e](n))
r e a l i z i n g S e Y n @ ( n , ( e ) ( n ) ) . Now from t h e hypothesis
V n [ ( p l ( g n ) ) r @ ( n , p 2 ( g n ) )1.
we have
9.
(fg)
T
whose a p p l i c a t i v e p a r t i s t h e o r d i n a r y recursion-theo-
T
It i s t o be shown f o r any
r e t i c model of 111.3. VnSm@(n,m) i n t o
Let
t h e f o l l o w i n g argument.
+CA2
theorem f o r r 2 - r e a l i z a b i l i t y h o l d s f o r considered.
I n addition,
(A-r)
2 r2 -
S. FEFERMAN
212
9.2
The ED property.
Suppose
i s any theory f o r which we have soundness
T
of r - r e a l i z a b i l i t y e.g. any of t h e t h e o r i e s j u s t indicated. 2
t h e r e i s a term T
have
such t h a t
T
@(p27). Thus
9.3
The
DP
T
and
1 Tr(3x@(x)).
T
properties.
EDm
197.2 0 )
soundness of
r - r e a l i z a b i l i t y of 2
Let
that i f
T
T
/-
For t h e s e we need a s p e c i a l argument
o
(7
in
T.
T
7,
lary that
(p2 7 ) o m A @(p27 ) .
N) then f o r some n,
T
has t h e
Hence
within
r
T.
Inconsistency of
It w a s
T.
mC
1- ( 7 =ii).
(The proof
it follows t h a t f o r
property; it i s a corol-
EDm
property.)
DP
i s t o formalize t h e p r o p e r t i e s of
10.
APP
I t i s a l i t t l e more work t o o b t a i n c l o s u r e under Church's r u l e .
Remark. T
T
T /-3n@(n)
has t h e
T
f o r which we have
Assume a l s o t h a t
made use of t h e model NF of normal terms.) Now i f some
T i - ) + CAL-)
b e a subtheory of T
we
r2
enjoys t h e e x i s t e n t i a l d e f i n a b i l i t y property.
due t o Beeson
s t a t e d i n 111.8.2
Then i f T 1 3 x @ ( x )
Hence by c l a u s e ( v i ) ' f o r
with AC. -
*o
corresponding p r o p e r t i e s f o r
To.
b i l i t y i s not sound f o r
since
To,
- realizability 2
One way
of any f i n i t e subtheory of
We next t u r n t o t h e question of obtaining This s e c t i o n shows t h a t i s i n c o n s i s t e n t with
To
r -(or r2-) realiza1 AC. This w i l l l e a d
us t o consider a new r e a l i z a b i l i t y i n t e r p r e t a t i o n (which does not v e r i f y f u l l A C ) . The proof of c o n t r a d i c t i o n of given f o r
To + Dv + A C ) .
using t h e conventions i.e.Xn(xx
= n).
Let
E
X
E
N
N Aff d f f
E
E
ll.l The theory
1
(f
C f To
.
TZ
X).
E
Note t h a t Vx[xxeN
.
The language
E
A
T
X
9".
If
P. i n t o C* T
TZ
A)
i s existentially
i s obtained by r e f i n i n g t h e
+ J
.
If
@ i s any formula of
i s any theory i n t h e language
i s denoted
(x
E
of
A)
, @*f o r
It should be noted f o r comparison with Beeson
mL-1
f0
ACX
I n t h i s language we t a k e
we mean t h e theory whose axioms a r e exactly t h e
and
X]
Hence
* . of
P.*
( x €,A).
edef 32 (x EZ
which gives a t r a n s l a t i o n of
m
AC
In
+ X E
IN.
as follows: i n s t e a d of t h e two-placed r e l a t i o n x
i t s t r a n s l a t i o n by
TL-)
v i a a refinement
To
we now have a 3-placed r e l a t i o n
noted
Hence i f
N). It follows t h a t ~ ( f Ef N ) . But t h e n by l o g i c f f oIl\T +ff
11. R e a l i z a b i l i t y f o r
20)
n].
which i s a c o n t r a d i c t i o n t o our o r i g i n a l assumption, namely t h a t
defined, which i s not p o s s i b l e i n
language
f
using definition-by-cases on
It i s of course e s s e n t i a l f o r t h i s argument t h a t
holds.
Recall here t h a t we a r e
@ ( x ) ) and (xx o w ) eEy(xx r: y Ay o m ) ,
v X f 3 3 n [ x x o m +xx
so
n fxx
N + xx f n ) ) .
E
A
Vx E X [ ~ X E IAN( x x oIN +xx f f x ) ] .
such t h a t
f
holds s i n c e we can always f i n d f
3n(xx
T r i v i a l l y by d e f i n i t i o n
particular, f o X + f f
so
(XI
3 n @ ( n ) c, & ( x
i s assumed we can f i n d an
l ( x o X ) + l (xx
i s by an argument due t o F r i e b a n ( o r i g i n a l l y
To+AC
X=
m.
2
of
a l l axioms
197 t h a t
TL-)
d: we denote To
, by
@ of
T*
T.
i s t h e r e de-
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES T r a n s l a t i o n of
11.2
simple way.
1"
into
With each formula
of
213
1. This i s accomplished i n t h e following S* i s a s s o c i a t e d a formula $ of 2 ,
-
which i s obtained by replacing each atomic formula
( x cZA) by
[ ( x , z ) c A]
which, except f o r some changes of constants, i s otherwise unaffected.
and
Each of t h e
- --
k, s,d,p,pl,p2,0, sm ,pm i s unchanged, but t h e c l a s s f o r c k , j , i a r e replaced by new constants c k , j , i as w i l l be ex-
combinatory constants mation constants
plained i n t h e next section.
"r- r e a l i z a b i l i t y .
11.3
which w i l l make a l l of expressed i n
w r Q) ( x )
CA
such t h a t from any
and conversely.
cn = '$(x,y,Z)'
of
cn,j,i
3X[x c X c, o ( x ) ] as
1 i . e . as
with
!B[3z((x,z)cx) ct $ ( x ) ] we can f i n d a
(x,z) c X
w e ' l l have
w with and
It i s here where t h e change of constants e n t e r s ;
-c n =
-8
(p,x)r
$(p,x, y,?)'
.
I,-r e a l i z a b i l i t y i s defined as follows; t h e t r a n s l a t i o n s
For t h i s purpose
a r e given by a simultaneous i n d u c t i v e d e f i n i t i o n .
Q)
fr
t h e c l a u s e s defining
z =w
The simplest way t o achieve t h i s i s t o t a k e
X = ((x,w)lwr $ ( x ) ) .
if
z
1)
S* (and thence of
The i d e a t o r e a l i z e
and t h e n t r a n s l a t e d back i n t o
.C*
i s t o produce an X
thus t o take
This i s an i n t e r p r e t a t i o n of
realizable.
@ i n S*
for
F i r s t we w r i t e down
rl
e x a c t l y l i k e those f o r
in $2.
The only d i f f e r e n c e appears i n t h e f a c t t h a t we now have atomic formulas i n p l a c e of t h e o l d
2,
(x
A ), so we read. [ f r ( x c Z A ) ] = ( x e Z A ) .
E
^r where
t o be
(f?Q))
-
($$),
=
i . e . we t r a n s l a t e t h e r e a l i z a b i l i t y i n t e r p r e t a t i o n of
"
( x cZA) Then we t a k e
*
C
j u s t described.
Now
a r e chosen t o s a t i s f y t h e following:
cn, j, i
r
-
Q)(x,Y,z) , c n ( y , z ) -1
(i)
if
n=
(ii)
if
V x ~ A [ C t ( f x ) ] then
(iii)
z(A,R) = i ( q , R 1 ) and
= ((x,z)Iz~Q)(x,y,~)l
j ( A , f ) =(((x,y),(z,w))I(~,~)~AA(y,w) e f x ) , an8
where A1 = ( ( x , ( z , f )
)I (x,z)eAAvy,w t ((y,x),w)c R-rf(y,w)J 1 1
R1=(((Ytfyw),(x,(z,f)))/((Y,x),w)
E
Rl.
The choice of t h e constants i s made i n such a way t h a t f o r each of t h e axioms from T
CAl
7 Q) i s
(or CA,),
J and I G as expressed i n
g*
we can f i n d a
T
4
such t h a t
provable from t h e corresponding axiom ( o r axiom schema) as expressed i n 2 .
It follows t h a t f o r any theory
T
over
which i s based on some combination
APP
of t h e s e axioms and schemata p l u s t h e axioms N
theorem f o r Fl - r e a l i z a b i l i t y of ~w r i t t e n out i n
S", i s t r i v i a l l y
2
:T + (CA )* + (A-r)
(1)
2
Hence any consequence of translated into
T*
2".
(A-r)
T
.
F1 - r e a l i z e d . in 1
o r NF, we have a soundness Once more t h e scheme
(A-r),
as
It i s a c o r o l l a r y t h a t
2 consistent.
i s c o n s i s t e n t with
T o + (CA ) 2
regarded as
It i s simpler t o stu& such consequences i n t h e l a t t e r
S. FEFERMAN
214
To g e t f u r t h e r consistency with Church's
we simply couple t h i s with t h e r e c u r s i o n - t h e o r e t i c model as described
CTo
in $8.
Note:
G.
language than t o pass through thesis
Consequences ~f " t o a s s e r t i s t o r e a l i z e "
12.
EM: + ( A - r )
r
where
is
r1
for
1 2 . 1 Dependent choices. i n g way.
Suppose
Then t h e r e e x i s t s
i s derived from t h e s e assumptions i n t h e follow-
DC
E
g
which r e a l i z e s t h i s statement, so f o r each
A Ey E A $(x,y) holds, i . e .
provides us with a t r i p l e
xo
f i x some
A
We assume a t l e a s t
Vx
g(x,z) E
I.
g
throughout t h i s s e c t i o n .
S*
with
z
x
E O
%
(y,w,u) A.
Vx,z[x E
such t h a t
Then using
~ + A Ey,w(y~~AAJi(x,y)~
x,z
with x c Z A ,
ycwAAur$(x,y).
Given
we d e f i n e a sequence (xn,zn,un)
g
g ( x ,z ) 1 ( x ~ + z~ ~, + un) ~ , and x E A an3 n n zn u r $ ( X ~ , X ~ f+o~r )each n. Let f = A n . x . Then passing from t h e r i g h t t o t h e
by r e c u r s i o n such t h a t
l e f t s i d e of ( A - r ) we have
fO = xo A Vn[ f n
E
AA
Ji
(fn,fn' ) ] .
A s a c o r o l l a r y of t h i s and 11.3 we have t h a t
To + CA2 + DC
(1) Remark.
consistent.
The argument here b r i n g s out t h e reason why
s t r u c t i v e l y even where
AC
trx E ~ m b ( x , y ) w r i t t e n i n S* clude
can be d e a l t with con-
DC
c a n ' t i n t h e presence of f u l l comprehension.
as
~ x , z [ xs
Z f V x , z [ x ~ ~+ A@ ( x , f ( x , z ) ) ]
from
From
Z +~~ y ~ ( x , y )we l can merely c o i This i s t h e r e s u l t which w a s
(A-r).
r e f e r r e d t o i n 1.4.7. 12.2
Canonically r e a l i z a b l e c l a s s e s (choice b a s e s ) . We w r i t e
C(A)
for the
following formula: %[gz(X A
EZA) + X
sgxA
1.
i s c a l l e d canonically (or s e l f - ) r e a l i z a b l e i f C(A)
lent t o using
ACA g
,
i . e . t h e scheme of choice with base
and t h a t
Vx E A 'Jy b ( x , y ) .
Yx,z[x c Z A + @ ( x , f ( x , z ) ) ] . ACA
holds then from
conclude
C(W)
because
C(A)
For suppose
Then as j u s t remarked we f i n d
It follows t h a t
Vx eA@(x, f ( x , g x ) ) .
i s equivaC(A)
holds because
Conversely i f
ACm
Vx Q @ ( x , y ) + 3fVx$(x,fx)
i s a consequence of by . ( A - r ) .
DC.
Also
then a l s o
C(AXB)
NU
C(A + B )
hold and i f
C([x c A I @ ( x ) ) ) . This gives consistency of
c o l l e c t i o n of i n s t a n c e s of AC
and
for all
AC.
C(V)
holds
Furthermore, t h e p r o p e r t y C
formulas, f o r which we have found canonical r e a l i z e r s by then
S*) we
C(A).
closed under c l a s s c o n s t r u c t i o n s which can be defined by e s s e n t i a l l y C(A) and C(B)
holds
such t h a t
f
Vx E A B z ( x E ~ A ) (which i s t r i v i a l by d e f i n i t i o n i n
3gVx E A ( X egXA), i . e .
Now
A.
holds.
(The consistency of
u EFTS was noted by Beeson 1977.)
$3.
In particular, i f
@ ( x ) i s e s s . (V,B)-free
T + CA2 To
is
(V,R)-free
with
with an extensive (AC)FT
, i.e.
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
h:
215
12.3 The presentation axiom (Aczel). Call (A:h ) 5 presentation of A if A ; this is called a g g presentation of A if c(A*) holds. In-
tuitively, in a presentation each element x more than one way) by
* E A*
x
formation" that "verifies" x
of A
is represented
(in possibly
such that h(x*)=x ; x* contains "additional inE
A.
When we have a fliLl presentation, no further
information need be added. The presentation
PA
is the statement that
every A there exists a f u l l presentation (A*,h)
of
A.
for
This was introduced
(in a slightly different form) by Aczel in unpublished notes; he observed that it serves to derive the various mathematical consequences of framework, PA is a trivial consequence of and h(x, z )
=
(A-r);
(A-r).
In the present
we simply take A*=((x,z)~xE~A)
x.
12.4 Having your cake and eating it too with
(A-r) as an implement. In the
informal discussion of 1.15.3 the attempt to have one's constructive cake and eat it too was taken to be a matter of being casual about showing the witnessing information required by the official definitions. Here we can provide a theoretical framework to justify such practices simply by assuming
(A-r)
for r =rl
in
d.
In effect, the informal definition of a class A in the form A=(xj@(x)) gives rise to A* = ((x,z) I zr '$(x)) , which corresponds to the official definition. By $(x)
Rz(zr '$(x))
of CA
in 2.*
we have x
E
A c, S z [ (x,z)
E
allows us to take vX[x e Z A <-, zr @(.)I,
exactly the same as the full presentation of A
A*
1.
A realizable refinement
so that this A*
is
described in 12.3. F o r example,
if we define
~ ' = ( x I x ~ I R A ~ > O ( x ~1 > n ) ) wthan e have (IR+) *=((x,n)lxslRhn>OA(xn>-)1 ), just as required by the official definition. Using AC
in its weakened form
we can conclude that an inVx ~ A X y @ ( x , y +?If ) Vx,z[(x,z) ,A* +$(x,f(x,z))] + * verse function is defined on (IR ) knowing that Vx E IR 3y ~IR(x*y=l). Remark. T*+ (A-r) provides an alternative way of reading Bishop which is in some respects simpler than by
To, since one can formalize the informal mathematical
(Note: The same ends can be achieved by the presentation (A-r).) It is not meant by this that Tzf (A-r) is in direct accordance with Bishop's views; (that is open to discussion).
arguments more directly. axiom instead of
13.
C l o s u r e properties of
T o . In order to obtain the ED and EDm
(and hence DP)
-
properties for To, Beeson 1977 introduced a kind of combination of r-and qrealizability. His definition (loc.cit.pp.281-282) is complicated by the requirement to have a doubling (X,X*) of class variables, where the new variable j(*
is to correspond to the class of all
(x,z) such that zr(xsX). In addition,
Beeson also doubled individual variables. This does not seem to be necessary, and the following simpler definition is proposed for the same purpose. With each formula ~(xl, ...,xn,yl, ...,Ym) of L=2.(To)
is associated a formula
S. FEFERMAN
216
Remark (added in proof): Beeson has pointed out real difficulties with the proposed realizability of To which are met when looking for suitable reinterpretations of the constants. It is thus not known whether his definition can be simplified in any essential way to serve the same purposes.
14. Applications to continuity properties. Beeson 1977 has used the consequences of realizability such as for the theories T considered in the and T = T ~ ,to prove local continuity preceding sections 21) rules cf the following form, where A , IB are any closed terms for classes: LCR( A , IB).
If T proves A
is a complete separable metric space and
is a separable metric space an3 and Vx Vx Here "y
E
E
A Yy
then T proves that IB [Q (x,y) A "y is stable for x" ]
E)
are understood to be those Q(x,y) Q(X1,Yl)
A
and 3. A
x1 =Ax2 AY1 =BY2
if
Equality in
under the same hypotheses
b2' Y,). T proves that F A , IB
then
continuous. (Note that the hypothesis means F: A +lB Beeson's method of proof of
.
is the metric of /A
e 3. A (x1'x2) = 0 ; extensional properties
for which
As a corollary of LCR(A, B) one has: to IE
.
stands for V c > 0 36 > 0 V u E N ~ ( x ) 'JveN~(y)[~$(u,v)],
is stable for x"
a metric space is defines by
A
IB
is an extensional property
W Y y eB@(x,y) E
where N A (x) = ( u e A / d A ( x , u ) <
from -
I$
LCR(A,IB)
spaces with countable dense subset D
a function
T proves that F
2
and x x +F(x ) = F(x 1-A 2 i m 2 ~
uses the representation of complete metric
in the form of the Cauchy sequences from D.
21) The matter is actually more complicated: one must formalize within the T considered the corresponding results for all finite subtheories of T.
CONSTRUCTIVE T H E O R I E S OF N N C T I O N S AND C L A S S E S
T h i s allows one t o push t h e problem back t o v e r i f i c a t i o n of
LCR(34
(T?+ N ) Em @(x,m) i s proved, e x t e n s i o n a l , t h e n f o r each s p e c i f i c g E (W -3 ZN ) we can prove
Roughly, t h e i d e a i s t h a t i f
w e can f i n d an
EDm
E
This r e q u i r e s only a f i n i t e p a r t
TUDiag(g). by
Vx
m s.t.
(g(O),
...,g(n-l))
@ ( g , h ) i s proved from t h e same.
t h i s argument one g e t s t h e d e s i r e 3 r e s u l t .
217
--f
IN, I N ) . @ is
where
(g,m)
of
from
g. F u r t h e r
By f o r m a l i z i n g
Beeson a l s o has r e s u l t s on l o c a l
uniform c o n t i n u i t y r u l e s f o r compact spaces an3 a number of c o n s i s t e n c y and independence r e s u l t s concerning c o n t i n u i t y s t a t e m e n t s .
He has f u r t h e r extended t h e s e
t o o t h e r f o r m a l i s m s such as t h o s e o f Myhill and F r i e h a n , as p r e s e n t e 3 i n h i s c o n t r i b u t i o n t o t h i s volume. Discussion.
I n a sense, Beeson's r e s u l t s confirm Brouwer's i d e a s t h a t we should
be a b l e t o prove t h a t every r e a l f u n c t i o n on
IR ( r e s p . [O,l]) i s continuous
But t h e p r e s e n t r e s u l t s have t h e advantage t h a t t h e
(uniformly c o n t i n u o u s ) .
systems t o which t h e y apply a l s o have a s e t - t h e o r e t i c i n t e r p r e t a t i o n . be s u r e t h a t i f an e x i s t e n c e proof
Vx E /A 3y
E
So one can
jB @ ( x , y ) can be formalized i n BCM
t h e n it y i e l d s s t a b i l i t y o r c o n t i n u i t y of s o l u t i o n s which a r e t r u e i n t h e c l a s s i c a l sense.
But t h e
Often such r e s u l t s can be o b t a i n e d d i r e c t l y by ad hoc arguments.
c o n t i n u i t y results d e s c r i b e d c o n s i d e r e d as a p a r t of g l o b a l ( o r s y s t e m a t i c ) cons t r u c t i v i t y may f i r s t p o i n t t h e way t o what can be o b t a i n e d f x s p e c i a l problems.
I n o t h e r words, t h e g l o b a l r e s u l t s can s e r v e as t h e s t i m u l u s and p o i n t of d e p a r t u r e f o r mathematically i n t e r e s t i n g l o c a l r e s u l t s .
(Indeed t h i s has been t h e c a s e w i t h
Beeson's s t u d i e s of s t a b i l i t y phenomena i n t h e P l a t e a u problem.) Question.
EDm
The p r o p e r t y
depends e s s e n t i a l l y on n o t having
Dv.
But one
cloesn't s e e why t h e c o n t i n u i t y r e s u l t s should be d i s t u r b e d by i t s presence. Beeson's r e s u l t s on
extend t o
LCR
TotDV
R e l a t i o n s with subsystems of a n a l y s i s .
V.
1. I n t r o d u c t i o n an3 summary of r e s u l t s .
I n t h i s p a r t ( e x c e p t f o r t h e s p e c i a l $2)
we d e s c r i b e r e s u l t s which e s t a b l i s h t h e equivalence of c e r t a i n subsystems w i t h subsystems
of
Do
by some o t h e r arguments?
c l a s s i c a l 2nd o r d e r a n a l y s i s .
of
It i s assumed h e r e t h a t t h e
TO
r e a d e r i s f a m i l i a r w i t h t h e d e s i g n a t i o n s of v a r i o u s of t h e l a t t e r such as
(nE-C A ) ,
( B I ) of
1 1 (A, - C A ) , (El - A C ) ,
bar
induction. 22)
(A,
1
When
- CA), 'r'
1 (E2 - A C ) ,
we mean t h a t t h e p r i n c i p l e of f u l l i n d u c t i o n on induction. to 22)
T
2
We w r i t e
T
1- 2
( i . e . i f Con(T2)
t o mean t h a t
implies
as w e l l a s with t h e p r i n c i p l e
i s used f o l l o w i n g d e s i g n a t i o n of a theory
Con(T )
1
T1
IN
i s r e p l a c e d by t h e axiom of
i s proof-theoretically reducible
by a f i n i t a r y argument) and
T1= - T'?
For d e s c r i p t i o n s of t h e s e and some i n f o r m a t i o n about t h e i r i n t e r r e l a t i o n s h i p s c f . Feferman 1977.
S. FEFERMAN
218
if T1-( T2 and T2
5
T1
.
In connection with the following results one also has
much information about which sentences are conserved in one direction or the other; however, for simplicity we do not mention such for the most part. PA denotes classical Bano's arithmetic, HA=Heyting's arithmetic. (1)
m0r
HA, in fact EM
(2)
EMor L J
(3)
EMo
+
(4)
EMo!\
2J
(5)
mo
+
(6) To
=
1 ( Z -AC)r 1
E
is a conservative extension of HA.
PA
(<-AC)
J
J
E
r
+ IGP +
=
IGr
EM, + J
(<-AC)I 1 (C2 -AC) 1
+ IG
E
1 (II,-CA)r
5 (c2 -AC) +
(BI).
In all of these except the conservation result of (l), we can also include classical logic and the axiom Dv on the 1.h.s. Conjecture. T~
1
(c,-Ac)
The exact relationship in (6)is unsettled.
+ (BI).
Credits. The conservation result in (1)is due to Beeson 1979, by a Kripke-model argument outlined in the next section. The = in (1)comes simply from (2) an&
HA. Conservation of (Z$-AC)P over PA has been established by Barwise-Schlipf 1975 using recursively saturated models; it is also stated by 1 1 Friedman 1975 where conservation of (C2 -AC)P over (Ill- CA)P (for a certain class of sentences) is announced as well. (The method of recursively saturated the fact that PA
E
models has a l s o been extended to prove the latter in unpublished notes by myself.) PA and (C1 -AC)r (Ill-CA)P 1 have The proof-theoretical equivalences (<-AC)P 2 been established by Sieg. The result (3) is due to Aczel (unpublished); a new method of proof was found by myself (Feferman 1976 c). there to establish
2
in (2) and (4)-(6).
This method was also used The relations 2 in (4) and (5) are
due to Sieg 1977. Only outlines of the various ideas involved are given in the following. Detailed presentations of the proofs of these and related results will be found in the chapter by Feferman and Sieg in the projected volume "Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies" (for the Lecture Notes in Mathematics Series) which is to consist of contributions by Buchholz, Feferman, Pohlers and Sieg. 2. EMor is conservative over HA. We first describe the proof of an easier re-
sult from Feferman 1976 a : EM r\
2
classical -logic is conservative over PA. are formally modelled in PA by
To begin with, the axioms APP (even with D ) taking App(x,y,z) e+(x](y) applicative structure 9J.
= z.
v
Any model 9ll
of PA thus determines an
This is used to build a model
(a,
Ca, E )
of EMo/'
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
219
by the method of 111.1, but leaving off the clauses for i and j ; now the process closes off at w . and x c 0 N
c,
x = x , C&n+l
mentary and al, ..., am
E
u
u Cln, E = c n where CLo=(JN] n<w n<w C.lnU (ck(y, 5 ) I k = ‘$(x,g, 2 )I and $ is ele-
Namely, C.l =
C.ln 1
=
,
with x E
~
(U, Cln, cn) I= $(x,y,a).
<+
ck( +’, ~ 5 )
is coextensive with the domain of 101.
Note that ’sn may be non-standard and 34
It may be seen that for each A cC& there exists a formula $(x,g)
of arithmetic
such that for some choice of parameters y
$
Hence the induction axiom
.
in 101, Vx[x
E
A
<-t
‘D1
I=
( I N P ) is verified, and we do indeed have
(x,y ) 1.
(U, Ce,
E )
To conclude, conservation holds by the completeness theorem for the classical predicate calculus: if 8 is a sentence of arithmetic such
a model of EMo? that
(EMop )I 8 but
P A P 8 we can choose
7
8
and get a contradiction.
Now Beeson 1979 has shown EM p conservative over HA by an adaptation of this argument to Kripke models, using the completeness theorem for intuitio= ((!Ill) p P € P ’ Z) (U,CEn, cn) as just described to a ~. construction of ((Up, c & ~ , E~ ,~ ) ,, 5~ ) for each n and thence of a Kripke model ((a,,cep, Ep)pEp, 5 ) of mor
nistic logic in terms of the latter. Given any Kripke model 101
of HA one mo3ifies the construction of
.
Discussion. The significance of this result is given by 1.15.5, according to which EM
r
is adequate to essentially all of BCM except for the theory of
ordinals and Bore1 sets. A corresponding result had previously been obtained by Friedman 1977 (conservation of
Over H A
for fl>
sentences, strengthened
to f u l l conservation by Beeson 1979). Thus this portion of BCM does not really take advantage of the strong constructive principles implicitly accepted by Bishop; on the other hand it is of foundational interest that it is justified by the most elementary of these.
3.
mop +
J
5
(C;-AC)P
,
EMo + J
5
1 C1-AC. The proofs of these results from
Feferman 1976~ are given by formal models which verify classical logic and
Dv .
We start again with the recursion-theoretic interpretation App(x,y,z) + (x](y)=. z .
Now, instead of defining C&
in transfinite stages, one defines it simply to be 1
1 . 1 A, indices. That is, let P,(e,x) (e=0,1,2,... ) be a standard 1 enumeration of all ll1 sets (predicates of one argument x); then 1 1 - enumeration of all sets. Tve put e sl(e,x) .a1pl(e,x) induces a
the Set of
in
c.l
<
if the pair of indices (e)o
, (e),
<
determines a A:
set, i.e.
n1-
S. FEFERMAN
220
model reduces to showing that if @(x,r,z)
is elementary an5 we substitute A:
definable sets Di for the Zi, the result is also A:
(with index e uni1 formly recursive in given indices di for the Di ) ; this is by the A l - sub-
stitution theorem of Addison-Kleene-Schoenfield. Formalization of the latter 1 (Cl-AC). So far, the argument serves to give a model of EM in 1 AC) ; next it is seen that only restricted ( E l - A C ) r suffices if one starts
makes use of
(g-
with m o p . T o complete the proof, J
is verified as follows. Suppose Cl(a)
and Vx ~aCl((f] (x)), i.e. that 1 vx(Pl((a)o,
1 x) +-fYIPl(((fJ(x))o,Y)
4
Then we easily obtain a Remark.
1
(Z,-AC)r
By
5
1 <+sl(((fJ(x))l,Y)lJ
.
index for j(A ,f) where a is the index of A.
, this
HA
shows that J
is really of no use without
unrestricted induction. That was already noted informally in 11. 11.3, where transfinite types were shown to exist in EM
4.
E N o r + J + I G I . ~ ( C 21- A C ) I
,
+
J - but not in EMOF + J .
E M o + J + I G i l ( C 2 -1 AC)
and T o Z ( $ - A C ) + ( B I ) .
The proofs (again from Feferman 1976~) all use the same idea, which simply follows 1 1 that of $ 3 one level up. Let P (x,e), S (x,e) enumerate the I$ , resp. 2 2 1 1 1 sets. Take CJ(a) .r,Vx[P2(x,(e).) (hS, (x, (e)l)] and X E a e P2(x,(a)o). Now one applies the A-K-S substitution theorem for A> predicates, which is 1 1 proved using Z , - A C . This serves to show EM + J modelled in ( C , - A C ) and
EMo? + J in
.
($- A C ) p
again: if A,R are A;
4
T o verify
IGr in :his mo3el we simply apply A-K-S
then the set i(A,R)
which is :I
in A,R
is also
(with index e uniformly recursive in the indices a,r of A, R resp.).
The induction axiom of I G P follows immediately by 3efinition of i ( A , R )
as
the least set satisfying the given closure conditions. T o obtain the full principle of induction for IG one must apply full (BI), which gives the final result: T~
=
5. (XI 1- AC)r
m0 +
J + IG
1
< P A , (E?
5
($-Ac)
- AC)P 5
+ (BI).
1 (111 - C A ) P ; consequent reductions into To
.
As was remarked in the survey of creciits in $1, one has proof-theoretical argu-
ments due to Sieg for these first two reductions, corresponding to earlier conservation results of Barwise-Schlipf and Frie3man. Now PA 5 HA by the negative
(11) translation as is well known, an3 HA 5 EM I' so this completes the re1 lations in § 1 ( 2 ) . Next (II,-CA)' can be interpreted in the corresponding in1 tuitionistic system (111 - CA)' (i) by the negative translation, an3 the latter is directly contained in EMor + IGI' . This completes the chain in $1(4).
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES
6. (< - AC)
5
Mo + J.
5
(Ily
<
i n levels RA(i’
< €0
c0 )
A s i s f a m i l i a r , (Ily-CA)
1977.)
l i n e d i n Feferman
@.< Eo <
, and
i s contained i n
IN ) .
inciuction on
(Xi-AC)
5
RA(i)
< €0
=_
a < so
EMo + J + I G P .
1 (Ill - C A )
classes
<
ID<
Qa f o r
CI
<
0
E
(ramified analysis Finally,
5
ID
ID?)
.
f o l l o w s from f u l l
5 l(3).
in
E
1970,
By F r i e d m n
< Eo
a
(full i n d u c t i o n up t o
5
($-AC)
(Ili-CA)
a<
.
E
(@), i . e . t o t h e i n t u i t i o n i s t i : €0
T h i s has been e s t a b l i s h e d by S i e g
( Q ) i s contained d i r e c t l y i n
and
< €0
< Eo where t h e l a t t e r i s a c l a s s i c a l t h e o r y
of i t e r a t e 3 f i r s t - o r d e r i n d u c t i v e d e f i n i t i o n s up t o any s t e p i s t o show
RAcEo
by t h e n e g a t i v e t r a n s l a t i o n .
< €0
T h i s completes t h e
by Fefermari 13’70,
< €0
by Friedman € 0
EMo + J , u s i n g J o i n t o t r a n s f i n i t e l y i t e r a t e t h e ramified
h i e r a r c h y up t o each o r d i n a l
ID(i)
- CA) <
(That used a model-theoretic argument; a p r o o f - t h e o r e t i c a l one i s out-
1970.
7.
(Xi - AC)
To begin with,
22 1
EMo + J + IGp
.
The main next
t h e o r y of t h e
1977.
Finally,
I n t h i s way t h e
=
i n Sl(5)
i s completed.
Remark.
R e s u l t s c l o s e l y r e l a t e d t o t h o s e of Sieg
1977 have
been obtained in3e-
pendently by P o h l e r s an3 Buchholz, by more complicated methods, b u t which a l s o g i v e more d e t a i l e 3 i n f o r m a t i o n .
P r e s e n t a t i o n an3 comparison of a l l t h i s work w i l l
b e found i n t h e forthcoming j o i n t volume r e f e r r e 3 t o i n
8.
1.
Questions and c o n j e c t u r e s . ( i ) The c o n j e c t u r e
(Xk - A C ) +
(BI)
5
To
has a l r e a 3 y been s t a t e d
i n Cj 1 .
What i s missing up t o now i s t h e p r o o f - t h e o r y analogous t o t h a t i n d i c a t e d i n (ii) W e have shown
EMo + I G + POW
c o n s i s t e n t i n 111.12.
s t r e n g t h o f t h i s system an3 v a r i o u s of i t s s u b s y s t e m s ? cannot be u s e 3 v e r y e f f e c t i v e l y with t h e s e axioms. Pod
$7.
What i s t h e
It appears t h a t
I conjecture t h a t
POW
EMor +
= HA.
( i i i ) T h e s e t - t h e o r e t i c a l model of
So+POW
model of s e t of
by e s s e n t i a l l y u s i n g
for any s e t
A
A.
What i s t h e s t r e n g t h of
consistent
(i 0,1)?
(iv) i s a33e3?
[0,1IA
Now p r e s e n c e of
effective.
111.6 can be modifieci t o g i v e a
in
To
J
So+POW?
as a r e p r e s e n t a t i v e of a power makes t h e axiom Further, i s
POW
much more
So+POF+CTi
What a r e t h e s t r e n g t h s of t h e v a r i o u s t h e o r i e s considered when
CA2
Stanforci U n i v e r s i t y
S. FEFERMAN
222
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1971
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S.
Formal t h e o r i e s f o r t r a n s f i n i t e i t e r a t i o n s of g e n e r a l i z e d i n d u c t i v e d e f i n i t i o n s and subsystems of a n a l y s i s ,
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A l g e b r a and Logic, L e c t u r e N o t e s i n “ a t h . ,
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450,
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223
CONSTRUCTIVE THEORIES OF FUNCTIONS AND CLASSES constructive mathematics)
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1 9 7 6 c On t h e l e n g t h o f s o m e c o n s t r u c t i v e s y s t e m s o f Mathematics' 1977
'Explicit
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of M a t h e m a t i c a l L o g i c ( N o r t h - H o l l a n d , A m s t e r -
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Recursion t h e o r y and s e t t h e o r y :
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H.
I t e r a t e d i n d u c t i v e d e f i n i t i o n s a n d 2'
1970
2
-
E,
in
I n t u i t i o n i s m and P r o o j T h e o r y ( N o r t h - H o l l a n d ,
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1975
S y s t e m s of s e c o n d - o r d e r reduction
1977
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Uber e i n e b i s h e r noch n i c h t b e n i i t z t e E r w e i t e r u n g d e s f i n i t e n Standpunktes, Dialectica,
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A
counterexample
+ M Martin-LGf, 1971
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179-216.
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Logic CoZto-
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Myhill, J. 1975
C o n s t r u c t i v e s e t t h e o r y , J. S y m b o l i c L o g i c ,
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Prawitz, D. 1971
S. FEFERMAN
224
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Scott, D. 1970
Constructive validity, in Symposium o n A u t o m a t i c
Demonstration, Lecture Notes in Maths., 1 2 5 , 2 3 7 - 2 7 5 . Sieg, W. 1977
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.
Dissertation,
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PhiZosophy of S c i e n c e III (North-Holland, Amsterdam), 185-199.
Troelstra, A. S . 1973
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LOGIC COLLOQUIUM 78 M . Boffa, D. van Dalen, K. McAloon (eds.1 0 North-Holland Publishing Company, 1979
O N PARTIALLY CONSERVATIVE EXTENSIONS O F ARITHMETIC
P e t r Hdjek Math.
Inst.,
FSAV,
115 67 Prague,
Czechoslovakia
Introduction. L e t PA b e P e a n o a r i t h m e t i c .
Following Guaspari [ 2
1
we c a l l a
s e n t e n c e q n o - c o n s e r v a t i v e ( o v e r PA) i f e a c h n o - s e n t e n c e )I p r o v a b l e k k i n (PA t q ) ( i . e . i n PA w i t h t h e a d d i t i o n a l a x i o m 9 ) i s p r o v a b l e i n PA.
r
Similarly for
n ko
"T-nonconservative". then 3
b e i n g r e p l a c e d by X E or a n y r e a s o n a b l e c l a s s
We w r i t e r - c o n
of sentences.
i s r-con
for " r - c o n s e r v a t i v e "
f o r each
r;
and r-non
for
i f p i s a s e n t e n c e p r o v a b l e i n PA
Obviously,
but i f q i s unprovable and,
s a y , no-con k
t h e n q i s n o t a IlE-formula.
The e x i s t e n c e o f p a r t i a l l y c o n s e r v a t i v e f o r m u l a s was f i r s t n o t e d by K r e i s e l ; t h e p r o t o t y p i c a l example i s lCon(PA) which i s an independent proof"
7,'-sentence
1
and s e e [ 6
1
which i s no-con,
1
f o r a "model t h e o r e t i c "
techniques used t o construct
s e e [ 7 ] f o r a "modal one.
The t w o p r i n c i p a l
f u r t h e r p a r t i a l l y c o n s e r v a t i v e formu-
l a s a r e GGdel d i a g o n a l t e c h n i q u e a n d ( u s u a l l y " t o g e t h e r w i t h " ) partial truth definitions,
see [ 4
1,
[ 5 ] : f o r each k
>
1, t h e r e is
a I I O - f o r m u l a t r k ( x ) s u c h t h a t f o r e a c h n o - s e n t e n c e 9 , we h a v e k P A F qC-*trk(y) a n d s i m i l a r l y f o r X o i n p l a c e o f Ilk". Kreisel's k e x a m p l e c a n b e g e n e r a l i z e d a n d t h e s e n t e n c e 3 x ( t r k ( x ) r r lCon(PA + x ) )
i s n O - c o n f o r k 2 1. On t h e o t h e r h a n d , i f 7 s a y s " b e n e a t h e a c h k p r o o f o f me t h e r e i s a p r o o f o f a f a l s e C o - s e n t e n c e " , t h e n q i s a k t r u e Xo-formula which i s u n p r o v a b l e and no-con. Similarly, if 3 k k s a y s " b e n e a t h e a c h p r o o f o f me t h e r e i s a p r o o f f r o m me o f a f a l s e Zo-sentence" k
sentence.
then q
is a true,
unprovable and Z;-conservative
We s h a l l o m i t t h e p r o o f o f t h i s f a c t , e v e n i f
no-k
it is not
d i f f i c u l t ; v a r i o u s p r o o f s of s i m i l a r and s t r o n g e r f a c t s a r e c o n t a i ned i n t h i s p a p e r . T h e p a p e r i s o r g a n i z e d a s f o l l o w s : I n s e c t i o n I , we i n v e s t i a n d s h o w t h a t i t is I l o - c o m p l e t e { q ; 9 i s no-con k 2 T h i s p a r t i a l l y a n s w e r s a p r o b l e m o f G u a s p a r i ; v e r y l i t t l e i s known gate the set Ik
225
226
P.
HAJEK
I n s e c t i o n 2 we s t r e n g t h e n G u a s p a r i ' s f o r Ilo b e i n g r e p l a c e d b y E z . k s e n t e n c e which i s r e s u l t s c o n c e r n i n g A: s e n t e n c e s : we f i n d a r-con
r
for
sentences.
b e i n g t h e s e t o f B o o l e a n c o m b i n a t i o n s o f Zo ( a n d k I n s e c t i o n 3 we c l a s s i f y i n d e p e n d e n t sentences.
Ei
Each
s u c h s e n t e n c e i n c l a s s i f i e d a c c o r d i n g t o ( 1) i t s t n u t h / f a l s i t y ,
i t i s c o n s e r v a t i v e a n d ( 3 ) how n u c h i t s n e g a t i o n i s
( 2 ) how much
conservative.
We e x h i b i t 2 1 n a t u r a l n o n - e m p t y
independent
sentences.
Ei
d i s j o i n t c l a s s e s of
Main r e s u l t s o f t h i s p a p e r w e r e c o m m u n i c a t e d a t t h e L o g i c ' Colloquium 78.
Thanks a r e due t o K .
McAloon f o r a n i l l u m i n a t i n g
discussion t h a t helped t o complete t h e r e s u l t s . S e c t i o n 1. n o - c o n s e r v a t i v e k
sentences.
I t is easy t o see that a sentence vely interpretable,
i.e.
i f
ip
i s no-con
1
i n PA i n t h e s e n s e o f [ 1 0 ]
.
t i v e sentences i n terms of
interpretability.
w i t h i t s GiSdel n u m b e r a s u s u a l . PA:
i s PA
f o r m a l l y d e f i n i n g Tr(Ilo) f o r e a c h IIi s e n t e n c e
k
k
B e f o r e we f o r m u l a t e
Each f o r m u l a i s i d e n t i f i e d
(1) PA
8)
is P A P n , i . e . t h e set of
is the s e t of a l l true k ( 3 ) t r k ( x ) i s t h e no f o r m u l a k and mentioned i n t h e i n t r o d u c t i o n , i . e . ,
a l l a x i o m s o f PA l e s s t h a n n . sentences;
it is r e l a t i -
Guaspari c h a r a c t e r i z e d no-conserva-
h i s r e s u l t l e t u s f i x some n o t a t i o n .
II;
iff
(PA t 9 ) h a s a r e l a t i v e i n t e r p r e t a t i o n
( 2 ) Tr(n
U Tr(IIt).
9 trk(F). Note that this schema i s a particular case of a more general schema concerning all Ilo formulas, not only sentences; namely, for each no-formula k -~ k $(xl, ..., x we have _ (* ) P A F i p ( x l , ..., x ) I trk(ip(xl, ,.. , xn))
7 we h a v e PA
( c f . [ 5 1 schema 1.5). Similarly for zi formulas. (The E Z truth definition for 1; sentences w i l l be denoted b y tr'(x).) We s h a l l c a l l t h e s c h e m a ( * ) t h e " i t ' s s n o w i n g " - i t ' s
snowing
schema remembering T a r s k i ' s d e f i n i t i o n s a y i n g t h a t t h e s e n t e n c e I, i .t ' s s n o w i n g " i s t r u e i f f i t ' s s n o w i n g .
(4) n ( x ) i s t h e n a t u r a l b i n u m e r a t i o n o f PA i n P A ; TI ( x ) i s Y < y ; P k ( x ) i s TI ( x ) v t r k ( x ) ; n k ( x ) i s r ( x ) v t r k ( x ) e t c . k Y Y k ( n n t j) a b b r e v i a t e s t h e f o r m u l a TI ( x ) V x = $. We w r i t e C o n ( n )
a(x) 8 x
i n s t e a d o f Con
F)
f o r the formal consistency statement;
thus e.g.
i s no proof o f a c o n t r a d i c t i o n from t h e 1 t h e o r y whose a x i s m s a r e 11) s e n t e n c e s x s u c h t h a t n ( x ) and 1 2 )
Con(ni + the
sa;s
sentence
z.
: there
ON PARTIALLY CONSERATIVE EXTENSIONS OF ARITHMETIC
I o f PA i n PA i s p r o v a b l y n o - f a i t h f u l
A relative interpretation
i f f o r e a c h I I z s e n t e n c e J, we h a v e P A E p r e t a t i o n o f P A i n PA i s p r c v a b l y II; Lemma 1. [ 2 ]
227
k
J,' 9 . Note t h a t e a c h i n t e r -+
faithful.
The f o l l o w i n g a r e e q u i v a l e n t
>
( f o r each k
1) :
( i ) q i s n,O-con, ( i t ) (PA t q ) h a s a p r o v a b l y Ilk" f a i t h f u l r e l a t i v e i n t e r p r e t a t , i o n i n PA,
( i i i ) For e a c h n , P A F C o n ( r r ; - l
t
F).
Solovay proved [ 9 ] t h a t t h e s e t {q
;(PA t $ 1 has a r e l a t i v e i n t e r p r e t a t i o n
i n PA1
is ~~O-cornplete. put 2
Ik =
{ q ; q
i s no-con)
k T h e n S o l o v a y ' s s e t i s I1.
Jk = I $ ;
and
q
i s ZO-con)
k
.
E a c h I k a n d e a c h Jk i s e a s i l y s e e n t o b e
G u a s p a r i a s k e d i n [ 2 ] w h e t h e r e a c h Ik a n d e a c h J k i s
a IIi s e t .
Ilo-complete. 2
I n t h i s s e c t i o n we p r o v e t h e f o l l o w i n g
>
T h e o r e m 1 . E a c h Ik (k
1) i s n o - c o m p l e t e . 2
This g i v e s a p a r t i a l answer t o Guaspari's question.
Some
r e m a r k s c o n c e r n i n g Jk a r e p l a c e d a t t h e e n d o f t h i s s e c t i o n . u t i l i z e one of
S o l o v a y ' s p r o o f s o f nO-completeness
w i l l be p r o b a b l y d i s t i n c t from
2
of
We
I1 w h i c h
the proof t o be given i n [ 9
I;
the
author is indebted t o Professor Solovay f o r h i s permission t o use h i s proof. tions.
Let Tot be t h e i n d e x s e t o f a l l t o t a l r e c u r s i v e func-
We s h o w t h a t a f u n c t i o n S o l o v a y u s e s t o r e d u c e T o t t o I1
i n f a c t r e d u c e s T o t t o Ik f o r a l l k
>
Definition
be t h e u s u a l Kleene p r e d i c a t e .
(Solovay).
Let T ( e , n , y )
1.
Set T
=
e ,n
{ q ; q
a n a x i o m o f PA a n d ( v y G 9 ) 1 T ( e , n , y ) l ,
let T ( x ) be t h e c o r r e s p o n d i n g f o r m u l a s u c h t h a t T (x) binue ,= e >n merates Te , n , i . e . T
e,z
(X)
( w h e r e T(-,-,-) 0;
=
n ( X )
8
(VY G
X )
1 T(e,Z,y)
b i n u m e r a t e s T) and s e t ( V z ) ( c o n ( s k, )
-+
con(Te,,)).
P. H ~ E K
228 Remark ( S o l o v a y ) e E T o t i f f
$2
Claim 1
E Iktl i f f
(Vn)(PA k C o n ( ~ ~ , ~ ) ) . The f o l l o w i n g sums up a r e l a t i v i -
e E Tot.
z e d v e r s i o n of t h e S e c o n d I n c o m p l e t e n e s s
Theorem
and w i l l be used
t o p r o v e C l a i m 1 , g e n e r e l i z i n g S o l o v a y ' s p r o o f o f t h e n 2o - c o m p l e t e n e s s of I , .
>
F a c t 1 T h e r e e x i s t s n o s u c h t h a t for a l l n
no,
(1) t r k ( x ) b i n u m e r a t e s T r ( I I o ) i n P A k . ( 2 ) n k ( x ) b i n u m e r a t e s PAk i k n PA k n
.
( 3 ) 1; P A E v I l ( 3 x ) ;rf ( 4 ) P A : - - Con( n: ) v. n:
(Y:x),
then P A E F v .
+
For each n, P r o o f of C l a i m 1 : Assume O k E Iktl. Con(n:) s o f o r g i v e n n t h e r e i s m s u c h t h a t PA PAm+
Con(.r
02
).
e ,n b y Lemma 1 a n d t h u s PA
PA
I-
Con(T
1, s i n c e
Con(re
).
S i n c e Ok E i Con(nf
PA^
and so
mwe have P A F Con(ni t
k t l t Con(T
1Con(T
P A F C o n ( nkn ) ;
)
)).
e ,n Pr
This
Tt)
implies
(-Icon(ie,n)).
--t
e .n e ,n lrk F o r t h e o t h e r d i r e c t i o n , b y F a c t (1) a s m f o r m a l i z e d i n P A , w e k k have P A E Con(nn t l C o n ( n n ) ) . A s s u m e e E T o t . T h e n for e a c h m ,
PA^
= (PA, .t
Put S
s,F
i axn n 2 n o .
~
t l C o n ( n kn ) ) ;
PA^
We p r o v e
Qke .
we p r o v e S n k
Con(n:
t
7;)
.
I n d e e d , for e a c h
s i n c e ~ o n ( r . ) i s a t r u e no s e n t e n c e ; t h i s e,l k T h i s p r o o f o f S n k Oe t o g e t h e r w i t h S n k l C o n ( n kn ) , g i v e s S n k O e .
i
<
n,
f o r m a l i z e s i n PA ,
s i n c e we h a v e P A
trl(Con(r
PA p r o v e s t h e c o n s i s t e n c y o f S n b y F a c t
(1).
)) for each i and e ,i Con(n: + T k i T h u s PA
T h i s c o n c l u d e s t h e p r o o f o f o u r claim, t h u s Iktl We h a v e p r o v e d t h a t e a c h I k i s n o - c o m p l e t e . "dual"
s e t s Jk
=
{p;
e v e r y t h i n g we k n o w
q
i s XO-con},
k
2
i s I l2o - c o m p l e t e . Concerning the
ft
t h e following proposition is
:
P r o p o s i t i o n 1.
J1 i s n e i t h e r a 2 ;
Proof
T be t h e Kleene p r e d i c a t e and l e t K ={n;(3p)T(n,n,pj
: (1) L e t
s e t n o r a IlT s e t
x
b e t h e u s u a l Zo-complete set. We r e d u c e ( t h e complement 1 t o J1. L e t T b i n u m e r a t e T i n PA. Indeed, n E K iff Nb(Vz i f f q n E J1 w h e r e 9
is the formula ( v z ) 1
s e a t r u e s e n t e n c e i s Xo-con; T h u s J 1 i s n o t .'X
1
1
( 2 ) We r e d u c e K t o J1.
n E K iff
a f a l s e II; Indeed, n
( 3 z ) ~ ( n , n , z )E J ~ .
E
T ( ; , ~ , Z ) .
o f K)
)i~(F,ii,z)
This is becau-
s e n t e n c e i s Xo-non
1
by [ 2
K iff PAb(3z)T(n,n,z);
I. SO
229
ON PARTIALLY CONSERATIVE EXTENSIONS OF ARITHMETIC Section 2.
sentences.
A:
i s A:
( o v e r PA) i f f t h e r e i s a L o s e n t e n c e q 1 k sentence q 2 such t h a t PAF(q q l ) & (q v2).
A sentence 9
a n d a :I q
i s e s s e n t i a l l y A:
nzT1
PA) nor a
but is neither a
i f q i s A:
formula
>
G u a s p a r i shows [ 2 ] t h a t f o r e a c h k essentially Ao
k t l
a n d i s Ao-con
For e a c h k
is true,
B(Co)-con k I c o m b i n a t i o n s of :
>
i s a q which i s
1 there
( a n d b o t h 1:-non
k
We a r e g o i n g t o p r o v e Theorem 2 .
formula (over
(over PA).
1, t h e r e i s a A:t1
a n d IlE-non).
sentence y such t h a t y
( w h e r e B(Co) d e n o t e s t h e s e t o f a l l B o o l e a n k s e n t e n c e s ) and 1 y i s no-con ( b u t Z i - n o n ) .
k
t h i s i m p l i e s t h e (known) f a c t t h a t t h e r e i s a Ao ktl s e n t e n c e ( o v e r P A ) n o t e q u i v a l e n t i n PA t o any e l e m e n t o f B(c;). Note t h a t
P r o o f o f Theorem 2 . [ 11.
i.e.
I n s i d e PA,
is a proof
o f x"
S a y i n g "y
i s a proof
o f x f r o m z " we mean P r f
i.e.
a proof
s a y i n g " p r o o f " we mean
"y
from n,
i s Prfn(x,y) i n the notation of
(ntz)(X .y) " y i s a p r o c f o f x f r o m n w i t h t h e a d d i t i o n a l a x i o m 2".
Koughly,
our y w i l l b e s u c h t h a t
nok
of a f a l s e a B(Zo) k
s e n t e n c e from y,
PA^
formally, tVp,q,t
Q
l y says : t h e r e i s a proof
s e n t e n c e from i y such t h a t
n:
y) (p is
E q is
ck
7y)(x,y~ E x is
n:
+
tr (p)
k
y of
More ltrk(x)
E
E Prf(nt U j p v q,t)
y
<
is true.
t h e s e n t e n c e p r o v e d by t
(3x , y ) ( p r f ( % +
7y
f o r each proof t
+
v
tri(qD).
( I n t h i s f o r m a l v e r s i o n , we r e s t r i c t o u r s e l v e s t o d i s j u n c t i o n s o f a Z i s e n t e n c e a n d a IIo s e n t e n c e o b s e r v i n g t h a t e a c h B ( Z o )
k
k
sentence
is e q u i v a l e n t t o a c o n j u n c t i o n of s e n t e n c e s of t h e former form). O b v i o u s l y , y e x i s t s by t h e s e l f - r e f e r e n c e
(1)
is Aitl
y
: as written, l y
lemma.
i s s e e n t o be a IEtl s e n t e n c e .
II:
B u t 1 y is e q u i v a l e n t t o t h e f o l l o w i n g s e n t e n c e
( 3y ) ( V y) i y
(y i s a proof
of a f a l s e
( y i s t h e l e a s t proof
* (vp,q,t < y)
This is
d
(.
..
o f a f a l s e Ilz s e n t e n c e f r o m
a s above
. . .) ) .
c o n j u n c t i o n w h o s e f i r s t c o n j u n c t i s Z;
a proof of a f a l s e
s e n t e n c e f r o m 1 y " i s I;).
"y i s t h e l e a s t p r o o f
of a f a l s e
t i o n o f a I;
:
s e n t e n c e from 1 y ) E
ni
sente?ice from
f o r m u l a a n d a IlE f o r m u l a , t h u s A E t l ;
1
(*)
( s i n c e "y i s
The f o r m u l a
ly" ic
s conjunc-
and t h e p a r t
HAJEK
P.
2 30 (\dp,q,t
y)(
Q
... )
is also AEtl.
f o r m u l a ( * ) i s IIkOt1 Akot1-
(2)
y is true.
t h e r e i s a proof
Thus t h e s e c o n d c o n j u n c t o f t h e
and s o i s t h e whole s e n t e n c e ( * ) . 1 y is true.
Suppose t h a t
n ok
of a f a l s e
sentence
v
of PA a r e t r u e a n d 7 y i s t r u e s o t h a t
v
from
T h u s y is
Then i t i s t r u e t h a t But a l l axioms
l y .
must a l s o b e t r u e , a c o n t r a -
diction. (3)
i s nz-con
1 y
v
be the proof of
i n (PA t 1 y ) .
2
there is a t Q
Then by y ,
v
q from
thematically,
y
,
p
.
Let d Let u s
Thus assume l t r k ( " .
such t h a t t is a proof of a f a l s e d i s -
p b e i n g 1;
...,
ul,
l e t plV
i s II;.
LP
We s h o w (PA t y )
I f t r k ( F ) , we a r e d o n e .
p r o c e e d i n (PA t y ) . junction p
: S u p p o s e (PA t 7 y ) k p ,
and q b e i n g
ch
p h V uh
c ko .
A r g u i n g metama-
be a l l d i s j u n c t i o n s of t h e
r e q u i r e d s y n t a c t i c form h a v i n g a p r o o f
i n (PA t y ) o f t h e l e n g t h
<
(piVui)
d.
Then,
ltri(oi)), $0
yh(lpi
i n (PA t y ) ,
:
but yklirk(pi) &
&lui), a contradiction.
i.e.
i n (PA t y ) . (4)
we h a v e
Obviously,
i s Zo-non
l y
k
: the
We h a v e p r o v e d
(*I
first conjunctof
is
1 y which i s f a l s e by ( 3 ) , h e n c e u n p r o v a b l e i n
a C E consequence of PA.
( 5 ) y i s B(Zo)-con. L e t (PA t y ) k p V u , p b e i n g n o a n d u k k , l e t d b e t h e c o r r e s p o n d i n g p r o o f . We p r o v e p v u i n
being C E
(PA t l y ) . f a l s e 1;
The s e n t e n c e
s a y s t h a t t h e r e i s a proof y of
~y
s e n t e n c e f r o m -(y
a
such t h a t something holds f o r a l l t 4 y.
F i r s t o b s e r v e t h a t t h i s y must b e b i g g e r t h a n d ( i n f a c t , y must be non-standard). of
0
if
Indeed,
f r o m 17 t h e n we h a v e 9
Hence t h e r e i s a y b i g g e r
of a d i s j u n c t i o n p trk(p)V trk(q).
v
p
v
T h i s proof due t o S o l o v a y .
than
d such t h a t , f o r each proof t
IIi
a n d q i s Z;
In particular, take d for t
v u.
7 is a proof 1~1, thus trk(F).
i s a numeral and i f
q from y where p i s
t r k ( p ) V t r k ( a ) , thus p t h u s PA
c
( s i n c e we a r e a s s u m i n g
We h a v e p r o v e d p
u and y i s B(Zo)-con. k
y
we h a v e
:
: we o b t a i n
v u
i n (PA +
TY),
This concludes the proof.
i s a n a l o g o u s t o t h e p r o o f o f Theorem 2 . 7
4
i n [ 2 1.
We p r e s e n t e d a d e t a i l e d p r o o f s i n c e v a r i o u s fur-
t h e r proofs w i l l be q u i t e analogous;
t h e n we s h a l l p r o c e e d m o r e
quickly. O b s e r v e t h a t Theoi'em
2 cannot be
b o t h y and l y s h o u l d be B(Zo)-con. k
i m p r o v e d by r e q u i r i n g t h a t
We h a v e t h e f o l l o w i n g e v i d e n t
231
ON PARTIALLY CONSERATIVE EXTENSIONS OF ARITHMETIC Proposition 2. : Let
Proof
A f a l s e Ilz+l
s e n t e n c e i s Eo-non. sentence; thus ( 3 x ) l v ( x ) is
( v x ) y ( x ) be a f a l s e IIz+l
t r u e a n d t h e r e f o r e t h e r e i s a n a t u r a l number a s u c h t h a t 1 7 ( a ) i s Thus @ ( a ) i s a f a l s e E o
true.
q(a) is not provable
Section 3.
consequence of
4
i n PA.
(vx)P(x); being false,
sentences.
Zi s e n t e n c e s
I n t h i s s e c t i o n , we s h a l l i n v e s t i g a t e i n d e p e n d e n t ( o r i n d e p e n d e n t II; Obviously,
sentences,
i.e.
negations of t h e former ones).
" i n d e p e n d e n t " means h e r e " n e i t h e r p r o v a b l e n o r r e f u t a b l e " .
There are a t least two r e a s o n s f o r t h i s i n v e s t i g a t i o n p e n d e n t 1;
: f i r s t ,
inde-
s e n t e n c e s a r e t h e s i m p l e s t s e n t e n c e s 9 s u c h t h a t knowing
m e r e l y t h a t q i s i n d e p e n d e n t we d o n o t know w h e t h e r y i s t r u e or f a l s e : t h e r e a r e t r u e i n d e p e n d e n t E; d e n t Z;
sentences,
s e n t e n c e s and f a l s e indepen-
w h e r e a s e a c h i n d e p e n d e n t 1;
sentence is false.
S e c o n d , t h e r e a r e i n d e p e n d e n t IIo s e n t e n c e s o f m a t h e m a t i c a l ( c o m b i natorial) content,
2
n o t c o n s t r u c t e d u s i n g s e l f - r e f e r e n c e and n o t
referring t o formal proofs.
Such a s e n t e n c e was e x h i b i t e d by P a r i s
a n d s i m p l i f i e d by H a r r i n g t o n ,
1.
see [ E
Notably, t h e i r sentence is
e q u i v a l e n t i n PA t o a c o n s i s t e n c y s t a t e m e n t , n a m e l y ,
1
1
t i o n o f S e c t i o n 1, t o Con(n ) ( w h e r e n ( x ) i s n ( x )
Ei
t h e negation is a f a l s e independent s e r v a t i v e E;
s e n t e n c e ( w h o s e n e g a t i o n i s EO-con) 2
c e w h i c h i s i n d e p e n d e n t a n d Ilo-con
Ei,
i.e.
2
i n d e p e n d e n t 1;
r
t o n X; tence.
2
true
Ei
senten-
More g e n e r a l l y , e a c h
s e n t e n c e c a n b e c l a s s i f i e d a c c o r d i n g t o (1) i t s ( 2 ) for w h a t
i t s negation is
r
it i s
r-conservative,
r-conservative.
Here e . g .
( 3 ) for
t h e Paris-Harping-
sentence obtains another c l a s s i f i c a t i o n than Solovay's senWe h a v e t h e f o l l o w i n g
Proposition 3.
T h e E;
i t s n e g a t i o n i s E;-con Proof.
is also false.
(Such a s e n t e n c e is e s s e n t i a l -
n o t e q u i v a l e n t t o a iIo s e n t e n c e ) .
t r u t h or f a l s i t y , what
?
V t r l ( x ) ) ; thus S o l o v a y ' s Ili-con-
Can t h e r e b e a
T h i s s u g g e s t s t h e following q u e s t i o n : ly
sentence.
i n the nota-
sentence
lCon(rr
1
)
i s f a l s e a n d Ilo
2-C0n;
b u t IIo
l-non*
Concerning t h e negation,
i.e.
1
t h e s e n t e n c e Con(n ) ,
1
it
i s t r u e a n d c o n s e q u e n t l y 2'-con. But Con(n ) i s n o t i n t e r p r e t a b l e 1 1 i n PA ( s i n c e C o n ( n ) i s n o t , s e e [ 1 I ) , a n d h e n c e C o n ( n ) i s no
l-non.
HAJEK
P.
2 32 To p r o v e t h a t
lCon(n
1
i s IIi-con
)
it s u f f i c e s t o note t h a t each
c o u n t a b l e m o d e l o f PA h a s a n o - e l e m e n t a r y e n d - e x t e n s i o n t o a m o d e l 1 1 o f ( P A t 1 C o n ( n ) ) , s e e [ 2 1 , N o t e s t o 6 . 5 a n d 6 . 6 a n d / o r [ 5 1,
4
Theorem 1 . 7 . Theorem 3 .
s e n t e n c e w h i c h i s Ilo-con 2
T h e r e i s a t r u e E;
n e g a t i o n i s nO-con.
a n d whose
1
Our formula is a modification
Proof.
Let y be such t h a t ,
o f Theorem 2 .
(3y) ( y is a proof
1y R +
t h e formula from t h e proof
of
i n PA,
o f a f a l s e I I ~s e n t e n c e f r o m
( V y ) ( y i s t h e l e a s t proof ( V t c y ) ( t is not a proof
of a f a l s e
of a f a l s e
This can be w r i t t e n c o n c i s e l y as f o l l o w s for
( 3 y ) ( y i s a proof
s i m i l a r l y f o r Fls [
1~
n o2 ,
in the notation of [ 2 Now,ly
i s I:,
1y)
sentence from
i y
n o2 s e n t e n c e f r o m y)). L e t Fls [ R y , l y 1 s t a n d
o f a f a l s e Iiy s e n t e n c e f r o m l y ) a n d
.
y ]
TY I
Fls [ f l y ,
:
no1
Then
<
Fls [ni,y
I
1.
i.e.
.
y is
Similarly a s i n t h e proof
of
T h e o r e m 2 we p r o v e t h a t y i s t r u e , l y i s I l o - c o n
b u t Zo-non;the proof 1 1 o f y is f u l l y analogous t o p a r t ( 5 ) of t h e
of Il:-conservativity proof
4
o f Theorem 2 b u t i s s i m p l e r .
We c a n now t u r n t o t h e c l a s s i f i c a t i o n
o f Co s e n t e n c e s mentioned 2 o above. To a n s w e r t h e q u e s t i o n f o r w h i c h r a g i v e n C 2 o r Iio s e n t e n c e 2 i s c o n s e r v a t i v e a n d f o r w h i c h i t i s n o t , w e m u s t c h o o s e some c a n d i dates.
Let us choose simply t h e f i r s t l e v e l s o f t h e a r i t h m e t i c a l
hierarchy,
i.e.
r = Z 1o ,
f o r an independent
Ily,
Ei,
Ci sentence
(-)
i t i s IIO-non
a n d CO-non,
( E l )
i t i s Zo-con
b u t no-non,
1
1
n; .
This gives f i v e p o s s i b i l i t i e s
:
1
A
(Ill) i t i s Z o - n o n b u t Il - c o n , 1 1 (B) i t i s b o t h XO-con a n d n o - c o n 1 1 ( I I Z ) i t i s nO-con.
b u t Co-non 2
and no-non, 2
2
(-1 and
I f t h e s e n t e n c e i n q u e s t i o n i s t r u e t h e n it i s Co-con,
thus
(Ill) are impossible.
s e n t e n c e we
Similarly,
have f i v e p o s s i b i l i t i e s ,
1
f o r a n i n d e p e n d e n t IIo
namely ( - ) ,
such a sentence i s t r u e then
(-)
i s f a l s e t h e n , by P r o p o s i t i o n 2 ,
( E l ) ,
(nl),
(B)
2
a n d (Z2); i f
and ( I l l ) are impossible
(El),
and i f
it
(B) a n d ( C 2 ) a r e i m p o s s i b l e .
233
ON PARTIALLY CONSERATIVE EXTENSIONS OF ARITHMETIC T h u s , with each independent Z i sentence its characteristic c(q) = c,(ip)
E {true,false)
and c , ( i p )
<
cl(q 1,
= X1, c2(v)
c2(ip),
racteristic o f
TCon(n
1
)
we can associate where
>
{(-),(El),(~l),(B),~nZ)]
E
= X3.
((-),(~l),(nl),(B),(~2)}
E
ip
c3(ip
is < false,(nZ),(El)>
is simply any triple from X1
x
X2
x
X3.
= X2
For example, the chaA characteristic
A characteristic is
admissible if it is not impossible by the remarks above (e.g. <true,(-),(-)>
is inadmissible).
There are 21 admissible charac-
teristics visualized by the following two tables
ip
Theorem 4.
TRUE
ip
FALSE
There i s an independent E i sentence o f each admissible
characteristic. The proof i s windy and we shall omit it.
Re fe Pence s [ l ] S. Feferman, Arithmetization of metamathematics in a general setting, Fund. Math. 49 (1960), 35-92. [ 2
I D . Guaspari, Partially conservative extensions o f arithmetic, t o appear in Transactions of the A.M.S.
[ 3 ]
P. Hzjek, Remarks on partially conservative extensions o f arithmetic, abstract, Logic Colloquium Mons. 1978.
[ 4
] G. Kreisel and
A. Levy, Reflection principles and their use
for establishing the complexity of axiomatic systems, Zeitschr.
f. Math. Logik 14 (1968), 97-142. [ 5 ] K. McAloon, Completeness theorems, incompleteness theorems
and mogels o f arithmetic, T.A.M.S.
239
(1978), 253-277.
[ 6 ] K. McAloon, Consistency statements and number theories, Collo-
que International de Logique, ed. M. Guillaume, C.N.R.S.
no 2 4 9
P.
234
HAJEK
(1977), 199-207. (7
1
A. Macintyre and H. Simmons, G6del's diagonalization technique and related properties of theories, Coll. Math. 2 8 (1973).
[ 8
1 J.
Paris and L. Harrington, A mathematical incompleteness in
Peano arithmetic, in : J. Barwise, e d . , Handbook o f Mathematical Logic, Amsterdam 1977. [9 ]
R. Solovay, On interpretability in set theories, to appear.
[lo] A. Tarski, A .
Mostowski and R.M. Robinson, Undecidable theo-
ries, Amsterdam 1952.
LOGIC COLLOQUIUM 7 8 M. Boffa, D. van DaZen, K . McALoon (eds.) 0 North-Holland Publishing Company, 1979
WEAKLY SEPARATED SUBSPACES AND NETWORKS A. HAJNAL and I. JUHhSZ Mathematical Institute of the Hungarian Academy of Sciences, Budapest
Abstract In this paper we give (consistent) solutions to two problems of M.G. Tkarenko C61, namely we produce under CH o r by adding lots of Cohen reals a regular space X with
<
R(X)=w
and using M A ( w , ) + O w
nw(X) = 2w,
Hausdorff space X
>
such that nw(X)
w
but
2
(E),
where
nvfYj=w
E={&w2:cf(a)=w),
a
KX, IYI=w I '
whenever
AMS Subj. Class: Primary 54A25 54620
Secondary 02K05 Key words and phrases: weakly separated subspace,network
Introduction In L61, Problem 3,M.G.
x
space
with
CH there is a
R(XJ
TkaFenko raised the problem whether there exists a
< nw(Xf. In this paper we are going
0-dimensional T2
space X with R(X)=w
b) in a model o f set theory obtained by adding mensional
T2
space X with
inequality nw(x) 5 2 RfX)
R(X)=w
for
!rj
and
K
t o show that
and
nw(X)=wl,
T
3
a) under and
Cohen-reals there is a O-di-
nw(X)=u,
consequently the obvious
spaces cannot be improved. We shall show that
f o r Hausdorff spaces the consistency of the existence of a much stronger example
is also provable, namely of a Hausdorff space X nw(YJ=w
whenever
-
KX
and
IYl=wl.
We recall that (cf. 161) a space map
y
uy
way that if
Y
R ( X ) = sup
then either
{I Yl : ruC
>
w
but
is called weakly separated if there is a
associating a neighbourhood yI#y2
such that nw(X)
This answers Problem 2 of C61.
Uy
z ~ , ~ V y , or
with every point y2bUy1.
is weakly separated].
Horeover
y€Y
in such a
A. HAJNAL and I. JUIdSZ
236
Since a right or left separated space is clearly weakly separated implies that X Finally nw(X)
R (XI =K
is both hereditary K-Lindelof and hereditarily K-separable. is the smallest cardinal of a network for X.
Our set theoretical notation and terminology are standard, as e.g. in C41. The "graph" topology
Let X
be a set and
f : [XI2
is considered an edge iff
-
2
be a map (i.e. a graph on
X,
where
{x,y)E11X12
We shall put
f({x,y)j=O).
and
uf
=
(+x
: f({x,y))=l).
(In accordance with this we shall write study the topology
on
T
(LJ: :
2
Then every If
is
T
f
X
f({x))=O
fix,
-clopen, hence
U
Clearly, [Us :
E€x(x))
= n(UEfx'
to
nw(X)=w
Z),
then we put
:fiD(s)].
i s a basis for
Assume CH. Then there is a "graph" and
We are going to
is 0-dimensional.
T~
T
f-
Theorem 1.
R(X)=w
EX.)
iE2).
(=all finite functions from X
sEH(x)
for each
generated by the family
f : Cw '1
I
-
2
such that X A w
r , ~isl T2 #
I'
Proof. Let us fix some notation first. L I
denotes the set of all limit ordinals in
w
Using CH, we can put
and
B where E
= ( E X : ?,ELf),
denotes the set of all maps of
wxw
into ~ ( w , ) . We can assume that
I'
237
WEAKLY SEPARATED SUBSPACES AND NETWORKS
(a,,
8 )CX
and
x
We let
xk,
E (n,m)EH(h)
h
holds for each
,ELl
and
(n,dEwXw.
denote the set of all disjoint members of [ [ w , I k l w
: MU\ {O)).
X=(Xk
and
:
Using CH again we can write
Next we put
x y
= {Z8€Xk
: B
<
a
&
uz 5C,).
Before we can start the transfinite construction of f
: [w]
2
-
2 we need a
technical lemma.
Lerruna Let
z-u(zk
a be a countably infinite ordinal, MH(a),
:
where each
Mw\{O3),
is a countable family of disjoint countably infinite subcollections of let S be a countable subfamily of I H f a ) l S W gEx(a)
some member of
S is compatible with
such that for every SES q. Then there is a map
zk
c a l k , and
and
F :a
-
2
such that I ) hCF;
2) every S S has a member compatible with 3 ) if k€w\ { O ) , EZk and
denotes the set of denoting the
~
such that
a€Z
€
2then ~
F(ni)=E(i)
F;
( ~ ( 2 ,€)/=a,
where
holds for each
i
Y(Z,
EJ
< k, with
ith member of a (we shall abbreviate this by writing
“i
Fkaz~).
Proof of t h e lerrnna. We put
where every member of the left-hand side appears for infinitely many over let
s={~: , Ew).
Then we define a sequence (h
by induction as follows. We put then consider there is & Z Q ,
( Z p,
E ~ );
a=(ni
:
h =h. Now if h
since hn i
<
is finite and
k), with
: r?Ed
E u , more-
of members of
H(a)
has been defined and n=211, ZQ
is disjoint and infinite,
anD(h 1 4 . Thus ye can put
If, on the other hand, n=211+1, we consider SQ. By our assumption there is compatible with hn. Consequently we can define hn+lEH(a)
by
oESQ
238
A. HAJNAL and I. JUHASZ
It is obvious then that if
F
is any extension of u{hn
to
fiw)
:
a , then
F
is as required. f. Suppose that
Now we return to the construction of the graph has already been defined. We then first put Accordingly we define h E H f A )
X
and
f( ( a A , h1 )=O
D ( h A ) = ( a A ,f i x ) ,
with
and
ELl
f(
ff[AI
2
B A , A) )=I. h ( 6 )=I.
hh(aA)=O and
A
Next we put for each n f w
h
and
sA
{S, : v + H ( A ) 3 a E S n
=
s.t.
is compatible with
a
g].
Finally we put for k E w \ (01
zy
...,u
= {Y(Z; ao,
&
where
5.
with
0; T)
Y(Z;
Ztxy
= (a€Z :
rl*-lj
I . . . ,
<
vi
=
Y
:
!YI = a
,...,u r-1 EZk
& a.
r vj
being the jth member o f
3
-
no
< a.
k f f { E ,, 3
L $
no ,..., r l - l € A l ,
&
nil)
oi(j)),
=
Note that if
r=O
then
YfZ;-;-)=Z.
Then we can apply our lemma for a-A with h=hh, Zk=ZLA), S=SA and obtain F :A
described there. With this we put
2
for each aEA. We remark that this will imply Next we consider those n E w
for which
incompatible with every member of Now if m=n+l
=gnu(( A ,
Zk
1)
>
S,$sA (h=0
1 , if
= ZLE) =
(y(Z;
- -n j 0 ;
= y
clearly
u%u'=~. X A
{a,B]=(a
gnEX(A)
a=h+m
with h=
otherwise), S=0, and
.
(a)
IyI = a & EXk :
A
T
Zf
on
A+&Lr]:
af
& ;€a=).
& *(2k)'
A+m * 2 we put again
Now we have to check that the topology Tz, since if
there is
S,.
BEX+m. Let us note that this will yield us
is
6 s 1 whenever s ES,.
Sn+SA. For each such n
whenever S,$SA. This completes the construction of
T~
:
0 , we apply o u r lemma again for the ordinal
With the thus obtained function F each
AW{Uu
f
f((B,X+m])=F(B)
and :
[ w,]'
At&(Ua
-
2.
is as required. First of all
,B ] ~ [ a l ], then we have h
aEUy
and
BEU:
and
Next we show that R ( X ) = w . Suppose, on the contrary, that (nu : a E w l ] - with qa
<
qB
if
a
<
B
-
for
: 6Sn)
is a weakly separated subspace with
UEa
a basic
Tf-
WEAKLY SEPARATED SUBSPACES AND NETWORKS
na
neighbourhood of for each D(Ea)
<
for a
D(Eg)
finally that
for each aEwl
showing this. We can assume that
moreover that the D(E
a&;,,
E
<
a E w l , where
also be ascsumed that
denotes the
1
With all these assumptions Z=(D(En) some BEwl. Therefore we have each i < k the function ~
(a)) (‘lo
, we
Since EZk that
2and~ for all
€
I
for each
aEw
D(Ea).
It can
I‘
for
such that defined € 2 as ~ follows:
~
for each rn
ff{nr), n:’)))
< k,
= u,frn)
etc. Finally we get (infinitely many)
< k.
~ ~ ( n j ~ ) ) = ~ jis i)
a€w \ w 1
infinitely many
f
for every
also satisfy rn
rl
<
k,
f((nr), nka))
such that
nEw
But this implies then that
such
nEw
consequently we have
so infinitely many of these n
every rn, P
k, and
belongs
fiw)
have by our construction of
f({np),rAa)))=uo(m)
a E w l . Thus
ith member of
for a fixed j
rl = r l r a )
a
~
nja)
naCD(€*)
are pairwise disjoint, in fact
B, they have the same number of elements, say
for a fixed
ZE
the same for ea:h
239
=nfn)
and
n
)=oPfm)
=n(.a)EU
n 1 “ca 0. I En contradicting that these neighbourhoods yield a weak separation of the set
Ing
for
,
: SEw11.
Since R I X ) = w , we get that the topology
is
T*
>
LindelGf. Now if we want t o show that n w ( X )
T2, 0-dimensional and hereditarily
by the regularity of X
w,
it
suffices to show that no countable family of closed sets forms a network in X. Moreover every open set in X the form
Consider a family
Then
{Fn
m
€=EX
for some
claim that for no
:
nEo)
of closed sets in X, where for each
cn€H(wl). Let us define
: nfw),
X\F,-U{U E
is the union of countably many basic open sets of
WE.
ELl.
nEw
EE
by
Efn,ml=Ei
Consider the neighbourhood Uy
do we have
hence
A€FnCUy,
(F,
nEw
for n,mEw.
of this point
: nEw)
we have
X. We
is indeed not a
network for X. Suppose that
XEFn,
i.e.
XEFn,
i.e.
: nfw).
X$tJ{U
As
wqs remarked in our
E
m
construction this implies sn€{c;
:
rEw)BSA
then by our construction again we have
with the notation used there, and
(A+n+l)d
x
and
(Atn+l)@{U
: fiw), E
m
A. W N A L and I. JUHASZ
240
consequently F,$U;.
This completes the proof.
Now we turn to our next result.
Theorem 2 .
-
Suppose that
and
f
is Cohen-generic over
2
Cohen-generic over and
ZFC,
is a countable standard model of
M
:[ K ] ~
M.
M,
Then for the space
is a cardinal in
i.e. f*G,
K H ( [ r12)
where
~2
in N = M [ f l
X ~ K ,
M
S
we have
R ( X ) =w
nw(X)=K.
Proof. Let u s first show
-
R(X)=w.
Assume, on the contrary, that there is, in M f ] , an
cp : Y
HfKJ
and a map
kE[XIwf
such that if
and
x,@Y
:r
but either 4 : i p l Y i ' forces all this about Y and
$Urp(yl,
cp
y+uo(xJ. , a name for
and
Y
and
E EH(K)
5
and ( p 5 :
: 5Ewf)
( E~
p5EH([ K ]
2
EEwI)
with p 3 p
)
E U
rp f x l
three sequences
M
such that
a EK, a f a
5
such that for each
5
respectively.
rp,
Now, by an easy transfinite induction, we can define in (a5 : 5 E o f ) ,
then
x#y
2
Let ~ € H ( [ K )] be a condition which
5 1 1
5Eol
if
C+q,
By suitably extending the p i s if necessary, we can assume that for each we have D ( p 5 ) = [ a 1
2
for some
5
a5€[
K]
<w
, moreover
that
a5Ea5
and
5Eof
D ( E )Ca
5
Then by a suitable "thinning out" of o u r sequences we can achieve that the form a A-system, i.e. each a =dubc, where the family
~~r a5
We can also assume that Note that since the many
5Ewl.
~~1a
and p $ ( a l
2
a ' s are distinct we must have
5 Thus let E#n
extension q
=
be such that
a Eb
of p 5 u p r l as follows: for every
and for UEb nD(E 1
we put
5
5
a Eb
n n
.
Sewf}
=E
\a these equalities will actually hold for all
q ( ( p , an})=€
5
is disjoint.
f o r all but finitely
Now we define an q f ( a ,v])=
Note that since
(u). *D(E
5
)
5,nEwl.
whenever
VEbnnDfEn) we put
(v)
5
(b5 : p$[aI2
a Eb
and
=E
5
=
5'
at's
and
5
E
5
la=
~ E D ( E ~ ) ,
respectively. That this is a valid definition follows from the fact that
l C E u ; - , hence if a
( a )=O, and similarly if 5 5 B"ED(E~) then E ~ ( C Y " ) = O as well. It is clear from all our assumptions that
p5!& qll-
of
EU-
5
En
&
, hence
Eu-
n
5
E5
we must have
obviously q
?, which is a contradiction, as
Using that
F={F
a'
ED(E
5
: <EX)
R(X)=w
with
and X h
5 network in X. Every
< F5
K
p
E
forces that
is not a weak separation
forces its negation.
is regular we show nw(X)=K
, is a family
$
by proving that whenever
of closed subsets of X then
can be written as
F
is not a
WEAKLY SEPARATED SUBSPACES AND NETWORKS F
with
fC[A]
'1
Since
=N'.
U{U
fiw)
:
€5
m
Consider the map
E$H(K).
well-known (cf e.g. [ 51 ) EEM
X \
=
5
E :
XXW *
K]
theorem to force with hf 1 K]
' ) = H I [ A]
2
2 \[A] )
2
)XH([
defined by
H(K)
that there is an
H([
KI
2
\[A]
2
tions forcing Let
F=K \ U{UE
and
K]
2
";;BF v
\ [A]
2
2 . :
-
m 81
:
I , we
can use the product
( E
: fidEN'
m
~ E \K A
fiw), then for every
i.e.
;&;
pl+
z+F
then we are done.
for all nEw. But clearly this is m
only possible if for each
fiw
Let us note that since p
is finite, there are only finitely many distinct v
Now take any
TI€K\A
there is a
is a sequence
the set of condi-
'\[A] .
has an extension that forces
k &,;
m
over N ' .
is dense in H([ w]
If p
).
If not then we have p
. It is
E ( 5 , m)=E5
such that we have already
A€[ K ] ' ~
First we will show the following statement: whenever from H ( K )
24 1
m
which is n o t "mentioned" in p , i.e.
q ( { n , v m ) ) = l - ~(v,)
M a ' ) , and put
with p ( ( a , v m ) ) = f - E (v,).
vmEDkm)
Obviously, then p U q f H ( [ K ]
z \ m[A]
n'&Dfp)
(in particular,
fib, furthermore q t { a , q ) ) = l .
for each and
PUqlk
A s a consequence of this we see that, in
N, if
kfn U i ,
consequently Puq!i-&!U?
and our statement is proven.
have either a$FC family
or
F5$U:,
hence as
U,"
&K\A
is a
T
f
.
SEX
then for each
-neighbourhood of
we the
a
F is not a network for X. This completes the proof of theorem
-
a '
2.
Now we turn to a stronger form of Tkarenko's problem (cf [6], Problem 2). To show how it connects to the previous one let us recall that the property R(X)=w space X
is decided by its subspaces of size
have R(X)=w
provided that
nw(Y)=w
of a
wl. Thus, in particular we obviously
holds for each
I.zu with
IYI=w
1'
Consequently a natural extension of TkaYenko's previous question is this: Does there exist a space X
such that nw(Y)=w
for each
E[XI
but nw(X)
>
w? In
what follows we show that i t is at least consistent with the usual axioms of set theory that the answer to this question be affirmative within the class of Hausdorff spaces. It is interesting to note that if we replace the net-weight function nw with the weight
w, then the situation changes completely: as we have recently
shown in 1 3 1 , for an arbitrary topological space X LE[ XIswf then
wfX)=w
as well (for regular X
if
wfY)=w
whenever
this was proved in [ 61 ) .
Before we formulate our result, however, we shall give a kind of "graph-theoretic" characterization of the property of having a countable network. For this we recall ) , where a V ] ' , is said to be w-chromatic iff there is a -,w such that for each nEw the set f - I ( { n ] ) is independent, i.e.
that a graph :G=( V, E map
f
: V
A. HAJNAL and I. J d S Z
242
[ f-'
( n )) I
0.
=
8
For an arbitrary topological space X with the open base
G(x, B)=G
G 4 V,€
as follows:
v
)
we define a graph
, where
= {(B,x )
:
xEBE8)
and
Lema .
x with any fixed open base B we have nw(Xl=w
For an arbitrary space only if the graph
G(x,BJ
if and
is w-chromatic.
Proof.
-
Suppose first that f
:
V
N=[A
:nfw}
is a network for X. We can define the map
in such a way that if ( B , d € V
w
clearly shows that
valid. But this f
Now we assume that G(x,B)
and
G(X,Bl
is o-chromatic and
f(( B , x ) )=n, then
f
:
Let us put for each 60 A
We claim that hence xEAn,
{( B,x),
( C,y)
3x [ f ( ( B , x ) I
(An : 60) is
f((B,x))=n
i.e.
= n(B :
A CB
be
V
-
establishes this.
w
n]}.
a network for
is defined. Since
f((C,y))=n
=
xfAnCB
is w-chromatic.
x. Indeed, let
xfEB. Then ( B , x ) E V ,
is trivial, it remains to show that
implies E C . But this is immediate from the fact that
I$€.
Now let us turn to our promised result. A s was mentioned above this is a consistency result, proved under the rather unusual combination of two well-known set-theoretic hypotheses, namely =(A€w2
:
cf(h)=w] (see
e.g. [ I ] ) .
cf e.g. 151 and
MA(wl)
Ow ( E l , where
E=
We do not go into proving $he joint consistency
of these two statements, but mention only that in the "usual" models of M A ( w z ) we also have 0 ( E l . It is interesting though that
W. Weiss has recently consi-
*2
dered this same combination for a completely different purpose.
Theorem 3. M A ( w z l and
Suppose X
such that n w ( X l
>
w
but
0 w2
nwfYl=w
(El
hold. Then there is a Hausdorff space
whenever
K X , IYl=wl.
WEAKLY SEPARATED SUBSPACES AND NETWORKS
24 3
Proof. Let us consider on
wz
an arbitrary Hausdorff topology
8. We shall define a topology
base
define for each VEE
an
-
Ef
w E
=
:
is stationary in wz. Then for each in such a way that in
svnf-'((n])#0 v
G\U[Sv
To show n w ( x )
>
: v€a), where
we pick the
VEE
spanned by the vertices
already
H
fails to be
Indeed, ("identifying" f-'({n))=Fn,
us put
u{Fn
:
( X
\
G(Y.,T')
{(X \
we choose all sets of
T'
a€[
is not
w-chromatic.
v)
with :
:
,
v)
To see
and consider the subgraph H
E
v)
Sv,
I={n€w
Sv,
moreover
+w
let
IF,I=w,).
of
1. We will actually show that 2 f
-
: w2
w
be arbitrary. Let
There is an aEw2
such that
nEw\I)Ca. Now the set
(where F'
=
n[F' n
: nEr)\a
denotes the set of all limit points of VECnEf. Since
w2, hence there is a But
f;'(
In))
since u$AS,,
hence
and denote by
T'
consequently
( A S v , v)
and
f, they are connected by an edge in
f does not yield an
w-coloring of
Now, to show that the small subspaces of X aEwz
f(v)=nEI,
is cofinal in v , therefore we
uESvnFn. In other words, although the vertices
have the same "color" under
p)
F ) is closed and unbounded in
v 2 a , we have
implies then that
f)v=f
must have a
(Asp,
cu-type cofinal set S Cv
w-chromatic.
c
vEF;.
and
GET
=0 for each VEw2 \
S
To do this we
so that whenever
frv=fv)
w we will prove that
this let us first put
w
be valid whenever the set f;'( 6)) is c:final
G(x,T')
any
fv : v
v. This is clearly possible. Now, as a basis for
the form
-
X~W~,T') We. first
such that US :v.
a map
VEE
and put
w2
w-type subset SvCv
0 ( E ) by picking for each wZ f : w2 w the set
use
on
T'>T
that has a countable
T
H,
H.
do have a countable network, consider
the subspace topology on a
induced by
T'.
Clearly,
the family 8
= ((B\U(Sv
is a basis for
T,',
: vEa))%=U(B,a)
as for v
to prove that the graph that f o r every fixed
HB
= {(
Ll(E,a),d
:
>
a
:
the set
BE8
& aE[a+l]<W)
S R x
is finite. Thus it will suffice
G(a,Ba) is w-chormatic. This will follow if we show
BE8
&[a+l]<W
the subgraph of & +U(B,a))
Gla,B )
spanned by the set of vertices
is w-cEromatic.
To this end we first define a partial order < P , S )
as follows:
A. HAJNAL and I. J L d S Z
244
where
s(aJ
that ( P , 5
=
u(s
: v€a),
and ( a , b
iff
5 (a',b')
)
has the CCC. Indeed let
)
<
F = s ( a ) . (Let u s note that the order type tp(FV) w w to show is that there are w,pEwl with v # p such that
E m l
F nb =@=F nb V
U
U
V
k. Then we write
FV,
bw
)':6(
=
:
i
<
<
6:"'
k } , where
for each fixed
i
<
the sequence
k
(
Ulv'
Let us put then, for w E w l , f(v)=(UEwl for each
w3
<
5 tp(Fw)
tp[fi(w)]
f
:
wI -[w,]
I
-
V
U
02,
<03
p
hence obviously
that
Now, since
and
p+f(vj
IPi 5
w1
i
<
j . NOW, using
is strictly increasing in
f(v)=.&
tp[ffw)]
v.
We claim that the order type fi(v), where
f.fwj=
<
w
3
.
that
But then for the set
we can apply a result of ErdGs and Specker from [2] o f , in particular then also two
4f(p), which was to be shown.
we can apply
MA(ol),
( P , < ) is o-centered, i.e.
to obtain that
( a , b ) , (a',b')€P,
if
8:')
is strictl; increasing, we se:
B!')
which assures us an independent set of size U,W
a Ua bVUb )EP, i.e. v u' i!
times, we can also assume that
: vEw )
: F n b #@}.
w. Indeed we can write
Since the map
: BjU)'EFw).
mapping
(
b v form a A-system, consequently we might assume
the Erdos-Dushnik-Miller theorem (see e.g. [ 4 ] ) k
=[p€wl
We claim
are actually disjoint, and all have the same number o f elements, say
that the bV
<
bX'.
and put for each w . ) Then what we have
. By the A-system lemma we can assume that the
whose intersection then is disjoint from each
tp[ f ( v ) ]
and
a3a'
: VEW I C P ,
[ ( a V , b w)
then
( a U a ' bEb')EP.
or rather its well-known consequence
P=ngw P,
and for any fixed n
Let us define the map
q :
HB
-
w
if as
follows: g ( ( u(B,aj,u ) ) =
minIn
: (a,
[UI)€P~I.
(Since p $ S ( a ) , this is a good definition.) Now, if = q((UfB,a'),
and
q(( U f B , a J ,
u
)
I
=
u ' ) 1 , then by the above we have ( a U a ' , [!J,u'))EP, i.e. 6 U f B , a ' ) showing that the above two vertices of H, are not connected by
p'EU(B,a),
an edge. Consequently g establishes an
o-coloring of
H B . This completes the
proof o f theorem 3 . The problem whether a similar result could be proved for regular spaces seems to be very difficult. References [I]
K.J. Devlin, Aspects of Constructibility, Lecture Notes in Mathematics, Vol. 354 (Springer, Berlin)
[2]
P. ErdGs and E. Specker, On a theorem in the theory of relations and a solution of a problem of Knaster, Colloq. Math. 8(1961),
19-21.
WEAKLY SEPARATED SUBSPACES AND NETWORKS
13)
245
A. Hajnal and I. JuhAsz, Having a small weight is determined by small sub-
spaces ( i n preparation) [4]
I. JuhAsz, Cardinal Functions in Topology, Math. Centre Tract. Vol. 34 (Math. Centre, Amsterdam).
[5]
R.M. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin's problem, Ann. Math. E(1971), 201-245.
[6]
M.G. Tkarenko, Chains and cardinals, DAN SSSR =(1978),
546-549 ( i n Russian).
LOGIC COLLOQUIUM 78 M . Boffa, D. van Dalen, K . McAloan (eds.1 0 North-HoZland Publishing Company, 1979
E Q U I V A L , E N C E RELATIONS,
L.
PROJECTIVE
H a r r i n g t o n and R.
AND BEYOND
Sami
were f i r s t
P r o j e c t i v e e q u i v a l e n c e r e l a t i o n s o n IR ( = ' w ) s t u d i e d b y S i l v e r who s h o w e d i n [ S i ] : I f E i s a !Il
1
e q u i v a l e n c e r e l a t i o n o n IR
e l s e E a d m i t s a non-em r e a l s (hence IB/E
I
=
2
then
B / E ' i s c o u n t a b l e or
t y perfect s e t of pairwise inequivalent
R, . )
T h i s w a s t h e p a r a d i g m for s e v e r a l s u b s e q u e n t r e s u l t s . abbreviate t h e second a l t e r n a t i v e p e r f e c t l y many c l a s s e s ,
E E
1
E, Ei
and c o a r s e
*
and coarse
*
I I
=.
1
Let u s
: E has
o n t h e o t h e r h a n d c a l l I: c o a r s e , i f i t
f a i l s t o have t h i s p r o p e r t y . E E
i n S i l v e r ' s t h e o r e m by
The f o l l o w i n g w e r e p r o v e d
IR/E
1 < K1 ( B u r g e s s , [ Bu 1 <&; , a s s u m i n g PD
B/E
1 < N1 ,
IR/E
:
1) (Harrington,
unpublished. ) E E
g:
and c o a r s e
1)
[ Ke 2
a s s u m i n g AD(L [ IR
1) ( K e c h r i s
I n 4 1 , u s i n g d e t e r m i n a c y h y p o t h e s e s we g e n e r a l i z e t h e s e r e s u l t s t o a l l l e v e l s o f t h e p r o j e c t i v e h i e r a r c h y , -:his i s Theor e m 5 , w h e r e a l s o we p r o v e
1
: E E TI2
and c o a r s e
*
1
IR/E
I n T h e o r e m 4 we d e a l w i t h i n d u c t i v e a n d c o - i n d u c t i v e relations. In
Kl.
An i m p o r t a n t t o o l i s T h e o r e m 1 w h e r e we p r o v e a s p e c i a l
c a s e o f a c o n j e c t u r e o f M a r t i n on c o - A - S o u s l i n tions.
1 <
equivalence
equivalence rela-
5 2,we s t u d y t h e r e l a t e d q u e s t i o n o f t h e s i z e o f A / E
w h e r e A C IR, E i s
C
r e s u l t s o f J. S t e r n .
int2
o r inductive, our r e s u l t s generalize
We w i s h t o t h a n k A . S .
K e c h r i s who p o i n t e d t o u s a s i m p l i f i -
c a t i o n o f t h e a r g u m e n t i n Theorem 5 , PD f o r a l l l e v e l s o f
a l l o w i n g u s t o s t a t e i f from
the projective hierarchy.
med s t r o n g e r h y p o t h e s e s b e y o n d t h e f o u r t h l e v e l .
247
O r i g i n a l l y we a s s u -
L. HARRINGTON and R. SAM1
248 §
0.
Preliminaries E a l w a y s d e n o t e s an e q u i v a l e n c e r e l a t i o n on t h e B a i r e Space
IR i s e q u i p p e d w i t h t h e s t a n d a r d t o p o l o g y i n d u -
w h i c h we d e n o t e IR.
c e d by t h e c o m p l e t e m e t r i c d(a,B) = ( n t l l - ' ,
:
where a # 5 and n = p k ( a ( k ) # B ( k ) ) .
The c l a s s o f a b s o l u t e l y i n d u c t i v e s e t s a s d e f i n e d i n [ M o 2 ] w i l l b e d e n o t e d by I N D ,
whereas HYP d e n o t e s t h e c l a s s o f a b s o l u t e l y The i n d u c t i v e a n d h y p e r p r o j e c t i v e s e t s w i l l
hyperprojective s e t s .
be d e n o t e d by X D and E P r e s p e c t i v e l y . We w i l l d e r i v e our r e s u l t s f r o m PD determined,
A D ( L [IR
t h e "demonstrably
1
: a l l games i n L
false"
( s e e [ Ma
: a l l p r o j e c t i v e games a r e
[IR ] a r e d e t e r m i n e d , a n d A D ,
1) Axiom o f D e t e r m i n a c y .
i s b e s t a v o i d e d by r e l a t i v i z i n g a l l s t a t e m e n t s t o L [ I R ] assumed t h r o u g h o u t
.
t h a t DC h o l d s i n t h e u n i v e r s e ,
V,
and
Falsehood
.
It w i l l t h a t DC
h o l d s i n L [IR]
A s e t o f r e a l s A C IR i s A - S o u s l i n [Ke 3
1 for
the definitions).
( a r n , f j ' n ) E T I
The c o m p l e m e n t o f s u c h a s e t w i l l b e c a l l e d c o - A - S o u s l i n .
T
t ,v
v E nA
x
T is
If
we s e t
= { ( t ' , ~ E ~ )T
I
t' 2 t t'
c
t
& V' 8 v'
2
c
S i m i l a r d e f i n i t i o n s a r e made for s u b s e t s o f wk
(see
We w r i t e
A = p [ T I = { a E I R 1 3 f V n
as a b o v e , t E
it i s
(A, an o r d i n a l ) i f
t h e p r o j e c t i o n o f t h e s e t o f b r a n c h e s o f a t r e e T on w x A
V ,
or
vl IRk
and t r e e s on
A.
L:
gi s e t s a r e w - S o u s l i n , sets 1 Assuming PD, $2ntl sets are A 1 12n+1 S o u s l i n where A 2ntl is an o r d i n a l < 6 . X -2nt1' -2nt2 sets are '2ntlSouslin. Assuming Det ( K P ) , s e t s i n Z D a r e 5 - S o u s l i n where : It is a classical result that
are N -Souslin
K
(Shoenfield.).
= sup {rank
(5) 1
<
i s a prewellordering of
(These r e s u l t s a r e due t o Moschovakis, a n d [ Mo 2
s e e [ Ke - Mo 1 ]
IR, 3 E %PI
, [ Ke
I).
The f o l l o w i n g w e l l - k n o w n
r e s u l t i s v e r y u s e f u l ( s e e [My
i t i s t r u e for a r b i t r a r y P o l i s h s p a c e s .
2
I);
1
EQUIVALENCE RELATIONS, PROJECTIVE AND BEYOND
249
Proposition 0 If E i s a meager
subset of
IR
x
IR t h e n E h a s p e r f e c t l y many
classes. Corollary 0 If E h a s t h e p r o p e r t y o f B a i r e and e v e r y c l a s s o f E i s meager t h e n E h a s p e r f e c t l y many c l a s s e s .
Proof : U n d e r t h e h y p o t h e s e s E i s m e a g e r . Kuratowski-Ulam
4
theorem ( s e e [Ox
T h i s f o l l o w s from t h e
I).
1 The m a i n t h e o r e m s
The t h e o r e m s of
S i l v e r and Burgess mentionned i n the introduc-
t i o n r e n d e r p l a u s i b l e t h e f o l l o w i n g c o n j e c t u r e of I f E i s C Q - X - S O ~ S( X~ ~a n~ o r d i n a l
IIR/EI
> w)
Martin:
and c o a r s e t h e n
a.
G
Theorem 1 AD
If E is co-a-Souslin,
=)
w Q h C
&, a n d E i s c o a r s e t h e n
IIR/E) < A . B e f o r e g i v i n g t h e p r o o f we s h a l l s t a t e a t h e o r e m o f H a r r i n g t o n and K e c h r i s which i s i n s t r u m e n t a l h e r e . Let 5 be an o r d i n a l ,
W C IR, 7
:
on20
5 a n o r m , g i v e n A E wr,,
t h e s t a n d a r d game G A i s d e f i n e d a s u s u a l , w h e r e a s t h e c o d e d game G Z
is defined as follows: P l a y e r s I a n d I1 a l t e r n a t e p l a y i n g r e a l s
ao,
If
al, a 2 ,
vi
: ai
T h u s GA'
...\,
E W,
i.e.
is none o t h e r t h a n G
Theorem 0 ( a ) AD GA'
:
W, A.
A
*
f o r s o m e A*
IR i s w e l l - o r d e r a b l e t h e n
(resp.
11) w i n s G~
iff
I (resp.
( Harrington-Kechris
[ Ha
- Ke
2
loses.
CUB. GA i s e q u i v a l e n t
to
11) wins G ~ ' .
1)
I f W E E D a n d p i s a n i n d u c t i v e norm t h e n ,
f o r any A
5
i s de'kermined.
( b ) PD GAP
+
: I
$
< w ) E
t h e f i r s t p l a y e r who p l a y s a i
then : I wins i f f ( q ( a i ) l i
Note t h a t i f G~~
:
=)
If W,
A*
is determined.
a r e p r o j e c t i v e and
@
i s a p r o j e c t i v e norm, t h e n
wr,,
L. HARRINGTON and R. SAM1
250
'The p r o o f we now g i v e t r a c e s b a c k t o [ H a 2
1
w h e r e a new p r o o f
o f S i l v e r ' s t h e o r e m i s g i v e n ; s e e a l s o [ L ] f o r an e l e g a n t w r i t e up and s e e [ S t 3 ] f o r an account of S i l v e r ' s o r i g i n a l p r o o f . More i m m e d i a t e l y i t i s p a t t e r n e d a f t e r H a r r i n g t o n ' s p r o o f
for the
Proof o f t h e t h e o r e m 1 Let E be co-h-Souslin,
r
A class
r
6.
of subsets of t h e spaces An
an a u x i l i a r y c l a s s f o r E i f ( 0 )
<
w 4 h
=, #
contains
x
IRm ( n , m
>
0 ) is called
:
and i s c l o s e d under c o m b i n a t o r i a l s u b s t i t u t i o n s
( p e r m u t a t i o n and i d e n t i f i c a t i o n
of variables,
a d d i t i o n o f "dummy
variables") ( i )
r
i s c l o s e d u n d e r s u b s t i t u t i o n of
t h e graph of a p a i r i n g function
r
( i i )
h x
c o n s t a n t s from h
X and contains
h
-*
i s closed under A , V
( i i i )
r
(iv)
is h - p a r a m e t r i z e d
r
:
i s closed under
(vl E E 2 IR - E =
r
, vIR
a n d normed
and i n f a c t , t h e r e i s a t r e e T on w p[T
PLTS,t,V I E Here
V h , 3'
?
1 ,
and f o r any k ,
x
w x
any s , t E kw, u E 'A,
X such t h a t we h a v e
F
denotes the dual c l a s s of
r.
usual A =
As
r
n
From t h e s e c o n d i t i o n s o n e c a n d e d u c e b y s t a n d a r d a r g u m e n t s ( v i ) There a r e I L h i n that
r ; D,;
8 h x
IR
r,
in
y. :
r e s p e c t i v e l y such
:
- I f S & IR i s i n A t h e n S = D for s o m e 5 " 5 - If E I then D = D (hence D E A ) 5 5 5
?
(vii)
E
I.
has the separation property.
Claim 1 : T h e r e i s an a u x i l i a r y c l a s s f o r E . Proof
:
Let T w i t n e s s t h a t E i s co-A-Souslin
l e s t admissible s e t containing I R , subsets of An f r o m {IR, T , That
r
x
A,
T.
IRm w h i c h a r e X 1 - d e f i n a b l e
Let
and l e t M be t h e smal-
r
consist of those
over M with parameters
h l U A.
posseses
( 0 )
t h r o u g h ( v ) i s well-known
[ B a 11, a s t o ( v ) n o t e t h a t
:
( s e e for i n s t a n c e
EQUIVALENCE RELATIONS, PROJECTIVE AND BEYOND
2 s, 6 1
if a
where,
T
S , t , U
t
(a,B) =
:
IT
E
d f t T
2
v or T
c
w E i f n = length (T)
t h e n ( a F n,E
Now d e f i n e
25 1
P n,r)
E
TI.
:
Q = U IS EIR
and S i s c o n t a i n e d in
I S E A
a single class
of E l . Case A
= IR.
: Q
I
I n t h i s c a s e , i t i s e a s y t o s e e t h a t IIR/E tion H
: IR
I <
X , via the func-
X d e f i n e d by :
-*
least 5 E X
H(a)
:
5 E I and Dc # 0 and D
5
is contained
i n t h e equivalence c l a s s of a.
= IR - Q # 0 .
Case B : U
H e r e we w i l l s h o w t h a t E h a s p e r f e c t l y many c l a s s e s . We n o t e f i r s t t h e f o l l o w i n g . Claim 2
: U E
?.
Indeed :
Q(a) - ( 3 5 ) ( 5 so that Q E
E I
r.
7
Claim 3 : If A E
8 a E Dc
8 V6(E E
and A i s a non-empty
more t h a n one c l a s s o f E .
Indeed,
x,
=.
a E 8));
s u b s e t of U t h e n A meets
suppose A is a s described,
i f
i t w e r e t h e c a s e t h a t A E Z f o r s o m e Z E IR/E t h e n n o t i c e f i r s t
r,
that Z E
-
for :
a E Z
v B ( B E A * B E a),
n e x t , by s e p a r a t i o n f o r
?
f i n d C E A such t h a t A C C
E Z, but then
= IR - U , a c l e a r c o n t r a d i c t i o n .
C C Q
We now s e t X
:
= {A C U
I A E
7
and A # 0)
a n d c o n s i d e r t h e game G ( E , r ) , I *O
EO
I1
I
A1
A2
B1
Bp
as follows :
.,
...
...
P l a y e r s I a n d I1 a l t e r n a t e p r o d u c i n g p a i r s ( A , , B , ) ( 2 ) If
vi
:
Ai 2 Ai+l
E
Bi
2. B i + l ,
I
I
o f members of X :
E Bi 2 B i + l , l o s e s . A. 1 1+1 t h e n I1 l o s e s u n l e s s f o r a l l i
( 1 ) T h e f i r s t p l a y e r who f a i l s t o m e e t
: A.
2
L. HARRINGTON and R. SAM1
252 t h e diameters of A2i+l,
are < l / ( i t l )
B2itl
( t h e diameter is rela-
t i v e t o the standard metric). ( I f a p l a y e r l o s e s b y ( 1 ) or ( 2 ) we s h a l l s a y
: he
loses f o r tri-
vial reasons). If n e i t h e r p l a y e r has l o s t f o r t r i v i a l r e a s o n s t h e n s e t t i n g
(3)
-
{a) =
,
Ai
i w i n s iff a E 8.
-
(The
means t o p o l o g i c a l c l o s u r e ) .
We w i l l s h o w b e l o w t h a t
I h a s no w i n n i n g s t r a t e g y i n G ( E , T ) .
I i has a winning s t r a t e g y i n G(E,T),
If
then t h i s w i l l give rise
s e t of pairwise inequivalent reals.
t o a perfect
" E r , which i s A-parametrized;
Note t h a t C
<
w e l l o r d e r e d i n t y p e ,c
A.
l e t W E E D and p
: W
now A D ,
OT0 <
'<.
Z
x
Z
can be
can l i t e r a l l y be Now s i n c e X
<
b y t h e t h e o r e m o f H a r r i n g t o n a n d K e c h r i s menA t
t h i s p o i n t we s h o u l d
switch
t o t h i s g a m e , b u t t o a v o i d a n o t a t i o n a l mess we k e e p t o G ( E , T ) , as if
5,
b e a n i n d u c t i v e norm.
is determined.
t i o n n e d a b o v e , Gpi
hence
T h u s t h e game G ( E , T )
v i e w e d a s t h e s t a n d a r d game G X , f o r some X Assume
:
Bi
=
do
i t w e r e d e t e r m i n e d a n d a s k t h e r e a d e r t o make t h e t r a n s l a t i o n
(Gne c a n a l s o i n v o k e a f a m i l i a r f o r c i n g a r g u m e n t w h e r e b y , t h e r e
a generic extension of V, w e l l ordered.
I n t h i s e x t e n s i o n GX'
is determined,
G(E,r)
w i t h n o new r e a l s , a n d w h e r e
is
IR c a n b e
is s t i l l determined hence
t h e conclusion t o be derived from t h i s f a c t
easily relativizes to V).
C laim -
4
I h a s no w i n n i n g s t r a t e g y i r i G ( E , T ) .
:
Proof : 13e p l a n t o d e s c r i b e t w o r u n b o f G ( E , I ' ) follows a given s t r a t e g y
5
i n which p l a y e r I
and t h i n g s are so drranged t h a t
:
- p l a y e r I1 d o e s n o t l o s e for t r i v i a l r e a s o n s
-
two p a i r s of r e a l s a r e t h u s p r o d u c e d ( a s p e r ( 3 ) ) s a y , and ( 2 , g ) and a
= 2, 6
(a,6)
$ g.
T h e c o n t r a d i c t i o n now a r i s e s f r o m p l a y r r 1's h a v i n g f o l l o w e d a
: a E
6 and
E
8,
hence 6 E
8.
T h e t w o runs o f t h e g a m e a r e d e s c r i b e d i n D i a g r a m 1 . s u h j e c t t o t h e following cooventions
-
an arrow X
X
-
1/2
-t
-*
:
Y indicatas X = Y
Y i n d i c a t p s t h a t Y i s o b t a i n e d i n s o m e s t a n d a r d way
from X s o a s t o have
:
0 # Y C X and Diam(Y) < 1 / 2 Diam(X).
We w i l l w r i t e Y = 1 / 2 X .
-
X
-
nl
+
Y
Y indicates that
= n , ( ~ )=
S i m i l a r l y for n 2 .
:
lst projection
of X .
EQUIVALENCE RELATIONS, PROJECTIVE AND BEYOND
253
I
I1
I A
I1
-
I
I1
I
-
I1
I
Diagram
1
-
Our d e s c r i p t i o n i s i n s t a g e s , t h e a c c e n t
i n d i c a t e s moves on
t h e 2nd b o a r d .
-
Stage [ 0 ] : -
- -
Player
(Ao,Bo).
-
Set S
I , using u produces
(Ao,Bo),
These p a i r s a r e i d e n t i c a l . 2 = (Bo x Bo) f- (IR - E). N o t e h e r e t h a t b y c l a i m 3 , Bo
m e e t s more t h a n o n e c l a s s ,
I n d u c t i v e l y now,
-
- The moves up t o ( A 2
- A s e t S k C 1R , s e q u e n c e s s such t h e following hold
) = n2(S ) =Bd 1 0 Stage [ k ] w e have
: II I S
E
kX
:
-
I1 a n d I1 h a v e p l a y e d s o a s n o t t o l o s e f o r t r i v i a l r e a s o n s .
(bk) A 2 k C A 2 k . (c,)
# 0 i n fact
hence S
s u p p o s e t h a t a t t h e e n d of
described :
(a,)
player I s i m i l a r l y produces
0 # Sk C p [ T
( d k ) rl(Sk) C B 2 k ,
Stage [ k t l ] : F i n d n.m.
E
I .
-
Sk.tk>vk n ( S ) = BZk.
2
such t h a t
k
:
254
L. HARRINGTON and R. SAM1
- L e t I1 p l a y
= 1/2
A2ktl
,
Azk
- Let I r e p l y (AZkt2,Bzkt2) by Set
-
-
s;
IR) n
x
= j’2kt2
’:+l L e t I1 p l a y
- Let I r e p l y ( A Z k t 2 , B z k t 2 ) , b y Set
S k t l = (IR
x
-
=
T~(S;+~).
0.
.
SCtl
BZkt2)
i s s t r a i g h t forward t o check t h a t ( a k t l )
It
We u s e o f c o u r s e t h e p r o p e r t i e s (O),
hold.
- -
verify (aktl).
-
i,
a = a, b e c a u s e
d,
because
f = U v
k
b’k
: Azk
v k
:
-
t-
6
-
2 Azk.
= sk,
k
(i),
through (dktl)
now
(ti),
t o
( v ) of
-
s e t t i n g a = p A i , 6 = r i? Bi
k = tk and ( a , B , f ) E
6
r
-
One t h u s g e t s { ( A , B i l l , {(Ai,B.)) -i and s i m i l a r l y f o r a, 6 one s e e s :
6
‘
.
Bzktl
AZkt2,
: AZktl-=
’1‘ ’it1)
‘2ktl = 0.
[Tlwhere
k’
T o f i n i s h t h e p r o o f o f T h e o r e m 1, i t s u f f i c e s n o w t o s h o w :
Claim 5 : Suppose I h a s a winning s t r a t e g y T, i n G ( E , T )
then E has
p e r f e c t l y many c l a s s e s . Proof
:
l e t hn
Denote by (A:
i
t
<
j
A:
$
=
<
<
i
k
for a l l relevant i , j .
2 A],
for a l l j
f’
least k
and s e t t i n g s . 1 i.
I
P.
1
we h a v e
T ( P ~ ,
:
iJ
S
...,
I.
:,”?
Hn
, h = u h
.
kIsl.
<
and s
t (lex.) then l e t t i n g
t(k))
= s (mtj), = h(s.,t.) = (A
<
n, say,
(s(k) f
=
: H
o f members o f Z s u c h t h a t
)
Is/
(lex.)}
t
2 ~j
I s 1 =‘It/
( c ) whenever
m
<
n n We a r e g o i n g t o d e f i n e a f a m i l y
Is1 t h e l e n g t h o f s .
=. *
s
kn be a b i j e c t i o n and s e t
--*
s E O2, 0
(a) s
(b) i
= ~ ( s , t ) /s , t E n~
L e t H~ : Hn
=
t j
t
k
(mtj)
<
for 0
),
j
<
n-m
I
pjel)
,
1P.
1
<
for 1
<
k
define Ai
j
n-m.
We p r o c e e d b y i n d u c t i o n on I s ] .
k o = 0 , d e f i n e A’
B
Having d e f i n e d A’
= U. for
Is1
<
n,
i
i n d u c t i o n on i a s f o l l o w s . Set first (si,ti) - A: = k n - l
= h(i)
Aubn-l
- Given it1 < k n ;
for
i
I S I ’
<
kn.
for u E “ 2 b y
EQUIVALENCE RELATIONS, PROJECTIVE AND BEYOND If
u i s o n e o f s i , t i t h e n A':
,
A it1 ) t
'i"
w h e r e m a n d F 1' r e p l a c e d b y sl, t
.
I f u # si, ti
set
It
=
i s g i v e n by
...,
r n-m-1'
T ( P ~ ,
Aii)) t
:
f o r f E w2
F(f)
(Aii,
a r e d e f i n e d as i n ( c ) above with s , t
Pn-m-1
i s now s t r a i g h t f o r w a r d t o v e r i f y
D e f i n e now,
255
:
= t h e u n i q u e member
::
of
(a),
(b),
(c).
A"fn
( T h i s w i l l make s e n s e i n a m o m e n t ) . If
f,g
...
<
are distinct, say f
E w2
n e s P 1, P 2 ,
g (lex.)
then,
i f one d e f i -
f , g r e p l a c i n g t , s it is appa-
( c ) above w i t h
as i n
r e n t t h a t t h e s e c a n b e v i e w e d a s 1's m o v e s i n a r u n o f G ( E , r ) , where
I1 f o l l o w s T a n d I d o e s n o t
ther,
t h e r e a l s F(f),
l o s e for t r i v i a l r e a s o n s .
F ( g ) w i l l b e t h e e n d p r o d u c t s o f t h i s r u n of
It F ( g ) .
t h e game, h e n c e F ( f ) F is continuous,
clearly,
pairwise inequivalent reals.
and thus F ( w 2 ) i s a p e r f e c t
We p o s t p o n e for a m o m e n t d e r i v i n g c o n c l u s i o n s c l a s s e s , r e s u l t s o f [Mo 1 1 riant)
: a E A
if
similarly, domain
a set A
if A C
IRn.
is invariant
A set A
.
*
A function f
the notation
denotes the closure of
is called invariant,
if
if
:
r
If
"i
U [A)
is a class,
A
c
IRn
u n d e r f i n i t e b o o l e a n combina-
If A = 0 , w e j u s t w r i t e B ( r ) f o r B ( T , A ) .
Let
Suppose E
:
r
b e o n e o f t h e c l a s s e s ll E
1
2 n t 1 ' ~ i n t 2 3I N ' * and l e t A C IRbe E-thin and i n v a r i a n t t h e n
Det - B ( I ' , A )
its
it does not have a p e r f e c t
inequivalent elements.
t i o n s and continuous pre-images.
Tt,eorem 2
inva-
a n d f f u ) d e p e n d s o n l y on a / E .
L IR i s c a l l e d E - t h i n
We i n t r o d u c e
(or j u s t ,
f? E A ,
subset c o n s i s t i n g of pairwise B(r,A)
from Theorem 1,
for t h e p r o j e c t i v e
IR i s c a l l e d E - i n v a r i a n t
8 a E 6
set of
+
t o p r o v e a c o d i n g theorem which g e n e r a l i z e s , Given E ,
Fur-
A E
:
L. HARRINGTON and R. SAM1
256 Proof
: A game G
is played
I1
I
n
0
kO
nl
kl
c1
I1 w i n s i f f
:
: a E
-
A
-
>
y E A(6).
C l e a r l y i f I1 h a s a w i n n i n g s t r a t e g y (Y
E A
T
[ a 1 )1 E A ( ( T
(1
t h e n A E r(T), f o r t h e n
:
[ a
Claim : I h a s no winning s t r a t e g y i n G. Suppose towards a c o n t r a d i c t i o n , a were such an o b j e c t . l o s s o f g e n e r a l i t y assume E E
= a
F(y)
Note t h a t { y
Define
[ < u,y
IF(y) 3 ‘ A}
be found w i t h : y < u ,yo >
*
-
r
and d e f i n e
> I
is countable f o r , otherwise a r e a l yo can
B A ( u ) a n d F(yo) B A a n d I1 p l a y i n g t o p r o d u c e
would d e f e a t I p l a y i n g a c c o r d i n g t o u .
-
: a E
6
Without
:
-
F(a) E F ( 6 ) . E
because i f t h e r e i s P ,
E
-
r(a), clearly.
E is coarse
a perfect set of pairwise E-inequivalent
t h e n b y t h e p r e v i o u s o b s e r v a t i o n we c a n f i n d s u c h a P w i t h
reals,
F(P) C A , F b e i n g c o n t i n u o u s t h i s e a s i l y c o n t r a d i c t s t h a t A i s E-thin. Thus, by C o r o l l a r y 0 , E h a s non-meager
z then
I
= Iy
-
classes, let
:
Y/E i s non-meager)
:
(i)2
r ( a ) , t h i s f o l l o w s from t h e r e s u l t s i n [Ke 11
E
(for
r =
(ii) y E Z
*
IND, a d i r e c t c o m p u t a t i o n i s e a s y ) . F(y) E A,
indeed A being E-invariant
r i a n t and hence from t h e argument
F
-1
(A)
above F ( y ) 4 A
-
-
i s E-inva-
y/E i s
c o u n t a b l e , hence meager. C l e a r l y Z i s non-meager, ( s e e [Ke 1 ]
r = Il 21 n + l , for r ) .
for
b a s i s theorems
hence i t c o n t a i n s a r e a l Y
Again I1 p l a y i n g t o produce < a , y o t o u-a
i n A(U)
the other cases are just the ordinary >
defeats I playing according
contradiction.
T h i s a r g u m e n t was i n s p i r e d f r o m [ H a - Ke 1 ]
.
We w i l l a p p l y
EQUIVALENCE RELATIONS, PROJECTIVE AND BEYOND Theorem 2 i n 9 2 , Corollary 3
meanwhile
the following is useful
Suppose E i s c o a r s e and
r
i s as a b o v e ,
and A i Z IRn i s E - i n v a r i a n t
(b) D e t - B ( E D )
then
( a , @ ’ )E Claim
:
2
-
(a), take n = 2 f o r simplicity,
(5,B’) : w2
+
-
2
E 6 &
(I
E’ i s c o a r s e .
# 5 * H(a) E
1 E is in A “2ntl
define
5’
E
2
Oefine,
f o r a,B E w 2
i s c o a r s e h e n c e by C o r o l l a r y 0 , E l
El
c l a s s , whence a p e r f e c t P C w 2 c a n be
of this class) a,5
E
We now r e p e a t
such t h a t
*
P
nl(H(a))
=)
n2(H(a))
E nl(H(5)).
HPP,
T 2
replacing w2, H,
P’
Now A C
is E -thin,
=)
2
E n2(H(B)).
hence by t h e p r e v i o u s theorem A E ; ’I
‘vie w i l l m a i n l y u s e
(a) a b o v e w i t h A C IR’,
We a r e now i n a p o s i t i o n
a prewellordering
t o prove
:
:
A s s u m e D e t ( L [ IR
1).
( a ) I f E E IAD i s c o a r s e ,
then
I
IR/E
a Z P p r q w e l l o r d e r i n g o f IR. ( b ) If E E C O - I L D
simi-
4
L.
IR, i n d u c i n g E .
Theorem 4
?il
H((I) E 2 H ( 6 ) - a c o n t r a d i c t i o n .
Hence a , 5 IR2 2
has a
( a subset
P’ E P such t h a t
E P’
I R -A E
fourjd
:
t h i s argument with P ,
to get a perfect a,5
:
T ~ ( H ( ~ )E ) T ~ ( H ( B ) ) .
S i n c e E is c o a r s e , non-meager
of
(I’
IR b e c o n t i n u o u s a n d s u c h t h a t
H(5).
a El B
larly,
h.
(b) a n d ( c ) f o l l o w f r o m ( a ) by t a k i n g t h e r e A = E .
If not l e t H (I
A E
=)
.
:
*
: Det - B ( r , A )
If E E I L D then E E K P . 1 ] then ( c ) PD * I f E i s i n l12n+l [ r e s p . Z ’ -2nt2 1 [resp. A -2nt2 1
Proof
to notice
:
(a) I f E E
To p r o v e
257
is coarse,
then
I <
I IR/E 1
5 a n d E i s i n d u c e d by Q
K
.
L. HARRINGTON and
258 Procf
: Work
i n L [ IR
1,
R. SAM1
AD i s t h u s a v a i l a b l e .
( b ) i s a n immediate consequence of Theorem
CO-ED
1 (recall that
sets are co-~-Souslin). ( a ) follows from ( b ) ,
indeed a coarse E i n E D w i l l be i n f a c t i n U s i n g (b) E i s i n d u c e d b y a p r e w e l l o r d e r i n g
HYP by C o r o l l a r y 3 . b . .
rv
5
of
<
i n v o k i n g now C o r o l l a r y 3 . a . w e g e t E S P (being an 2 E - i n v a r i a n t s u b s e t o f IR ) a n d h e n c e t h e l e n g t h o f <, i s < 5 .
IR
The d e t e r m i n a c y h y p o t h e s i s i s a b i t tance an examination of would show t h a t
the proofs
D e t - B ( E )
extravagant here,
f o r ins-
Theorem 0)
( i n c l u d i n g t h a t of
s u f f i c e s for ( a ) a b o v e .
4
T h i s i s not v e r y
I n t h e n e x t t h c o r e m we c o n s i d e r p r o j e c t i v e e q u i v a -
satisfactory.
lence r e l a t i o n s and though t h e conclusions a r e easy consequences o f our previous results,
i s needed t o weaken t h e h y p o t h e s i s
some work
t o PD. Theorem 5
:
A s s u m e PD : L e t E b e a c o d r z e e q u i v a l e n c e r e l a t i o n . 1 1 E is p [resp. E 21it21 t h e n W E [ resp. 22nt1 2ntl 1 IR. and E i s induced. by a 2n+l [ r e s p . p r e w e l l o r d e r i n g of 1 (b) I f E is -211t2 Ill then IR/E Q $ 2 n t l . ( a ) If
Proof If
a
: Assume A D ,
1
I
I
a;nt2]
I
momentarily
'
I
:
i
E E l 1 2 n t 2 t h e n E i; c 0 - 6 ~ ~ ~ ~ - S o u sa nl di n t h u s w e g e t f r o m Thm 1
t h a t E b e i n g c o a r s e i s i n d u c e d by a p r e w e l l o r d e r i n g of 1 yields : l e n g t h Q 6.2nt2, t h i s with Corollary 3.a
(b') I f E
E lI:r,t2
iz c o a r s e t h e n I I R / E I
a projective prcwellordering of R e v e r t i n g t o PD o n e c a n d e r i v e 1
1
Q &2ntl
IRof
is induced by
and E
IR. (a) f r o m ( b ' )
:
S u p p o s e E E iJr a r i d is c o a r s e t h e r : f r o m ( b ' ) E is induced by i n t l a p r o j e - t i v e p r e w e l l o r d e r i n g o f IR , b y C o r o l l a r y 3 . a t h i s p r e w e l 1 h e n c e h a s l e n g t h < -62 n + l hence, lorceririg is i n f a c t 4 2ntl
n;nt2,
i s c o a r s e t-hen b y C o r o l l a r y 3 . b E E 1 c u o ' ~( b ' ) a b o v e a r d C o r o l l a r y 3 . b t o get W E &2nt2 1 2nd E i s ir.ducEd by a 4 p r e w e l l o r d e r i n g o f E. 2n I t reniainz t o prove b ' ) f r o m PD. W h i c h we a s s u m e f r o m h e r e
dgain
on
we
I
I
. F c r n o t a t i o h d l s i m p 1 c i t y s u p p o s e C E IIint2a n d s e t 6
=
1 6 -2nt1'
EQUIVALENCE R E L A T I O N S ,
E
P R O J E C T I V E AND BEYOND
259
is co-6-Souslin We f o l l o w v e r y c l o s e l y t h e p r o o f
only d i f f e -
o t Theorem 1, t h e
r e n c e b e i n g t h e c o n s t r u c t i o n o f a " b e t t e r " a u x i l i a r y c l a s s for E . 1 onto 1 F i x W a c o m p l e t e I12ntl set, q : W + 6 a Il 2 1 l t l norm. Let A be a p o i n t class. Given S C g k x IRn, set code(S,v)
...,
= {(ul,
...,
(9(u1), S
...,
is called A i n the codes
Set
r' =
:
{S
t o c o n s i s t of
gktl
U
I
am)
1
E
I
-
<
s o m e 5,
Claim :
r
Proof
(Sketch)
1
(relative t o q ) if
6 one has
code(S.9)
is an auxiliary claim.
- Ke 1 1 c a n b e s t a t e d a s
:
S C 6
If
A.
U(S0,F,U).
x IR i s p r o j e c t i v e i n t h e c o d e s a n d VC < 6 1 t h e n t h e r e i s a -A2 n t l p a r t i a l function F, such t h a t :
(*)
E
:
The main t e c h n i c a l r e s u l t of [Ha follows
and
u i E W S a . E IR
SI.
i s ~l i n the codes] fii.ally r is defined 2nt3 t h o s e s u b s e t s S C g k x IRm s u c h t h a t f o r s o m e
scr,u, :
a
s
IRm,
x
...,
uk, a l , al,
V ( U k ) '
3a
:S(c,a)
- Dom F 2 W - vu E W : S ( q ( u ) , F ( u ) ) From t h i s (**) If
{(u,v)
1
is not too difficult
it
and $
V E
u E
w
8 v E
v
: V
+
t o show
i s a II;ntl
6
:
norm t h e n
We now g o t h r o u g h ( 0 ) t o ( v ) o f
the definition:(o),
( i f 1 are t r i v i a l except f o r t h e p a i r i n g function p
:
6
x
6
+
6 a bijection 2
1
:
T~ q ( u ) = j l ( v ) ~ i s A '2 n t 2 '
i n the codes.
( i i i ) Closure o f
r'
of
r)
under
I t s u f f i c e s now t o v e r i f y c l o s u r c o f
7'
VA,
vIR
under
t/"
i n [ H a - Ke 1 1 w h e r e i t i s s h o w n : 1 Let : s 1) b e a A 2 r , t l norm, i f U
is obvious This is essen-
tially
t h e c o d e s , r e l a t i v e t o $, proof
then
s e e T h e o r e m 3'7 o f [ H e The same p r o o f ,
WA.
(VF, <
.
9 )
C
9
S (C,.)
x
'
i s E J r , t L ir. 1 i s E2"+'. ( F o r ' ;i
IR
1).
mutatis nutandis,
( i v ) We f o l l o w [ Ke 4 ] t r i z e d and normed.
( i ) and
takes
l a G6del and u s e s (**) t o v e r i f y t h a t
graph p i s A 2nt2
(hence,
: one
y i e l d s c l o s u r e of
We a r e g o i n g t o s h o w t h a t
This w i l l sufficc.
Let
U" C w
r' x
7'
undcr
i s w-paranie-
IR
x
E he
L. HARRINGTON and R. SAM1
260
A theorem o f Solovay states t h a t for a Z
s e t is invariant 1 2n+3 norm' Theorem 2 . 1 ) . This yields
i t a d m i t s a il
e q u i v a l e n c e r e l a t i o n o n IR t h e n
invariant f o r t h i s relation ( i e c [Ke 4 1 6 such that : a n 2 n t J n o r m : t" : U '
n 21n t 3
a
If
:
1,
+
U'(n,u,a)
ui,ai
give rise t o
s e t and a r'-norm
u C
w
Say
-
A(a)
-
t ,v
]
x
x
IR , tl
x
U
E
7.
3 5 E(cr,B) with 3
i l e f i n e T a t r e e on
Set
-
Q(CC,U)
6,
+
r'
an o-universal
iii
6 by
x
{((n
,...
Vir
( u ) E ~
1
let
fl,l,t,
E
{an]
:
8
EIR i n
Ill
I$ 1
1 b e a I12n+l
(see [Ke
(aP i,(q((u)
P
2n+2
,
scale
- Mo 11).
- _ _ .,p(mk,Ck)))l(n,m,C)
,nk),(p(m,,Eo),..
w
given A
t h a t w h e n e v e r t E kw, u E k &
6 such
x
o n 8 , l e i - T' b e t h e t r e e a s s o c i a t e d t o
T
= @'(n,v,a).
n,u,a)
s i n i p l i c i t y we shah t h a t ,
t h e r e i s a t r e e T on w : p [ T
~ ( v =). 0 '
respectively.
( v ) For n o t a t i o n a l then
=
g p(u)
T'I
E
, . . . ,P ( ( U ) ~ - ~ ) ) )E
T
I.
One c a n now v e r i f y u s i n g ( * * ) t h a t Q E A 2 n + 3 .
(co, . . . ,
Now i f u a
p[Tt,"
-
Ck-l)
1
then
a 2 t
E,
( 3 u ) ( Q ( a , u ) 8 (u)o = C 0 ,
p [ T ] is i n A . t >v R e f e r r i n g a g a i n t o t h e proof o f T h e o r e m 1 , i t
. . . , ( u ) k - 1 = 5 k-1)
Tt.iis
t o check t h a t
H
:
IR
-t
if
is projective
6 ,
v i e w e d a:
t h e game G ( E , T )
IR.
t h e s t a n d a r c ganr:
If
C a s e E! h o l d s t h e n we a r e i n
a s i n our d i c c u s s i o n t h e r e t h i s c a n b e
G x for s o m e X
w6.
Again it
straiglltforward (but rather
tedious) to verify that
i r i t h e o b v i o u s way. t h e n t h e
set
-
-X(u)
df
vi
:
:
i:) t h e c o d e s , h e n c e E i s i n d u c e d b y a
p r o j e c t i v e p r e w e l l o r d e r i n g sf presence of
is s t r a i g h t f o r w a r d
the function H defined there
Ciise A h o l d s ,
is
i f t h i s is done
:
( u ) . E II a n d < ~ ( ( u ) ~ )
is projective. We c a n now i n v o k e
( b ) of
T h e o r e m 0 T O s e e t h a t PD i s e n o u g h
i
t o a s c e r t a i n t h e d e t e r m i n a c y o f t h e c o d e d g a m e G' X' C o r o l l a r y F.
:
Assume t h e A x i o n o f C h o i c e a n d P D , 1 Q K 2
E E r12 * jIR/5
1
iet
E be c o a r s e
:
EQUIVALENCE RELATIONS, PROJECTIVE AND BEYOND E E
gt
1
or II,
Proof
: By
6'
H3.
-3
Q
I
=)
1
IR/E
H ~ .
Q
a theorem o f Martin ( s e e [Ke 3
4 2. Sundry questions
1 1, u n d e r t h e h y p o t h e s e s ,
:
2 IR, b y d e f i n i t i o n A/E
Let A
is {a/E
I
J . S t e r n p r o v e s t h e f o l l o w i n g : PD * I f E E t i v e E-thin
26 1
set then
< N1.
IA/EI
a
E
gi
A}.
I n [ S t 11,
and A is a projec-
We g e n e r a l i z e a n d s t r e n g t h e n
this, here. Theorem 7
:
A s s u m e PD. E-thin
then
C 12 n t 2
Let E E
lA/EI
Q
1
a n d suppose A i s p r o j e c t i v e and
c52ntl.
Proof : C o n s i d e r f i r s t t h e c a s e w h e r e A i s i n v a r i a n t t h e n b y Theorem 2 ,
Cknt2.
A E
F u r t h e r more w h a t i s shown i n e f f e c t i n 2
t h e proof of C o r o l l a r y 3 . a is t h a t : A i s E-thin * A x A i s E - t h i n . 2 2 H e n c e ( A x A ) n (IR - E ) i s E - t h i n , s o a g a i n b y T h e o r e m 1, t h i s
set is i n
zL
-2nt2' I f we now d e f i n e
a E
A
B - a B A
6 B A
&
a E 5
or
t h e n c l e a r l y E A i s c o a r s e a n d t h e p r e v i o u s c o n s i d e r a t i o n s show A 1 1 E h e n c e b y T h e o r e m 4 : IIR/EAI Q i z n t l . A But IA/EI t 1 = [IR/E ,hence the conclusion.
E
I
If
-
now A i s n o t E - i n v a r i a n t ~
~ ~
=
~~
B{
E
consider
:
aA
E
-
:
~
B
~
.
-
An e a s y a r g u m e n t u s i n g u n i f o r m i z a t i o n s h o w s t h a t A i s E - t h i n . 1 T h u s l A / E / = IA/EI Q f i 2 n + l .
-1
Assuming A D , s u b s e t o f IR. Theorem 8 AD
A c o u l d b e t a k e n to b e a n a r b i t r a r y i n v a r i a n t
A similar result
c a n b e f o r m u l a t e d for E E E D :
:
* If E E E D and A
c
IR i s E - i n v a r i a n t ,
then
IA/EI
Theorem 7 c a n i n f a c t b e d e r i v e d from t h e f o l l o w i n g :
"&.
L. HARRINGTON and R. SAM1
262 Theorem 9
:
I f E is a coarse projective equivalence relation 1 1 E is C t h e n /EVE QS2n+1. 2n+2 ( b ) Assume AD(L[IR]). I f E i s a c o a r s e i n d u c t i v e or c o i n d u c t i v e 1 c l a s s e s t h e n ]?&/El <62nt1. equivalence r e l a t i o n with X -2nt2 ( a ) Assume PD.
1
and every c l a s s of
Proof
:
( b ) Work i n L[ I R 1.
stated as
: AD
1
d i s j o i n t -Z2 n + 2
IR/E
s e t s then
is well-orderable
We s h a l l n o t p r o v e
can be proved,
A theorem of Kechris
<
* If ( A c 1 5 1)
q )
1
is a
< fi2n+2.
i n [ Ke 2 ] c a n b e
1-1 s e q u e n c e
of pairwise
.
Now if E i s a s g i v e n t h e n
( b y Theorem 4 ) , h e n c e t h e c o n c l u s i o n .
( a ) h e r e , we j u s t mention t h a t Kechris'theorem
for t h o s e s e q u e n c e s ( A E
assuming only PD,
which a r e " p r o j e c t i v e i n t h e codes".
1
F,
<
q )
This uses the techniques
of
[Ke 5 1 .
TG c o n c l u d e w e l i s
(1) C o n j e c t u r e
/ W E /Q A
(w
( 2 ) Conjecture
a few open q u e s t i o n s a n d c o n j e c t u r e s .
( M a r t i n ) : If E i s co-A-Souslin
<
and coarse then
A)
(Solovay
: AD
*
If E is coarse then
W E
is w e l l -
orderable. ( 3 ) What
a r e t h e minimal determinacy hypotheses needed s a y , f o r One s h o u l d b e a b l e t o d e r i v e t h e r e s u l t s f o r 1 1 Note t h a t ( a n d p e r h a p s j 1 2 n + 2 ) from D e t
Theorem 5 .
1 c2n+l, C:ntl
Cint2
is different;
(azn).
from t h e r e s u l t s o f [Ha 1 ] one can quote
r e l a t i v e c o n s i s t e n c y r e s u l t s to t h e e f f e c t t h a t a lence relation
"can"
x
Va :
<
have any number o f c l a s s e s
(4) G e n e r a l i z e t h e f o l l o w i n g r e s u l t o f S t e r n [ S t 2
y 1 < H1 * If
E E
4;
C1
6 2
equiva-
1
and a y l t h e c l a s s e s o f E ( e x c e p t
p o s s i b l y a c o u n t a b l e n u m b e r ) a r e Bore1 o f b o u n d e d r a n k t h e n IIR/E
1
.
EQUIVALENCE RELATIONS, PROJECTIVE AND BEYOND
263
References Barwise
K.J.
[ Ba 1 1 A d m i s s i b l e
sets and s t r u c t u r e s ,
Heidelberg-New-York, [ Ba 2
1
Springer-Verlag,
Berlin-
1975.
Handbook o f M a t h e m a t i c a l
Logic,
North-Holland,
Amsterdam,
1977. H.
Becker
1
[Be J.P.
Infinitdry
dissertation,
L.
i n [Ke
- Mo 2 1 .
Burgess
1
[ Bu
Partially playful universes,
languages
Univ.
and d e s c r i p t i v e
Calif.
Berkeley,
s e t tiieory,
Doctoral
1974.
Harrington
[Ha 11
Long p r o j e c t i v e
well-orderings,
Ann.
Math.
Logic,
12 (1977),
1-24. [Ha 2
1
A powerless
proof
of
a theorem of
Silver,
mimeographed,
November 1 9 7 6 . L.
H a r r i n g t o n a n d A.S.
[Ha
- Ke 1 ]
ngraphed, [Ha
tiechris
Ordinal quantification
January
and t h e models
L[T 2 n + l l ,
mime-
1977.
- Ke 2 1 O r d i n a l g a m e s a n d t h e i r a p p l i c a t i o n s , t o a p p e a r .
A.S.
Kechris
1Ke 1 1 Ann.
Measure and c a t e g o r y i n e f f e c t i v e d e s c r i p t i v e
Math.
[Ke 2
1
[ K e 4
I 1
5
(1972/73),
$
equivalence relations,
AD and p r o j e c t i v e
set theor-y,
337-384.
On t r a n s f i n i t e s e q u e n c e s o f p r o j e c t i v e
cation to
[Ke 3
Logic,
ordinals,
sets,
w i t h an appli-
mimeographed. i n [tie
- Mo 2 I.
Countable ordinals
and t h e a n a l y t i c a l h i e r a r c h y ,
Another proof
L[TZntl]
11, t o
appear. [ Ke 5
1
mimeographed, A.S.
Kechris
[ K e - Mo 1 1 [Ke
- Mo 2 1
berg-New-York,
that
n IR = C
January 1979. and Y . N .
i n Z F t D!:
+
PD,
Moschovakis
N o t e s on t h e t h e o r y o f Cabal Seminar 76-77, 1978.
2nt2
scales,
i n [tie
Springer-Verlag,
-
Mo 2
1.
Berlin-Heidel-
L. HARRINGTON and R. SAM1
264
A. Louveau [ L ] Relations d'gquivalences coanalytiques (suite), in Sgminaire de Thgorie des Ensembles, Publications Mathgma,tiques de l'Universit6 Paris VII, to appear. D.R.
\!aTtin
[ M a 1 Descriptive set theory, in [ B a 2
1.
Y. N . Mos chovak i s [Mo 1 1
Determinacy and prewellorder,ings of the continuum, in
Mathematical logic and foundations of set theory, Y. Bar-Hillel ed., North-Holland, Amsterdam, 1970. Inductive scales on inductive s e t s , in [ K e - Mo 2 ]
[Mo 2 ]
.
J. Mycielski
[My]
Independent sets in topological algebras,
Fund. Math., 5 5
(1964), 139-147. J.C. O x t o h y [ Ox
1
Measure and Category, Spriiiger-Verlag, Berlin-Heidelberg-
New-York. 1971.
J.H. Silver
si
I
II~ equivalence
relations, to appear.
J. Stern [ S t 1 1 Perfect set theorems f o r analytic and coanalytic equivalence reiations, to appear. [St 2
1 Definable families o f Bore1 sets of bounded r a n k , mimeogra-
phed, 1978. [ St 3 ] Relations d'gquivalences coanalytiques, in SQminaire de
Thgorie des Ensembles, Publications Math6matiques de l'Universit6 Paris V I I ,
t o appear.
LOGIC COLLOQUIUM 78 Boffa, D. van Dalen, K . McAloon (eds.) 0 North-Holland Publishing Company, 1979 M.
PROJECTIONS OF LAWLESS SEQUENCES I1
G.F. van der Hoeven and A.S. Troelstra Department of Mathematics University of Amsterdam Amsterdam, The Netherlands
Contents 1.
Introduction
2. Description of the model 3 . Patterns, dressings and restrictions
4 . Comparison of restrictions
5. The overtake-property 6 . Assignment of values
7. Permutability conditions 8 . Schemata valid in the model
I. INTKODUJ
This paper may be regarded as a sequel to
1.1.
[ T 3 ] and [D,T], which explains
the title. The conceptual basis of choice sequences has been discussed at length in earlier publications, to which we refer the reader for detailed information (see e.g. [ T 4 ] , especially appendices B and C). We are indebted to D. van Dalen and
G. Rreisel f o r criticism of earlier drafts. 1.2. We recall motivation and results of [ D , T ]
and [T2]. The simplest concept of
choice sequence is that of a lawless sequence; analysis of this concept leads to a convincing axiomatization
(E in IT&],
Chapters 2 and 3; originally due to Kreisel)
which via an elimination theorem completely characterizes the properties of lawless sequences LS relative to (the properties of) lawlike objects and relative t o a suitable language. Lawless sequences however exhibit two peculiarities which might be regarded as limitations (though in certain contexts they are an asset!): (a) LS is not closed under any non-trivial continuous operation such as e.g. CY
-+
Ax. 2 a x ;
(b) no element of LS satisfies any non-trivial overall-restriction on its values,
not even if the restriction does not contain choice parameters. For example, no lawless sequence satisfies Ra = 3xVy>x3z(ay
=
2 2 ) , and a fortiori.the law-
like sequences are not even extensionally contained in LS, i.e. VaVa(a # a). 265
266
G.F.
VAN DER HOEVEN and A . S .
TROELSTRA
I n o r d e r t o d i s c o v e r u n i v e r s e s of s e q u e n c e s n o t s u b j e c t t o s u c h l i m i t a t i o n s , w e may e i t h e r (1)
c o n s i d e r more g e n e r a l pr
(2) c o n s t r u c t u n i v e r s e s w i t h t h e d e s i r e d p r o p e r t i e s from l a w l e s s s e q u e n c e s and lawl i k e objects.
(Examples: t h e u n i v e r s e s U, U* i n C h a p t e r 4 o f [ T 4 ] ) .
Here and i n t h e s e q u e l t h e t e r m " u n i v e r s e "
B
refers, rather loosely, e i t h e r t o a
c o l l e c t i o n o f s e q u e n c e s which c o r r e s p o n d s t o some s p e c i f i c ( i n f o r m a l l y d e s c r i b e d ) c o n c e p t , o r t o a c o l l e c t i o n of s e q u e n c e s which c a n s e r v e as " c h o i c e sequences'' i n
2
some i n t e r p r e t a t i o n o f a f o r m a l t h e o r y o f c h o i c e s e q u e n c e s ( s u c h as
07
E ) , i.e.
t h e o b j e c t s t h a t t h e c h o i c e v a r i a b l e s r a n g e o v e r . Thus e . g . w e may s p e a k a b o u t a CS-universe
m
f o r a c o l l e c t i o n o f c h o i c e s e q u e n c e s s a t i s f y i n g t h e axioms o f
E.
S i n c e c o n t i n u i t y is, h i s t o r i c a l l y as w e l l as c o n c e p t u a l l y , a k e y n o t i o n i n t h e t h e o r y and p r a c t i c e of c h o i c e s e q u e n c e s , w e w i l l be p r i m a r i l y i n t e r e s t e d i n u n i v e r s e s s a t i s f y i n g c e r t a i n c o n t i n u i t y schemata. Possibility ( 1 )
mentioned above h a s b e e n i n v e s t i g a t e d f o r example i n [ T l l , [ T 2 ]
( c f . a l s o [ T 4 ] , Appendix R),
and m a i n l y c o n c e r n s f a i r l y s t r a i g h t f o r w a r d
general-
i z a t i o n s of LS where a c h o i c e s e q u e n c e i s viewed as a p r o c e s s o f c h o o s i n g a t e a c h s t a g e a v a l u e and a ( l a w l i k e ) r e s t r i c t i o n on f u t u r e c h o i c e s of v a l u e s ( l a w l i k e means t h a t t h e r e s t r i c t i o n s do n o t c o n t a i n c h o i c e p a r a m e t e r s ) ; t h e r e s t r i c t i o n s a r e t a k e n from a f i x e d c o l l e c t i o n
R.
U n i v e r s e s of t h i s t y p e do p e r m i t n o n - t r i v i a l
o v e r a l l r e s t r i c t i o n s , but are s t i l l
s u b j e c t t o l i m i t a t i o n ( a ) , and might t h e r e f o r e r i g h t l y be termed dn<'i,)&ta:*:/ ("anti-social"
.
1
i n [D,T]).
I f t h e c o l l e c t i o n R of r e s t r i c t i o n s o f a n i n d i v i d u a l i s t i c u n i v e r s e A i s enumera b l e , t h e n i t seems n a t u r a l to a t t e m p t t o i m i t a t e e l e m e n t s o f A by p i c k i n g numeri c a l v a l u e s and r e s t r i c t i o n s i n a l a w l e s s way. To b e p r e c i s e , i f a i s a l a w l e s s s e q u e n c e , we u s e t h e v a l u e ax t o c o d e b o t h a n u m e r i c a l v a l u e and a r e s t r i c t i o n from I?,
s u c h t h a t i f ryx c o d e s a n a d m i s s i b l e s e q u e n c e o of v a l u e s and r e s t r i c t i o n s
( i . e . a p o s s i b l e i n i t i a l segment f o r a s e q u e n c e from A ) , A) f o r each y I*x
*9
codes a n a d m i s s i b l e c o n t i n u a t i o n a
B) each a d m i s s i b l e c o n t i n u a t i o n of
CJ
i s coded by
L e t Il be t h e mapping a s s i g n i n g t o e a c h a;
n
ib
E
i s c a l l e d t h e p r o j e c t i o n mapping, and A'
K .;
* F
*
then <<x,R>>
of
0,
and
f o r some y .
LS t h e s e q u e n c e of values coded by =
Ula
:
u
t
LSI t h e s e t of p r o j e c t -
i o n s i m i t a t i n g A. The s t u d y of such p r o j e c t i o n s c o n s t i t u t e s t h e s u b j e c t of [D,T] and [T31. Note t h a t t h u s a p p r o a c h ( 1 )
l e a d s t o t h e s t u d y of a v e r y s p e c i a l c a s e of app-
r o a c h (2). To e x t e n d t h e i m i t a t i o n by p r o j e c t i o n s t o c a s e s where
R
cannot be in-
dexed by N, one n e e d s t o c o n s i d e r l a w l e s s s e q u e n c e s w i t h v a l u e s i n o t h e r domains (e.g. lawlike sequences).
PROJECTIONS OF LAWLESS SEQUENCES I1
267
As n o t e d i n [D,TI, t h e u n i v e r s e o f p r o j e c t i o n s A ' w i l l a s a r u l e be a s a t i s f a c t o r y i m i t a t i o n of A o n l y i f w e r e s t r i c t o u r s e l v e s t o a s u i t a b l e l a n g u a g e ( e . g . t h e l a n g u a g e of L S , w i t h A' as t h e i n t e r p r e t a t i o n of t h e l a w l e s s v a r i a b l e s ) . The need f o r s u c h a r e s t r i c t i o n s t e m s from t h e f a c t t h a t a s i n g l e i n i t i a l segment of a n A-sequence,
( c o n s i s t i n g of p a i r s o f v a l u e s and r e s t r i c t i o n s ) may b e coded by
d i s t i n c t i n i t i a l segments o f l a w l e s s s e q u e n c e s , t h e r e b y i n t r o d u c i n g i n e s s e n t i a l " i n t e n s i o n a l information".
Because of t h e r e l e v a n c e o f t h i s phenomenon f o r t h e
rest of t h i s p a p e r , w e r e c a l l a v e r y s i m p l e i l l u s t r a t i o n : as a p r o j e c t i o n model i m i t a t i n g lawless 0-1 s e q u e n c e s ( v a r i a b l e s
A = I r a : a €LSI w i t h We have VF3x3n(r =IIa
A
aO=x)
~i =
E,~I),
w e may t a k e
hahx.sg(ux).
i n A ' , b u t x c a n n o t b e found c o n t i n u o u s l y i n t . On
t h e o t h e r hand, f o r t h e p r i m i t i v e c o n c e p t o f l a w l e s s 0-1 s e q u e n c e s V d x - c o n t i n u i t y
i s always v a l i d . However, r e l a t i v e t o t h e l a n g u a g e o f 0-1 s e q u e n c e s )
1.3.
A ' s a t i s f i e s a l l axioms o f
IS-
E"
( c f . [T41,
( t h e t h e o r y of l a w l e s s 2.18,
3.23;
[T31).
I n d i v i d u a l i s t i c u n i v e r s e s are s t i l l more o r l e s s d i r e c t g e n e r a l i z a t i o n s o f
L S , b u t social (= n o n - i n d i v i d u a l i s t i c ) u n i v e r s e s which a r e n o t s u b j e c t t o l i m i t -
a t i o n ( a ) , p r e s e n t new problems. An example of s u c h a c o n c e p t , i n t e n d e d a s a model f o r t h e c h o i c e sequences of before i n [TI], [T2]
o f [K,T]
i s s k e t c h e d i n a p p e n d i x C of [T4] (and
1.
We c a n n o t a p r i o r i e x p e c t t o f i n d a p e r s p i c u o u s a x i o m a t i z a t i o n t o g e t h e r w i t h a
c o n v i n c i n g c o n c e p t u a l m o t i v a t i o n f o r t h e axioms, as i n t h e c a s e o f LS. T h i s p r e s e n t s u s w i t h a s t r o n g p r a g m a t i c argument f o r u s i n g t h e t e c h n i q u e o f p r o j e c t i o n s o f LS. The p r o p e r t i e s of t h e p r o j e c t i o n s a r e d e r i v a b l e from LS, and inasmuch t h e p r o j e c t i o n s t r u l y i m i t a t e t h e p r i m i t i v e c o n c e p t s t u d i e d , w e c a n u s e them t o b a c k up o u r a n a l y s i s o f t h e p r i m i t i v e c o n c e p t . Note t h a t e s t a b l i s h i n g t h e c o r r e c t n e s s of t h e i m i t a t i o n c a n b e d e l i c a t e o r d i f f i c u l t . E i t h e r o n e h a s a s t r o n g c o n c e p t u a l i n s i g h t i n t o t h e n o t i o n under c o n s i d e r a t i o n , i n which c a s e i t a l s o becomes p o s s i b l e t o j u d g e t h e s u c c e s s of t h e i m i t a t i o n , o r o n e h a s a number of p l a u s i b l e key p r o p e r t i e s (axioms) which t h e i m i t a t i o n h a s t o s a t i s f y . I n t h i s c a s e o n e may hope t h a t t h e s t u d y o f p r o j e c t i o n s e n a b l e s u s t o improve on t h e c o n c e p t u a l a n a l y s i s o f t h e c o n c e p t t o be i m i t a t e d . T h e p r o j e c t o f i m i t a t i n g p r i m i t i v e n o t i o n s o f s e q u e n c e by means o f p r o j e c t i o n s
of L S , and t h u s i n a s e n s e reducing t h e s e p r i m i t i v e n o t i o n s t o l a w l e s s s e q u e n c e s may b e compared t o a s i m i l a r r e d u c t i o n i n s e t - t h e o r e t i c f o u n d a t i o n s , where i t i s shown t h a t t h e s t r u c t u r e s of a c t u a l m a t h e m a t i c s (which do n o t a p r i o r i p r e s e n t themselves a s s e t - t h e o r e t i c
s t r u c t u r e s ) c a n b e i s o m o r p h i c a l l y r e p r e s e n t e d by means
of s e t s . The a n a l o g y i s n o t c o m p l e t e : t h e s e t - t h e o r e t i c r e p r e s e n t a t i o n i s r e g a r d e d
as f u l l y i s o m o r p h i c ( o f c o u r s e w . r . t .
extensional p r o p e r t i e s only) t o t h e o r i g i n a l
s t r u c t u r e , w i t h o u t r e s t r i c t i o n t o p r o p e r t i e s e x p r e s s i b l e i n a p a r t i c u l a r language, though o b v i o u s l y l i m i t e d t o e x t e n s i o n a l p r o p e r t i e s . R e s t r i c t i o n s t o a p a r t i c u l a r
268
G.F. VAN DER HOEVEN and A . S . TROELSTRA
language p l a y a r o l e i n s e t - t h e o r e t i c d e s c r i p t i o n s of e . g . h y p e r a r i t h m e t i c s e t s , I where o n e c o n s i d e r s e . g . A -comprehension. 1
1 . 4 . We now t u r n t o t h e t a s k of a c t u a l l y r e p r e s e n t i n g s o c i a l n o t i o n s o f c h o i c e
sequence by means o f p r o j e c t i o n s o f LS. L e t u s f i r s t d e s c r i b e t h e c o n c e p t s t u d i e d i n t h i s p a p e r , which may be r e g a r d e d a s a t y p i c a l example.
*
L e t IFn : neN1 b e a c o u n t a b l e s e t o f " l o c a l " c o n t i n u o u s mappings i n t h e form
*
(F,C)(X)
=
Fn(~x,x)
*
a l a w l i k e two-place f u n c t i o n d e t e r m i n i n g F ).
(F
*
R e l a t i v e t o IFn : nrN1 we now d e s c r i b e t h e u n i v e r s e of c h o i c e s e q u e n c e s U as f o l l ows. An a r b i t r a r y e l e m e n t E = F
*
E
o f U i s g e n e r a t e d i n s t a g e s : a t s t a g e 0 , we know t h a t
; w e choose v a l u e s
E~
0 x + I we may d e c i d e t o make 1
E~
F
, c O 1, ...
0 0
a t stage 1,2,
...
dependent o n a n o t h e r s e q u e n c e
; a t a certain stage
by c h o o s i n g f u r t h e r
E]
v a l u e s of c O a c c o r d i n g t o
E ~ ;a t s t a g e X I + I , E] itself 1 h a s n o t y e t b e e n made dependent on a n o t h e r s e q u e n c e , b u t a t a l a t e r s t a g e X,+l > t x2+l,Lc. may i n t u r n b e made t o behave l i k e F E~ f o r a l l s t a g e s X xl+l
i.e.
f o r s t a g e s xl+y+l t x l + l
E~
b e h a v e s l i k e F:
*
"2
etc.
I f we now a t t e m p t t o g e n e r a l i z e t h e method o f p r o j e c t i o n s from t h e i n d i v i d u a l i s t i c t o t h e s o c i a l c a s e we meet w i t h a n o b s t a c l e : i n o r d e r t o i m i t a t e t h e g e n e r a t i o n of a n
E E
U , d e s c r i b e d above, by a p r o j e c t i o n of a l a w l e s s s e q u e n c e a , a must
n o t o n l y t e l l u s what v a l u e s t o c h o o s e f o r e O O , c O 1 , . . . , b u t when f u r t h e r c h o i c e s are t o b e made d e p e n d e n t on dependent o n
E,,
In fact, on
T),
CWJ
o r q on
E]
i n t u r n i s made
e t c . Since a should c o n t a i n a l l t h e information on
E,,...
n e c e s s a r i l y a l s o c o n t a i n a l l i n f o r m a t i o n on E ~ . F ~ , E ~ , . . . E
a must a l s o i n f o r m us
and when
two s e q u e n c e s c , q of E,
o r both
E
and
q
U
.
E,
a must
a t some s t a g e may become d e p e n d e n t , e i t h e r
on a t h i r d s e q u e n c e
E',
and t h u s a lawless a
used i n t h e p r o j e c t i o n s h o u l d c o d e i n f o r m a t i o n a b o u t a l l p o s s i b l e p r o j e c t e d s e q u e n c e s , i . e . a s i n g l e a s h o u l d be c a p a b l e o f g e n e r a t i n g t h e whole p r o j e c t e d universe. The s o l u t i o n of t h e d i f f i c u l t y i s s u g g e s t e d by t h e o b s e r v a t i o n t h a t i t i s poss i b l e t o construct a countable E - u n i v e r s e
from a s i n g l e l a w l e s s s e q u e n c e a ; t h i s
universe is
I n * ( a ) n : ncN1, where f o r ( a ) we may t a k e > y . a j ( n , y ) o r Xy.(af),
( c f . [T4],
3.19-22).
In t h i s universe i t i s possible t o r e f e r t o individual
s e q u e n c e s "by name" : t h e i n d e x n s e r v e s a s t h e name of n * ( a ) n . This suggests t h a t we look f o r p r o j e c t i o n s a the universe U
a
TI
such t h a t f o r any s i n g l e l a w l e s s
= { n n : ncN} i m i t a t e s t h e s o c i a l u n i v e r s e
n
U
s k e t c h e d above.
PROJECTIONS OF LAWLESS SEQUENCES I1
269
1.5. P r i n c i p a l r e s u l t s . F o r t h e u n i v e r s e [I d e s c r i b e d i n t h e p r e c e d i n g s u b s e c t i o n i t i s indeed p o s s i b l e t o c o n s t r u c t s u c h p r o j e c t i o n models ial"
ua,
which c a n b e d e s c r i b e d i n a v e r y " p i c t o r -
way ( s e e s e c t i o n 2 ) a n d which c a n b e shown t o b e a d e q u a t e i n t h e f o l l o w i n g
sense: ( i ) it is i n t u i t i v e l y evident that the U
really imitate U ;
( i i ) r e l a t i v e t o s t a t e m e n t s i n t h e l a n g u a g e L of LS, Ua c a n be shown ( i n _ ? ) to
s a t i s f y a set o f axioms which p e r m i t s e l i m i n a t i o n o f v a r i a b l e s r a n g i n g o v e r More p r e c i s e l y , l e t E , F ' , ~ b e used f o r c h o i c e v a r i a l r l e s o f L, and l e t a,t?
ua'
be d i s t i n c t v a r i a b l e s ( n o t i n L ) r a n g i n g o v e r LS. L e t A b e any s e n t e n c e o f L , and Aa i t s i n t e r p r e t a t i o n i n
ranging over
u
uoi
(i.e.
t h e c h o i c e v a r i a b l e s of L a r e i n t e r p r e t e d a s
) . The s e t o f axioms d e t e r m i n e s a t h e o r y
2 over
L, f o r which an
e l i m i n a t i o n mapping u may b e d e f i n e d s i m i l a r t o t h e e l i m i n a t i o n mappings f o r and
( c f . [T41, 3.13, page 40 and 5.3, T + u(A)
N
Then a l s o
g I-
CI
o(A)
c)
LS), h e n c e
T-t-
A
*
-
IDBl
2 was
(because
A"
e l i m i n a t i o n theorem f o r m
page 70) s u c h t h a t
A. :
ElI-
o(A)
c)
.r(VuA")
t h e p r o p e r t i e s of
ua
(T
t h e r e f o r e by t h e
t h e e l i m i n a t i o n mapping f o r
+ u(A). ( T h i s method of p r o o f i s a t l e a s t as s i m p l e a s
2 would
a d i r e c t p r o o f o f t h e e l i m i n a t i o n theorem f o r t o L,
ua) and
valid for
have b e e n . )
Thus, r e l a t i v e
are f u l l y c h a r a c t e r i z e d r e l a t i v e t o t h e p r o p e r t i e s o f
lawlike objects. There i s o n e schema shown t o b e v a l i d f o r f o r U, a n analogue of Va3B-continuity f o r (1)
V E ~ ~ A ( E ,+~ ) 3ecKVEA(c,F* e(F)
u
g: (E))
which i s n o t immediately o b v i o u s
.
I f we t h i n k o f a u n i v e r s e c l o s e d under some s e t o f c o n t i n u o u s o p e r a t i o n s {Yi : i r I } t h e a n a l o g u e would become
(2)
VE~~A(E,T+ ) ) 3e€K3SEN-.IVEA(E,r5(e(E))
(E))
.
I f { r i : i e I } c o n t a i n s ui.7 i n d u c t i v e l y d e f i n e d c o n t i n u o u s f u n c t i o n a l s , t h e schema
( 2 ) i s e a s i l y s e e n t o b e e q u i v a l e n t t o Va36-continuity. The p r o o f o f t h e v a l i d i t y o f ( I )
l e n d s s u p p o r t t o t h e remark i n [T4](A6, page
137) on t h e i n c o n c l u s i v e n e s s of M y h i l l ' s c o u n t e r e x a m p l e t o V a 3 6 - c o n t i n u i t y
: the
c o u n t e r e x a m p l e t a c i t l y assumed t h e u n i v e r s e o f c h o i c e s e q u e n c e s t o b e c l o s e d under c e r t a i n h i g h l y non-extensional
o p e r a t i o n s ( i n t r o d u c e d by t h e a c t i o n s o f t h e "crea-
t i v e s u b j e c t " ) , whereas o u r r e s u l t seems t o p o i n t t o t h e c o n c l u s i o n t h a t a l l cond i t i o n s o f c l o s u r e u n d e r c e r t a i n c o n t i n u o u s o p e r a t i o n s r e a l l y have t o b e b u i l t a p r i o r i i n t o t h e c o n s t r u c t i o n of t h e u n i v e r s e . Furthermore w e observe ( i i i ) The c o l l e c t i o n o f models
{ua
: a
E
LS} c a n b e made i n t o a s h e a f model o v e r
Baire s p a c e ( s e e 8 . 7 ) . ( i v ) I t l o o k s a s i f t h e p r o o f s g i v e n i n t h i s p a p e r do n o t r e a d i l y g e n e r a l i z e t o more complex c a s e s w i t h o u t l o s i n g t h e i r i n t u i t i v e a p p e a l , b e c a u s e of t h e i n c r e a s i n g
270
VAN DER HOEVEN and A . S .
G.F.
technical complexities.
TROELSTRA
However, t h e i n t u i t i v e d i r e c t n e s s o f t h e p r e s e n t a p p r o a c h
and t h e p a r a d i g m a t i c v a l u e of U make a n i n d e p e n d e n t t r e a t m e n t o f t h i s s p e c i a l case (The f i r s t a u t h o r i s a t p r e s e n t w o r k i n g on a n a l t e r n a t i v e t r e a t -
appear worthwhile.
ment which i s less d i r e c t b u t p r o m i s e s much w i d e r a p p l i c a b i l i t y . ) ( v ) Our p r o o f s do n o t e s s e n t i a l l y depend o n b a r - i n d u c t i o n f o r LS; t h e e x t e n s i o n p r i n c i p l e ( [ T 4 ] , 2 . 1 1 , page 2 0 ) i s enough. (The e x t e n s i o n p r i n c i p l e s t a t e s t h a t e a c h c o n t i n u o u s o p e r a t i o n d e f i n e d on LS i s a l s o d e f i n e d on a l l s e q u e n c e s . )
U N is d e s c r i b e d i n t h e n e x t s e c t i o n ; i n 2.8 , t h e r e a d e r w i l l f i n d a n o u t l i n e of t h e r e s t of t h e p a p e r .
2. DESCRIPTION OF THE MODEL 2. I .
The models U ,
Crn
=
lawlike projections
oi
: n a N l t o b e d e s c r i b e d below are o b t a i n e d by a p p l y i n g
t o a s i n g l e l a w l e s s s e q u e n c e a. As a matter o f c o n v e n i e n c e
71
however, we s h a l l u s e t h r e e p a r a m e t e r s a,B,y
i n s t e a d ; we may t h i n k o f a,C;,'y
either
a s d i s t i n c t l a w l e s s s e q u e n c e s o r a s t h r e e i n d e p e n d e n t s e q u e n c e s e x t r a c t e d from a s i n g l e l a w l e s s s e q u e n c e 6 ( e . g . a c c o r d i n g t o 6 = LI ( n , B , y ) = Xx.v ( a x , b x , y x ) , "3 3 3 c o d i n g N3 o n t o N) ; r e l a t i v e t o t h e l a n g u a g e c o n s i d e r e d t h i s y i e l d s e q u i v a l e n t t h e o r i e s . Thus we w r i t e U s t r u c t e d from a,B,y.
( = U ) f o r t h e u n i v e r s e of p r o j e c t i o n s t o b e con-
6
a,B,Y
CI s e r v e s as a s o u r c e o f n u m e r i c a l v a l u e s , %R,Y ' U g o v e r n s t h e i n t r o d u c t i o n of r e l a t i o n s o f dependence between d i f f e r e n t e l e m e n t s o f
2.2.
I n t h e c o n s t r u c t i o n of LI
t h e u n i v e r s e , and y r e g u l a t e s t h e a p p l i c a t i o n of c o n t i n u o u s mappings from a s e t
*
{Fn : n ~ N 1 . Below w e s h a l l f i r s t i n t r o d u c e t h e "carriers", and d e s c r i b e i n p i c t o r i a l terms t h e i r b e h a v i o u r . The c a r r i e r s t h e m s e l v e s behave l i k e a model o f t h e " f r e e s e q u e n c e s "
( [ T 4 ] , C6, p.158),
e x c e p t t h a t we have t o g u a r a n t e e t h a t a l l i n i t i a l segments c a n
o c c u r ; t h i s may be done a s f o l l o w s : i f {un : neN1 i s t h e set of c a r r i e r s ( d e f i n e d below, w e t a k e a s o u r i m i t a t i o n (by p r o j e c t i o n ) o f t h e f r e e s e q u e n c e s
from ii,l3,y)
t h e c o l l e c t i o n { A n ~ m: n,m
{
=
E
N}
, where
A
i s t h e o p e r a t i o n g i v e n by
(n)x i f l t h ( n ) > x
cx
otherwise
We a b b r e v i a t e nl
t
-def
A
~
E
F r e e s e q u e n c e s a r e such t h a t i n d i v i d u a l l y t h e y behave l i k e l a w l e s s s e q u e n c e s ; howe v e r , t h e y c a n have a n o n - t r i v i a l
r e l a t i o n s h i p t o e a c h o t h e r , namely c o i n c i d i n g
from a c e r t a i n p o i n t onwards. Thus we t h i n k of a f r e e s e q u e n c e g e n e r a t i n g v a l u e s FO, EI, c 2 ,
...
w h i l e a t any s t a g e @ + I )
E ( X + Y )c o i n c i d e w i t h n ( x + y ) f o r a l l y ,
E
a s a p r o c e s s of
w e may d e c i d e t o l e t
a n o t h e r f r e e s e q u e n c e , and so o n .
PROJECTIONS OF LAWLESS SEQLRNCES I1
27 1
L a t e r , by introducing "dressed c a r r i e r s " constructed from the c a r r i e r s , we guarantee c l o s u r e under t h e operations of {F:
: ncN).
2.3. C a r r i e r s . "Val" is a f u n c t i o n which serves t o e x t r a c t many values from a s i n g l e number, e.g. we may take val(x,y) = ( Y ) ~ Any sequence 5 gives r i s e t o an i n f i n i t e sequence of sequen-
("value of y a t x"). ces v i a "Val" :
(5)
:XY.(
We c a l l the sequences
5
Xy.val(x,Cy).
(a),
,n
E
N b a s i s sequences. Below we s h a l l introduce a
f u n c t i o n r ; r e l a t i v e t o t h i s f u n c t i o n we d e f i n e
-
un :Xx.val(r(n,8(X++l)),ax) t h e sequences u
,n
E
N ;
a r e c a l l e d t h e carl-iers; u
i s t h e nth c a r r i e r , c a r r i e r n or
c a r r i e r with index n. 2.4.
Root, root-function, bundle. L e t p be a p a i r i n g f u n c t i o n from N 2 onto N-{O},
with inverses p I , p 2 (e.g. p(x,y) = 2 x ( 2 y + i ) ) . The a c t i o n of t h e root-function r may now be described a s follows r(n.0) = n r(n,m*<x>) =
r ( p x,m) i f x # 0 { r(n,m) 2 otherwise.
A
r(p,x,m)=r(n,m)
I n t u i t i v e l y r(n.8x) = m expresses t h a t a t s t a g e x and a t a l l l a t e r s t a g e s t h e val-
un a r e going t o be i d e n t i f i e d with t h e values of um. Let us t r y t o develop
ues of
an i n t u i t i v e p i c t u r e of the behaviour of t h e c a r r i e r s f i r s t . ( a ) A t any s t a g e * I ,
t h e value of on a t x coincides with t h e value of a c e r t a i n
b a s i s sequence, namely t h e b a s i s sequence ( a )
r(n,Z(x+l)) ' The c a r r i e r with index r(n,gx) i s c a l l e d t h e root of c a r r i e r n at- stage x; the
c a r r i e r s with t h e same r o o t a t s t a g e x form an x-bundle. Note: I f Bx # 0, then a t s t a g e x+l t h e x-bundle containing u p,Bx w i l l be joined
(b)
t o the x-bundle containing uPZBX; the r o o t of t h e r e s u l t i n g (x+l)-bundle i s t h e r o o t of u
P2BX at stage X * (c) With each passage of a s t a g e t o t h e next, t h e s e t of indices of r o o t s can
onlydiminish
. With
each passage, a t most one bundle g e t s assigned t o t h e r o o t of
another bundle. A s a r e s u l t , it i s now easy t o develop some p i c t o r i a l i n t u i t i o n a s t o how a typical
i d e n t i f i c a t i o n p a t t e r n over a number of s t a g e s proceeds. Take e.g. a 8 s t a r t -
ing with
G.F.
212
88
VAN DER HOEVEN and A.S. TROELSTRA
=
We can now make a p i c t u r e ( f i g . I ) i n which t h e v e r t i c a l l i n e s correspond t o b a s i s sequences; h o r i z o n t a l l i n e s r e p r e s e n t s t a g e s ; b a s i s sequences, where not coinciding with a c a r r i e r a r e indicated by i n t e r r u p t e d v e r t i c a l l i n e s . STAGE
J. =o=
-1-
-2-
-3-
-q-
- 5 -
- 6 -
-7-
-8The c a r r i e r s coincide over p a r t s of t h e i r length with b a s i s sequences, but may jump from one b a s i s sequence t o another. No c a r r i e r ever jumps back t o a former b a s i s , nor does i t jump t o a b a s i s deserted before by another c a r r i e r ; t h e c a r r i e r s once i n the same bundle s t a y t o g e t h e r . In our p i c t u r e , a t s t a g e 8 t h e c a r r i e r s with indices 0,1,2, bundles { 0 , 1 , 2 )
,
{3,4,5>,
{a}
2.5. Closure under continuous f u n c t i o n a l s .
IF:
: naN}
...,6
a r e grouped i n
( t h e root-index of each bundle i s underlined).
i s a s e t of f u n c t i o n a l s which a c t s pointwise:
PROJECTIONS OF LAWLESS SEQUENCES I1 Fa:
273
,
= hx.Fn(ccx,x)
c o n t a i n s t h e i d e n t i t y and which i s c l o s e d u n d e r c o m p o s i t i o n :
Fn*rma We want U
U,B,Y
=
F:
0
F:(a)
=
FE(F2).
t o b e c l o s e d u n d e r t h e F*, and t h e A n .
s o r b e d t h e A n s i m p l y among t h e F:
, b u t since the A
Of c o u r s e w e m i g h t have aba r e of a s p e c i a l n a t u r e (they
c h a n g e i n i t i a l segments o n l y ) w e found i t s l i g h t l y more c o n v e n i e n t t o d e a l w i t h them s e p a r a t e l y .
2.6.
The f u n c t i o n d ; d r e s s e d c a r r i e r s . d(m,n,O,O)
= m,
d(m,n,v*z,w*$)
=
{
d(m,n,v,w)
i f r(n,v)
d(m,n,v,w)*Y
=
otherwise
r(n,v*G),
.
d s e r v e s t o d e t e r m i n e t h e i n d e x o f t h e c o n t i n u o u s f u n c t i o n a l from t h e s e t {F:
: ntN1 t o b e a p p l i e d t o t h e a n ; t h e r e s u l t i n g s e q u e n c e s
dressed carriers and a r e g i v e n by pj
p
n.m
are called
(n,m) ( a . 6 , ~ )(x) :P n,m (a,B,v) (x) :
' F d ( n , m , ~ ( x + l ) , y ( x + l ) )( v a l (r(n,B (x+ 1 ) ) , a x ) ,X) .
W e s h a l l o f t e n simply w r i t e p n or p
i n s t e a d of p n ( a , 6 , y ) ,
n,m
~~,~(a,B,,y).
I b e l o n g t o t h e same c a r r i e r a n . A t each s t a g e and p n,m n,m a r e g o i n g t o b e o b t a i n e d by a p p l i c a t i o n o f a w e know t h a t f u r t h e r v a l u e s o f p n,m c e r t a i n F; t o t h e c a r r i e r a ; when t h e c a r r i e r n jumps a t a l a t e r s t a g e t o a b a s i s
Two d r e s s e d c a r r i e r s P
sequence n ' ,
t h e F;
i s e x t e n d e d t o F;*kr
t a i n e d by a p p l i c a t i o n of F*k*k, t o on,
,
= FE o FEr and v a l u e s o f p
u n t i l t h e n e x t jump.
n,m
now are ob-
A s l o n g as t h e r e i s no jump, t h e Fk c a n n o t b e e x t e n d e d ! Note t h a t (d) a t s t a g e 0 i t i s a l r e a d y s p e c i f i e d t h a t F*m h a s t o b e a p p l i e d i n any c a s e
t h i s i s done i n o r d e r t o e n s u r e c l o s u r e o f U
-
u n d e r F* n' a,6 .Y W e c a n make a p i c t u r e of t h e h i s t o r y o f d r e s s e d carriers i n a s i m i l a r way a s be-
f o r e ; now w e l a b e l t h e nodes w i t h t h e i n d e x of t h e o p e r a t i o n s t o be a p p l i e d a t t h a t s t a g e , t h e "dressing". The d r e s s i n g i s governed by b o t h B and y and r e g u l a t e d v i a t h e d - f u n c t i o n .
G.F. VAN DER HOEVEN and A . S . TROELSTRA
274
( i = 0,1,2,3)
A t y p i c a l diagram, r e p r e s e n t i n g t h e h i s t o r y over 4 s t a g e s of p ni,m.
--
i s given i n f i g . 2 . i n i t i o \ drcsrinq
carrier indca
-*
"0
ma
n*
"0
-3
"a
"r
5TR GE
J.
A t s t a g e 4 we have a grouping i n bundles In1,n2,n31
{no]
with corresponding d r e s s i n g s
Imlty2,
{mol
m31
m2*y0*y2,
.
The d r e s s i n g s of sequences i n t h e same bundle may d i f f e r because t h e v a r i o u s sequences j o i n e d t h e bundle a t d i f f e r e n t s t a g e s .
2 . 1 . The p r o j e c t i o n s . The elements T I , ( u , % , ~ ) ( T n f o r s h o r t ) of TI
n
E A.
31"
(p.
J2"
u
a, B,Y
a r e given by
( u ,B , y ) ) .
This guarantees t h a t a l l i n i t i a l segments w i l l occur among t h e
TIn.
O f c o u r s e , we
, any dressed f (n) would a u t o m a t i c a l l y have an i n i t i a l segment n. On t h e o t h e r
might have assumed t h e An t o occur among t h e F* ; then i f An = F * carrier p
u , f (n)*m hand, a s remarked b e f o r e , t h e A 2.8. {H,
p l a y a s p e c i a l r81e.
O u t l i n e of t h e r e s t of t h e p a p e r .
The r e a d e r should compare t h e u n i v e r s e
: ncN} thus c o n s t r u c t e d w i t h our d e s c r i p t i o n of t h e u n i v e r s e i n t h e introduc-
t i o n ( 1 . 4 ) ; t h e u n i v e r s e o f p r o j e c t i o n s may indeed be s a i d t o i m i t a t e t h e u n i v e r s e
275
PROJECTIONS OF LAWLESS SEQUEXCES 11
described t h e r e . I n order t o give a p r e c i s e meaning t o t h e term "imitate", we need f i r s t of a l l f o r n-tuples of p r o j e c t i o n s a s u i t a b l e concept of r e s t r i c t i o n o r
available information a t some s t a g e .
...,
+
Consider a sequence of p r o j e c t i o n s n ( 6 ) =
...,
+ a v a i l a b l e information on p o s s i b l e continuations of II a t s t a g e x c o n s i s t s of (lo)
t h e grouping of the c a r r i e r s u m.
(mi = j 1j 2n i f o r 1 < i
< p)
underlying
IT^.
1
i n t o bundles, and the dressings, i.e.
(Z0)
t h e continuous f u n c t i o n a l s t o be applied from s t a g e x
onwards. Note t h a t i n x , and t h e r e s t r i c t i o n a t s t a g e x on t h e f u t u r e of
- - -
determined by ax,Bx,yx
, or
6x (with 6 = v (a,B,y)). 3
p r e s s i o n "the onluc: of i n x (or
is completely
Therefore we can use t h e ex-
y for y < x) a t v = 6x" without ambiguity; s i m i l n
TI
" the r e s t r i c t i o n of xn a t v", meaning t h e r e s t r i c t i o n on (continuations o f )
arly, TI
TI^
(6) a t s t a g e l t h ( v ) f o r any 6
E
v.
P r e c i s e d e f i n i t i o n s of i d e n t i f i c a t i o n p a t t e r n s , r e s t r i c t i o n s and e q u a l i t y between r e s t r i c t i o n s a r e given i n s e c t i o n 3 . Section 4 introduces a n a t u r a l ordering between r e s t r i c t i o n s
more information, than R 1 ion v*w
R
.
I
(R2 i s a s t r o n g e r r e s t r i c t i o n , i . e . contains
R1 < R2
, and
it is shown t h a t f o r a sequence
a t v it i s always possible t o continue v s o a s t o l e t %
I f f i n i t e sequences i? and
3'
+ IT
with r e s t r i c t -
reach R2 a t some
of p r o j e c t i o n s obey t h e same r e s t r i c t i o n s a t v
and v' r e s p e c t i v e l y , then t o each v*f t h e r e i s a y such t h a t t h e r e s t r i c t i o n of i'i a t v*Si c o i n c i d e s w i t h t h e r e s t r i c t i o n of it' a t v'*F. Also, t o each extension
if
U Inn} we can f i n d an extension
3 U IT^}
a t v and if' U
1 1 ~ a~t 1v'
t'
U { I T ~such I t h a t t h e r e s t r i c t i o n s of
coincide, and n n ( l t h v) a t v = n m ( l t h v ' ) a t v ' .
The next c r u c i a l property i s t h e "overtake-property"
??=RO,?l,RZ, some
TI
...
i n s e c t i o n 5. Let us c a l l
an admissible sequence i f t h e r e i s some sequence 5 such t h a t f o r
-
Rx i s the r e s t r i c t i o n a t c ( x + l ) f o r a l l
X.
276
G.F. VAN DER HOEVEN and A . S . TROELSTRA I f R - < K O , w e can f i n d a n o t h e r a d m i s s i b l e sequence P,'
w i t h Vx(R'x
5
Rx),
R'O=R, R ' z = Rz f o r a l l s u f f i c i e n t l y l a r g e z : R' s t a r t s a t a weaker r e s t r i c t i o n b u t overtakes 1z ( t h e overtake-property). S e c t i o n 6 d e a l s w i t h t h e assignment of v a l u e s t o t h e p r o j e c t i o n s . I n s e c t i o n 7 , t h e r e s t r i c t i o n on t h e of p r o j e c t i o n s from 6 , 6
anguage considered becomes e s s e n t i a l . L e t if,??' be n-tuples r e s p e c t i v e l y . Suppose:
(a)
s a t i s f y t h e same
r e s t r i c t i o n a t s t a g e x, x g r e a t e r than t h e l e n g t h of an) i n i t i a l segment s p e c i f i e d i n advance f o r 3 o r ?f', ( b ) i f
T ~ n, n ,
a r e corresponding members from
p e c t i v e l y t h e n m(x = )n n l ( 6 ' ) (x) ; under t h e s e assumptions table, i.e.
if
and
t'
resa r e permu-
f o r a l l formulae A of t h e language considered
V ~ E A(?F(a)) ~ X
ct
Va'rg'x
A(*' ( a ' ) ) .
Compared w i t h t h e p r i m i t i v e n o t i o n b e i n g modelled, t h e p r o j e c t i o n s may c o n t a i n e x t r a ( i n t e n s i o n a l ) i n f o r m a t i o n a t s t a g e x s i n c e a n i n i t i a l segment of values
n
-
(x) and a sequence Kx
= cRO,??l,.
..,R(x-I)>
can be obtained by p r o j e c t i o n from
d i f f e r e n t i n i t i a l segments of lawless sequences; i n g e n e r a l , o p e r a t i o n s on t h e p r o j e c t i o n s might p o s s i b l y r e f e r t o t h i s i n t e n s i o n a l i n f o m a t i o n ( i . e .
t o t h e law-
l e s s sequence b e i n g p r o j e c t e d ) . A l s o , f o r t h e p r i m i t i v e concept a s w e l l a s f o r t h e p r o j e c t i o n s i m i t a t i n g t h i s concept, o p e r a t i o n s which can be a p p l i e d a t s t a g e
X
might depend not only on t h e f i n a l r e s t r i c t i o n R(x-I), b u t on t h e whole sequence
-
px. The theorem on permutable n-tuples shows t h a t r e l a t i v e t o a s u i t a b l e r e s t r i c t ed
language both p o s s i b l e i n t e n s i o n a l e f f e c t s do n o t occur. The f i n a l s e c t i o n 8 i s then devoted t o t h e d e r i v a t i o n of s e v e r a l schemata f o r
t h e model described above; t h e e s s e n t i a l t o o l s b e i n g t h e "overtake property" and t h e theorem on p e r m u t a b i l i t y . The preceding s k e t c h of t h e c o n t e n t s of s e c t i o n s 3-7 should enable t h e r e a d e r t o follow t h e arguments i n s e c t i o n 8 ( g r a n t i n g a few d e t a i l s )
-
s o i f he is i n t e r -
e s t e d i n t h e main i d e a s only, he may c o n t i n u e d i r e c t l y w i t h t h e f i n a l s e c t i o n .
2.9. Connection w i t h f o r c i n g .
For a r e a d e r f a m i l i a r with f o r c i n g b u t n o t w i t h t h e
theory of l a w l e s s sequences, i t might be h e l p f u l t o keep t h e following connection i n mind. Lawless sequences a r e c l o s e l y r e l a t e d w i t h s t r o n g f o r c i n g , where "Va~vAcl"
PROJECTIONS OF LAWLESS SEQUENCES I1
277
corresponds t o " t h e f i n i t e information (on a s e q u e n c e a ) , coded by v, f o r c e s A", and i n d i v i d u a l l a w l e s s sequences play t h e r z l e of g e n e r i c sequences. The study of p r o j e c t i o n s then amounts t o a r e d u c t i o n of c e r t a i n o t h e r types of f o r c i n g t o s t r o n g f o r c i n g . Note however, t h a t o u r treatment i s e n t i r e l y i n t u i t i o n i s t i c .
3 . PATTERNS, DRESSINGS AND RESTRICTIONS Definition
3.1.
of p a t t e r n .
A p a t t e r n P of length n i s (given by) an equivalence r e l a t i o n on { O , ] ,
...,n-I}.
W e s h a l l w r i t e E ( i ) f o r t h e equivalence c l a s s t o which i belongs. W e use P
P,P',P",P,,
...,Q,Q'...
f o r p a t t e r n s . I f P i s a p a t t e r n , we d e f i n e "n has p a t t e r n
P" by ncP zdef l t h ( P ) = l t h ( n )
A
V i , j < l t h ( n ) ( j c E ( i ) ++ ( n ) i = ( n ) . ) . P J
Each f i n i t e sequence n has a p a t t e r n denoted by P(n). We say t h a t Q is cazrser than P, o r P 2 Q i f each equivalence c l a s s of P is cont a i n e d i n an equivalence c l a s s of Q, and P,Q a r e of equal l e n g t h . Let X be a f i n i t e l y indexed s e t of c a r r i e r s , w i t h i n d i c e s v O , v I .
...,vn
; let
v =
-
i=(v,ix) = < r ( v o , 6 x ) ,
-
...,r ( v n , 8 x ) > -
f o r t h i s p a t t e r n we w r i t e P(v,gx).
;
We put
e(gx,i,v) = {v : r ( v i , i x ) = r ( v . , i x ) 1 , j J E(gx,i,v) = { j : r ( v i , ~ x ) = r ( v . , ~ x ) l . J
3 . 2 . D e f i n i t i o n of d r e s s i n g and r e s t r i c t i o n . A dressing D i s nothing b u t a f i n i t e sequence of ( i n d i c e s o f ) elements of {F*, : ncN).
A r e s t r i c t i o n i s a p a i r
278
VAN DER HOEVEN and A . S . TROELSTRA
G.F.
with
...,v > = v,
w > = w
Cwo,...,
,
-
t h e p a t t e r n a t s t a g e x i s given by P ( v , B x ) ,
t h e dressing i s s p e c i f i e d by a f i n i t e sequence D(v,w,;x,;.x)
- -
- -
satisfying
D
E q u a l i t y of r e s t r i c t i o n s is given by < P , D =
Zdef
... f o r
(P=P')
A
(D=D')
dressings, R,R1,
...,R ' , ...f o r
restrictions.
3.2. Successors of p a t t e r n s and r e s t r i c t i o n s . A p a t t e r n P' i s s a i d t o be a successor of P ( v , i x ) , i f P' = P(v,Bx*y) f o r some y.
P' i s e i t h e r equal t o P, or i s obtained from P' by i d e n t i f i c a t i o n of two equivalence c l a s s e s . I f a t t h e t r a n s i t i o n from s t a g e x t o s t a g e x+l t h e r e i s no jump, o r a bundle X with elements i n v jumps t o a bundle o u t s i d e v, t h e p a t t e r n
i s un-
changed; i f on t h e o t h e r hand a bundle with elements i n e ( i x , i , v ) jumps t o a bundle with elements i n e ( z x , j , v ) , the corresponding equivalence c l a s s e s a r e i d e n t i f i e d . A successor of a r e s t r i c t i o n < ~ ( v , B x ) ,D ( v , w , ; ~ , ; ~ ) > c o n s i s t s of a p a i r
P'
a successor of P , and D'
r e l a t e d t o D = D(v,w,Ex,;x)
a s follows:
i f the elements of e(Ex,i,v) jump on t h e t r a n s i t i o n from s t a g e x t o s t a g e x + l , t h e r e i s a z such t h a t ( D ' ) . j
J
= (D).*z f o r a l l j
J
4 E(Ex,i,v); i f t h e r e i s no jump, In fig.3
E(;x,i,v),
(D'). = (D). f o r J J
= D.
we have i l l u s t r a t e d two p o s s i b i l i t i e s f o r successors t o a p a t t e r n
0
D'
E
...,m5>.
279
PROJECTIONS OF LAWLESS SEQUENCES I1
Note:
The restriction R at stage x for a finite sequence X of dressed carriers
- -
(of length n say) generated from 0,B.y is determined by Bx,)x; if another sequence of dressed carriers Y of length n is generated from a',B',y', tion R' at stage x w e have R
=
and for the restric-
R', then the successor-restrictions of R and R'
-
-
are in one-to-one correspondence, and to a continuation ~ x d ,y m 0 determining a successor R, of R for X we can always find effectively u',v' such that Z ' f l G ' ,
7 ~ 0 determine ' R at stage x+l for Y (in fact, one can take v' 1
=
v).
3.3 Extendability.
Assume
=
with v w
...,nP> = <wo ,...,w > P =
<no,
, v'
=
, w'
=
...,n'> , P <wo' ,...,w'> , P
<no',
lth(t) = lth(t'), and let
R
=
,
D(wi~Z~,v*?~,t,S)>.
Then we can find ul,u2 such that
R =
,
D(w'*~~,v'*GI ,t',~':>
as follows. a) Assume that
at stage lth(t) does not belong to a bundle with elements in
p,
1*,2
v. Then choose u1 such that u 1 does not belong to any bundle with elements in v', and such that r(ul,t') = u lth(t),
let u2 be equal t o the dressing of z , at stage
;
i.e. u2 = d ( z 2 , z l , t , s ) .
b) If on the other hand p z
l*z2
at stage lth(t) does belong to a bundle with ele-
ments in v, say r(zl,t) = r(vi,t)
,
take u 1 = r(v!,t)
and again u2 = d(z2,zl,t,s).
Thus given two finite sets of dressed carriers with equal restrictions, possible extensions of the sets correspond one-to-one w.r.t. restrictions.
280
G.F.
VAN DER HOEVEN and A . S . TROELSTRA
4. COMPARISON OF RESTRICTIONS 4.1.
D e f i n i t i o n of e x p a n s i o n .
Let
D 1 , D2
D
I
b e two d r e s s i n g s .
...,dP>,
i s o f t h e form <do, D2\DI
=
and D
i s c a l l e d t h e difference o f D2 and D 4.2.
t o D2 (D1 exp D2)
expands
DI
2
of t h e form <do*d;),...,d
if P
*d'>. P
I n t h i s case
1'
D e f i n i t i o n , (Comparability of r e s t r i c t i o n s ) .
We s a y t h a t t h e r e s t r i c t i o n R l
i s weaker than R2 (R,
than R1)
iff a ) R I =
,
R2 =
,
P
I
< P2 '
b) D 1 exp D2, and c) PI
5
P(D2\DI).
4 . 3 . Convention.
We s h a l l use s c r i p t l e t t e r s
P,D,R f o r i n f i n i t e s e q u e n c e s of
patterns, dressings, r e s t r i c t i o n s respectively. F i n i t e s e q u e n c e s o f p a t t e r n s , d r e s s i n g s o r r e s t r i c t i o n s may t h e n be i n d i c a t e d as Pz,%,Ez.
4.4.
Definition.
Fz ( Fz ) i s s a i d t o b e ndmissible i f ( e f f e c t i v e l y ) we c a n ob-
A sequence 4.4.
=.
P( R If R
) is said t o
1
(R y
) a s a s u c c e s s o r t o Py
t a i n F ( y + l ) ( %!(y+l)
5
R
2
) f o r e a c h y s u c h t h a t Osycz.
8
be admissible i f pz (
z) i s a d m i s s i b l e f o r a l l
Z.
t h e r e i s a n a d m i s s i b l e p ( z + l ) w i t h z 5 2 1 t h ( R I ) , Ro=R],
Rz=R2. Proof. -
We d e s c r i b e t h e c o n s t r u c t i o n i n f o r m a l l y . The i d e a i s t o
s t a r t from R
1
=
( P I , D l ) and t o expand f i r s t D 1 t o D2 i n a number o f s t e p s , then t o t r a n s f o r m (P , D ) i n a number o f s t e p s t o (P , D ) = R2. 1 2 2 2 Assume D I # D2
, and
I n d e t a i l , we proceed a s f o l l o w s .
l e t i b e t h e least j s u c h t h a t (D \D ) . # 0. L e t P I = 2 1.l
P ( v , t ) , and c o n s t r u c t a s u c c e s s o r b y l e t t i n g t h e c a r r i e r w i t h i n d e x ( v ) . jump t o a r o o t o u t s i d e v , and add (D2\Dl)i
t o t h e d r e s s i n g (Dl)i
carrier, and s i m i l a r l y f o r a l l c a r r i e r s (v)
.i
with j
E
E
PI
of
this
(i). The r e s u l t i n g res-
28 1
PROJECTIONS OF LAWLESS SEQUENCES I1
t r i c t i o n h a s t h e same p a t t e r n , b u t i t s d r e s s i n g d i f f e r s from D2 a t most f o r a j > i . We may t h u s c o n t i n u e u n t i l w e have r e a c h e d a p o s i t i o n w i t h (P 1 , D 2 ) ; t h i s req u i r e s a t most l t h ( P I ) s t e p s , s i n c e D2 c a n d i f f e r from D l a t most a t l t h ( P 1 ) p l a ces. Now we k e e p t h e d r e s s i n g c o n s t a n t and s t a r t c h a n g i n g t h e p a t t e r n . L e t
If
j = mini[Ep ( i ) # E ( i ) ] 1 p2 k = mini[Ep ( i t # E p ( j ) A E ( i ) = E (j)] 1 1 p2 p2 P I = P ( v , s ) , w e c o n s t r u c t a s u c c e s s o r P' by l e t t i n g t h e b u n d l e of c a r r i e r s
with elements i n E
PI
( j ) jump t o t h e b u n d l e w i t h carriers i n E
PI
( k ) ; P' h a s l e s s
e q u i v a l e n c e c l a s s e s t h a n P I , t h e d r e s s i n g i s k e p t c o n s t a n t . Thus w e r e a c h (P2' D 2 ) from ( P I , D 2 ) i n a t most l t h ( R 1 ) s t e p s . o
5. THE OVERTAKE-PROPERTY Our n e x t aim i s t h e f o l l o w i n g
5.1.
Proposition.
L e t R be a d m i s s i b l e , R a r e s t r i c t i o n s u c h t h a t R S R O , t h e n w e
may c o n s t r u c t , p r i m i t i v e l y r e c u r s i v e l y i n R,
an
a d m i s s i b l e sequence R ' such t h a t
a) R ' O = R b) Vx(R'x < Rx) c ) R ' z = Rz f o r a l l z s 3 ( l t h RO)'
= 3 ( l t h R)2.
This i s c a l l e d t h e "overtake-property". a t m s t 3 ( l t h R)'
A
R
in
s t e p s . We need a number o f d e f i n i t i o n s and lemmata.
5.2. D e f i n i t i o n s . c h g , m ) :-0
R ' s t a r t s "below" R , b u t o v e r t a k e s
We p u t
R (mAl) # Rm
("R changes a t m") C H e z ) 2 c a r d i n a l o f {m : ch02,m)
cl(R,n,z)
("changeability
of f z " )
n
(DO),=(Dz),
A
e h e r e R x = < P x , P e ,for a l l
X.
A
m
cl(R,n,z),
"n i s c l o s e d up t o s t a g e z"
no d r e s s i n g i s added between s t a g e 0 and s t a g e z a t t h e nth p l a c e . I f w e s a y t h a t R h a s a r e p e t i t i o n a t m.
, 1
means t h a t c h @ ,m),
282 5.3.
G.F.
Lemma
VAN DER HOEVEN and A . S . TROELSTRA
( l o w e r i n g of CH). L e t R h a v e a c o n s t a n t p a t t e r n , and l e t R ( z + l )
a d m i s s i b l e . Then t h e r e i s a n R' a ) R'O = RO,
R'z = Rz, Vx
b) R ' ( z + l ) a d m i s s i b l e
such t h a t 5
Rx),
,
c ) C H ( R ' ( ~ + I ' I ) Sl t h ( R 0 )
- cardinal I n : cl(R,n,z)}
.
P r o o f . For e a c h n < l t h R , a l l changes i n R ( z + l ) , which o n l y o c c u r i n t h e d r e s s i n g , a r e postponed t i l l t h e l a s t change i n t h e d r e s s i n g a t t h e nth i n d e x o c c u r s , and t h e n t h e combined a d d i t i o n s i n t h e d r e s s i n g a r e made a l l a t o n c e ; ing sequence i s R ' ( z + l ) .
We i l l u s t r a t e t h e i d e a p i c t o r i a l l y i n f i g s . 4 and 5.
L e t R 7 d e s c r i b e a p i e c e o f t h e h i s t o r y of t h e d r e s s e d c a r r i e r s p
P
"2.9'
.
see f i g . 4 .
the result-
nO,mo'
,ml,
PROJECTIONS O F LAWLESS SEQUENCES I1
283
w\ mr (x3
atr w5
R6
*
5 . 4 . Definition.
-
L e t m be a change i n R ( z + l ) , i . e .
ch(R,m)
A
msz. Then
d e p t h ( m , i ( z + l ) ) ?number o f r e p e t i t i o n s k i n R ( z + l ) w i t h m
E c a r d i n a l I k : ?ch(R,k)
m
A
We p u t
depth(R(z+l))
5
Z{depth(m,R(z+l))
: d z
A
ch(R,m)l
.
G.F. VAN DER HOEVEN and A . S . TROELSTRA
204
A s an example see fig. 6.
*
indicates the points of change,
o the repetitions.
depth(7.6) = depth(l,?6) depth(3,?6)
+
= 3+2 = 5.
R3
5.5. Lemma (shifting changes upwards).
Let R ( z + l ) be admissible. We may construct
primitive recursively in ? ? ( z + l ) , an admissible a)
Ro
=
R'O , R'z
b) CH(?'(z+l))
=
( z + l ) with
Rz,
= CH(??(z+l))
c) Vx~z(R'x 5 Rx) d) depth R ' ( z + l ) = 0. Proof. We transform R via a number of intermediates R=R" 1 -
, Ry,
...,!?i=R'
into
R'
;
all intermediates have the same number of changes up to stage z, but with ever lower depth. Assume!?." to have been constructed; determine the maximal change m < z in ??i'(z+l), followed by a non-change m+l. Postponing the change made at m to l ) lower depth but the same number of changes.0 m+l yields ~ ~ l ( z + with
5.6.
Let R be admissible and let e.
RO
=
and let n be the number of
equivalence classes in P (the w;'?th of P ) , and assume z t mn. Then R ( z + l ) contains a subsequence of length (m+l), Proof.
...,R(k+m)>
with constant pattern.
Any change of pattern diminishes the width. Assume all subsequences with
constant pattern to have length I m, then there can be at most nm elements in i(z+l)
(maximum number of changes in width multiplied by the maximum length of
subsequences with constant pattern), and this is contradictory.~
PROJECTIONS OF LAWLESS SEQUENCES I1
5.7.
Proof o f p r o p o s i t i o n 5.1.
P u t zo
R' i s constructed i n t h e following stages:
3 ( l t h R)'.
=
a ) d e t e r m i n e (lemma 5.6) a s u b s e q u e n c e
.,R (k+m%
of E(z+l) with c o n s t a n t
pattern, n t 3(lthR).
R"
b) lower t h e number o f changes i n t h i s s u b s e q u e n c e ; t h i s y i e l d s (lemma 5 . 3 ) w i t h a subsequence
...,R" (k+m) =R(k+m)
R" (k+1 ) ,
Rk.=R"k,
w i t h a t l e a s t 2 ( l t h R) + I r e p e t i t i o n s . Then p u t R"x
Rx f o r x < k , x > k+m.
=
c ) s h i f t changes upwards ( 5 . 5 ) ; t h i s y i e l d s a n 5 ' " w i t h Q"'0 = R"O = RO,
a)+
R"z = Rz, w i t h r e p e t i t i o n s a t t h e f i r s t 2 ( l t h d) u s i n g t h e lemma of s e c t i o n 4 ,
R"'z
=
I arguments.
d e t e r m i n e R*(y+l) w i t h R*O = RO
=
R, R*y = R O ,
y 5 21th(R). e ) s i n c e R*y
-
RO
=
=
R"'(y+l), w e may t a k e R*O
,...,R L y ,
R"'(y+l)
,..., R"'(z)
for
R' ( z + l ) .o
6. ASSIGNMENT OF VALUES This s e c t i o n is completely straightforward.
6.1.
Definition.
A p - t u p l e of f u n c t i o n s
x I , ...,x
P
conforms to a s e q u e n c e o f
patterns P iff Vx(Px
c l , ...,c
P
x I , . ..,xF
5
. .. ,xpx>) ) .
P (<x x ,
meet a s e q u e n c e of r e s t r i c t i o n s R w i t h ?& =
<,,...,Sp
meet
R via x l ,
...,xF)
which conform t o p
, and
such t h a t Vx(cix
=
F ( h ) i ( ~ i ~ , ~ ) ) f o r 1 i i 5 p.
These d e f i n i t i o n s c a n a l s o b e e x t e n d e d t o p - t u p l e s o f f i n i t e s e q u e n c e s of t h e s a m e l e n g t h i n a n o b v i o u s way.
(I)
P
,...,o
g e n e r a t e d from *,B -n P hx.P(
Note t h a t c a r r i e r s
0
conform t o
"1
5
and t h a t t h e d r e s s e d c a r r i e r s p "1 ' m l
,...,p
"P'"P
g e n e r a t e d from a,B,y
meet
286
VAN DER HOEVEN and A . S . TROELSTR4
G.F.
R
(2)
(n = < n l ,
Ax.
...,nP>, . Let
6.2. __ Lemma
Proof. -
x.
=
ti
= pn.
via u
,...,a
"I
P
( I ) , ( 2 ) be a s i n 6.1, and l e t
meet R v i a
C1,...,Sp
...,mP>)
m = <ml,
x ],...,xP -
x I,...,x
P
conform t o P, and l e t
; then t h e r e is a n 8 such t h a t
Xx.val(r(ni,6(x+l)),Bx)
a
=
n . (8,6), a n d
f o r I 5 i 5 p.
(e,B,y)
i'mi
Observe t h a t i f P ( n , s ) 5 P ( < x l ,
...,xP> ) ,
it i s easy t o i n d i c a t e a t such
that val(r(ni,s),t) let k
=
maxIr(ni,s)
for j
8 Ir(ni,s)
Now o b s e r v e
...,xP>
<xl,
6.3.
Lemma.
=
(t)
= x . f o r I < i s p :
r(ni.s)
1
t =
: Isisp},
..,yk>
with yr ( n i , s )
=
xi' Y j = O
: I
P(n,i(x+l))
,...,xpx>),
P(<xIx
5
,..., x Px>
and t h u s w i t h < x I x
we may t a k e ex e q u a l t o t h e t i n d i c a t e d a b o v e .
L e t 6 = Ax.
l y f o r ji',a',fi',y',n',m',
and l e t
-
- -
n =
= <ml,
=
0
...,mP>
and s i m i l a r -
-
R(n,m, Bx,yx) = R(n:m: B ' x , y ' x ) . There is a p r i m i t i v e r e c u r s i v e X Y = < X , Y * X*Y,
-
R(n,m,Ex
X3Y>9
*
-
x
such t h a t i f w e p u t y
=
R(n',m:B'x
then
y2,-ix
* 9 3)
-
-
*
<x2y>,y'x
*
<xy>
=
*
<x3y>)
and P n . ,m. (SX
for I
Proof.
5
i
1
5
1
*9*
5) ( 4
p and a r b i t r a r y
= Pn!,m!
t,S'.
1
CPIIX
1
*
5')
(x)
Combine t h e remark a t t h e end o f 3 . 2 w i t h t h e o b s e r v a t i o n i n t h e p r o o f o f
t h e p r e c e d i n g 1enuna.o
6.4. Lemma. let p l
__ Proof.
,...,p P
Let
R I , R 2 b e s e q u e n c e s of r e s t r i c t i o n s s u c h t h a t Vx(Rlx
meet R2,
Let p 1
then p
,...,p P meet
R~
1
,...,p P
a l s o meet R , .
= ~ x . < P ~ x , x>, i)
2
via x ],...,xP,
i.e.
5
R2x), and
287
PROJECTIONS OF LAWLESS SEQUENCES I1 p.x = F
U3
Let
=
,).(x;x,x)
(Dz
1
Ax. (D,x\i),x),
xi
Z
1
for
2
i
2
p.
and p u t
,
1 x . F (D,x)i(xix,x)
1 2
i 5 P
then PIX 2
P(D,x)
,
P I X 5 P2x 5 P(<XIX, ".,X
h e n c e P x
P
=),
,...,xP' w ) , a n d t h e
pi meet
R via the 1
xi .
0
7 . PERMUTABILITY CONDITIONS 7.1.
A s a l r e a d y mentioned i n t h e i n t r o d u c t i o n , e q u a l i t y o f r e s t r i c t i o n s w a s meant
t o g u a r a n t e e t h a t two n - t u p l e s
o f p r o j e c t i o n s s a t i s f y t h e same f o r m u l a e of a s u i t -
a b l y r e s t r i c t e d l a n g u a g e , i . e . c a n b e i n t e r c h a n g e d w i t h r e s p e c t t o t h i s language. S i n c e o u r d e f i n i t i o n of "equal r e s t r i c t i o n s " p r i m a r i l y a p p l i e d t o dressed c a r r i e r s , not t o t h e p r o j e c t i o n s ( i . e .
s p e c i f i c a t i o n s o f i n i t i a l segments were d i s r e g a r d e d )
w e have t o make a s l i g h t a d a p t a t i o n i n o r d e r t o o b t a i n a " p e r m u t a b i l i t y c o n d i t i o n " on p a i r s of n - t u p l e s
of p r o j e c t i o n s .
F i r s t of a l l w e need t o s p e c i f y t h e l a n g u a g e . S i n c e o u r t h e o r y c o n c e r n s sequences which are n o t c l o s e d under a l l c o n t i n u o u s o p e r a t i o n s , b u t o n l y under some, t h e
l a n g u a g e of
(of [K,T]) i s not s u i t a b l e , s i n c e e l m w i l l i n g e n e r a l n o t b e a
l e g i t i m a t e c h o i c e f u n c t i o n . We t h e r e f o r e p r e f e r t h e l a n g u a g e L o f
E , which
i s non-
c o m m i t t a l as t o c l o s u r e under c o n t i n u o u s o p e r a t i o n s . W e c a n c e r t a i n l y express i n c l o s u r e under an o p e r a t i o n e l
: Va3pVx3y(e(%+:y)
L
= f i x + ] ) , and may i n t r o d u c e
eta = 5 a s an abbreviation. Also, i f t h e need a r i s e s , one may add f u n c t i o n a l c o n s t a n t s F . t o t h e l a n g u a g e o f LS
M
. From now on, w e s h a l l u s e a,a'
( i n s t e a d of 6 , 6 '
as i n t h e p r e c e d i n g s e c t i o n s )
t o i n d i c a t e t h e lawless s e q u e n c e s e n c o d i n g a l l i n f o r m a t i o n on which t h e p r o j e c t i o n s depend.
7.2.
Definition.
L e t a b e a s e q u e n c e c o d i n g a t r i p l e B,y,6 and l e t
{nn(U)
: n.€N}
VAN DER HOEVEN and A.S.
G.F.
288
TROELSTRA
i n d i c a t e o u r u n i v e r s e o f p r o j e c t i o n s g e n e r a t e d from B,y,6. Let $ o , $ l , $ 2 , . . .
enumerate t h e c h o i c e v a r i a b l e s o f L
,
and l e t A($i
) b e any P a n d o n l y odd-numbered numeri-
f o r m u l a of L w i t h c h o i c e v a r i a b l e s among $ i , . . . , $ i , I P Then Aa i s o b t a i n e d from A by c a l variables v1,v3,.
.. .
,...,$i
1
in A by n v
a ) r e p l a c i n g each occurrence of $
( a ) , and 2n by Vv2n,3v2n r e s p e c t i v e l y .
b) r e p l a c i n g q u a n t i f i e r s V$n,3$n
A s a t a c i t c o n v e n t i o n , we s h a l l u s e , whenever A($i
,...,$i ) h a s b e e n i n t r o d u c e d , 1 P A a ( ~ l , . . . , ~ p ) f o r t h e f o r m u l a o b t a i n e d from Aa by s u b s t i t u t i n g nk f o r v 2ik'
7.3. D e f i n i t i o n .
The f o u r - p l a c e
a b i l i t y condition f o r the Per(cnl
(I)
TI
a
p r e d i c a t e ?er(u,,v;u2,w) if
,...,nP >,v;<ml ,...,mp>,w)
... , n-P
v a c v A~(;,,
is s a i d t o b e a permut-
-,
... , m-P
) H Vasw
)
For t h e set of p r o j e c t i o n s we a r e s t u d y i n g , w e may d e f i n e a p r e d i c a t e "Per" a s follows. L e t ~ , y , 6 b e coded by a, s u c h t h a t a x = v 3 ( 8 x , y x , 6 x ) ,
d; l e t
a r l y coded by el,y1,6'
,...,vn
TI
P ( i . e . TI = TI (%,y,6), n. n.
be p r o j e c t e d from 8,y,6
"1
TI'
m;
Then ? e r ( < n l ,
...,nP>,v;<m,, ...,mP>,w)
(a) l t h ( v )
l t h ( w ) , s a y l t h ( v ) = x;
=
J
= n ' (%',y',&')).
, and
86
-,..., m
ml,
...,mp,a'x -
( c ) ; n . ~= 7 b . x
P
n; J
holds i f
-
(d) l t h ( v )
=
and
are t h e same; for I s i s p ;
1
1
from
TI'
1
(b) t h e r e s t r i c t i o n s a t s t a g e x d e t e r m i n e d r e s p e c t i v e l y from n I , . . . , n p , a x
-
be s i m i l -
and l e t 8',y',6'
lth(w)
L
lth(ni),lth(m.)
for I 5 i
5
p.
Now w e s h a l l f i r s t s t a t e and p r o v e a theorem s t a t i n g s u f f i c i e n t c o n d i t i o n s f o r a p r e d i c a t e Per t o be a permutability c o n d i t i o n .
7 . 4 . Theorem.
L e t P e r ( u l ,v;u2,w) be a f o u r - p l a c e p r e d i c a t e , $ ( u l , v , u 2 , w , x )
f u n c t i o n from N5 t o N ( $ ( x ) f o r s h o r t ) , E s s ( n , v )
("v i s e s s e n t i a l f o r
two-place p r e d i c a t e , n ( n , v ) a f i n i t e s e q u e n c e ( i . e .
TI
a
") a
a n a t u r a l number) s u c h t h a t
PROJECTIONS OF LAWLESS SEQUENCES I1 (i)
Vn3eeKVv(ev#O
(ii)
V a ~ v ( n ~ aE n(n,v)),
+
Ess(n,v)),
(iii) VnVx3eeKVv(ev#O + lth n(n,v) > x), (iv) Per(ul,v;u2 ,w) e, Per(u2,w;ul,v), (v)
Per(ul,v;u2,w)
(vi)
Per(ul,v;u2,w)+ vx Per(u1,v*<+(x:>;u2,w*Z),
(vii) Per (ul,v;u2,w)
+
+
~x(~ss(x,v) + jy Per(ul*ii,v;u2*F,w)),
Vi
( (u,)
,w)) ;
then Per is a permutability condition. Note that because of (i) it is no restriction to assume Ess(n,v) (simply replace in (i)-(vii)
Ess(n,v) by ev # 0, for an e
E
to be decidable
K validating (i));
and
also note that (vi) yields a + : $[ul,v,u2,w,x] such that Per(ul,v;u2,w) and a $'[ul,v,u2,w]
(Take $x
=
-
-
VxVx Per(ul,v*$x;u2,w*xx),
such that
Per(u ,v;u2,w)
I
+
+
Vw' Per(ul,v*$'w';u2,w*w')
0 (u1 ,v*~x,u2,w*~x,xx),and $ ' (m*^x) = $'m*$(ul ,v*$'m,u2,w+m,x)).
Proof. We have to establish -
(I)
in 7 . 3 by induction on the logical complexity of A.
...,$,
Let A have its choice parameters among sl,
i.e. A
2
A(cI,
..., P) . At each E
step our hypotheses are
...,nP >,v,<ml,...,mp>,w) ; we
(2) Per(
(3)
let
( 4 ) Induction hypothesis : Per satisfies (1)
of 7.3 for all formulae of complexity
less than A.
We shall assume our formulae to be rewritten such that all prime formulae containing variables
-_---Case I. ( 3 . I)
Let a
E
A
E. S
are of the form ~ . =t s , t and s not containing choice variables. (~.t=s). Hypothesis ( 3 ) becomes in this case
Vc%~v(n (a) (t)=s). -. "i w, $ E $[ul,v,u2,w,Xz.a(lth(w)
+ z)].
With (2) and (vi)
(5)
Vx Per(u,,v*;x;u2,~(lth(w)
+ x)).
G.F. VAN DER HOEVEN and A . S . TROELSTRA
290
( i i i ) g u a r a n t e e s t h e e x i s t e n c e of a n xo w i t h
(6)
> t , n ( m i , a ( l t h ( w ) + x,))
n(n.,v*zxo)
> t
(lsisp).
Apply ( v i i ) t o (5) w i t h x = xo t h e n = n(mi,"1th(w)
n(ni,v*GxO) and t h e n w i t h (3.1), i l
m.
Case 2 . Case 3.
(3.3)
(a)(t) A < B
(Isisp),
x,))
(ii), (6)
+ x,))
n(mi,a(lth(w)
=
t
= n(ni,v*~xO)t = s.
C : trivial.
A
. H y p o t h e s i s ( 3 ) now becomes ~ " ( ,..., 2 ~ n )). ~ a t v ( ~ " ( ,..., n] n ) -P -P A s B
C
-B
-t
We have t o show
(ml,. ..,q) v a t w * w '
vwrw+w' B"
-B
t h e r e f o r e assume (7)
+
C"
(m, ,. .., m-P
);
(ml,. .., E ~ ) .
Va~w*w' B"
By (2) and ( v i ) we f i n d a
v'
such t h a t
P e r ( u ,v*v' ;u2,w*w'), I and t h u s w i t h ( 4 ) , ( 7 ) Vurv*v'
Ba(zl
,...,-Pn
( a g a i n w i t h ( 4 ) ) Vatw*w' Ca(m1,.
hence Case 4 . -_-----
+
B)
A
.. ,uIP).
(x#o
-B
)
C)).
Case 5. A -_---__
z Vx Bx, Va B a , V e Be ; t r i v i a l .
Case 6 .
E
A
,...,-nP
i s reduced t o o t h e r c a s e s by t h e u s e of
A i B v C
A c r 3x((x=O
) ; w i t h ( 3 . 3 ) V a ~ v * v ' Ca("]
3 x Bx, 3 a Ba, 3 e Be.
S i m i l a r t o , b u t s i m p l e r t h a n Case 8 t r e a t e d
below.
-____-Case 7 . A (3.7)
. Hypothesis B"(II,, . ..,Ep,,n).
z VEBE
Vatvvn
We h a v e t o show
L e t m,
C~EW
( 3 ) becomes ( 3 . 7 )
VmVaew B ' ( I ~ ~ , . . . , E ~ , ~ ) .
be a r b i t r a r y , and l e t
E s s (m,, ( l t h ( w ) + x ) ) With ( v i ) ,( i v ) ,(v)
9 be a s b e f o r e . By (i) t h e r e i s an x w i t h
. we find successively
a
P e r ( u l , v*Gx; u 2 , (1t h (w)+x) ) ; P e r (u,
,.
( l t h (w)+x) ;u 1 ,v*Gx) ;
P e r ( u 2 * % , a ( l t h ( w ) + x ) ; u 1 *R,v*Gx)
f o r s u i t a b l e n;
,
29 1
PROJECTIONS OF LAWLESS SEQUENCES I 1 hence w i t h ( i v ) , Ba(ml
------Case 8 . (3.8)
( 3 . 7 ) and ( 4 )
,. - .,%,$
A E
~ E B E . H y p o t h e s i s ( 3 ) becomes
Varv3n Ba(nl,.. . , E ~ , E ) . Va'cw3n B a ( r a l , . . . , ~ p , ~ ) . T h e r e is a n
We h a v e t o show
Vv' (fv'#O
(8)
+
We may assume t h a t Vv' (fv'#O
+
K
such t h a t
.
also satisfies
Ess(fv''1
,v*v')).
(To s e e t h i s , l e t t h e e n be e l e m e n t s o f env#O
E
Varv*v' . ) ) ' v f B, a~(~z -l ,~. . . f
+
f
K satisfying
Ess(n,v)
as p o s t u l a t e d i n ( i ) ; r e p l a c e a n f Xv'.fv'.sg(ef
s a t i s f y i n g (8) by
v,.I(v*v'))
which i s e a s i l y s e e n t o be a n e l e m e n t o f K . ) Let
5 be such t h a t f (*'W')#O
((v)of 7 . 4 )
+
Per(ul*
v*$'w';
u2*
($'w')'l)>,
w*w');
t h e n w i t h (8) Vw'(f($'w') h e n c e by
Remark.
+
+
3nVa'cw*w' ~ ~ ( E ] , . . . , ? ~ , f i ) ) ,
Varv*($'w')Ba(~l,
f($'w')'l)),
(4)
vwt ( f ( $ ' w ' ) # o and t h u s
...,-P'n
0
Vaew3n Ba(ml,
...,mp,z)
.o
Assuming o u r l a n g u a g e t o b e e x t e n d e d w i t h o t h e r s o r
of c h o i
variables,
which a r e t h o u g h t of as s e q u e n c e s c o m p l e t e l y u n r e l a t e d t o t h e r a n g e of t h e l a w l e s s v a r i a b l e s , w e c a n s t i l l e x t e n d o u r theorem c a r r y i n g s u c h p a r a m e t e r s a l o n g ; e l e m e n t s o f K t h e n have t o b e i n t e r p r e t e d a s e l e m e n t s of K l a w l i k e r e l a t i v e t o s u c h p a r a meters.
7 . 5 . Theorem.
The r e l a t i o n
Per
f i l l s a l l the conditions (i)-(vii)
i n 7.3 d e f i n e d by t h e s t i p u l a t i o n s ( a ) - ( d ) f o r a p p r o p r i a t e n(n,v),c$,Ess
ful-
and i s hence a
permutability c o n d i t i o n f o r t h e p r o j e c t i o n s s t u d i e d i n t h i s paper.
G. F. VAN DER HOEVEN and A. S. TROELSTRA
292 Proof -
E s s ( n , v ) :l t h ( v ) t l t h ( j l n ) ; i . e . ,
(A).
- - -
Bx,yx,Gx,
and
TI
as a code o f
v
, then lth(Ex) = x = l t h ( v ) > l t h ( j , n ) guarantees 2" f o r y 2 x a r e d e t e r m i n e d by a , i . e . A has no l o n g e r any
= jlnlo. J
TI^
t h a t values of
i f w e t h i n k of
j
effect. (B).
For
n(n,v)
we take
-
nn(lth v)
a(n,v) =
i f lth(v)21th(jln)
,
j n otherwise. 1
Let
(C).
ion
w*?
Then 6
Per(ul,v;u2,w).
a c o n t i n u a t i o n v*<$*,
indices
u2
=
<ml,
such t h a t t h e r e s t r i c t i o n s on t h e p r o j e c t i o n s w i t h
...,mP> , a t
jections with indices
i s a f u n c t i o n which a s s i g n s t o any c o n t i n u a t -
, are
w*^x
u1 =
t h e same as t h e r e s t r i c t i o n s on t h e pro-
...,n P> a t
v*<$x>
,
main c o r r e s p o n d i n g a s w e l l . The e x i s t e n c e o f s u c h a (D)
and s u c h t h a t t h e v a l u e s ref o l l o w s from 6 . 3 .
With t h e s e s p e c i f i c a t i o n s , t h e p r o p e r t i e s ( i ) - ( i v ) , ( v i ) , ( v i i )
Property (v) expresses t h a t provided lth(j,x),
u2*9
is essential for
w e c a n f i n d t o each e x t e n s i o n
u *? 1
of u 1
TI^ ,
i.e.
lth(v) 2
a corresponding extension
, f o r a s u i t a b l e y , such t h a t t h e p e r m u t a b i l i t y p r o p e r t y i s pre-
o f u2
served.
v
become o b v i o u s .
To s e e t h i s , l e t
...,n P> , v ; < m l , ...,mp>,w),
(1)
Per(
(2)
lth(v) t l t h ( j l x )
Then c e r t a i n l y
((Z), d e f i n i t i o n of n ( x , v ) )
lth(n(x,v))
(31
.
=
lth(v)
lth(w) ;
=
also Ess(y,w)
,
(4)
y = j ( n ( x , v ) ,m)
-+
(5)
y=j(n(x,v),m)
-,n ( x , v ) - n ( y , w ) .
The d r e s s e d c a r r i e r s u n d e r l y i n g t o t h i s t h e r e corresponds a t
n
such t h a t
P
,ny
a t w (lth(w) = l t h ( v ) )
have i n d i c e s
j 2 n ] , . . . , j 2np,j2x;
a p a r t i c u l a r r e s t r i c t i o n R.
j 2 m l , . . . , j 2mp,y'
obey t h e same r e s t r i c t i o n R; i f w e t a k e
,...,TI
ml
, . . . , T I TI^
"1 P v (stage l t h ( v ) )
A s n o t e d i n 3 . 3 we c a n f i n d a y '
lth(w)
TT
y
=
at stage lth(v)
j(n(x,v),y')
are permutable w i t h
, then
,...,n n
,irX
T T ~
1
=
P
a t v.0
293
PROJECTIONS OF LAWLESS SEQUENCES I1
8. SCHEMATA VALID I N THE MODEL T h i s s e c t i o n i s d e v o t e d t o e s t a b l i s h i n g t h e v a l i d i t y o f v a r i o u s schemata i n
8.1.
o u r model. W e s h a l l assume t h e FE t o c o n t a i n t h e Am
F5 = A.
assume
,
= identity.
I n o r d e r t o s i m p l i f y t h e n o t a t i o n , w e s h a l l o f t e n use
r , r ' , r i...
and
n for the projection
a s s y n t a c t i c a l v a r i a b l e s f o r v a r i a b l e s ranging over
Almost s e l f - e v i d e n t Theorem
and i n p a r t i c u l a r , w e may
{F:
nn
,
: ncN}.
is the following r e s u l t
I n o u r models
Ua
vEvr3n(rE = TI).
8.2.
Theorem. AE
.+
In o u r models 3 r ( 3 q ( E = rq)
%
AVE'A(TE'))
and more g e n e r a l l y
where u r a n g e s o v e r mappings from Proof. ___
{ I , ...,p
C o n s i d e r any s e q u e n c e o f p r o j e c t i o n s
n. = j ( n i , j ( n y , n y ) ) . Assume a l s o
essential for put
m.
are
pi,qi
segment
r.
X.
zi h a s
-
such t h a t
n. ( l t h ( v ) ) a t v
~i
r.
1
= A
VEcv(n. -1
Now l e t v
=
'
qi
,
rimi)
5
r(v,ny)
i
5
v
,
,
i.e.
p
= nf
i s completely u n r e s t r i c t e d a t
OF* ) ( n m , ) 'i
a t v ; here
p.
i s t h e d r e s s i n g of
IT,,,
v
is
v
, and
there
is the i n i t i a l
1
F*
at
V.
There a r e
1
S
p
,
and t h u s
1 5
i
<
p.
i
and l e t
a t v ; i f we
1
qi
1 5
for 1
at
By t h e axiom o f open d a t a f o r l a w l e s s s e q u e n c e s , t h e r e
).
such t h a t
essential for
r 1,...,r
and
OF*
., ., -Pn
Aa("],
~
v
Pi
,
Pi
}.
with root
mi
r n , = (A
X =
lth(v) t lth(nl)
carrier
then 1
such t h a t
is a
,
j(O,j(nf,O))
=
1 into { I , ...,p
P
E
, Vpcv
V
A
6 (1
II~,...,-nP
such t h a t
{n(l)
],...,q
.
) ,
By t h e p r e c e d i n g remark, we c a n f i n d u ,
,...,n ( p ) )
=
{I
,..., ql
,
and
ml,
...,mq'
VAN DER HOEVEN and A . S .
G.F.
294
VBEV(II~(E)
m l , ...,--(1 m
and
m
rn!,...,
fll,..., 4 r and l e t s ( i )
at v
-q
,
and t h u s
(p) )
. S i n c e m l , ...,-m1
s i s q)
ion on
...,r&
(
C o n s i d e r any s e t v (1
(6))
completely u n r e s t r i c t e d a t v
VOcv AB (r
(1)
I'i%(i)
=
b e t h e i n i t i a l segment o f m. 1
a r e completely u n r e s t r i c t e d a t v
i s weaker t h a n t h e r e s t r i c t i o n on
a t v ; and t h u s we c a n a p p l y t h e o v e r t a k e - p r o p e r t y (i)
lth(v*v')
z l ,...,-r- 4 '
( i i i ) t h e r e s t r i c t i o n of
3
the restrict-
A s ( l ) r l , "',As(q)5q
t o f i n d x,v'
such t h a t
and
I I I ~ , . . . m,
i s e q u a l t o t h e r e s t r i c t i o n of
a t v*v'
--4
A s ( l ) ~ l , . . . , ~ s ( q ) ~a t a x .
El
,
at
x,
=
( i i ) &x e s s e n t i a l f o r
Therefore
TROELSTRA
-..,q a t
v*v'
and
z l ,...,+ r I
at
-
ax
are p e r m u t a b l e , and thus
from ( I )
8 . 3 . Theorem.
u
I n our models
VE
P
3 x A(E I , . . . , ~ p ~ ) 3eVE + I...~p A ( € ,
Similarly VE, +
Proof. -
is a
...E P3 a
,...,
3egbVc l . . . ~
P
Assume
v
ACE l , . . . , ~ p , a )
Vnl
...n P3x
,...,
E ~ , ~ ( E
-.
~ ~ , ( b ,...,€ ) ~ ( )~) .~ P
,...,n+x)
,
t h e n by t h e axiom of open d a t a t h e r e
such t h a t a
v,
VOcvVn I . . . n
,...n P
3 e V ~ E VA
E
P
6 3 x A ( ~ l r - . . , ~ p p ,,~ )
and t h u s Vn Let
tions
lth(m) f. (I
lth(v) = y
= S
i 5 p)
of
5
, mi
,...,c p , e ( B ) ' l ) . =
ky(m)
.
T h e r e are f i x e d p r i m i t i v e r e c u r s i v e func-
v and m s u c h t h a t
PROJECTIONS OF LAWLESS SEQUENCES I1
' f l ( m , v ) 1 * * a 9f np (m,v) a r e completely f r e e a t
v
,
-
and n f i ( m , v ) ( y )
=
mi.
Let
295
em b e t h e e l e m e n t o f K
such t h a t
B VBEV A ( f l ( m , v )
(1)
and l e t fl(m,v),
b e a p r i m i t i v e r e c u r s i v e f u n c t i o n s u c h t h a t a t v*C(m,n)
c(m,n)
...,f P (rn,v) "fi
,...,fp(m,v),em(5)'1).
a r e s t i l l completely u n r e s t r i c t e d , while
( l t h ( v ) + Z t h ( n ) ) = mitkp(n)
(m,v)
l t h ( n ) =lth'F,(m,n)). Determine
f
fn = 0 f
(m*n)
-
em(v*C (m,n) )
Quite obviously, fn'#O
fn' # 0
Assume n'
=
, lth(m)
m*n
f.(m,v)
E
i o n of
f )
f
E
K
V;lck!n'
, pi =
.
prove Aa("
a t v*C(m,n) v*C(rn,n)
fi(m,v)
Then
,,...,-P'n
fn''1).
. Since f n ' m,n . C o n s i d e r
( 1 5 i 5 p)
l t h ( v ) for c e r t a i n
. Continue
a(lth(v+v'))
.
. We s h a l l ...Vn-P ckPn' P kyn'
E
kpn' = kP(m*n)
r e s t r i c t i o n of t h e
v*v'
lth(n) < lth(v),
if =
'
a s follows:
K
E
,
( 1 5 i 5 p)
and
t o v*v'
a t v*v'
z ],...,-Pn
,
# 0 , lth(n')
t l t h ( v ) , and
the
; they s a t i s f y
fi(m,v)
em(v*C(m,n)) = f n '
(by t h e d e f i n i t -
such t h a t (overtake-property)
the
i s t h e same as t h e r e s t r i c t i o n o f t h e
a t a(lth(v*v'))
and
fl(m,v)
,...,f P (m,v)
a r e p e r m u t a b l e , and t h u s from
V ~ E V * V ' A6 ( f l ( m , v )
,...,f P (m,v),fn"l)
w e may c o n c l u d e
8 . 4 . Theorem. Vc ] . . . E
I n o u r models
ua
~~A(E],...,E~+ , TVE ~ )]...E 3 i j ( O S j S p P P
A
A(€, l , . . . , ~ p , F * ~ . ) ) 1 J
fii a t at
VAN DER HOEVEN and A.S.
G.F.
296 Proof. -
We f i r s t c o n s i d e r a
v
,.. .
VBE v3m A'
TROELSTRA
such t h a t
m)
,
r l ,...,-nP which a r e c o m p l e t e l y u n r e s t r i c t e d c1,...,-nP . There i s a n e t K s u c h t h a t for
Vn(enZ0 Let
v*v'
ed
at
(a)
n'
+
,
and such t h a t
ev"1
=
II~,..., -Pn
I n t h i s case there is an
i
VBEV*V' (F+n.(6) = 1-J
,v
essential for
.
ev'#O
a r e s t i l l completely u n r e s t r i c t -
Then t h e r e a r e two p o s s i b i l i t i e s .
h a s t h e same c a r r i e r as
and n'
v
..,zp,e e ) ) .
~ ~ c v *A' n
be a c o n t i n u a t i o n such t h a t
V*V'
at
n. -J
a t v*v'
j.
f o r some
such t h a t
fl' (6)).
E' h a s a c a r r i e r w i t h r o o t d i f f e r e n t from one of t h e c a r r i e r - i n d i c e s o f
(b)
II,,...,II~ a t v*v'
. Now
v*v'
extend
v+v'
,n"
E' = F.n"
; say
1-
t o v*v'*?
n o t s u b j e c t t o any r e s t r i c t i o n a t
such t h a t ( t h e c a r r i e r o f )
jumps t o
n
-1
V B ~ v + v ' * j i ( n ' = F I zl).
s a y , n o d r e s s i n g added, and t h u s
T h e r e f o r e i n b o t h c a s e s w e f i n d some c o n t i n u a t i o n 3ijVBcv*v"[A 6
r"
( r l ,..,%,FSn.) .
A
1-J
v+v"
of
v such t h a t
ISjSp].
,..., P3nA(c1 ,...,~
We s h a l l u s e t h i s o b s e r v a t i o n below. Assume
V E ~
E
~ , ni n) o u r
model, i . e .
t/nl...n 3m ~ P
~
,..., ( r p , 2m ) .
~
By "open d a t a " t h e r e i s a v s u c h t h a t acv, tlnl...n
,...,-pn m) .
VBEv3m ~ ' ( 2 ~ P
Consider a n a r b i t r a r y s e t
Let at
(a)
, and
1 s i 5 p.
v for v*v'
be a c o n t i n u a t i o n of
a(lth(v*v'))
=
, ri
v*u'
f o r some i , j ( 1 s j
5
determine
v*v'
v such t h a t
IT
n.
...,np
nn . ( l t h ( v ) )
lth(v*v') a t
completely u n r e s t r i c t e d a t
such t h a t
nl,
v*v'
= nm.(lth(v)) 1
v+v' =
,
IT
m.
lth(v*v')
and s u c h t h a t
p)
V ~ E V * V ' AB ( y l , .
Extend
...,-Pm
a r e n o t s u b j e c t t o any r e s t r i c t i o n s a t v ,
"l,...,n-P at
ml(a), -
1
t o v*v'*v''
..,n -p'
F? (n.)) 1 -J
s u c h t h a t t h e r e s t r i c t i o n s on
c o i n c i d e w i t h t h e r e s t r i c t i o n s on
g l , . . ,, m
-P
a t v*u'*u"
II~,...,n -P
a t v*v'*v"
= Llth(v*v'*v")
(over-
PROJECTIONS OF LAWLESS SEQUENCES I1 take-property).
Then "1
,...,--P n
at
V*V'*V"
ml,...,-mP
and
297 at v*u'*u"
are
permutable, and thus
B (ml,.
~~~v*u'*u'' A
. .,m-p' rtm.) 1-J
and hence also A~(E,,.
..,mP,r;mj).
8.5. All schemata derived above concerned formulae A in our standard language
;
in contrast, we note the following Theorem. Assume
{rz
to include operations with constant (lawlike) values,
: ncN}
X such that
then we have in our model, for any E=E'
A
x=x'
X(E,X)
A
X(E',X')
+
the validity of VE~!X X(E,X)
+
3eVEX(E,e(E))
We omit the proof here ; it is similar totheproofs of V~3!x-continuity for projections in [D,T]
,
and for choice sequences in [TZ].
8.6. Elimination. The schemata derived in 8.1.-4
for our projections suffice
to define an elimination mapping similar to the ones given for
and
(see e.g.
[T41).
For example, the schema of 8.2. is used in the form
,..., P )
VE~...E~(A(E~
E
vr ]...r P r\(vE l...E O
A(TIEo(l P
VE
+
B(el
,...,s P ) )
)...rPEO(P)
B ( T ~ E ~ ( ]) . . . r
P
P
E
CI
) +
dd)'
and the schema of 8 . 4 . is combined with 8.3; the schemata then enable us to push strings of quantifiers VE l...~p inwards, to be replaced ultimately by lawlike quantifiers in front of prime formulae. (Cf. 1.5, (ii) in the introduction to this paper.) 8.7. Extension to higher types, sheaf model. It is possible to include sets and relations in our language and comprehension axioms in our theory, as follows. We simply put down as a n extra axiom
G.F. VAN DER HOEVEN and A. S . TROELSTRA
298
VWn1...n Vml...m [Per(cnl,..., n >,v;<ml,...,m >,w) P P P P + (Vcr~vX(zI, n ) + Vacw X( r n l . . . . , m ) ) I ;
(1)
...,--P
--P
let us abbreviate this axiom as VX Per(X). duced to the theory without ( I ) , {X : Per(X)l.
-t
The theory with ( 1 ) added can be re-
simply by relativizing the quantifiers VX,3X
In the theory with axiom ( I ) ,
to
all o u r arguments for the various
schemata given above remain valid. Similarly for the theory extended to a language with relations of all finite types. This opens the possibility of re-interpreting our collections of models as a sheaf model over Baire space (topological model) for a theory with finite types, since the model depends ultimately on a single lawless parameter, which may be interpreted as ranging over Baire space (cf. [T4], 7.16, 4.17;[F,H]). we obtain an example of a sheaf model satisfying Va3x-continuity.
Thus
Seen from the
outside, the model is very thin: there are only countably many functions.
References D. van Dalen and A.S. Troelstra (1970), Projections of lawless sequences, in: Intuitionism and proof theory (A.Kino, J.Myhil1 and R.E.
Vesley, edi-
tors). North-Holland, Amsterdam, 163-186.
M. Fourman and J.Hyland (1976), Sheaf models for analysis.
To appear in:
P _ r o c e e d _ i n g s _ o f _ t h e - ~ ~ ~ ~ - ~ ~ ~ ~ ~ ~ _ ~ ~ ~ e
-theory ---- _--_-to Logjk,-Algebra
and Analysis (D. Scott, editor). Springer, Berlin.
G. Kreisel, A.S. Troelstra (1970)
, Formal
systems f o r some branches of
intuitionistic analysis. Annals of-Mathematical Logic I , 229-387. A.S. Troelstra (1969), Principles of intuitionism. Springer, Berlin. (1969A), Informal theory of choice sequences. Studia Logica 25, 31-52. (1970), Notes on the intuitionistic theory of sequences I1 indag. math. 32, 99-109. (1977) , Choice sequences, a chapter o f intuitionistic mathematics. Clarendon Press, Oxford.
LOGIC COLLOQUIUM 78 M. Boffa, D . van Dalen, K. McAloon leds.) 0 North-Holland Publishing Company, 1979
Linear o r d e r s i n
( w ) ~ u n d e r e v e n t u a l dominance
R i c h a r d Laver Department of M a t h e m t i c s U n i v e r s i t y of Colorado B o u l d e r , C o l o r a d o 80309
, t h e set of functions
Let dominance.
.
< ttl
In t h e a b s e n c e o f
embeddable i n with
;to 2
I t i s w e l l known t h a t
( w ) ~?
, be o r d e r e d b y e v e n t u a l
u) + u)
, which l i n e a r o r d e r i n g s of p a y e r
CH
n o t embeddable i n
w2
f :
( w ) ~ embeds e v e r y l i n e a r o r d e r i n g ,of power
Under t h i s o r d e r i n g ,
a stronger r e s u l t i n t h i s direction. o r converse w e l l o r d e r i n g of payer
2"
,
i s embeddable i n
can be a r b i t r a r i l y l a r g e
Now
implies t h a t every w e l l ordering
MA
it 2
i
, and e v e r y l i n e a r o r d e r i n g of power
, b u t Kunen
( w ) ~
(w)~ .
MA
CH
See a l s o Solovay ( [ 4 ] ) f o r
[Z] h a s shown t h a t ;to
MA
i s con-
which i s n o t em-
2
s i s t e n t w i t h t h e e x i s t e n c e of a l i n e a r o r d e r i n g of p a y e r beddable i n
are
( f o r example s t a r t w i t h a ground model o f
and add Cohen r e a l s , Sacks r e a l s , o r Solovay r e a l s ) .
<
i"
i
The problem w i t h a n a t t e m p t t o i n d u c t i v e l y embed, u s i n g
so
, a n y l i n e a r o r d e r i n g of power
into
2
( w ) ~
, i s t h e p o s s i b i l i t y of ([l]) , t h a t i s ,
c r e a t i n g i n t h e c o u r s e o f t h e c o n s t r i c t i o n a "Hausdorff gap"
( w ) ~ , of l e f t and r i g h t c h a r a c t e r
a c u t i n a l i n e a r l y o r d e r e d s u b s e t of which c a n n o t be f i l l e d i n
o r i n a n y e x t e n s i o n of
V
ito
In t h i s p a p e r i t i s prov. d c o n s i s t e n t t h a t orderi ng of c a r d i n a l i t y
i
SO
2
i s embeddable i n
2
which p r e s e r v e s
V
w
and every l i n e a r
> ;Il
( w ) ~
.
' 1
'
'This q u e s t i o n was
r a i s e d by S o l o v a y ( u n p u b l i s h e d ) i n c o n n e c t i o n w i t h h i s and Woodin's r e s u l t s on homomorphisms of Banach a l g e b r a s For
f,g
E
( w ) ~
, let
[S])
"41,
f < g
.
mean t h a t
lim(g(n) - f ( n ) ) =
a countable t r a n s i t i v e m o d e l of
ZFC
cardinal satisfying
I t i s shown t h a t t h e r e i s a
extensi on of
a0
power
;lo 2 = K'2
i s embeddable i n
2
The e x t e n s i o n i s by the iteration; steps.
.
2cK = K
i n which
T o g o from
= K
((IU)~ ,<)
ccc i K
let
T,
to
q,+l
v,
: v
<
in
K)
TK
.
Let
<
be
ccc
forcing
, which w i l l p r o v e t h e theorem. ([31)
a p a r t i a l ordering
, o r d e r e d by
+m
i s a r e g u l a r uncountable
.
There a r e
K
steps in
be t h e model o b t a i n e d a f t e r t h e f i r s t
a n d from t h e r e s u l t i n g g e n e r i c set a n (f,
K
a n d t h e s a t u r a t e d l i n e a r o r d e r i n g of
iterated forcing
01
for
i n which
Q,
ET,
u
i s forced w i t h ,
f, E (m)" is d e f i n e d . The f i n a l set , i s t h e s a t u r a t e d l i n e a r o r d e r i n g of t h e
theorem. Let
.
q0.=T
o r d e r e d by
<
.
Suppose q ', In
q,
i s d e f i n e d , i n which
, define
Q,
a s follows.
,
(f, If
:
p<
a = 0
is linearly
, Qa
s t a n d a r d p a r t i a l o r d e r i n g of f i n i t e c o n d i t i o n s f o r a d d i n g a g e n e r i c If
a
7
0
, pick a cut
C,
in
((fp :
< a ] ,<)
299
.
is t h e g :
u1 + u1
Then a c o n d i t i o n i n
Q.
300
R. LAVER
is a
(i)
L,R
(ii)
E
s
such t h a t
GtO [a]
E
,
fy < C ,
, m<
(w)'"
, and,
w
, Ca < fs
E L)
(u
letting
( 6 E R)
be t h e g r e a t e s t member o f
t(s)
,
dom(s)
+
(iii) f y ( n t
Order
2 m s f6(n)
,s c f
+
-
m < s ' ( n ) < f (n)
6
ma , Q,)
be
,G
< n).
s(s)
, R' , s ' , m' )
(< means i n i t i a l segment h e r e ) ,
s'
(n)
Y
E L ,6 E R ,
(Y
(L,R,s,m) < ( L '
by t h e r u l e :
Q
R i R'
Let
,
q = (L,R,s,m)
-
generic.
m
Then
m
i f and o n l y i f b
m'
(Y
E L , 6 E R , t(s) < n <
f
=
*
member of
U
defqEG, (Y,6 < a
sq
L
8
L'
,
, and ~ ( s ' ) )
.
i s by g e n e r i c i t y a
,
(UI)~ s a t i s f y i n g f < fa < f 6 , f Y < C, ,C < f a ) . Y The theorem w i l l b e proved upon showing t h a t f o r a n y s e q u e n c e o f c h o i c e s o f
cuts
ccc
i n t h e c o u r s e of t h e i t e r a t i o n , t h e a s s o c i a t e d o r d e r i n g s
C,
.
Namely, when t h i s i s d o n e , t h e u s u a l t o be c h o s e n s o t h a t i n
C,'S
a < K
then t h e r e is a n
The o r d e r i n g s
,
Qa
cy
m y be
inductively characterized a s follows.
domain
a
.
@a+1= ( p
:
B1 =
pT
( p : p ( 0 ) € Qo]
E Q, , p ( a )
LY
and
: a < K]]
A < B
.
, s u c h t h a t B,
> 0
have t h e
Q,
bookkeeping method a l l o w s t h e
A , B E [[f,
, if
MK
A < fa < B
with
ccc
gives the extension
p < q
o r d e r e d by
1 +aa ,
i s a s e t of f u n c t i o n s w i t h
B,
Q
.
p(0) < q(0)
a t e r m i n t h e f o r c i n g l a n g u a g e of
E Qa] ,
B,,k(a)
< q(a) . Finally, for x a l i m i t o r d i n a l , 6' = ( p : p r c v E B, f o r a l l a < A , and f o r a l l b u t f i n i t e l y A many a < X , p ( o ) is t h e c a n o n i c a l t e r m f o r $1 , o r d e r e d by p < q o f o r a l l a < A , p r Q < q T CY . We p r o v e by i n d u c t i o n on y a s t a t e m e n t which o r d e r e d by
implies t h a t For
p < q
pTa s q r a
e)
has t h e
QY
c a n o n i c a l term f o r
.
0)
q ra#p(u)
(1 < Y
ccc
, supp(p)
p €6',
and
If
p
E 6',
( i p , p , R p , p , Sp,p, ip,p) Call p d e t e r m i n e d i f f o r e a c h
P E
b e e n d e c i d e d , t h a t i s , t h e r e i s an
rP
tt(ip,kp.sp.mp)
i f t h e r e is a n
t
P determined c o n d i t i o n Lp U Rp
i supp
(p)
= p
Each
$ ,n
supp (p)
(ip,^Rp,;p,Gp)
LY
: p(p)
supp ( p )
i s not the
then
p(p) =
, t h e d e n o t a t i o n of
.
€1
A determined c o n d i t i o n
P E
,
supp ( p )
p(p)
has
such t h a t p
i s uniform
.
A ( s ) = fi A P P h a s c l o s e d s u p p o r t i f f o r every p E supp (p) ,
Let
.
P E
(Lp,RP,sp,mp)
such t h a t f o r a l l
c l o s e d s u p p o r t , and l e t Lemma 1.
and
{P <
( i P , ~ , s p , m p ) i f the context permits.
, written
p
, which w i l l p r o v e t h e t h e o r e m .
K)
5:
i s t h e f i n i t e set
U
I+
Y ,n
b e t h e set o f d e t e r m i n e d , u n i f o r m
= (p E
i s d e n s e i n 6'
Y
$
:
'
J p z n]
,
p E
By
with
LINEAR ORDER I N ( w ) By i n d u c t i o n on
proof.
.
Y
WDER EVENTUAL DOMINANCE
A t l i m i t ordinals
y
30 I
t h e l e m m a f o l l o w s from t h e
i n d u c t i o n h y p o t h e s i s and t h e f i n i t e s u p p o r t p r o p e r t y o f t h e c o n d i t i o n s . Suppose now t h a t Y = a + 1 a n d t h a t p E By L e t q = p T a . Choose
.
E ??I
(La.R,,Say.m,) q2
Pick an
so that
q1
B
,
ii t n E
s
then
and L,
.
(L,,R,,s
q4n
.
Let
43 As = A I
.
.
2
P
, 1
%
) '
.
s
As
2
=
q3
5
b e a member o f
B, ,q
^
^
(?,,i,&,k,)
.
be a member of
q2
.
:u,K
A
q4 (L,,R,,s,,m,) w i l l b e a n e x t e n s i o n of p i n
.
94
q4
ql#p(a) Let
, such t h a t
,map
,
The e x i s t e n c e o f s u c h a n
41 E
such t h a t
q
U R, G s u p p (q2)
0,
with .
q1
Choose
.
U,
We c l a i m t h e r e i s
< q4 U
-..
.
(L,,R,.s
1
.
.ma)
;
a s desired.
cu+l,n i s g u a r a n t e e d by t h e f a c t s t h a t La U R, G supp q4, n.. . , a n d q4 (L,,R,,s,,m,) is a c o n d i t i o n (extending
..
L
We work o n l y w i t h d e t e r m i n e d c o n d i t i o n s from now on, and s u p p r e s s t h e
A
notation.
P E
some o f t h e
supp p
Let
Lemma 2 . (i)
means t h a t
,
= s
s
P9P supp (p)
.
p E Uy
-
For e v e r y
r
i s o b t a i n e d from
r
with
6
, and
*r
p
, and
by r a i s i n g
such t h a t f o r each
, p<
p < *r
with
p#fp
< f6
( p t t f 6 < fp)
.
P E
such t h a t
a l i m i t o r d i n a l t h e l e m m a follows from t h e i n d u c t i o n h y p o t h e s i s .
.
Y = a + l
(9)
, %,p's t o include
L 's P,P
L (P E R r , 6 ) r,b We p r o v e ( i ) and ( i i ) s i m u l t a n e o u s l y , by i n d u c t i o n on y.
Proof. -
p
.
P,6 E SUPP ( P I
If
there is an
q
'Ihen
t h e r e i s an
m
, supp (p) = supp
p < q
(thus
).
, mp,.
SUPP ( r ) (ii)
p < "q
P>P P>q and i n c r e a s i n g some o f t h e
's
m
new members o f
B E
$ ,
p,q E
For f o r each
.
then
If y i s Assume
.
I f p E L, and 6 E ,R then , Card La ' Card R, times t o P s u c h t h a t whenever P E La , 6 E ,R , then e i t h e r g e t a q1 w i t h q < "ql P E L or 6 E R , d e p e n d i n g on w h e t h e r p < 6 o r 6 < p . Now p i c k a 9 6 q2 wlkh q1 < *q2 "!uch t h a t f o r e a c h p E dom q2 , m > 2E . Thus q2 92 f o r c e s t h a t i f p E L, , 6 E R, , and n > P, , then f (n) i 2 s c f (n) . P P 6 Then ( i ) is s a t i s f i e d by t a k i n g r = q (L,,R,,s,," 1 . (i)
We a r e g i v e n
pT a#f
.
< f6
p
E
UY
Let
q = pTa
Apply, t h e n , t h e lemma f o r
(1
713
(ii) Then
By symmetry i t i s enough t o c o n s i d e r t h e c a s e t h a t
p
r6
b p < f6
w e l l assume every
u E R,
lemma for
a
6 =
(Y
, s i n c e t h e c u t C i s d e t e r m i n e d by Q . We may a s 6 6 , l e s t w e be d o n e by i n d u c t i o n . L e t q = p r u . For
, q p P
, Card R,
fu
(since
q F t f p < fa
times, t o g e t a
q1
with
u E R,)
and q
5
*ql
.
Apply t h e
such t h a t f o r a l l
LAVER
R.
302
, p E L
u E R,
or u E L , d e p e n d i n g on w h e t h e r P < u o r ql' q la q < "q2 s u c h t h a t f o r e a c h a E s u p p (q2) , m
2 2m,
. .
(ii)
is
u <
p
2 .
Pick
with
q2
Then f o r e a c h
E
u
, q2&fn
R,
Lemma 3 .
If
,
.
By i n d u c t i o n on
for
a l i m i t o r d i n a l , s o assume
s u p p (q
r a) #
I f , say,
4
u
U
Lq,,
E
Y6
r2 w i t h
"g,,
1
E
Y =
(Y
+
1
Lr 2 ,6
6
Rr
r2 i "r3 s u c h t h a t f o r a l l r = r3 A (Lp,, U L q , a
.
.
Assuming
a s d e s i r e d , g i v i n g t h e lemm. The lemmas imply t h a t
,
supp(pTa)
rl
pT
,
LY
be t h e o t h e r ) .
n
.
supp ( q )
W e have t h a t
f < f 6 ( s i n c e r1 f o r c e s f ( f ) t o P 6 By r e p e a t e d a p p l i c a t i o n s o f Lemma 2 ( i i ) t h e r e
).
.
2
p E
Rp , a
.
kP=Er
U Rq,,
C,
E
,
"
E LP,a il Lq,,
By Lemma
2(i)
, all 6 E R
t h e r e is a n
supp r
.
r3 w i t h
, mr3,? a 2mp,, , 2mq R q , a ' S p , a ' max(m P,Q mq,J) ,
By h a s t h e
ccc
(1 < Y < K )
U
P ,ff
.
Take
Then
r
,ff
.
: a
References Mengen, Fund. Math. 26 ( 1 9 3 4 ) , 241-255
111
H a u s d o r f f , F . , Summen von
[2]
Kunen, K
[3]
S o l o v a y , R . , and S . Tennenbaum, I t e r a t e d Cohen e x t e n s i o n s and S o u s l i n ' s problem, Annals o f Math. 94 ( 1 9 7 1 ) , 201-245.
[4]
Solovay, R . ,
[5]
Woodin, H . ,
til
, p r i v a t e communication.
p r i v a t e communication. p r i v a t e communication.
is
Namely, g i v e n
< wl) c Qy , c h o o s e , by Lemma 1 , (q, : a < ul] Z I+ w i t h pC U ~ Q , t h e n t h e u s u a l A - s y s t e m argument g i v e s N q ' s which p a i w i s e s a t i s f y t h e l a h y p o t h e s i s o f Lemma 3 , and which a r e t h u s p a i r w i s e c o m p a t i b l e . {p,
,
s a t i s f i e s t h e lemma, and (p)
rl s *r2 s u c h t h a t f o r a l l or
, q s r
p s r
with
be g i v e n by a p p l y i n g t h e lemm t o
Re,,
be l e f t ( r i g h t ) of t h e c u t
is an
E Uy
r = rlnq(a)
then
Then
.
, a n d f o r e v e r y (3 E s u p p ( p ) n s u p p ( 9 )
, s o assume a E supp
q
.
i fo(n)
2m,
, q T a h a s empty s u p p o r t , l e t
supp (p)
Lp,,
P
R,.s,,m,,
Again, t h e i n d u c t i o n h y p o t h e s i s g i v e s t h e r e s u l t
E U,
r1
pTa
s y m m e t r i c a l l y for
rltc;Yp E
Y
, let
0
( i f one of
qrQ'
{P),
= P. P P then there is an r
Proof. Y
+
> !,(p)f ( n )
, 8
E Uy
p,q
s P , p = Sq,P
u
r = q2n(LLu
s a t i s f i e d by t a k i n g
q2 ,a
'
LOGIC COLLOQUIUM 78 M. Boffa, D . van Dalen, K . McAloon (eds.) 0 North-Holland W l i s h i n g Company, 1979
HYPERMEASURABLE CARDINALS
WILLIAM MITCHELL
The R o c k e f e l l e r U n i v e r s i t y If
11
i s a normal measure on
then
K
L(p)
i s an i n n e r model i n which
is
K
measurable; and t h e e x i s t e n c e of t h i s L-like model has been v i t a l t o understanding measurable c a r d i n a l s . a
K
+-supercompact
For l a r g e r c a r d i n a l s t h i s v i t a l t o o l i s missing:
cardinal
K,
measurable c a r d i n a l , no such i n n e r model i s known. a normal measure on
P,(K+)
even f o r
t h e weakest b a s i c l a r g e c a r d i n a l s t r o n g e r than a
+
The problem i s t h a t i f
p ( P K ( ~) f l L) = 0
then
p o s s i b l e approach t o f i n d i n g an i n n e r model f o r a
L(p)
so
is j u s t
+-supercompact
K
is
p
L.
A
c a r d i n a l would
be t o d e f i n e l a r g e c a r d i n a l p r o p e r t i e s i n t e r m e d i a t e i n s t r e n g t h between measurabili t y and
K
+-supercompactness
and f i n d i n n e r models f o r t h e s e "hypermeasurable"
c a r d i n a l s , and then t o work up through t h e s e models t o c o n s t r u c t t h e needed memb e r s of
+
P K ( )~.
This paper i s a f i r s t s t e p i n t h i s program.
We w i l l say t h a t a c a r d i n a l
K
i s u-measurable i f t h e r e i s an elementary em-
bedding
M
is a
j:V
moved, and
+
M
such t h a t
U. = { x C K : K L j(x)}
3 t h e r e i s such an embedding w i t h
E
M.
transitive class, A cardinal
K
K
is
i s the f i r s t ordinal 2 P (K)-measurable i f
2 P ( K ) C M . Although t h e s e c a r d i n a l s a r e i n t r o -
duced a s s t e p p i n g s t o n e s t o supercompactness, we hope t h a t they w i l l prove i n t e r e s t i n g i n themselves. as provisional:
Many of t h e d e f i n i t i o n s given h e r e should be regarded
more study may show how they can be modified t o be more general
and more i n f o r m a t i v e . The paper i s divided i n t o 3 s e c t i o n s .
The f i r s t s e c t i o n d e f i n e s hypermeasures
and t h e i r p r o p e r t i e s . The second s e c t i o n i s t h e major p a r t of t h e paper: i t de2 f i n e s and s t u d i e s i n n e r models f o r u-measurable and P (K)-measurable c a r d i n a l s . This s e c t i o n i s s e l f c o n t a i n e d b u t a knowledge of [ 4 ] w i l l be h e l p f u l .
The f i n a l
s e c t i o n d i s c u s s e s some a c t u a l and p o s s i b l e e x t e n s i o n s of t h e e a r l i e r m a t e r i a l . Some of t h i s s e c t i o n assumes some knowledge of [5]. 1.
Hypermeasures
Unlike measurable c a r d i n a l s and
K
+-supercompact
c a r d i n a l s , t h e hypermeasurable
c a r d i n a l s do n o t seem t o have an u l t r a f i l t e r c h a r a c t e r i z a t i o n which i s , i n general, equivalent.
We p r e s e n t a n o t i o n which seems t o be r e l a t e d and which i s
303
MITCHELL
W.
304
e q u i v a l e n t i n t h e models t o b e p r e s e n t e d i n t h e n e x t s e c t i o n . Hypermeasures are m o t i v a t e d by t h e f o l l o w i n g a p p r o x i m a t i o n of a n a r b i t r a r y elementary embedding Define
and
Xa
Xa
j:V
M
-f
by an i t e r a t e d u l t r a p o w e r .
{ j ( f ) ( x ) : f f V and
=
The d e f i n i t i o n c o n t i n u e s f o r a l l each
Ua
Now l e t
a.
= {X C K & x
P ( K ~Ma) ~and M
E
k :M a a Ma and
K
j
.
=
id.
act
k ION
Hence
or until
a EON
E ka(x)J.
K~
Ma+1 = MaKa/Ua.
I f we take
is a f i n i t e s u b s e t o f
x
O N C Xa.
Then
Ua
Indeed w e have
a'
t o b e t h e l i m i t of t h e
jION = j_lON
i s a normal u l t r a f i l t e r on
M
=
V
and
j a ' s then
ja:M
ity
w,
M
then
a
j = jm.
One e x p e c t s , f o r example, a n i t e r a t e d
t o h a v e o r d i n a l s which are r e g u l a r i n
b u t t h i s d o e s n o t happen.
-f
o
j = k _ j m and
and i f t h e r e i s a d e f i n a b l e w e l l o r d e r i n g o f t h e
T h i s i s a t f i r s t s i g h t somewhat s u r p r i s i n g . Ma
for
Xa< M and
Xa
u n i v e r s e ( a s would b e t h e c a s e , f o r example, i n a n i n n e r model) t h e n
ultrapower
.
:a'
Then
ja is t h e e l e m e n t a r y embedding,
If
jm
{K
b e t h e t r a n s i t i v e c o l l a p s e of
Xa
= ulta(Mo,(Ua,:a'
j =
a:
by i n d u c t i o n on
K~
Ma
b u t r e a l l y of c o f i n a l -
The r e a s o n i s t h a t u n l i k e i n t h e c l a s s i c a l
i t e r a t e d u l t r a p o w e r s ( s e e [I] and [ 3 ] ) t h e u l t r a f i l t e r s are n o t i n g e n e r a l Mau l t r a f i l t e r s (indeed, of M o - u l t r a f i l t e r s .
may b e a s u c c e s s o r c a r d i n a l i n
K
M ) and are n o t images
Thus, i n c o n t r a s t t o a c l a s s i c a l i t e r a t e d u l t r a p o w e r , where
t h e i n d i v i d u a l u l t r a p o w e r s are t o t a l l y i n d e p e n d e n t , i t i s p o s s i b l e t o h a v e o n e ultrafilter
U
i n t h e s e q u e n c e i s o m o r p h i c t o r(Ua, :a'
are so i n t e r r e l a t e d a s t o form a h y b r i d between a s i n g l e u l t r a p o w e r and a c l a s s i Our d e f i n i t i o n o f a hypermeasure i s an a t t e m p t t o c a p t u r e
c a l i t e k a t e d ultrapower. this interrelationship.
We f o l l o w Kunen [ 3 ] i n r e p r e s e n t i n g a n i t e r a t e d u l t r a p o w e r a s a s i n g l e u l t r a -
power by a n u l t r a f i l t e r on t h e s e t o f s u b s e t s of
w i t h f i n i t e s u p p o r t , where
&K
6 i s t h e number of i t e r a t i o n s and t h e u l t r a f i l t e r i s on 'K
Y
has. support 6 K:afx
= {a E
For example i f
x C 6 E
i f t h e r e is a set
X'
f o r some
K
x',
" a r x E Y"'.
F u n c t i o n s on
sets o f
having a f i n i t e support.
'K
&K
6
P f ( a)
6
on
Y
x c x ' c 6 , w e w i l l w r i t e "a
a
Y
of
and
Y'.
such t h a t
are t r e a t e d s i m i l a r l y .
An i t e r a t e d u l t r a p o w e r of l e n g t h t h e Boolean a l g e b r a
A subset
W e w i l l not g e n e r a l l y d i s t i n g u i s h between
Y'}.
a E
Y ' c x K
K.
6
Pf( K)
E
Y" f o r
i s t h e set o f sub-
i s defined t o be an u l t r a f i l t e r
which s a t i s f i e s t h e f o l l o w i n g 4 c o n d i t i o n s :
F
on
305
HYPERMEASURABLE CARDINALS (1)
< v'
V V
<
6
{a:av
V A < a (2)
I
{a:X < a
E
F
E
F
and
and a l l
(normality) f o r a l l
v
f ' w i t h support i n
v such t h a t
(3)
(nontriviality) for a l l
(4)
( c o u n t a b l e completeness) i f
u
f, if
{a:f(a)
E
(a:f(a) = f ' ( a ) }
and a l l
f
1
a E
E
t h e n t h e r e is a n
F
F.
with support i n
{Xn:n~wl C
F
then
4 F.
v, { a : f ( a ) = a )
flnswXn
+
Q.
One example o f a n i t e r a t e d u l t r a f i l t e r i s , o f c o u r s e , t h a t g i v e n by a c l a s s i c a l i t e r a t e d ultrapower. for
x
C
6 define
If
i s an i t e r a t e d u l t r a f i l t e r o f l e n g t h
F
F[x] = F O Pf(x6).
6
on
then
a
We a t t e m p t t o e x p r e s s t h e d e s i r e d i n t e r -
6
r e l a t i o n s h i p by d e f i n i n g a hypermeasure o f l e n g t h
on
t o be a n i t e r a t e d
a
u l t r a f i l t e r such t h a t (5) If
v < 6,
for all
F[v]
ult(V,F).
E
i s an i t e r a t e d u l t r a f i l t e r then we w r i t e
F
i
F
f o r t h e e l e m e n t a r y embedding
V
-t u l t ( V , F ) and, i f f i s a f u n c t i o n with f i n i t e support, [f] for the FF e q u i v a l e n c e c l a s s of f i n t h e u l t r a p o w e r by F. L e t bF:6 + i (a) b e d e f i n e d F F by b ( v ) = [Xa a v I F . Then f o r a l l X , X E F 3 bF (iF)-' E i (X). T h i s ap-
p a r e n t dependence on port x for F F b o E i (X').
X , a map
1.1 P r o p o s i t i o n : then
If
function
F
&
x, and X' = { a
E
"(r:ao-l
f
with support i n
bF["]
=
v.
F bV
Then
X}.
6
t h e n by ( 2 )
X
X
F
L
iff
v < 6
and
= [fIF
I t f o l l o w s t h a t f o r any s u c h
In p a r t i c u l a r , t h i s holds for t h e functions
f o r some
f,
Xa a
for
n
E
v
bFrv.
Proposition:
If
F
i s a hypermeasure of l e n g t h
i s c l o s e d under s e q u e n c e s o f l e n g t h l e s s t h a n i n f ( a ~
E
i s a n i t e r a t e d u l t r a f i l t e r of l e n g t h
i s a n y o r d i n a l less t h a n
X
[fIF = [flFtv1.
1.2
o:n
hF["] = b F r v .
Proof:
so
If
c a n b e removed i f n e c e s s a r y by t a k i n g a f i n i t e sup-
(iF)-'
6
on
+, c f ( 6 ) ) .
o
then
ult(V,F)
( [ f T l ] : n < X ) b e a s e q u e n c e o f members of u l t ( V , F ) , w i t h h 5 a , and f n . Then ( iF (f,,):n<X) and ( i F ( x T l ) : n < X ) are i n l e t x,, b e t h e s u p p o r t o f F x c v . u l t ( U , F ) , a s i s i r u n < X x n . I f h < c f ( 6 ) t h e n t h e r e is v s u c h t h a t v n < X Tl We have F [ v ] L u l t ( V , F ) a n d , s i n c e a l l f u n c t i o n s i n v o l v e d a r e coded by s u b s e t s Proof:
Let
of
bF["]
a,
( [ f T] ~ F
=
i s t h e same i n u l t ( V , F ) F F F ( i ( f T l ) ( ( b r x , ) i ):,,
=
(iF(f,,) ((hF"U1/x,,)iF) :n
E
ult(V,F)
. 0
as i n t h e r e a l w o r l d .
Then
306
W. MITCHELL i s a hypermeasure on
o f l e n g t h more t h a n 1 t h e n
u-
Clearly, i f
F
measurable.
We do n o t know w h e t h e r t h e o t h e r d i r e c t i o n h o l d s , o r i f e i t h e r
0
a
is
d i r e c t i o n h o l d s f o r l a r g e r hyper-measures.
2.
The I n n e r Models
F
L ( F ) , where
The models w e are c o n s i d e r i n g a r e of t h e form
is an u l t r a f i l t e r
sequence.
2.1
definition:
F
F,
domain
E
6 = 6(a,B)
and
(normality)
F = F(a,B)
F
E
If
v < v' < 6
6 and
Ia:f(a)
v < v,
(3)
(nontriviality)
If
(4)
(countable completeness)
f'
E
f
E
L(Fr(a,B)) If
F
E
such t h a t
{a:a
+ON,
such t h a t f o r
Pf(6,)
f o r some
F
< 0 (a,)}
v'
t h e n f o r some
cF
then
f'
v
then
flnEuXn
#
F
{a:f(a)
Ia:f(a)=f'(a)}
= F/{(a',B'):(a'
< a
v
has support i n
{Xn:nEw)
E
E
F.
L(Fr(a,B))
E
OiF(F)(a)
=B
(a:f(a)=a )
4 F.
$. E
F
then t h e r e
F. and F r y = F/'(y,O).
(a'=a and B ' < B ) }
or
implies i n p a r t i c u l a r t h a t
Hence ( 5 )
OF:!LF
F.
i (F) r(a+l) = F r ( a , B ) , and i f
L(Fr(a,B))
Fr(a,B)
Here,
{a:f(a)=f'(a)
then
F
(coherence)
is
and
ON
E
and i f
with support i n
(5)
9.
i s a n u l t r a f i l t e r on
s a t i s f i e s (1) - ( 5 ) below.
F
(1) v h < a { a : h < a (2)
F
{ ( a , B ) : a < e F and B
form
each ( a , 6 )
F i s a f u n c t i o n w i t h domain o f t h e
An u l t r a f i l t e r s e q u e n c e
.
The second p a r t of (5) and
t h e s t r e n g t h e n i n g o f ( 2 ) and t h e weakening o f ( 3 ) from t h e d e f i n i t i o n of a n i t e r a t e d u l t r a f i l t e r are i n c l u d e d t o e n s u r e t h a t t h e p r o p e r t y o f b e i n g a n u l t r a f i l t e r
F i s a n u l t r a f i l t e r s e q u e n c e t h e n any
sequence i s a b s o l u t e i n t h e s e n s e t h a t i f
Fr(a,6)
is a n u l t r a f i l t e r s e q u e n c e i n any model
of t h e weakening o f (3) t h e F(a,B)'s
M
containing
Fr(a,t?).
Because
need not be i t e r a t e d u l t r a f i l t e r s b u t we
do have: 2.2
proposition:
F
If
is a n u l t r a f i l t e r sequence then e v e r y
F(a,6)
is iso-
morphic t o a n i t e r a t e d u l t r a f i l t e r .
Proof: L e t F
= F(a.6)
port i n
v {a:f(a)=av}
f o r each
v
u:6'
and s e t
2
y
E
6
u l t r a f i l t e r and
-
y
fv
F* F*
=
and
4 F}.
or
L(G)
=
{x C
6'
a:{au
-1
i s isomorphic t o
F
Let
G(a,6).
y = [v
E
6: for a l l
Then t h e r e i s a s e q u e n c e
has support i n
For t h e rest of t h i s p a p e r L(F)
6
or
v n y
:a
F
E
x} by
E
u
:v
E
and
(fV [a:av = f v ( a ) )
F].
Then
and
E
w i t h supsuch t h a t
E
F.
Let
i s an i t e r a t e d
F*
(fv:v
f
6-y)
6-y).
fl
G w i l l always be an u l t r a f i l t e r sequence i n
but n o t n e c e s s a r i l y , u n l e s s s t a t e d o t h e r w i s e , i n t h e r e a l world.
I n a d d i t i o n w e w i l l assume t h r o u g h o u t t h a t f o r a l l
F
a < 9.
,
F
0 (5) = 0
f o r every
307
HYPERMEASLIRABLE CARDINALS
B 5 OF(,). OF(,)
<
F such t h a t
We w i l l i n f a c t b e p r i m a r i l y c o n c e r n e d w i t h s e q u e n c e s
+
a*
1 in
Such s e q u e n c e s w i l l g i v e models f o r u-measurable and
L(F).
2 P (K)-measurable c a r d i n a l s .
The more g e n e r a l c a s e i s d i s c u s s e d f u r t h e r i n s e c t i o n
3.
A s i n [ 4 ] , t h e n e x t lemma i s t h e key t o t h e whole t h e o r y o f u l t r a f i l t e r s e q u e n c e s . Main Lemma:
2.3
i:L(F)
and
ry
iYF') = i ( F )
F'
F and
For a l l
L(i(F))
+
i':L(F')
f o r some
+
y
t h e r e are i t e r a t e d u l t r a p o w e r s
L(i'(F'))
The p r o o f i s a m o d i f i c a t i o n o f t h a t of l e m m a 2 . 3 o f [ 4 ] .
Proof:
i'(F')
such t h a t e i t h e r i ( F ) =
r
y
OI
ON.
E
As i n [ 4 ] w e
d e f i n e a s e q u e n c e of i t e r a t e d u l t r a p o w e r s iX:L(F)
L ( F ~ ) ,FA
-t
ii:L(F') by i n d u c t i o n on
?.
6
"(a))
c
sup(0
FA ( a ) , O F
i s p;firjed
i'xX+l
a x -> i n f ( l l
(ax,Bx) i s t h e least p a i r
A,
either
0
FX
X
then
I f f o r some
i = i
:L(Fx~ult(L(FX),Fx(aX,EX));
XX+1 (aX,Bx) is u n d e f i n e d o r
y = ax are as r e q u i r e d .
and
c o m p l e t e t h e prTof by showing t h a t t h e a s s u m p t i o n t h a t
F
ax < i n f ( i l F x , % I)
fur a l l ordinals
X
Fx(a,B) # F;(a,B).
or
either
X
i ' = i;
X'
i
such t h a t
(a,B)
FA ( a ) , O F i ( a ) )
6 = inf(0
and o t h e r w i s e
(a )
similarly.
,ll ')
x x
and =
,
= i'(F')
For each
BX
ixX+l = i d i f
Then
F'x
L(F;),
-+
i,(F)
=
We w i l l
( a X , B X ) i s d e f i n e d and
leads t o a contradiction.
The p r o o f w i l l
b e b a s e d on t h e f o l l o w i n g p r o p o s i t i o n , which w a s s u g g e s t e d by t h e r e f e r e e of [ 4 ] . Proposition:
2.4
f o r each
X
E
r
Suppose dX
x
E
i,(x)fl
E
r)
(which w e w i l l s t i l l c a l l
L(F), x'
r
L(F'),
E
i s a s t a t i o n a r y c l a s s , and
Then t h e r e i s a s t a t i o n a r y s u b c l a s s o f
i;(x').
such t h a t f o r
h < u
r
in
w e have
r
ixu(du) =
i i w ( d w ) = du.
Proof:- A t some
X'
d
E
any l i m i t ix,(x).
works f o r a l l
there is a
Then w e c a n s h r i n k
X
E
r.
r
A'
h < u
in
r
i'
i n s t e a d of
X
E
Proof o f 2.3,
A, v
for
E
r,
For
concluded: iXu(Fx)
=
BI < i n f ( o F x ( a x ) , O ' ( a , ) ) F i = Fi(a,lB,). bFx
#
bFi
r
w e now h a v e
such t h a t
and
aw >
# Fi
for
x
i
f u r t h e r so t h a t
=
i
x'x (2) f o r
ax s o
Fx(ah,Bx)
d i s t h e same -
i x u ( d X ) = i x u ( i X , X ( d ) )= i v ( d ) = d w . completes t h e proof.
iXh+l # id
# F;(aX,Bx).
We w i l l f i r s t show t h a t t h e c l a s s of FA
d
F i r s t w e apply p r o p o s i t i o n 2.4 w i t h iiu(aX) =
is nonstationary;
assumption t h a t
< X
t o a s t a t i o n a r y c l a s s s o t h a t t h e same
Then w e c a n s h r i n k
R e p e a t i n g t h e argument w i t h
for a l l
r.
r
in
h
Let
X
E
a 2.4
d X = (aX,BX). Then
and
r
# id.
Thus
FA = Fh(aX,BX) and such t h a t
t h e n w e w i l l c o n c l u d e t h e p r o o f by showing t h a t o u r
X
E
r
leads t o a contradiction.
30 8
r
Suppose w e c a n s h r i n k
1
I-.
E
Let
gFA
of
sFi
and
L(Fir(ah,Eh)) ('Ih,gh)
is g r e a t e r than
and
ii,(support
gh
and l e t
[ g X l F h = bf;;.
'Ih
+
1
nx +
# bF'f
1.
for
A t least one
r)
We can assume ( p o s s i b l y s h r i n k i n g
'Ih.
But gx)
has support i n
r
be i n
h,v
Now a p p l y p r o p o s i t i o n 2.4 w i t h F We h a v e i h v ( g h f ( ( b ' ) ( i h i ) )
=
Also
=
i h v ( s u p p o r t g h ) = s u p p o r t g, FA -1 (g,) C ' I h , (b ) ( i x v ) l rsupport(g,) =
and s o , s i n c e s u p p o r t
=gh(ar'Ih)}
A < v.
'Ih.
with
i h v ( g h ) = gv = i i v ( g h ) .
(b F'h ) ( i ~ v ) - l ~ s u p p o r t ( g v ) . Hence
ia:a
r
bFh
' I h and bFX < b F h i f sFh > rlh. Hence by d e f i n i t i o n 2 . 1 ( 2 ) and 'Ih 'Ih c a n be r e p r e s e n t e d i n t h e form [ g , l F h where g h E L ( F h r ( a h , B h ) ) =
( 5 ) , b:: =
bFh # bFi
t o a s t a t i o n a r y s u b c l a s s such t h a t
be least such t h a t
'IX
sFX >
that
dh
MITCHELL
W.
i i v ( g h ) ( ( bF i ) ( i i v ) - 1 ) = b F 'Ih
so
Fi, contrary t o definition 2.1 ( 3 ) .
E
'Ih
r
Hence we c a n s h r i n k Pick
so t h a t
Xh
d h = Xi.
Xi
F;
E
t o a stationary subclass so t h a t
F A and
Then f o r
i f f (bFX)(iiV)-'
r
in
h < v E
w e have
iiV(Xh).
Xi.
Definition:
2.5 L(F)
$but
$
F
of o r d i n a l s t h e r e i s a s e t such t h a t
Proof:
Let
i':L(F')
T:L(F')
(1)
Theorem:
-
Xv
=
Fx i f f
E
E
ix,Fh)
= i' hv (Xv ) ,
xh
r.
E
bA
and
= bf',
~c oin t, rary
E
and to
i s a s e n t e n c e such t h a t
$
F
i(F)
F
F'
and
and
r
F.
6 of
are $-minimal f o r t h e same
$
g i v e n by lemma 2 . 3 are a l s o $-
i(F')
Then f o r any y
a
definable i n
i s any o r d i n a l such t h a t
L(F)
in
x C a L(F)
and any class
from p a r a m e t e r s i n
+
x
X
where
i s minimal s u c h t h a t
m i n i m a l , so w e h a v e i t e r a t e d u l t r a p o w e r s
@
L(G)
with
i
y = ~ ( x ) .Q
and we c a n t a k e
2.7
A
a
vr
r
y f l a.
=
is a l s o
LV') and
x
xh
F i s $-minimal and
Suppose
0 (a') < a.
implies
for
Hence w e h a v e , as i n [ 4 ] :
i(F').
=
Proposition:
2.6
iAv(Xh)
f o r a l l p r o p e r i n i t i a l segments
then t h e i t e r a t e d ultrapowers
a' < a
F h i f f bFhiy:
E
F i s $-minimal i f
A sequence
~(F16)
i(F)
bFh = bFi
and a p p l y p r o p o s i t i o n 2 . 4 w i t h
2.3
This property is important because i f minimal, s o
Xh
But
so w e g e t
iXvpsupport(XX)= ii,rsupport(X,), t h e choice of
Xh
d i s a g r e e on
FI
F
(5) f o r some
ra
L(F):
is unique i n (u,B)
then
ra
= i'
F
If
id.
F
E
I f n o t w e c a n assume
fails.
Then w e h a v e i t e r a t e d u l t r a p o w e r s
Then
L(F)
6 < 0 (a) and
Proof:
F
=
and
and
E
F
L(G),
u r C X. +
so
satisfies
Then
L(G)
x
E
L(F')
2.1
F = F(a,B).
i s $-minimal where i
x
a
i:L(F)
c$
i'
asserts t h a t t h e t h e o r e m
HYPERMEASURABLE CARDINALS
We c a n assume fails.
F
Thus
are any
i s minimal i n t h e o r d e r i n g o f
F
ij(x) f i ' ( x )
such t h a t
x
This is impossible s i n c e
Since
F
r
F
(a,O
F
r
(a) = F
(a+l)
so
a
s u c h t h a t t h e theorem
then t h e l e a s t such
ij(x) = i(x)
2 . 1 (1)-(5),
satisfies
L(F)
L ( F ) , as are t h e maps i n (1). But t h e n i f t h e r e
is definable i n
L(F).
309
F
5
8
F
0 (a). I f
8 = 0 (a)
i(a) = ij(a) > a.
=
is definable in
x
f o r any d e f i n a b l e
x , so
i j = i'.
j(F)
ra+1
then
F
8 < 0 (a)
Hence
and
0
by t h e same argument as i n t h e end o f t h e p r o o f of lemma 2.4.
F = F(a,5)
=
The most i m p o r t a n t a p p l i c a t i o n of theorem 2.7 i s g i v e n by t h e f o l l o w i n g c o r o l l a r y .
F
Note t h a t i f
0
t h e o n l y sets 2.9 g
E
X
*+ 1
Suppose
F(a,5').
isomorphic t o
F(a,8)
6
for y C
2.9
=
0.
c a L (G)
Y
ILy(G)I most and
r
F
X
o:q
t h e i t e r a t e d ultrapower
8) #
L(F),
F.
some
F
Then
and l e t
11
y
E
F
be
s a t i s f i e s 2 . 1 (1)-(5) T h i s set
F.
E
holds i n
/$a
8 ' ; otherwise
and r e a c h a c o n t r a -
F(a,F)
= P(K)./) L(Fr(a,B)).
{a:yflaoEL(F~ao+l))
-
for
(a, 6 ' )
f o r some o r d i n a l
P ( K )L ~( F r ( a , B ' ) )
then
L(F).
L(Fr(a,O(F)) L(F)
But i f has support
=
L(Fr(a,E')).fl
for all
a , and t h e r e
x
E
=
+.
a
then we say and
x < x'
a , any
The p r o o f d i v i d e s i n t o two c a s e s
P(a)/JL(F) = ?(a)
x
L (G)
Y
an u l t r a f i l t e r sequence), E
=
iFIX1(F)(a,
and h e n c e i n
Every s u b s e t
collapsing we get
x
F(a,B)
We w i l l o n l y p r o v e t h e GCH; t h e r e s t of t h e theorem uses t h e same i d e a .
F
X.X'
6(a,B).
E
6 = 6 ( a , 8 ) , and X C 6 are such t h a t i f F bu t h e n u E X. Then F[X] i s isomor-
It is easy t o check t h a t
The GCH h o l d s in
0 (a) # 0 , t h e n
0 (a)
c.
F[Xl
We w i l l work i n s i d e
F
17
and
(a,6)
w e l l o r d e r i n g o f t h e reals.
,A!
Proof:
If
Now l e t
L(F r ( a , 6 ) )
Theorem:
is a
for a l l
K*
=
ra+1
iF['](F)
by
so i t is i n
X
in
[g],
be least such t h a t
F[X]
is i n
2
6(a,B)
In p a r t i c u l a r f o r e v e r y
i f w e know t h a t
(a,6') K
then
by p r o p o s i t i o n 2.2 i s a hypermeasure.
d i c t i o n w i t h theorem 2.8. isomorphic t o
L(F)
and
X
First note that
w e could let
in
F = F(a,B).
has support i n
p h i c t o some
Proof:
K
w i t h t h e s t a t e d c o n d i t i o n are t h e o r d i n a l s
corollary:
L(F)
5
(K)
x
of
=
i f f o r any s u c h
L
ILy(G)I = a, G
x < x'
L
G F
f o r some
y
and
ra
(F)
h a s a t most
a
so w e can assume
with
rl
r e g u l a r and by
x'
E
L (G) b (ZF-+ G i s G Y 0 ( a ) = 0. I f
such t h a t
r a and Y
(G)
with
i n t h e o r d e r of c o n s t r u c t i o n i n
Ly(G).
Ly(G), Since each
p r e d e c e s s o r s so t h e o r d e r t y p e of
<
is at
i s a l i n e a r o r d e r ; t h a t i s , t h a t i f L (G,) Y1 are two s u c h models t h e n t h e o r d e r i n g o f P ( a ) i n o n e o f them i s a n
We h a v e t o show t h a t
Ly (GZ) 2
4 ult(L(F),F(a,O))
is i n some
K
<
W. MITCHELL
310
By lemma 2.3 t h e r e are i t e r a t e d u l t r a -
i n i t i a l segment of t h a t i n t h e o t h e r .
il:LYl
powers
= c2 I Tl. Then 1
-
G
(G1)
+l(Gl)
-+
or(") L
+2 (c2). T h i s
f o r some
a
then
li(w)(
IP(a)/)
5
0 (v) < a
+.
5 "+*a 5
2'.6(w,B)
t h a t (say)
P(a)
and
Ly1
c o m p l e t e s t h e p r o o f o f c a s e 1.
If
and
B < a+
by c a s e 1.
a
+ , and
(z1
)
is
i:L(F)
-f
ult(L(F),F(v,B))
Hence which i s o f some i n d e p e n d e n t
lemma 2.10,
< i(v) < a
ult(L(F),F(v,B))I
(c2) s u c h
L
+
v < a.
F
T h e r e i s no problem i f
(G,)
i 'L
2' Y 2 y2 i2 p r e s e r v e t h e o r d e r i n g o f
and
il
a n i n i t i a l segment o f Case 2:
and
i n t e r e s t , w i l l complete t h e proof. 2.10 lemma:
If
and
v < a
ult(L(F),F(a,E))
a
f o r some
+
5
8 <
F 0 (v)
t h e n e v e r y s u b s e t of
X Ca
i s i n some
+ a .
Proof:
We f i r s t show t h a t e v e r y
L (G)
(zF-+ G i s a n u l t r a f i l t e r s e q u e n c e ) ,
y
+
, and e v e r y u i x ] . Let n
B < a
member of
a
be a r e g u l a r c a r d i n a l such t h a t
L (G)
Y
3
Since
X < L ~ ( F ) where
+
L (G)
and s o f o r any
f i e s t h e h y p o t h e s i s o f c o r o l l a r y 2.8.
B
=
U(Xn
F
1x1
0 ( v ) ) , and
= 0.
G
Thus
6 < a
so
X
+.
L (G)
2.3 t o d e f i n e i t e r a t e d u l t r a p o w e r s
L (G)
j(7)
other.
s o t h a t o n e of
i(G)
and
Y
and h e n c e f o r
L(T)
c o n s t r u c t i o n i s i n s i d e of L ~ ( G )E L ( i ( 7 ) )
since
and
L(F).
L(F).
1=
F
and
and
1 (v,B)
j(f)
of
j
x
X n 6(v,X)
L(1)
6
X.
satis-
where
U s e lemma
=
i s a n i n i t i a l segment o f t h e
i ( G ) ; i f i t w e r e w e could
t o g e n e r a t e indiscernible?.
i(G)
Hence
a C X
F
u l t ( L ( F ) ,F(v,B)).
E
c a n n o t b e a p r o p e r i n i t i a l segment o f
use an e x t r a measurable c a r d i n a l i n L(i(7))
Y
X II 0 ( v ) ,
E
Y
of
i
f o r some
(v,B)
from p a r a m e t e r s i n
P ( a ) C L ~ ( F )and t a k e
ra+
To f i n i s h p r o v i n g t h e lemma, w e show t h a t ult(L(F),F(v,B))
F ['
=
L (G)
i s t h e s m a l l e s t s u c h set w i t h
X
P(w) C X
v C a C X,
i s d e f i n a b l e in
Y
is i n
L ~ ( G ) such t h a t
ra+1
G
a
for
This i s impossible since the e n t i r e L
( i ( G ) ) E L ( j ( f ) ) and s o i(v) L ~ ( G ) i s t h e t r a n s i t i v e c o l l a p s e of t h e members t o
( i ( G ) ) which are d e f i n a b l e from p a r a m e t e r s i n a U {XI. But i(Y) / L y ( G ) ) / = a and j P ( a ) = i d , s o Ly(G) E L ( 7 ) , as w a s t o b e s h o w n . f f 2 . 1 0 , 2 . 9
L
r
The n e x t two r e s u l t s show t h a t , as c l a i m e d , t h e models f o r y m e a s u r a b l e and
2.11
Proposition:
i s v-measurable i n measurable i n
Proof:
P'(K)-measurable (i) L(F).
If
If
g i v e i n n e r models
cardinals.
6 ( a , B ) > 1 and
(ii)
L(F)
F
0 (a) > a
P ( a ) n L(F) c L ( F r ( a , 6 ) ) -I+
in
L(F)
then
a
is
then 2 P (a)-
L(F),
Part ( i ) is clear.
Because t h e GCH h o l d s i n
L(F),
PL(a)-measurability
HYPERMEASURABLE CARDINALS
+
is equivalent t o
31 1
tc ) ) f o r some 6 c a*, ult(L(F),F(a,E))C ult(L(F),F(a,a tc P ( a + ) C u l t ( L ( F ) , ( a , a ) ) and a i s P ( a ) - m e a s u r a b l e .
Theorem:
Suppose
G such t h a t
sequence
Proof: F i r s t l e t point
h a s t h e same p r o p e r t y i n
K
j:V
-+
W e w i l l use
K.
2 P (K)-measurable.
i s p-measurable o r
K
M
is i n
a
so
+
2.12
+
By lemma 2 . 1 0 , any s u b s e t o f
P(a )-measurability.
Then t h e r e i s a
L(@.
b e a n a r b i t r a r y e l e m e n t a r y embedding w i t h c r i t i c a l
F and p r o v e a g e n e r a l lemma
t o c o n s t r u c t a sequence
j
which w i l l l e a d t o t h e d e s i r e d r e s u l t when
witnesses t h a t
j
h a s o n e of t h e
K
stated properties.
F
i s d e f i n e d by i n d u c t i o n on
K
F
e q u a l t o some s u c h fails.
r
F
Once
sequences
bv
F
0 (a)
F.
tc
6
Let
0 (0 F (KI-MJ .
=
2 . 1 3 lemma: Proof:
If
F. J
either
4M
F.
L(F)
I
and
F
or K
F
then
F
2
=
F[11
J F[1] i n
E
U. J
6
1 (K,O F
M, so
6 1.2.
M,
L(F).
But t h e n
g i v e n by a f u n c t i o n from
6 > 2 L(F)
x
j
=
F.
satisfies
in
M
I
K
r
K)
K
+
1.
{xEPf(%):bj-'
F
K
j
If
Now d e f i n e
E
Fj
2 . 1 (1) - (5) f o r
F
t h e way
4 M.
Let
F
=
(K))
= F
was d e f i n e d i n
K
F
F
=
F
j'F
6
=
6(F)
i s p-measurable.
K
i s p-measurable i n
then
K
4 L(Fr(r,bl))
V,
+
K*
a 2.13
F,b M unless
F [ 1 ] = U . and t h e isomorphism i s i n M, s i n c e i t i s J i n t o K s o F E M , c o n t r a r y t o a s s u m p t i o n . If
F(lc,b ) = F [ 2 ] shows t h a t K i s p-measurable 2 But i f x E P ( K ) L(F) ~ and P(K)fl L(F)&L(Fr(r,b2)).
which i s a b s u r d .
then Hence
{a:x if
1
and
Since
L(F')
t h e n by p r o p o s i t i o n 2 . 1 1 unless
=
.
6
2
(K)
( K ) } by
witnesses t h a t =
.
j(x)}
L(F).
( F ' ) = domain (F) U { ( K , O
w i t h domain
F ' ( K , ~(K))
U.
F
P (K)-measurable i n
tc
F'
and
r
j(F
2
is
K
is d e f i n e d a t
F'
t s set
F(a,E)
OF(c)> K i n L ( F ) . Thus i f F . E M t h e n 0 3 2 i s P (K)-measurable i n L ( F ) by p r o p o s i t i o n 2.11.
and d e f i n e
Now s u p p o s e t h a t
then
such t h a t one o f t h e s e c o n d i t i o n s
0 and s e t
=
F o r t h e rest of t h e p r o o f w e c a n assume b = bF
E
v:
It c a n e a s i l y b e v e r i f i e d t h a t
(K,OF(K)). S i n c e in
M
E
i s defined, t h e r e i s an
has f i n i t e support}
0 (K)-M~
be least such t h a t
r (a,E)
L(Fr(a,E)),
F i s set e q u a l t o
by i n d u c t i o n on
Mw
Mw = { j ( f ) ( ( b r v ) j - 5 f
bv
in
is t h e l e a s t
is defined
K
and
F
if
(a,L3):
E 5 a
2 . 1 (1)-(5), and
satisfying
n aoEL(F(ao+l))} 6 > 2
then
a
E
F
is
but p
{ao:x
in
n ao~L(F~(ao+l))}~F[l],
measurable i n
L(F).
W. MITCHELL
312 P2(K) C M , s o
Now suppose
2 P (<)-measurable.
is
K
F = ~ ( F [ X ] : X ~ [ ~ so ] < ~F} ,i s e s s e n t i a l l y a t h i s would imply L(F')
F E M.
Hence
6 =
If
6 <
+-sequence
K * ~ ( ~ ) so
=
K*
K
then
K*
of members of K
is
P ( K ) . But
2
P (K)-measurable i n
by p r o p o s i t i o n 2 . 1 1 .
We now know t h a t
L(F)
i s an i n n e r model f o r U-measurable o r
2 P (K)-measurable
c a r d i n a l s , but i n what sense i s t h e r e a unique minimal such model? There i s no 2 M i n t h e sense t h a t Mk (ZFC+ K i s P (K)-measurable) and M C M'
1
minimal model
f o r every o t h e r such l a r g e below
s i n c e t h e f i r s t measurable c a r d i n a l can be a r b i t r a r i l y
M',
However by 2 . 1 1 we can r e s t r i c t o u r s e l v e s t o models
K.
F $-minimal f o r t h e d e s i r e d p r o p e r t y .
with
such a model which i s minimal i n t h e same sense a s t h e model when
L(p)
is minimal
i s a measure on t h e s m a l l e s t p o s s i b l e o r d i n a l .
LJ
Theorem 2.13:
I f t h e r e i s a sequence
minimal sequence a t e d ultrapower
Proof:
M = L(F),
The next theorem shows t h a t t h e r e i s
F
Let
c l a s s then
Xr
closed i f
r
cofinal i n
Y
F
i:L(F)
+
$ then t h e r e i s a $ -
F'
t h e r e i s an i t e r -
L(F').
F
X
i s t h e smallest c l a s s
T C X.
such t h a t
X(L(f)
i s a proper c l a s s and f o r some o r d i n a l
then
r
1
L(F)
such t h a t
be an a r b i t r a r y $-minimal sequence and l e t
Y
E
r
so t h e r e i s a c l o s e d F i x such a
F
such t h a t f o r any $-minimal sequence
and let
r.
= 11
6 , i f cf
.
r
If Call
r
y > 6 and
is a
r
is
Any i n t e r s e c t i o n of closed c l a s s e s i s a c l o s e d c l a s s
such t h a t n:L(f)
Xr
'= Xr.
61 L p + ( f ) C
Xr ,
W e w i l l show t h a t
r '.
f o r any closed c l a s s
F
is t h e required
sequence. Let
F'
be any $-minimal sequence.
Then by lemma 1.3 t h e r e a r e i t e r a t e d u l t r a -
filters i:L(F)
y
i':L(F')./ We have t o show t h a t lemma 1.3) and s e t F
6"
F
i' = i d
= 0 (a,)
F = F'(av,Bv).
Since L(F").
ult(L(G),F).
i t e r a t e d ultrapower
L(F*)
of
.
and hence ( t a k i n g t h e n o t a t i o n o t t h e proof of
for a l l
i s a n u l t r a f i l t e r on
not, neither is
L(G)
v.
L(F")
Let and
be l e a s t such t h a t
V
L(F')
W e claim t h a t
F ' (au)
6,$ < 0
have t h e same s u b s e t s of
ult(L(?},F)
i s w e l l founded.
a
U '
If
But then by a b s o l u t e n e s s t h e statement " t h e r e i s an L(F')
such t h a t
ded" i s t r u e i n L ( F ' ) . This i s impossible because Now consider t h e following t r i a n g l e , where
j
and
ult(L(F*),F) F
i s not w e l l foun-
i s countably complete inuF'). j '
a r e given by lemma 1 . 3 :
HYPERMEASURABLE CARDINALS
We c l a i m t h a t
r L+F,)
jk
L ~ ~ + ( F " ) is i n L(F).
-f
is definable i n
L(F)
j k ( a v ) > a"
=
r'.
from p a r a m e t e r s i n
r'
f i n a b l e from members o f j'(av)
a F,
so
0 "(a,)
so
A
r ?(F,)
j K
0
Corollary:
If
for a l l
(a)
a
T h i s f o l l o w s from
L(Fw).
= j ' r
B v , as w e l l .
>
F and F' then F = F'.
F
51
2.15
Corollary:
for a l l
i s de-
In particular
x
F'(av,B )
=
F = F (a v W*BW)>
a r e $-minimal f o r t h e same
F
0 (a) =
and
$
D
The c o r o l l a r y i s n o t t r u e w i t h o u t t h e a s s u m p t i o n o f $ - m i n i m a l i t y , 0 (a)
L ( F ) x v
But now w e c a n u s e t h e argument
(av,Bv).
c o n t r a d i c t i n g t h e d e f i n i t i o n of
F'
*1.
Thus a n y member o f
a t t h e end o f t h e p r o o f of lemma 1 . 3 t o show t h a t
2.14
i (A
=
L ( F v ) h a s t h e form i v ( f ) ( C ) where 5 E aw and h I f I" = { y : i ( y ) = j K ( y ) = j ' ( y ) = y l t h e n e v e r y s u c h f
t h e f a c t t h a t e v e r y member of f:a
-
~ L ~ ( F " )where ,
= j '
313
even i f
( s e e [51).
O.
Every e l e m e n t a r y embedding
j
:L(F)
-f
L(F)
definable i n
M
is
an i t e r a t e d ultrapower. Proof:
I f not, then
fines a
L 1- e l e m e n t a r y embedding.
t e r t h a t works, so
F'
is d e f i n e d by some f o r m u l a
j
j
ultrapower that
i:L(F)
# j(x)
i(x)
a l l definable
L(F).
is definable i n
is a l s o $-minimal.
By t h e o r e m 2.14, in
+ L ( F I )
L(F).
would b e d e f i n a b l e i n
Now
n
j = i:
L(F)
rl
such t h a t
We
de-
$
i s t h e smallest parame-
M = L(F')
applied i n
Then
a
X.
is an
W e c a n a l s o assume
n.
w i t h parameter
$
F $-minimal f o r t h e a s s e r t i o n t h a t t h e r e
can t a k e
f o r some
F ' , and
L ( F ) , t h e r e i s an i t e r a t e d o t h e r w i s e t h e least
and c e r t a i n l y
x
i(x) = j(x)
such for
T h i s c o r o l l a r y l e t s us c h a r a c t e r i z e t h e countably complete, uniform u l t r a f i l t e r s
L(F):
in
2.16
Theorem
F(a,B)
In
L(F)
F o r a l l ( ~ , B ) E domain ( F )
(i)
if
F ( a , B ) [ { v > ] ,a u n i f o r m u l t r a f i l t e r on
6(a,B) = w
1 then
The only c o u n t a b l y c o m p l e t e , u n i f o r m u l t r a f i l t e r s are t h e u l t r a -
(ii) filters
F ( a , B ) [ { v ) ] and f i n i t e i t e r a t i o n s of them.
Proof:
Let
L(Fr(a,B))
+
a.
F = F(a,B)
and s u p p o s e
and t h e r e are f u n c t i o n s
6 = 6(a,B) = u
fl
and
f2
+ 1.
Then
with support i n
F bv < a *
in
{w>
such t h a t
+
[ f l I F = a and [ f 2 I F : a+ b.: But f o r e a c h v ' < u t h e r e i s y < a such F and t h e r e i s a g w i t h s u p p o r t i n {Ol s u c h t h a t t h a t b", = [ f 2 I F ( y )
Y v'
=
[gylF.
Set
hvt(a,)
was a r b i t r a r y ,
= f2(av)(gy(fl(av)));
F[{v}]
F
then
{a:av,=hv,(av)}
and w e h a v e proved (i).
E
F.
Since
W. MITCHELL
314 Suppose
is a c o u n t a b l y c o m p l e t e u n i f o r m u l t r a f i l t e r on
U
ult(L(F),U).
-
X
Then i f
i t e r a t e d ultrapower. such t h a t
If],
f o r a l l of
Let
w
Suppose
X
then
x
x
ia:f(a)Exl
2.17 in
F
E
SO
Corollary: L(F)
a r e no
F
K
X
and l e t
=
F(a,8)'s with
f.
Then
x
Hence a s c l a i m e d
6(a,B)
a
successor.
F
f
be
g
E
L(0
i s a f i n i t e itera-
Also,
X
E
U :
The o n l y c a r d i n a l s w i t h u n i f o r m , c o u n t a b l y c o m p l e t e u l t r a f i l t e r s
+- s t r o n g l y
K
-*
i s an
i
i s a l s o a support
u l t ( L ( F ) , U ) , t h e n f o r some
compact c a r d i n a l s i n
Hence t h e r e
a
L(F).
is not even close t o b e i n g a n i n n e r model f o r
L(F)
compact c a r d i n a l .
F i s a s e q u e n c e o f measures t h e n
i t w a s shown i n [ 4 ] t h a t e v e r y c o u n t a b l y
complete u n i f o r m u l t r a f i l t e r i s i s o m o r p h i c t o a member o f t h e c l a s s
of f i n i t e
r.
i m p l i e s t h a t t h e r e i s a uniform u l t r a f i l t e r n o t i n Problem:
r
The e x i s t e n c e of a p-measurable c a r d i n a l , however,
i t e r a t i o n s of normal measures.
2.18
i:L(F)
Now by 2 . 1 5
a r e m e a s u r a b l e c a r d i n a l s and l i m i t s of m e a s u r a b l e c a r d i n a l s .
+- s t r o n g l y
If
i(X).
E
u. C7
W e w i l l see i n s e c t i o n 3 t h a t
a
x
U
be a support f o r
ult(L(0.F)
E
w = [glU = i(g)(x) = i ( g ) [ f I F = [g(f)lF.
t i o n and o n l y i n c l u d e s
E
u' be t h e associated i t e r a t e d u l t r a f i l t e r , let
F
and l e t
=
F:
[id ]
=
Does i t f o l l o w from t h e e x i s t e n c e o f a c o u n t a b l y c o m p l e t e , u n i -
form u l t r a f i l t e r n o t i n
r
t h a t t h e r e i s a n i n n e r model w i t h a p-measurable
cardinal?
3.
Loose Ends
I n t h i s s e c t i o n w e w i l l d i s c u s s t h e f i n e s t r u c t u r e of t o larger cardinals.
L(F)
L(F)
and t h e e x t e n s i o n of
S i n c e o u r i n t e n t i o n i s more t o p o i n t o u t l o o s e e n d s
t h a n t o t i e them up w e w i l l a t b e s t s k e t c h p r o o f s .
I n d e e d no d e t a i l e d p r o o f s o f
s e v e r a l of t h e r e s u l t s mentioned i n t h i s s e c t i o n h a v e been w r i t t e n o u t , so t h e s e r e s u l t s should be regarded a s preliminary. 2 The models i n t h i s p a p e r e x t e n d w e l l beyond P (K)-measurable c a r d i n a l s : They 3 can i n c l u d e , f o r example, P (K)-measurable c a r d i n a l s , PK(K)-measurable c a r d i n a l s and
A
P (K)-measurable
cardinals
Ramsey c a r d i n a l g r e a t e r t h a n
x
P (K)-measurable
They c a n , i n f a c t , i n c l u d e a n y t h i n g s h o r t of a
K.
c a r d i n a l where
X
is t h e next measurable c a r d i n a l .
l a r theorem 2 . 1 2 e x t e n d s t o t h e s e c a r d i n a l s : then t h e r e i s a sequence
F
i s t h e l e a s t weakly compact o r
where
K
such t h a t
K
If
K
h a s t h e same p r o p e r t y i n 2 P (K)-measurable
s t r e n g t h of t h e s e c a r d i n a l s i s n o t s o c l e a r as f o r If
K
is
x
P (K)-supercompact t h e n of c o u r s e
K
is
I n particu-
i s o n e of t h e s e c a r d i n a l s , L(F).
The
cardinals.
Ph+l(K)-measurab1e,
b u t on
t h e o t h e r hand i t i s e a s y t o s e e t h a t t h e l e a s t P(K)-supercompact c a r d i n a l i s n o t 3 P (<)-measurable. I t i s s t i l l open w h e t h e r P(K)-supercompactness i m p l i e s
even
315
HYPERMEASURABLE CARDINALS
3 P (K)-measurable c a r d i n a l s , b u t i t i s known t h a t i t i m p l i e s t h e
t h e existence of
e x i s t e n c e o f i n n e r models w i t h a l l of t h e s e h y p e r m e a s u r a b l e c a r d i n a l s . which i s o u t l i n e d below, o n l y assumes t h a t
+- s t r o n g l y
is
K
K
K(F).
t h e u s e o f t h e c o v e r i n g l e m m a f o r t h e c o r e model
+ K -supercompactness
The c o r e model
N o d i r e c t p r o o f from
i s known.
K(F)
F described in
is defined f o r t h e u l t r a f i l t e r sequences
t h i s paper j u s t a s i t i s d e f i n e d i n [5] f o r sequences of measures.
K (F)
about
F.
It i s t h e
F t o g e t h e r w i t h approximations, c a l l e d mice,
c l a s s of s e t s c o n s t r u c t i b l e from of extensi ons of
The p r o o f ,
compact and r e q u i r e s
We have been u n a b l e t o check d e t a i l s , b u t t h e b a s i c r e s u l t s
seem t o g e n e r a l i z e s t r a i g h t f o r w a r d l y t o hypermeasures.
The c o v e r i n g
lemma i s somewhat more d i f f i c u l t b u t w e are c o n f i d e n t t h e f o l l o w i n g theorem i s true: I f t h e r e i s no model
theorem:
3.1
X
where
L (F)
i s t h e n e x t m e a s u r a b l e c a r d i n a l above
F such t h a t
sequence
h = 6
Any e l e m e n t a r y embedding
i:K(F)
If
(ii)
6
i s a n i t e r a t e d u l t r a p o w e r of (iii)
in
K
+
i s a c a r d i n a l and
(i)
h P (K)-measurable
with a
K(F)
in +
cardinal,
L ( F ) , then t h e r e i s a then
M, with
M
6.
cf(h)
a transitive class,
K(F).
E v e r y t h i n g t h a t w a s proved f o r
i n s e c t i o n 2 i s t r u e of K ( F ) .
L(F)
Theorem 3 . 1 c a n b e u s e d t o show t h a t many of t h e c o n d i t i o n s which a r e proved i n
[51 t o i m p l y t h e e x i s t e n c e of a model w i t h 3.2
theorem:
which
is
K
O(K) =
are much s t r o n g e r .
K*
Any o f t h e f o l l o w i n g imply t h e e x i s t e n c e of a model
x
P (K)-measurable,
(i)
K
is
+- s t r o n g l y
in
L (F):
compact.
K
There i s an X2-saturated
(ii)
L(F)
i s t h e n e x t measurable c a r d i n a l i n
X
where
(iii)
(dependent c h o i c e o n l y ) C a n u l t r a f i l t e r and i = x2.
i d e a l on
X1.
The c l o s e d unbounded f i l t e r
C
on
x1
is
(x,)
Proof:
We w i l l p r o v e ( i ) ; t h e o t h e r s are s i m i l a r . K(F)
We w i l l l e t Since
K
such t h a t cf
X
=
K
+- s t r o n g l y + i(K ) > UiqtK+.
is or
K
1 =
Suppose t h e c o n c l u s i o n f a i l s .
b e g i v e n by theorem 3 . 1 and p r o v e a c o n t r a d i c t i o n .
K+;
i t e r a t e d ultrapower of
compact t h e r e i s a n e l e m e n t a r y embedding Let
X
= K
+
i n e i t h e r case K(F)
and so
c o n t r a d i c t i o n completes t h e p r o o f .
n
in
K(F).
i(A) > ( l i " X . i(h)
=
ui"X
i:K(F)
-t
M
Then by 3 . 1 ( i ) e i t h e r But by 3.1 ( i i ) since
X
= K
+
in
i
i s an
K(F).
This
I t s h o u l d b e n o t e d t h a t when a p p r o p r i a t e models become a v a i l a b l e f o r l a r g e r c a r d i n a l s t h i s p r o o f s h o u l d e x t e n d t o g i v e ( a t l e a s t f o r ( i ) and ( i i ) ) t h e b e s t p o s s i b l e
316
W. MITCHELL
r e s u l t , which i s p r o b a b l y a
K
+-supercompact
a n a l m o s t huge c a r d i n a l f o r ( i i ) .
3.3 an
theorem: L(F)
c a r d i n a l f o r ( i ) and ( s e e [ 2 ,
I f t h e r e i s a measurable c a r d i n a l
with a
5
< 0
F
such t h a t
(K)
1171)
I n c o n t r a s t , w e h a v e been u n a b l e t o s t r e n g t h e n
5
=
K*
K
with
ZK >
t h e n t h e r e is
K+
i n uIt(L(F),F(K,B)).
While 3 . 3 i s u n l i k e l y t o be t h e b e s t p o s s i b l e , t h e f o l l o w i n g problem i s w o r t h considering:
3.4
problem: Does t h e c o n s i s t e n c y of 2 P (K)-measurable c a r d i n a l ?
m e a s u r a b l e and
K
2'
>
f o l l o w from
K+
t h a t of a
I t d o e s , of c o u r s e , f o l l o w from t h a t o f a [ 2 , 1251).
K
+-supercompact
c a r d i n a l ( S i l v e r , see
In o n e c a s e w e are u n a b l e t o improve t h e r e s u l t of [ 4 1 and t h e f o l l o w -
i n g seems q u i t e p l a u s i b l e :
3.5
problem:
If
0
c o m p l e t e f i l t e r on
F K
(K)
=
K*
in
L ( F I , then i s i t t r u e i n
L(F)
t h a t every
K-
can b e e x t e n d e d t o a n K-complete u l t r a f i l t e r ?
References
1.
H. Gaifman, E l e m e n t a r y Embedding of S e t T h e o r y , i n T. J e c h , e d . , A x i o m a t i c S e t Theory, P r o c . Symp. P u r e Math. E ( 2 ) , 33-101.
2.
A. Kanimori and M. Magidor, The E v o l u t i o n of L a r g e C a r d i n a l Axioms i n S e t
Theory, p r e p r i n t ( 1 9 7 8 ) . 3.
K. Kunen, Some A p p l i c a t i o n s of I t e r a t e d U l t r a p o w e r s i n S e t T h e o r y , A n n a l s Math.
Logic 4.
W. M i t c h e l l , S e t s C o n s t r u c t i b l e from Sequences of U l t r a f i l t e r s , J. Sym.
Logic
5.
1.179-227. 3,57-66.
W. M i t c h e l l , The Core Model f o r Sequences o f U l t r a f i l t e r s , i n p r e p a r a t i o n .
T h i s work was p a r t i a l l y s u p p o r t e d by NSF g r a n t MCS78-09864.
LOGIC COLLOQUIUM 78 M. Boffa, D. van DaZen, K. McAloon (eds.) 0 North-Holland Publishing Comprmy, 1979
ON THE NUMBER OF EXPANSIONS OF THE MODELS OF ZFC-SET THEORY TO MODELS OF KM-THEORY OF CLASSES 2. Ratajczyk Institute of Mathematics
University of Warsaw Warsaw, Poland
Assume that E = (M,E) = ZF. We say that E is expandable to a model for KM iff there is a family F E P(M) such that F = ( F,M,E*), the model obtained by adding a new class universe, is a model for KM; where by KM we mean the Kelley-Morse class theory with full comprehension scheme and with no form of the axiom of choice. The following theorem is well known: If M is a countable, KM-expandable model for ZF, then M has continuum many expansions which are models for KM. This theorem is a direct consequence of the Mansfield Perfect Set 1 Theorem (see [ 5 1 , ch. 8, 8B), which states that every Z -relation 1 included in P(M) has the power 2 card(M) or contains a hyperelementary element. It suffices to observe that inside every model ( F , M , L * ) for XM, we can define a class which is not hyper-elementary with respect to M. In the case of uncountable models the situation is different. Basing ourselves on the proof of the Chang-Makai theorem 2],ch.5, 8 3 ) we can show that the Mansfield Theorem is true for special models. I do not know, however, if it is valid for a wider class of models, in particular for Kleene structures (i.e. such 8 that the completeness theorem is true for HYPM). Neither do I know if it is true for at least one structure of the form Ra, where cf CY = w,. Ix 2 w + w . By adapting the construction used in the proof of Mansfield theorem we show here that: If g is a KM-expandable model for ZFC whose height has the expansions to models cofinality w , then g has at least card(M) for KM+VX r.a (X) where the sentence VX r.a (X) states that all 317
318
Z.
RATAJCZYK
c l a s s e s a r e r a m i f i e d a n a l y t i c a l ( f o r a r e f e r e n c e see [ 4 1 ) .
M Marek a n d Mostowski [ 4 ) h a v e o b s e r v e d t h a t i f c f ( 0 n - )
>
and M i s t r a n s i t i v e t h e n e a c h e x p a n s i o n t o a model f o r KM i s a 6model. From t h i s i t f o l l o w s t h a t f o r t h e l e a s t a s u c h t h a t R e x p a n d a b l e a n d c f ( a ) = wl,
R
i s KM-
h a s e x a c t l y o n e e x p a n s i o n t o a model
f o r KM+ X r . a for
= R
.
( X ) . Hence t h e c o n c l u s i o n o f o u r t h e o r e m i s n o t t r u e M T h i s shows t h a t t h e a s s u m p t i o n cf(On-) = w i s e s s e n t i a l .
I n t h e f i r s t s e c t i o n w e s h a l l show a u s e f u l p a r t i t i o n t h e o r e m which i s a p p l i e d i n t h e n e x t s e c t i o n . I n s e c t i o n 2-d,
the
u s e o f r e c u r s i v e c l o s e d game f o r m u l a s and t h e i r i n f i n i t e a p p r o x i m a t i o n s (see B a r w i s e [ l ] , c h . V I , 5 6 ) i s e s s e n t i a l f o r o u r p u r p o s e s . This paper contains p a r t s of t h e a u t h o r ' s Ph.D-thesis, w r i t t e n under t h e s u p e r v i s i o n o f P r o f e s s o r W.
Marek and a c c e p t e d
by t h e U n i v e r s i t y o f W a r s a w .
1. A p a r t i t i o n t h e o r e m
E r d o s , H a j n a l and Rado c o n s i d e r e d , i n [ 3 ] , t h e f o l l o w i n g p a r t i t i o n property:
We s a y , t h a t a
( b , c ) , where a , b , c a r e c a r d i n a l s , i f f o r
+
a n y s e t S s u c h t h a t c a r d ( S ) = a and f o r a n y t w o - e l e m e n t p a r t i t i o n [ Sl
[XI
*2
= I U J there e x i s t s a set X
5I
5
S s u c h t h a t c a r d o ( ) = b and
o r c a r d ( X ) = c a n d [ X I 2 _C J .
The f o l l o w i n g theorem,arnong o t h e r s , i s p r o v e d i n [ 31 : ZFC I- I f a i s a s t r o n g l y i n a c c e s s i b l e c a r d i n a l , t h e n (Vb)+
a
-L
(a,b).
Here w e s h a l l p r o v e a c o r r e s p o n d i n g t h e o r e m , which i n -
formally, can b e expressed as follows: (ZFC) f
( V a ) On
-f
(On,a)
T o do t h i s we s h a l l a d j u s t t o o u r needs t h e r a m i f i c a t i o n
lemma, w h i c h i s t h e key p o i n t i n t h e p r o o f o f t h e a b o v e m e n t i o n e d theorem o f [ 3 ] . B e f o r e d o i n g t h i s , however, l e t us e q u i p o u r t h e o r e m w i t h a p r e c i s e and f o r m a l s h a p e : The t h e o r y ZFC(R) i s c o n s t r u c t e d a s f o l l o w s : w e add t o t h e l a n g u a g e o f ZFC a new r e l a t i o n symbol R . ZFC(R) i s t h e n ZFC p l u s
MODELS O F Z F C - S E T
319
THEORY TO MODELS O F KM-THEORY
t h e r e p l a c e m e n t schema f o r a l l f o r m u l a e o f t h e e x t e n d e d l a n g u a g e . Theorem 1.1. F o r a l l k E w t h e r e e x i s t s a n 4? E w s u c h t h a t t h e f o l l o w i n g i s p r o v a b l e i n ZFC(R):
( v a ) (3x1
*
On c a r d x
-
>
o &
(vy,zIx [ y E z
=*
-tp(y,z)l v
-
( 3 x ) { ( v y ) [ ( j , P ( x , y )* O r d ( y ) ) & ( 3 z ) ( r a n k ( y ) E z & P P $ “ X , Z ) ) & (VY,Z)[j, ( x , y ) & l j (x,z) & y E z cp(y,z)l}
where:
$
P ( x , z ) i s a u n i v e r s a l f o r m u l a f o r f o r m u l a s of classes X P
w i t h o n e f r e e v a r i a b l e a n d o n e p a r a m e t e r , and r u n s t h r o u g h a l l f o r m u l a s o f c l a s s Zk. P r o o f . L e t u s f i x t h e number k and t r y t o p r o v e t h e schema
*.
The number P w i l l r e s u l t from t h e p r o o f . Hence, assume t h a t t h e
second p a r t of t h e d i s j u n c t i o n i n ( v x ) I ( V y ) [ ( $ ( x , y )* O r d ( y ) )
P
P j, ( x , z ) ) ) l * ( 3 y , z ) [ j , -cp(y,z)l Let
K
*
does not hold, i.e. &
(ord y * (3z)(y E z
(X,Y)
& $
P
(x,z)
y E
&
& 2
&
1.
b e a n o r d i n a l . Now w e s h a l l d e f i n e a n o b j e c t , c a l l e d
by E r d o s , H a j n a l and Rado “ t h e r a m i f i c a t i o n s y s t e m ” . We s h a l l use d e f i n a b l e ZFC c l a s s e s . = i l a , ~ }: a , @ E On & a
Let
L e t u s d e n o t e by I,J t h e p a r t i t i o n
On]
+
B}.
d e s c r i b e d by t h e f o r m u l a 2
B E On & c o ( a , i 3 ) 1 , J = [ O n ] \ I . We s h a l l d e n o t e t h e e l e m e n t s o f t h e c l a s s Z G O n a by u . Now, by i n d u c t i o n w i t h r e s p e c t t o 0 < K , w e d e f i n e s e t s N C OnB s u c h rp,
I = {ia,Bl
:
a
E
8 -
that:
[i)
(VY
[.ii)
(vy
<
B) (Vu)[o B)[Lim(h)
NB
E &
+
(Va
O P Y E Nyl < A) (ola
E
Na)
* u
E N
and mappings S a n d F s u c h t h a t
[ i i i ) (Vy
<
K)
( V u E N Y ) [ S ( o )= F ( o ) U L J { S ( u ^ a )
where F ( o ) i s t h e u n i o n o f a l l maximal s u b s e t s x
XI
F ( o ) i s not defined.
I
: una E
( i ~ ) L i m ( 8 ) * (Vo E N B ) [ S ( u ) = a Q B S ( u n a ) rank such, t h a t [
Y
5 S(o)
N
Y+l
11
o f minimal
_C I , i f s u c h maximal s u b s e t s e x i s t , o t h e r w i s e
320
Z.
RATAJCZYK
Note, that the only mapping S satisfying conditions (iv) and (vi) is the mapping defined by the formula S(a) = I f 3E On : (Va E Dom
u) { B , o ( a ) } E J}
Substituting the above definition for S ( a ) we see that the conditions (i) + (vi) give us a definition by transfinite induction in set theory, provided we show the following: (Vg <
K)
* N
[ N B is defined
B
is a set].
We shall obtain this by proving, by simultaneous induction, that (Vg
<
K)
[ N
B
is a set
Vo E NB F ( a ) is defined].
&
Thus, we shall also show that induction defined by the conditions (i) + (vi) can be continued up to K. First, we prove, that (Va E N ) F ( o ) is defined. B This can be reduced to the proof of the formula: (Va E N B ) (3x)[ x
5
S(a)
&
[XI
5
I
&
x is the maximal set
possessing this property]. Assume the contrary, i.e. that there exists N
[Wiii
(VX)[X
5
S(o)
&
[XI2
I * (3yl(x
$
y
We define by induction the function G G(0) is any set x such that x
5
S(o)
&
[x]’
5 S(o) :
On
-+
5
such, that
&
[YI2
5
I)].
V such that
5 I.
From this definition we infer, that for every a, G(a) is the maximal 2 subset of U G(a) such that [G(a)] 5 I. From this and from (vii) it follows that (Valon( tp G ( a ) Assuming G = [GI2 C I.
a) . G(a) we obtain a proper class such that
32 1
MODELS O F ZFC-SET THEORY TO MODELS OF KM-THEORY
L e t ?! b e t h e s m a l l e s t P '
such t h a t t h e formula d e f i n i n g G
i s i n X i , . C l e a r l y , P d e p e n d s o n l y on k . So w e o b t a i n a c o n t r a d i c t i o n w i t h o u r i n i t i a l a s s u m p t i o n . Now w e s h a l l prove t h a t N
i s a s e t . The n o n - l i m i t case i s B t r i v i a l , w h e r e a s i n t h e case of l i r n ( 6 ) i t i s s u f f i c i e n t t o employ the fact that
W e h a v e , h e r e b y , shown t h a t f o r a l l B, moreover, t h a t ( v u ) [ u E corollary, we obtain
[uiii) U { F ( u )
-. u
N
U
B
%
=+
N
B
i s defined and,
F ( u ) i s a s e t ] . Hence, a s a
E B & N ~ } i s a set.
I n t h e s e q u e l w e s h a l l show t h a t t h e r e e x i s t s a u E OnK s u c h , that
(vg
<
upB
K)
E
NB
and
n S(uPg)
BCK
f @
L e t y b e a n o r d i n a l number n o t i n t h e s e t ( v i i i ) and l e t L
b e a maximal c h a i n i n t h e s e t { o E B v K N B : y E S ( u ) } , p a r t i a l l y o r d e r e d by i n c l u s i o n .
L e t u = U L. A s i s e a s i l y s e e n , Dom u E On, Dom u
end o u r r e a s o n i n g w e must show t h a t D o m converse, i.e. S(u^a)
,
t h a t Dom u = B
E K.
U
=
K.
K.
TO
L e t u s assume t h e
Since y E S(a)\F(U)
t h e r e e x i s t s a n a E F ( u ) such t h a t y E S ( O ^ a ) .
d i c t s t h e c h o i c e of L, s i n c e u"a
<
5 agF(u)
This contra-
i s t h e u p p e r bound o f e l e m e n t s o f
L not beloninq t o L.
T o c o n c l u d e t h e p r o o f , w e s h a l l show, t h a t c a r d ( R n q u ) =
and t h a t [ Rnq u]*
5 J,
K
w h e r e u i s t h e e l e m e n t whose e x i s t e n c e w e h a v e
j u s t p r o v e d . To d o t h i s , l e t u s n o t e t h a t from ( i i i ) a n d ( v ) w e g e t
[ix)
(VB
<
K)
( V u E NB) ( V y
<
B)[ u ( y )
K)
( V u E NB) ( V y
<
B ) ( V a E S ( o f y + l ) ) [ { c c , o ( y ) } E 51
a n d from ( v i )
(XI
(VB
<
E s(Uby)l,
It i s e a s i l y observed, t h a t u :
To show t h a t [ Rnq u ]
* 5J
1- 1
K~~~~
Rnq
(5.
l e t us assume t h a t a,B E Rng o , a # 6.
Hence, a = o(y), B = u ( 6 ) , w h e r e y,6
<
K
Thus, by ( i x ) , B = o ( 6 ) E S ( u P 6 )
( w e c a n assume t h a t y < 6). S ( u F y ) , w h i c h , by (x),
5
2 . RATAJCZYK
322
imp1ies that Remark
Note that in the proof, the fact that I , J is a partition of [On]2 was used in following cases: when defining the function G (it was significant for the class to be well ordered) and when choosing the element y (it was significant only that On be a proper class). Thus we obtain the following: Corollary. ZFC(R). If X is a well-orderable proper class (definable with a definable well-ordering) then for any definable partition: [XI2 = I+J there exists a definable proper class S such that [ S ] 5 I or V K (3X
2.
5
X) card(x)
K
&
XI
5X
5 J].
The perfect set theorem for the case cf
= w
The main goal of this section is the adaption of the construction of the Perfect set theorem to the case of cf = w . Here we restrict the class of Z 1 -predicates under consider1
ation to those characterizing the expandability to KM. A total open problem, not investigated here, is the generali1 zation of the Perfect Set Theorem to the case of arbitrary Elpredicates over models considered by Nyberg [ 6 1 . One of the essential properties of KM-expandability in the case of cf = w, which we intend to employ here, is the possibility of Characterizing it with the help of games expressed by recursive game formulas (see Barwise 1 1 1 , Ch. VI, 5 6 ) : a recursive closed game formulas is an infinite formula GZ q ( Z ) = Q ~ X ~ . . . Q , ~ ~ . . . ...iFwRi(~l,...,~ ) where each Qn is either 3 or V, and i + R. is n a recursive sequence of a formulas of a language K ww
.
Definition. The notion of a Kinning plan: assume that the recursive formula Q1x1...Q2~2...if?wRi(~1...~ n. game over model E .
describes a closed
st A subset C of n-&Mn is called a plan of the 1 player also called the 3-player if the following conditions are satisfied: ( 1 )
s E Z
&
lh(s) = k
&
Qk+l
=
V
=)
(Vx),(s"x
E
C)
323
MODELS OF ZFC-SET THEORY TO MODELS OF KM-THEORY s E Z & lh(s) = k & Qk+l = 3
(2)
* ( ~ x ) ~ ( s -Ex C)
A plan 2 of the first player is called a winning plan, for
him, if (VfIMu[ (Vn),(
fpn
E C
* M_ k i&Ri[
xl/f(1),
. . . ,xi/f(ni)1 1
From the proof of the Gale Stewart theorem it follows that if player (3) possesses a winning strategy ( i.e. the 2nd player (V) does not have a winning strategy) then Z = P, U { ( al,
...
... ran)
E k
:
Qn+lXn+ l...i~,Ri[al/Xl,...,an/xn,Xn+l,.'.'Xn
1 rn
i
E
u}
is a maximal winning plan of player (3). Moschovakis proves in [5] that if E is acceptable, then C (or, more precisely, the set of codes of the sequences belonging to is a coinductive set, For our needs it will be more convenient to define the set n Z: using approximation formulas u a , and the following theorem proved
Z)
in Barwise [ 11 , Ch. VI, 56: there exists a ATF Inf-operation u : wxOn
(I)
+
K
ww
such that
for every n and p
E w,
u
E
Kww,and if M is a structure for
K, then
C
(2)
Qn+lXn+ l...i ~wRi[al/X1 ,..',an/xnrxn+l,...,X
n .I
n
*EkCua (3
1
b 0 (HYPM)
KM
-
*
&
E b
Ct
un[ al/xl,.
. . ,an/xnl
?jt= Qn+lxn+l.. 'iEo MR.[al/" 1 l,...,an/xn,xn+l,...,~n,l 1
Now the maximal winning plan of the first player ( 3 ) can be defined as follows: n Z = {(al, an) : k MMua[al/xl ,an/xnl ,n E wl For
CY
<
..
K?
z
...,
,...
CY
let us denote
= {(al,.
.. ,an)
:
uz[al/xl,.
. . ,an/xn] ,n E
w)
Clearly, Z = a 2 K Z a . In the sequel, the investigation of plan Za which, as it should be noted, are elements of H Y P M (if K~ > 0 ) . -
C can be reduced to the investigations of sets
324
Z.
Lemma 2.1. Assume, that
(i)
RATAJCZYK
E
is a model for ZF.
(VF)IE = (F,M,E*) k KM+VXr.a(X) * (Def(E) ,M,E*) k KM],
c
where Def(E) denotes the family of all classes of the model definable by formulas of the language IKM with parameters from M. in the (ii) there exists a recursive closed game formula GG language containing only the symbols from LZF such that (a] is KM expandable =* C G; cp(G). M = w , then M_ has an expansion If g i= GG lo($) and cf(0n-) , and KM+VX r.a(X) F satisfying I = Def (I)
~(z)
.
Proof. In [8] we find the proof of the following theorem: 1 KMn+VXr.a(X) I- The Basis Theorem for Zn-formulas.
En
k KMn+VXr.a X we can express this as follows: In any model the family of all classes, A 1-definable by formulas with parameters 1 from M, denoted by Def"(L"), 9 s a base in En for Zn-formulea without class parameters. If (F,M,E*) b KM+ Xr.a(X), then by the reflection principle, see [ 4 1 , there exists a tower
Fn -
F1 + 1 f 2 -< 2.. 1 . such that n&Fn i= KMn+VXr.a(X).
_C
F and for every n
E w,
It is easy to see that Defn(E) = Defn(fn). Hence Def(i) = Defn(E) is a base for formulas without class parameters, which implies that Def(E)4 f. This concludes the proof of (i). To prove (ii), we note first that for every 1= (F,M,E*) I= l= KM+VXr.a(X), Defn(E) 4 E; This follows easily from the basis theorem. Moreover, let us note that for 2 < n E w, Defn(E) F b KMn-2+VXr.a(X) and that Def"(f) b (VX) (3x)(3y) (Vz)[z E X * tpn(x,z) * -tpn(y,z)], where (p" is a universal formula for ZA-formulas without class parameters (in short, Defn(E) I= A:-Def) Arguing as in the proof of lemma 1.1, [ 7 1 , we find a direct system qi: (Fi,Ri,E*) 4 1 (Fi+l,Ri+l,E*), i E w,
A
.
the elements of which are of the form aE = Ib
E
M
:
bEa), where
325
MODELS OF ZFC-SET THEORY TO MODELS OF KM-THEORY
M M a E M. In particular, 3 a . E On- (Ri = Ra,-). If cf E = w, it can be chosen so that its limit is a certainlexpansion of E to a model of KM+VXr.a(X) It differs from the system of lemma 1.1 [ 7 1 only in that 1 ( F . ,R.,E* ) = KMi-2+VXr.a ( X )+Ai-Def. 1 1 Moreover, we can assume, that qi"(Fi) = {X E Fi+l : X is AS-definable in ( Fi+l ,Ri+l,E*) by formulae with parameters belonging
.
to Ri} for i 6 w. Thus consider the following game between ( 3 ) and (V) : the nth move of (V) is an ordinal an of M; the nth move of ( 3 ) is a set of M of M gn - (hn)E; then the form (Fn,Bn,hn-l). Let Fn = (FnIE, Rn = (RB;)E, is a model of ( 3 ) wins if for all n E LO, an G En, ( F ,R ,E*)
.
n n < 1 ( Fn+l ,Rn+l,E*) KMn-2+VXr.a (X)+An-Def, 1 and qn : ( Fn,Rn,E*) Clearly this game is expressible by a recursive closed game formula
G z (o(f;) on M; and the above remarks imply that if E is expandable to a model of KM, then ( 3 ) has a winning strategy. Moreover, if cf(0 -)M = w and (V) chooses his sequence of moves, an, to be cofinal nM in On-, then the limit of the direct system constructed by (3) applying a winning strategy is an expansion of $ satisfying F = Def(E) and KM+VXr.a(X). This shows (ii) of the Lemma (comp. proof of Th. 2.1 in [ 7 ] ) . Theorem 2.2. If E = (M,E) is a KM-expandable model for ZFC M such that cf On- = u , then the family
{F has power
: (F,M,E*)
>
card(M)
Proof. Let
G2
XM+VXr.a(X), card(F)
=
card(M) 1
. Q(;)
be the recursive closed game formula of
the above lemma. in the game . In view of the assumption, M b G; ~ ( 2 )Thus, defined by G;)x ~ ( 2 the ) first player possesses a winning strategy. Let C be the maximal winning plan of the first player. To estimate the number of expansions of model M to models for
KM+VXr.a(X), we shall in the first place attempt to estimate the number of possible moves of the first player, consistent with the plan and defining different expansions.
....
Assume, that s E 2 , s = (al,F1,B1;a2,F2,hl,B2;
...;an,Fn,hn-l,Bn), where a1,Fl;a2,F2,hl;...an,Fn,hn-l are moves of tne first player, which constitute the direct system:
326
Z.
RATAJCZYK
which determine the level which has to be passed by the first player. M Let 6 < K- be a fixed ordinal. Let us consider two cases: Case 1. M_ is an w-model, so M_ is acceptable. Hence HYPM k KPU' + Infinity. In view of the definition of 6 (xl,. #Xn)In€ approximation formula, the sequence of formulas {on belongs to HYP thus Z6 as the union U on(M) belongs to HYPM. E' nEw 6 As is well-known, HYPM (under the above assumption) is-closed
..
with rF w c t to the operation Def. Hence Def (M;X6) = dfngwDef (M; Zg Mn) E HYPM. Let H be an element of HYPM n P(M)\Def(M_;X6) - the existence-of such an H follows from the fact that HYPM b " P ( M ) does not exist", as is well-known from the projectibility of HYP and the diagonal argument. From Moschovakis ! 1 [ 5 ] ch. 7 theorem 7.F.2 it follows that there exist a Z1-formula Q ( in the language LKM({a : a E MI) defining the above set H in any 1 1 model for A 1-comprehension (Al-CA) extending the model (M;a)aEM. Let al,..., am be the sequence of all constant occuring in the formula 0. M M M L e t y = max-( rank-(al), . ,rank-(am)). M Let R = { { ( a , f 3 ) , ( a 1 , R ' ) } - : (3F,h,Fl,hl)
..
[sm(a,F,h) E Z6
0'
e
E
sn(a',Fl,hl) E Z 6
&
M_ b
(39)( g : (F,Rfl,€)< ( a ' , B ' ) -Ml 1.
= osp(F1) &
(a,B)M
f
(6
=
osp(F)
E
(F1,Ral,€)E gFR, = id))
Note, that if S~(U,+~ ,Fn+l,hn) E Z 6 , then from the definition of approximation formula (see Barwise 1 1 1 , ch. VI 9 6 ) we get ~ ' - ( a , + ~ , F ~ + ~ ,Eh ~2:)0 and this means that M i hn : (Fn,Ra , € ) 4 1 (Fn+l,Rfln+l,€)and M_
((Fn,Ru , € ) n
n
=
VXr.a(X)+nn-Def). 1
Moreover we may assume, (cf. Remark in the proof of lemma 1 2.1) that g k [h;(Fn) = {X E Fn+l : X is An-definable in (Fn+l,Ran+l,€) by formulas with parameters from Ra 1 1 . n Thus, hn is uniquely determined by an,Fn,an+l,Fn+l. Indeed,
327
MODELS OF ZFC-SET THEORY TO MODELS OF KM-THEORY
if cpl(X),cp2(X) is a pair consisting of a C1 n and rIA formula, ,then the same pair defines respectively and defines h'(X) in F n n+l X in (Fn,Ran,E),which implies that the Cn-formula (Vx)(x€X @,(x)) E (Vx)(cp2(x) x E X) is satisfied in (Fn,Ran,€). So, by X 1-elementary of hn,it is satisfied also in (Fn+l ,€). This consequently means that hn (X) = ql((F n+l 1 ' ) ) = = h;l(X) i.e. hn is uniquely determined. Finally, let us note that Fn+l is uniquely determined by an+l and ~ s p ( F ~ +-this ~) is a consequence of the fact that Fn+, may be assumed to be standard and that ramified analysis is absolute with respect to B-models (see [ 41 ) M Let X = Dom-R. Obviously X E HYPM. Moreover, X is an unbounded subset of (On21-M in the sense o r ordering: M M M M , since it is a superset of the ( a , B ) - < ( a l , ~ l ) -* a ( - a l & (-Bl) set of all winning moves of the first player. Since R and X are typerelementary sets, they belong to every model for KM expanding model M, which implies (M;R,X) ZFC(R,?). From the corollary of Theorem 1.2, by the fact that X is unbounded in (On2)-,M we obtain: there exists a formula cp E -CzF(E,?) such, that cp((M;R,X)) is an unbounded subset of X and (VaIOn(3x)M [ xE -C X E ( 3 C card(x) a ) E ( V Y , Z ) ~(y#z* sR(y,z))l v E v cp((M;R,X)) 5 X E (Vy,z)[y,z E cp((M;R,X)) E y # z * R(y,z)].
.
Now we shall show that the second component of the above disjunction is not true. Suppose the contrary, i.e. that
(Vy,z)[y,z E cp((M;R,X)) E y < z +.R(y,z)l. Assume that M ( a , @ - E cp((M;R,X)), and let F be an element such that M = B!. Let ( a , f 3 )M- , ( a l , B 1 ) -M, ( a 2 , B 2 ) - M, . . (3h)(sn(a,F,h) E C E osp-(F) 6
..
be a fixed sequence of elements of cp((M;R,X)), unbounded and M increasing for < in (On2 )-. From the definition of R we infer that there exists a direct system : gi : ( F . ,R.,E*) < 1 ( Fi+l,Ri+l,E*); i E w 1
1
M (R,-)E, girRi = id Ri M and for every i E u , Ri = (R ) ai E' The direct limit of the above system is a relational system of the form (F,M,E*). By gim let us denote the natural monomorphism from the i-th model Fi = ( Fi,Ri,E*) into = ( F,M,E*) Obviously such that F1
=
(FIE, R1
=
.
328
Z.
RATAJCZYK
1 qim is a Z1-elementary monomorphis. Now we shall prove that F c A~-CA. 1 1 Let then cp(x,A) be a Z1-formula, $(x,A)- a nl-formula, respectively, where A is a parameter belonging to F. I= (vx)cp[ (x,A) * $(x,A)1 and that Assume also that A = g. (B), where io E o , B E F . . 101 10 The property of being Z -elementary extends to the family of 1 boolean combinations of l Z1-formulas. Moreover, adding the universal quantifier does not lead beyond the class of formulas invariant with respect to the passing to a fixed submodel. Hence
( F . ,R. ,E*) C (Vx)[q~(x,B) $(x,B)]
lo
lo
Thus, also the formula (3X)(Vx)[x E X * cp(x,B)] is satisfied in ( Fi ,Ri ,E*). 0
0
Since g. is a SL-elementary monomorphism, the same formula 10" 1 1 will be satisfied in (F,M,E*) and this means that k Al-CA. M Summing up, for every ( a ,4) - E c p ( ( PJ;R;X)) and for every F M = 6) there exists an such that (3h)(sn (a,F,h) E Z6 & osp-(F) extension (F,M,E*) of M which is a model for A 1l - c ~and such that M M (v$ E 2 ; ) (Val,..., as) (R~-)~((F)~,(R~)-)~,E*) c
9 (al/xl,. . . ,as/xs) * ( F,M,E*) C
@ (al/xl,.. . ,as/xs)
.
1
Since the set H is definable in every model for Al-CA (extending PJ) by a C 1-formula @(x) with parameters al,. . . ,ak beM 1 longing to (R -IE, the following holds for every a > y (a as above) : Y M H n (Ry-)E = {a E M : ( F,M,E*) I= $(a/x,al/xl,. ,a,/xk) 1 =
..
s"
(a,F,h)EZg &osp(F) = 4
. . . ,xn/an)1 j.
&
(F,Ra,€)
@(x/a,xl/al,...
Since X,R E Def(M,C6), then also H E Def(M_,Z6) which contradicts the choice of H. Thus, we have proved the second component of the disjunction
THEORY TO MODELS OF KM-THEORY
MODELS OF ZFC-SET
329
to be false. The first component, on the other hand, can be interpreted as follows: (!;a)aEM ( ~ c i ) ~ ~ ( 3 x On2 ) I2 a & (vB,B',Y,Y') [ ( B , B ' ) ,(y,y') E x (3F,h,Fl,hl)[s" (B,F,h) E
+
C6
& s"
(y,Fl,hl) E C 6 1
osp(F) = 0 '
&
osp(F1)
&
=
y'
&
idR 1 1 1 1 B where Z6 should be replaced by an appropriate approximation formula. Let us denote the formula appearing in the curly brackets ?.(3g)[ 9 : (F,R ,€)
B
(F1,R
Y
,d
&
glR
B
=
by cp6(a,x) = cp(a,x,Z6). By Barwise's theorem in [ 11 quoted before 2.1, the mapping 6 + cp6(a,x) is C1-definable over HYPM. M Let a E Onbe a fixed ordinal in E. Hence
1 (3x) 5 On2 c p 6 ( a , x ) .
(v6)
We shall show that the quantifiers V6,3x can be interchanged. Let us assume the contrary, i.e. (VX)~(PJb x _C On2
(36)
K !
(M;daEM -
k
%cp6(a,x))
Using Z -collection in HYP we obtain that there exists an ' M 3' ordinal number T < K- such that (vx)M [ -M i= x
5 On2
=+
(36)
On the other hand, since the formula defining t h e set C6 is positively included in the fomula cp 6'
( ~ 6 , 6 ~ ) [< 6 S1 (Val (VX) and by ( ! ; a ) (!!;a)aEM
-
On 2
(cp6
b (3x)
'
k
=+
(3x)
C
-
1
(a,x)
* cp6 (a,x))l
On2 cp, (a,x) we have: On 2 6<,
cp6(Cr,X).
This contradicts the previous conclusion. Thus, the possibility of interchanging the quantifiers has been shown (note that the above "trick" is closely connected with the notion of a recursive saturation appearing in Barwise [l]. By the same argument, the quantifier (V6)
can be inter-
changed with the remaining existential quantifiers,
hence, also
2 . RATAJCZYK
330
with (3F,H,Fl,hl). Thus, we finally obtain the following property: M There exists a set x of E such that Card-(x) > ci and such M E On--;F,h,Fl,hl E M if ( B , B ' ) E , ( y , y ' ) f i E x that for all B , y , B ' , y ' E' = y' and f ( y , y ' ) E and osp-(F) M = B ' , osp-(F1) M ( BtB sr. (B,F,h) E Z, sn (y,Fl,hl) E 2 then there does not exist 1 M ( (F1)E,(RE) ,E*) such that Y E g : ( (F)E,(Rz)E,E*) M qr(R~) = ~id.
Let us assume that s1 = sn (B,F,h) E C, B < y s 2 = s" (y,Fl,h ) E E . Moreover, assume that there exist complete 1 games (compatible with plan C) extending respectively s1 and s 2 , such that the models for KM determined by those games are identical and equal to = ( F,M,E*). Then there exists q :
F1 =
(
(F),,(RF)~,E*) M
(
,E (F1)E,(R-) M Y E
)
=
F
-2.
To show this, l e t us observe that by the assumption and = Def(c), is a model for KM and there exist lemma 2 . 1 ,
such, that
F by formulas with
-
M 1 , hi(F2) = {X E F : X is An+l1 parameters belonging to (R-) B E definable in by formulas with parameters belonging to M (R-) 1 . Hence, by E b 6 < y , hi(F1) 5 hi(F2). Y E Let h = hi' o hl. From the Znil-elementarity of hl and h2 it follows that h is 1 M a Cn+l-elementary monomorphism. Of course, also h I\(Rg)E = id M ( RB) holds. In view of the above fact, the previously obtained property can be formulated as follows: M For any s E Z and for any c1 E On- there exists a set x in M_ (one such set let us denote by m(s,a)) such that the elements of set xE are the moves of the first player ii) extending the game s and compatible with plan Z. M (4i) card-(x) >a
MODELS OF ZFC-SET
THEORY TO MODELS OF KM-THEORY
33 1
M M ( i i i ) for any (B,F,h)-, (B1,Fl,hl)- E xE the models for KM determined by any two games (compatible with plan Z ) extending respectively sr'
(
B,Fl,h) , s r .
(
B1,Fl,h1 ) are different.
To end the proof of the theorem for this case, we choose a sequence c1 l , a 2 , . . . of ordinals from E cofinal with the height of E . Let T _C ngwMn be the least transitive relation generated by the relation:
x
s E
&
t E Z
&
(3x)Ix E m(s,aph(s))E
&
t = scIxna Ph ( s ) 1
Evidently, T is a tree of height w . Every maximal branch of this tree is a game consistent with the winning strategy of the first player and determining, in effect, a certain extension of the model E to a model for KM of the form ( F , M , E * ) . is easily noted, different branches determine different extensions. Hence the power of the set of extensions of a model M to models of KM+V::r.a(X) is greater or equal to the power of the set of branches of the tree T i.e. it is >nzw(an)E. - $0 Thus it suffices to show, that n:w(an)s==== E M If there exists an [Y E On- such that = g, then, r. As
obviously, n&(q)E not exist. We have
=
(z)"
= (
So, let us assume that such an Ho = Z N 0
ngw(&)E)
CL
does
_.
Case 2 . E is not an w-model. Since is KM-expandable, there exists a class of satisfaction for M and all the formulas belonging to E which can be used as a parameter in the induction scheme. Hence follows that E is recursively saturated. Thus (see M [ 11)
K-
=
W.
Putting
M R = [(a,@)-
:
(3F,h,Flrhl)[s^(a,F,h) E
X = Dom R , where n
[ xE
5X
&
card-(x) M
xn
&
332
2. RATAJCZYK
Indeed, by Th. 1.1, the negation of (ii) would mean that M there exists an unbounded and definable subclass cp(E) of On-, such that (Va,8) cp(E)
E C (a E B
=)
R a d RB)
M When summing all the classes of satisfaction for (R-) a E with respect to all a E cp(M), we would obtain a definable class of satisfaction for model 5, which contradicts Tarski's theorem on the non-definability of the notion of truth. By the same argument as in the previous case we show that in (ii) the subformula xE _C X can be replaced by (VB)x
E
(3F,h)( s c (B,F,h) E
z).
M Let m(s,a) be such a set x of E that card (x)
>
a and
(VB
and
If B,y E m(s,aIE,8 f y and ( F1,M,E*) ,( F2,M,E*) are two expansions determined by complete games (compatible with plan 2 ) extending respectively s - (B,F,h) , s r ' (y,Fl,hl), then these expansions have different classes of satisfaction for Y. To prove this, let S1,S2 be classes of satisfaction for 5 in these expansions. Assume that S1 = S 2 and let 6 < y . Hence if uE(R o )-,M B c p [ u l ) iff ( c p , u ) E S 1 iff ( c p , u ) E S 2 iff then M_ I= ( R B R + Ry, which contradicts M C ( R y k cp[ w ] ) . This proves that M our assumption. Constructing an identical tree as in Case 1 we obtain the required mlower" estimation of the number of extensions to models of KM+VXr.a(X). Corollary. If E is a KM-expandable model for ZFC such that M cf On- = o , then the power of the set of expansions of this model to models = D e f (f) and KM+ Xr.a(X) is equal to fib satisfying (in particular, for a standard model of the form R with cf a = L O , this power flKo equals to 2').
333
MODELS OF ZFC-SET THEORY TO MODELS OF KM-THEORY
kNo<
P r o o f . From t h e p r o o f o f t h e p r e v i o u s t h e o r e m w e i n f e r t h a t t h e power of t h e f a m i l y o f e x p a n s i o n s F o f models
fying F = Def
(r) a n d
KM+VXr.a ( X )
.
5
satis-
U F where t h e F n ' s nEw n ' a r e c o d a b l e i n F , as f o l l o w s f r o m t h e p r o o f o f lemma 2 . 1 . Using M t h e f a c t t h a t c f On- = o w e c a n r e p r e s e n t F a s a c o u n t a b l e sum o f
b e a member o f t h i s f a m i l y :
Let
s u b s e t s of F ,
codable i n
p r o o f o f lemma 2 . 1
F
=
f by classes w i t h a s e t domain. From t h e
(see a l s o t h e p r o o f o f lemma 1 , [ 7 ] ) , w e d e d u c e
t h a t t h e r e e x i s t s a d i r e c t s y s t e m o f power N O , w i t h e l e m e n t s b e l o n g i n g t o M and t h e l i m i t b e i n g e q u a l t o
E. Thus
we obtain t h e
r e q u i r e d e s t i m a t i o n o f t h e power.
REFERENCES Barwis
,
K.J.,
A d m i s s i b l e se s and s t r u c t u r e s , S p r i n g e r ( 1 9 7
and K e i s l e r , H . J . ,
Chang, C . C .
T h e o r y o f Models, North-
Holland (1973) ErdBs, P . ,
Hajnal, A.,
Rado, R . ,
Partition relation for
o r d i n a l numbers, A c t a M a t h e m a t i c a e X V I / 1 - 2 Marek, W . ,
Mostowski, A . ,
On e x t e n d a b i l i t y o f models of ZF-set
t h e o r y t o models o f KM t h e o r y o f c l a s s e s , S p r i n g e r L.N.
5 7 3 (1976)
Moschovakis, Y . ,
I n d u c t i v e d e f i n i t i o n s on a b s t r a c t s t r u c t u r e s ,
S t u d i e s i n L o g i c , North-Holland Nyberg, A . M . ,
(1974)
A p p l i c a t i o n s o f model t h e o r y t o r e c u r s i o n t h e o r y
on s t r u c t u r e s o f s t r o n g c o f i n a l i t y w , I n s t . o f Math. Univ. Ratajczyk, Z . ,
Preprint Series,
of O s l o .
A c h a r a c t e r i z a t i o n o f e x p a n d a b i l i t y o f models
of models f o r ZF t o models f o r K M ,
t o a p p e a r i n Fund.
Math. Ratajczyk, Z . ,
On s e n t e n c e s p r o v a b l e i n i m p r e d i c a t i v e
e x t e n s i o n s of t h e o r i e s , t o a p p e a r i n D i s s . Math.
LOGIC COLLOQUIUM 78 M. Boffa, D. van Dalen, K . McAloon leak.) 0 North-Holland Publishing Company, 1979
A FINE STFIJCTUFE GENERATED BY PEFLECTION FORNULAS OVER P R I L I T I V E RECURSIVE ARITHMETIC
Ulf R . Schmerl M a t h e m a t i s c h e s I n s t i t u t d e r V n i v e r s i t a t Yiinchen
I n t h e f o l l o w i n g we s h a l l d e s c r i b e a f i n e s t r u c t u r e on h i e r a r c h i e s generated by r e f l e c t i o n formulas over p r i m i t i v e r e c u r s i v e a r i t h m e t i c ( P R A ) and some o f i t s e x t e n s i o n s .
L e t T be some r e c . enum. e x t e n s i o n of PRA and CJTI
a formula
e x p r e s s i n g t h e s e n t e n c e " e v e r y ni+l-formula p r o v a b l e i n T i s t r u e " . Then f o r e a c h n a t u r a l number n h i e r a r c h i e s o f t h e o r i e s (:) defined by
c a n be
where t h e o r d i n a l s a r e s u p p o s e d t o be e l e m e n t s of a w e l l o r d e r definable by a p.r. order predicate <.
The main r e s u l t o f t h i s p a p e r , t h e f i n e s t r u c t u r e t h e o r e m on t h e h i e r a r c h i e s ( n ) , r e a d s : For a l l n a t u r a l numbers n , p and a l l
The r e s u l t s r e p o r t e d h e r e a r e p a r t o f t h e a u t h o r ' s d o c t o r a l d i s s e r t a t i o n , H e i d e l b e r g 1978. Thanks a r e due t o J . - Y .
GIRARD
( H e i d e l b e r g ) , H.SCHVICHTENBEPG (Miinchen), and C.SMORYNSK1 ( C h i c a g o ) . ( P a r i s ) , O.H.MuLLER
335
336
U.R. SCHMERL
ordinals a, P
w h e r e -n means p r o v a b i l i t y o f t h e same n z + l - f o r m u l a s .
-
These
r e l a t i o n s h a d b e e n c o n j e c t u r e d by GIRARD.
The c o m p a r i s o n o f t h e h i e r a r c h i e s
(E)
or t h e i t e r a t e d r e f l e c t i o n
f o r m u l a s i n v c l v e d w i t h o t h e r p r o o f t h e o r e t i c a l c o n c e p t s and t h e a p p l i c a t i o n of t h e f i n e s t r u c t u r e theorem y i e l d a l o t of r e s u l t s ; s o r e o f t h e m ar e new, o t h e r s , a l r e a d y known, a r e much e a s i e r t o o b t a i n by t h e f i n e s t r u c t u r e t h a n b y t h e i r o r i g i n a l p r o o f s . I t t u r n s o u t , t h a t t h i s method i s a u s e f u l t o o l f o r d e a l i n g w i t h p r o b l e m s c o n c e r n i n g c o n s i s t e n c y and r e f l e c t i o n p r i n c i p l e s ,
transfinite
i n d u c t i o n , p r o o f s o f r e s t r i c t e d c o n p l e x i t y , a n d p e r h a p s some o t h e r concepts.
Although i n t h e s e q u e l o n l y systems b a s e d on c l a s s i c a l l o g i c are c o n s l f l z r e d , i t seems e v i d e n t t h a t t h e f i n e s t r u c t u r e r e l a t i o n s a p p l y t o i n t u i t i o n i s t i c systems too. F o r t h e s a k e o f s h o r t n e s s we s h a l l n o t g i v e c o m p l e t e p r o o f s ; some
a r e s k e t c h e d , n o s t o f t h e m h a v e b e e n d r o p p e d a n d a r e somewhat t e c h n i c a l anyway.
Let < be a p.r.
o r d e r p r e d i c a t e d e s c r i b i n g some t o t a l we11 o r d e r o n
t h e n a t u r a l numbers ( i n f a c t , u s e d ) . For t h i s o r d e r , p . r . t o <),
+,
* ,
expo,
E
o r d e r p r o p e r t i e s w i l l n o t even be
functions l i k e
(successor with respect
e t c . s h o u l d b e d e f i n e d and t h e i r u s u a l p r o p e r -
t i e s s h o u l d b e p r o v a b l e i n PRA. I n a d d i t i o n , l e t { } b e a t w o - p l a c e d p . r . f u n c t i o n which a s s i g n s t o each non-zero o r d i n a l u a c e r t a i n s e r i e s ({cx}n)n<w o f o r d i n a l s c o n v e r g i n g t o u i f D, i s a l i m i t o r d i n a l o r t h e c o n s t a n t series { u l n = p i f a i s t h e s u c c e s s o r o f B.
E x t e n s i v e u s e w i l l b e made o f t h e f o l l o w i n g n o t i o n c h a r a c t e r i z i n g p r o v a b i l i t y i n t h e formal systems we c o n s i d e r h e r e : L e t T be a r . e . e x t e n s i o n o f PRA a n d l e t P r , ( x )
b e a c a n o n i c a l (I;-) p r o v a b i l i t y
p r e d i c a t e f o r T . T h e n a f o r m u l a ~ ( x i) s c a l l e d r e f l e x i v e l y p r o -
337
REFLECTION FORMULAS OVER PRIMITIVE RECURSIVE ARITHMETIC
gressive (in x) with respect to T if Vxlvy<xPrT(rcp(gF
+
v(x)I
is provable in T. This characterizes provability in T: Theorem (reflexive progressiveness and provability): P formula is provable in T iff it is reflexively progressive with
respect to T. Proof: One djrection is trivial; for the other, suppose cp(x) is reflexively progressive with respect to T. Then
T t-
PrT(rvy<xcp(y7 )
+
Vy<xPrT(rcp(g7 )
+
rz<x+vy
T I-
PrT(rvy<xcp(yT )
-
VY<XCD(Y).
By LBB's theorem [ 6 1 follows: T k Vy<xcp(y). But then PrT(rvy<xv(yT ), and hence Vy<xPrT(rv($r ) , is also provable in T and another application of the hypothesis shows the provability of ~ ( x )in T. - We owe this simple and short proof to a hint by GIRARD (it replaces a much longer one we had before).
We can now give an exact definition of the iterated reflection By formulas needed for the construction of the hierarchies KLEENE's Primitive Fecursion Theorem 141 a primitive recursive function Fn exists such that
(E):
Fn(0)
=rVx.x["n+l(~)~Pr(x) + Tn+l(~)I1
Fn(a+l) =rVxI"n+l(X)APr(Fn(a)ix) Fn(X)
=rVyTntl(Fn({A}9))1
-.
Tntl(x)
for limit ordinals X
(here Pr reans "provable in PRA", "n+l(x) is a p . r . predicate for Gijdel numbers of Fz+jformulas and Tn+l (x) is a partial truth definition for these formulas). Apparently Fn(a) is a Godel number of a certain formula Cn(u): For every ordinal a , let
U.R.
338
SCHMERL
The C n a r e i t e r a t e d r e f l e c t i o n f o r m u l a s h a v i n g t h e f o l l o w i n g b a s i c properties:
P r o o f : ( 1 ) b y showing t h e r e f l e x i v e p r o g r e s s i v e n e s s of t h e f o r m u l a ;
(4), ( 5 ) f o l l o w f r o m ( 1 ) and t h e d e f i (6) i s t r i v i a l .
( 2 ) f o l l o w s from ( 1 ) ; ( 3 ) ,
n i t i o n of the Cn;
Now let, f o r a l l n a t u r a l numbers n and o r d i n a l s order
(E)
PRA
and
(:)
CY.
of t h e pre-fixed
= PRA U{Cn(f3): p < a }
.
For a b b r e v i a t i o n we n o t e
The f i n e s t r u c t u r e t h e o r e r ? s t a t e d i n t h e i n t r o d u c t o r y r e m a r k i s j u s t
a straightforvard generalization of the following Theorem ( f i n e s t r u c t u r e t h e o r e m ) : For a l l n and o r d j n a l s a , ii
("L1)
(1) (p:l)
(2)
(nil)
1'
In
=n
'$
)
(azn)
" )
(e+wcy
S k e t c h o f p r o o f : Apparently ( 2 ) i m p l i e s ( 1 ) . I n o r d e r t o prove ( 2 ) , two d i r e c t i o n s h a v e t o b e s h o w :
( a ) For e v e r y V~,l-formula (&)
c-
cp
we h a v e : I f
(p:l)U(nil)
rg
,
then
339
REFLECTION FORMLJLAS OVER PRIMITIVE RECURSIVE ARITHMETIC
(b)
(Byl
For e v e r y y < p + o a
t Cn(Y)
)U("a+l)
.
To ( a ) : C o n s i d e r t h e f o l l o w i n g f o r p a l i z a t i o n o f ( a ) :
-
def
v y v x l ~ ~ , ~ ( x ) ~n P) r" ( n + l ) ( x ) -+ P r n ) ( x ) l ( @,(a) ) . ( Y t l a (Y+ w a Then ( a ) c a n b e p r o v e n b y s h o w i n g t h a t ( p n ( x ) i s r e f l e x i v e l y p r o 0 p r e s s i v e i n x w i t h r e s p e c t t o (1) ( t h i s w i l l b e e x p l a i n e d l a t e r ) ; c l e a r l y , i f w n ( a ) h o l d s f o r a l l a, t h e n ( a ) h o l d s .
0
Vy
t-
+
cpn(o
i s t r i v i a l l y t r u e ; even
0 (1) t-
Lim(A)hVy<XPr 0 ( r v n ( $ ) l ) (1)
i s o b v i o u s . The s u c c e s s o r c a s e would be d o n e , i f
c o u l d be shown f o r a r b i t r a r y a . We s h a l l g i v e a n i n f o r m a l p r o o f of 0 t h i s by p o i n t i n g o u t t h a t u n d e r t h e a s s u m p t i o n (1) t cpn(a) t h e f o l l o w i n g h o l d s f o r e v e r y o r d i n a l y a n d e v e r y no - f o r m u l a J I : n+l
So s u p p o s e
(yyl)u(::t)
f o r m a l PRA-deduction
.
t- JI Then t h e r e e x i s t s a G e n t z e n - t y p e (the induction rule is restricted t o Tlz-
f o r m u l a s ) w i t h an endsequent
' n + l (a), C n ( Y )
c
$
a n d f r o m which a l l c u t s more c o m p l i c a t e d t h a n
TlE
have been
e l i m i n a t e d . As i t i s e a s y t o s e e t h a t t h e r e a r e a n o - f o r m u l a 8 E ( x , y ) and a Io-formula x n ( x ) such t h a t Y n Cn(Y) vxx ( x ) , Cntl(a)
-
+-+
V x 3 y Bna ( x , y )
Y
and
VxPr n + l ( r 3 y O i ( k , y ) 1 )
( a ) a r e p r o v a b l e i n P R A , we may demand w i t h o u t l o s s o f g e n e r a l i t y t h a t a l l o c c u r r e n c e s o f Cn+l(a) and C n ( y ) i n t h e g i v e n d e d u c t i o n a r e i n t h e f o r m ' v x 3 y P ~ ( ( . x , y ) a n d V x x n ( x ) . kfe a r e now g o i n g t o p r o v e t h a t Y f o r e a c h s e q u e n t ? + A i n t h e d e d u c t i o n t h e r e i s a n a t u r a l number k such t h a t t h e sequent
C , ( Y + ~ ~ . ~ ) r* , k A
w h e r e T* i s o b t a i n e d f r o m
r
i s a l s o p r o v a b l e i n PRA,
by d e l e t i n g a l l o c c u r r e n c e s o f t h e
U.R. SCHMERL
340
f o r m u l a s V x x c ( x ) , Vx3yfJE(x,y), 3 y B z ( t , y ) ( t a n a r b i t r a r y t e r m ) . is a nxtl-formula: This i s t r u e first f o r Note t h a t t h e n /Mr* + t h e e n d s e q u e n t ; b u t s i n c e t h e d e d u c t i o n h a s no n o n - t r i v i a l c u t s , i t h o l d s f o r a l l s e q u e n t s i n i t . The p r o o f p r o c e e d s by a n i n d u c t i o n o n t h e height of t h e sequents r F A i n t h e given deduction.
vh.
-
S u p p o s e h ( r l - A ) = O . Then r t A i s a l o g i c a l a x i o m or a p r o p e r PRA-
axiom a n d , t r i v i a l l y ,
Cn(y),l'*l-A
i s p r o v a b l e i n PRA.
-
Suppose t h e a l l e g a t i o n proven f o r a l l s e a u e n t s of h e i g h t 5 p . I n o r d e r t o p r o v e i t f o r s e q u e n t s o f h e i g h t p t l , we d i s t i n g u i s h d i f f e r e n t c a s e s a c c o r d i n g t o t h e l a s t i n f e r e n c e r u l e . Checking a l l
p o s s i b l e i n f e r e n c e r u l e s r e v e a l s t h a t t h e r e i s only one n o n - t r i v i a l c a s e , v i z . when t h e l a s t i n f e r e n c e i s o f t h e f o r m Ai(t,y),
rt-
A
3 y B i ( t , y ) , I'C A
(height p) (height p t l )
with a f r e e variable y not occurring i n
r,
Let xi,
A.
..., x
be t h e
f r e e v a r i a b l e s o c c u r r i n g i n t , l', A . By t h e i n d u c t i o n h y p o t h e s i s , t h e r e e x i s t s k such t h a t cn(Y+w"-k), ( e i ( t , y ) ,
r)*
-I
A
i s p r o v a b l e i n PRA. A s @ " ( t , p ) i s n o t one o f t h e " p r o h i b i t e d " , i.e. the ( Q i ( t , y ) , r ) * ys identical with On(t,y),r*
formulas, sequent
C,(y+wa*k),
@E(t,y),
r*
i s p r o v a b l e i n PRA. An i n f e r e n c e " 3 - l e f t "
k A
gives
cn(YtwO.k), 3 y e i ( t ( x l . . . x ~ ) , Y ) ,r * ( x l . . .xr) and t h e r e f o r e
i s p r o v a b l e i n PRA t o o . W i t h
t h e f o r m u l a 3 y A t ( t ( f l...Pr),y)
.
F A ( x ~ . x?)
c a n b e c u t o f f a n d we g e t
REFLECTION FORMULAS OVER PRIMITIVE RECURSIVE ARITHMETIC
34 I
0 l l n t l ( r l - * k fl) i s p r o v a b l e i n PRA, s o t h e a s s u m p t i o n (1) tc a n be a p p l i e d ar.3 g i v e s
wn(a)
f r o m what t h e p r o v a b i l i t y o f
f o l l o w s and h e n c e
C (y+w'-(k+l)), n
l-*
t- A ; q . e . d .
The f o r m a l i z a t i o n o f t h i s p r o o f n e e d s a n i n d u c t i o n o n some IT:-formula T h i s i s t h e r e a s o n why t h e r e f l e x i v e p r o g r e s s i v e n e s s o f w n ( x ) h a s t o b e p r o v e n i n PRA+CO(0) ( a n d n o t i n PHA): I f f o r a n o - f o r m u l a $
$J.
1
+ ( @ ) A V x [ $ ( x ) + + ( S x ) l is p r o v a b l e i n PRA, t h e n
Vx$(x) i s p r o v a b l e i n
PRA+CO(0). The p r o o f o f d i r e c t i o n ( b ) i s much e a s i e r : I t s u f f i c e s t o show t h e r e f l e x i v e progressiveness o f t h e formula
cnt
(x)
-t
vz Icn ( z ) + C n ( z+wX) 1
i n x w i t h r e s p e c t t o PRA; t h i s g o e s s t r a i g h t f o r w a r d .
The u s e f u l n e s s o f i t e r a t e d r e f l e c t i o n f o r m u l a s and t h e f i n e s t r u c t u r e becomes more e v i d e n t b y c o m p a r i n p
t h e Cn t o o t h e r p r o o f t h e o r e t i c a l c o n c e p t s a n d by s h o w i n g some a p p l i c a t i o n s . Here some e x a m p l e s :
1. I n d u c t i o n a n d i t e r a t e d r e f l e c t i o n f o r m u l a s
D e f i n i t i o n : F o r e v e r y f o r m u l a ~ ( x l) e t I n d ( q ) b e t h e f o r m u l a P(O)Avx[P(x)-m(Sx)!
+
Vxcp(x)
and l e t
n-Ind be
{Ind(w):
~€n:+~].
P r o p o s i t i o n ( n z + l - i n d u c t i o n and t h e C n l : (1)
PRAtn-Ind
(2)
PRA+Cnt2(0) t-
I- C n ( 0 ) n-Ind
Proof: ( 1 ) C o n s i d e r t h e f o r m u l a
wn ( z ) ' a
Vx [ nnt
( x )A 3 YPrf* ( Y
,x ) Ah (Y
+
Tnt ( x ) l
I
U.R. SCHMERL
34 2
where P r f * ( y , x ) s t a n d s f o r " y i s a number o f a PRA-proof o f t h e f o r m u l a w i t h number x and t h i s p r o o f h a s n o c u t s more c o m p l i c a t e d t h a n n:'' and h i s a p . r . f u n c t i o n a s s i g n i n g t o e a c h number o f a p r o o f i t s h e i g h t . Then c l e a r l y C n ( 0 ) Vzcpn(z) i s p r o v a b l e i n PRA.
-
F u r t h e r m o r e , we have PRA IA s cpn(z)
PRA
t
(on(0)
and
i s a lT:+l-formula, I- VZQ,( z)
an a p p l i c a t i o n o f
.
n-Ind
PRA I- V Z [ Q ~ ( Z- )w n ( S z ) l
.
n no + 1- i n d u c t i o n g i v e s
V z P r ( r + ( O ) ~ V x [ $ ( x ) ~ $ ( S x ) l--. + ( i ) 7 ) i s
( 2 ) F o r e v e r y f o r m u l a $,
p r o v a b l e i n PRA. I f $ i s ft3 and t h e r e f o r e PRA t-
t h e n $ ( O ) ~ V x [ $ ( x ) - $ ( S x ) l + $(z) C n + 2 ( 0 ) -, I n d ( $ )
.
is
The above p r o p o s i t i o n e n a b l e s u s t o compare Peano a r i t h m e t i c PA w i t h t h e PRA h i e r a r c h i e s . We h a v e t h e f o l l o w i n g Theorem ( P e a n o a r i t h m e t i c and t h e PRA h i e r a r c h i e s ) :
For e a c h n a t u r a l number n (1)
PRA
t
n-In?
" )
n
(2)
(00
P r o o f : (1) I f PRAtn-Ind
t
,
Q
PA
-n
" ) .
(€0
t h e n by t h e above p r o p o s i t i o n
( n i ? ) I- cp. I f cp i s a n z t l - f o r m u l a , t h e n I- Q , i . e . two a p p l i c a t i o n s of t h e f i n e s t r u c t u r e t h e o r e m y i e l d ) t- cp
PRA + Cnt2(0)
(wc
.
On t h e o t h e r s i d e , we h a v e PRA + n - I n d t- C n ( 0 ) a n d , b y a n i n d u c t i o n on k , PRA t n - I n d I- V z ~ C n ( z ) + C n ( z + m k ) ] f, o r a l l f i n i t e k . Hence for a l l
y<ww
PRA + n - I n d
!-
Cn(y)
.
( 2 ) T h i s r e s u l t s from ( 1 ) and b y i t e r a t e d a p p l i c a t i o n s o f t h e f i n e
s t r u c t u r e theorem. Note t h a t t h e u n i f o r m r e f l e c t i o n schema o v e r PRA, i . e . t h e s e t R F N ( P R A ) = {VxPr('cp(~)l),Vxcp(x):
i s e q u i v a l e n t o v e r PRA t o t h e s e t with the proposition
,
KREISEL and LEVY ( 5 1 :
(0
formula with only x f r e e )
{ C n ( 0 ) : ncwl
.
This, together
g i v e s a very s h o r t proof of t h e r e s u l t by PRA
t
RFN(PRA)
c
PA.
REFLECTION FORMULAS OVER P R I M I T I V E RECURSIVE ARITHMETIC
34 3
2 . T r a n s f i n i t e i n d u c t i o n and i t e r a t e d r e f l e c t i o n f o r m u l a s
D e f i n i t i o n : U s i n g a p a r t i a l t r u t h d e f i n i t i o n , d e f i n e T I n ( a ) t o be a formula expressing t r a n s f i n i t e induction ( w i t h r e s p e c t t o t h e pref i x e d o r d e r ) on no - f o r m u l a s up t o t h e o r d i n a l a. L e t n+ 1
[TI]:
=
PRA U { T I n ( B ) : B
Note t h a t b e c a u s e of t h e p r o v a b i l i t y of TIn(")
-+
for all f i n i t e k
TIn(ak)
i n PRA, we h a v e f o r a l l a > w [TI]: and
=
[TIIa
( i f a c w , t h e n o f c o u r s e [TI]: designate the E-nUmberS.
[TI]" p) w2
f o r some o r d i n a l Ec
ITIIEB
f o r some E-number
E
[T l a
=
E~
P R A ) . So i t i s c o n v e n i e n t t o
[TI]'$ o n l y w i t h w2-numbers and t h e
[TIIa
only with
The f o l l o w i n g p r o p o s i t i o n d e m o n s t r a t e s t h e c l o s e r e l a t i o n s h i p between t r a n s f i n i t e i n d u c t i o n and i t e r a t e d r e f l e c t i o n f o r m u l a s . P r o p o s i t i o n ( T I n and C F A ) : The f o l l o w i n g i s p r o v a b l e i n PRA:
(1)
~ Z I A A T I+ ~ (C~n )( a )
For t h e p r o o f we need a lemma which c o n n e c t s p r o g r e s s i v e n e s s t o r e f l e x i v e progressiveness. A formula
Q(X)
with f r e e variable x i s
s a i d t o be p r o g r e s s i v e i n x i f V x [ V y < x ~ ( y )--. v ( x ) l ( = Progx'P) h o l d s . T h e r e a r e t h e f o l l o w i n g r e l a t i o n s b e t w e e n p r o g r e s s i v e n e s s and r e f l e x i v e p r o g r e s s i v e n e s s ( a n d hence p r o v a b i l i t y ) : Lemma: L e t ~ ( x be ) a n:+l-formula.
Then
( 1 ) If i t i s p r o v a b l e i n PRA t h a t Q i s p r o g r e s s i v e , t h e n C n ( x ) - ' d x ) i s r e f l e x i v e l y p r o g r e s s i v e w i t h r e s p e c t t o PRA.
U.R. SCHMERL
344
( 2 ) If i t i s p r o v a b l e i n PRA t C n ( a ) t h a t cp i s p r o g r e s s i v e , t h e n C n ( a t x ) + ~ ( x i) s r e f l e x i v e l y p r o g r e s s i v e w i t h r e s p e c t t o PRA.
Proof: obvious. Proof o f p r o p o s i t i o n :
( 1 ) U s i n g a n a n a l o g o u s argument a s i n t h e p r o -
p o s i t i o n on ll:tl-induction PRA I-
we show t h a t
and t h e C n , TIn(w)
PRA I- T I n ( w )
-D
Cn(0)
-O
Va[Cn(a)
+
Cn(af)l
.
Hence, w i t h r e g a r d t o Lim(X)hVy
Cnt2(a)
+
t
Cnt2(0)
TIn(w,(a)) f o r a l l a .
The p r o p o s i t i o n y i e l d s t h a t , c o n c e r n ng p r o v a b i l i t y o f V:+l-formulas, t h e t r a n s f i n i t e i n d u c t i o n t h e o r i e s [TI]: c a n be embedded i n t h e PRAh i e r a r c h i e s and t h u s t h e f i n e s t r u c t u r e r e l a t i o n s f o r t h e l a t t e r i n d u c e t h e same s t r u c t u r e on t h e [TI]:. We h a v e t h e f o l l o w i n g Theorem ( [TI]:
and t h e h i e r a r c h i e s :):(
L e t a be a n w2-number
d i f f e r e n t from w . Then f o r a l l n a t u r a l numbers n
(3)
[TIICY
pn
(
EY
)
(
E
a~n a r b i t r a r y E-nUmber).
P r o o f : (1) A s c o w we have f o r a l l y<m[TIl: s i d e , l e t cp be a n:+l-formula.
F C n ( y ) . On t h e o t h e r
We d i s t i n g u i s h two c a s e s :
( a ) a=w2(y+l). I f [TIl:t-cp, t h e n 3 k < w PRAI- T I , ( u ~ ( ~ )-.~ cp) and s o , by t h e above m e n t i o n e d r e m a r k , P R A I - TI,(w2(y))+tp; from t h e p r o p o s i t i o n f o l l o w s P R A F Cnt2(y)-.cp
, hence
(nt2) I- cp. Two
)=(.,) F cp. a p p l i c a t i o n s o f t h e f i n e s t r u c t u r e t h e o r e m g i v e ( y? w2 ( Y + l 1 ( b ) a = w , ( X ) , where h i s a l i m i t o r d i n a l . Then cp i m p l i e s 3y
345
REFLECTION FORMULAS OVER PRIMITIVE RECURSIVE ARITHMETIC ( 2 ) and ( 3 ) f o l l o w from ( 1 ) and t h e f i n e s t r u c t u r e .
The r e l a t i o n s b e t w e e n t r a n s f i n i t e i n d u c t i o n a n d t h e i t e r a t e d r e f l e c t i o n formulas t o g e t h e r with f i n e s t r u c t u r e p r o p e r t i e s allow a
v e r y s i m p l e p r o o f o f ' t h e well-known GENTZEN r e s u l t i n [11 , t h a t i s t h e a n s w e r t o t h e q u e s t i o n how f a r t r a n s f i n i t e i n d u c t i o n c a n be p r o v e n by a d e d u c t i o n o f bounded c o m p l e x i t y . By a T I - d e r i v a t i o n we s h a l l u n d e r s t a n d a Peano d e r i v a t i o n w i t h Vy
a d d i t i o n a l i n i t i a l s e q u e n t s o f t h e form
with an
a r b i t r a r y term t and whose e n d s e q u e n t i s o f t h e form I- @ ( a ) , a a n o r d i n a l o f t h e p r e - f i x e d o r d e r . The c o m p l e x i t y o f a T I - d e r i v a t i o n i s t h e least k such t h a t a l l formulas o c c u r r i n g i n it are llE+l. the
-
s l i g h t l y m o d i f i e d and g e n e r a l i z e d
-
Then
Gentzen r e s u l t r e a d s :
Theorem ( G e n t z e n ) : T r a n s f i n i t e i n d u c t i o n on a l l l l E + l - f o r m u l a s up t o t h e o r d i n a l a i s d e r i v a b l e by a T I - d e r i v a t i o n o f c o m p l e x i t y n i f f
a < 'nt3-k' Proof: F i r s t note t h a t t h e r e i s a T I - d e r i v a t i o n o f complexity n f o r a formula Q up t o t h e o r d i n a l a i f f PRA t n - I n d t- T I n ( v , a ) . So it s u f f i c e s t o show t h a t f o r e v e r y ( ~ E l l i + ~ PRA , + n-Ind
( a ) L e t n,k;
vo(x)
-
a s SCHOTTE i n [ 8 1 ,
v(x)
and
vmtl(x)
t- T I ( v , a )
iff
we define ++
Vz[Vy
and show t h a t
PRA + ( m t k ) - I n d l-
and t h e r e f o r e
PFA
t
( m + k ) - I n d I- TI(comtlya)
hence
PRA
t
( m + k ) - I n d I- T I ( v m t l , a )
Trivially,
4
VY
ProgxcPmtl
-. +
,
TI(vm,wa) TI(vywm+l(a))
.
f o r a l l m and f i n i t e p . Thus we have
TI(co,,p)
PRA I-
Progxvm
.*
PRA t ( m + k ) - I n d C T I ( q ~ , w ~ , , ( p ) ) w i t h p < w , o r P R A + ( m + k ) - I n d t - T I ( v , a )
f o r a l l a<wm+3. With m=n-k, t h i s g i v e s one h a l f o f t h e p r o o f . ( b ) VcpEni+l
PRA
t
n-Ind
I-
TI(cp,a)
only f o r a<wn+3-k:
Suppose :PRA t n - I n d t- T l ( c ~ , w ~ + ~ -f ~o r) a l l f o r m u l a s UJETTE+~; t h e n PRA t n - I n d I- C k ( w n + 3 - k ) . But we a l s o h a v e
U.R. SCHMERL
346
and so
(w
nt3-k
)
I-
Ck(wn+3-k). This is a contradiction!
Remark: In [ 7 1 WINTS observed that transfinite induction up to w n+3 is not only not provable by a derivation of complexity n, there exists even a quantifierfree formula provable in PRA t TI up to wnt3, but not provable by a deduction of complexity n. This result is quite easy to obtain by using iterated reflection formulas: - By the above proposition:
PRA + TIo(wnt3) k CO(wnt3)
- By the fine structure theorem:
i.e. PRA + n-Ind W CO(unt3). CO(wnt3) is a ny-formula; so by dropping the universal quantifier of its prenex form, one obtains a quantifierfree formula (with a free variable) provable in PRA t TIO(wnt3) but not provable by a derivation of complexity n.
3 . Fine structure relations for other theories than PRA It is of course quite natural to ask whether a fine structure like that over PRA does exist for other theories too. The definition of iterated reflection formulas CA(a) over a theory T only requires (the Gddel number of) a formula PrT(x) characterizing provability in T T. Once defined the Cn(a), hierarchies over T can be defined by
(t)T
As a first example we consider hierarchies over basic theories of we already defined Pr n (x). These will be noted the form .I:():( The corresponding reflection( aformulas ) Cma (")(8) are linked to
(t);
the ordinary Cn by the following
Lempa: The following formulas are reflexively progressive with respect to PRA: (1)
(PfO)
REFLECTION FORMULAS OVER PRIMITIVE RECURSIVE ARITHMETIC
34 7
From this lemma we easily infer the Fine structure theorem for mixed hierarchies: For all natural numbers n, p and ordinals a ,
,f2
A more serious example is Peano arithmetic. We already saw, that for ) . So we may use as a characterization of provabilall n PA = ( Eo ity in Peano arithmetic the following formula
where d is a p . r . function which assigns to each Godel number x of a formula (0 the least k such that cp is I I E + l . With PrFA(x) the CKA(a) and the hierarchies over PA can be defined. The proof of the following lemma is rather technical; so we omit it. Lemma: P'or all n the formulas CEA(x) ++ Cn(~o-(ltx)) are reflexively progressive with respect to PRA. The lemma shows that - concerning provahility of llg+l-formulas the PA-hierarchies are embeddable in the PRA-hierarchies. An immediate consequence is that the fine structure relations for the PRA-hierarchies can be transferred correspondingly to the PAhierarchies. Thus we get the Embedding and fine structure theorem for PA-hierarchies: F a r all natural numbers n and ordinals a
(Ed("
)
(embedding)
n )PA ( a f o ) (fine structure)
(E:
ntl
,
U.R. SCHMERL
348
As i n t h e c a s e o f PRA, t h e u n i f o r m r e f l e c t i o n schema f o r Peano a r i t h m e t i c , RFN(PA), i s e q u i v a l e n t ( o v e r P R A ) t o I C z A ( 0 ) : n<w} follows t h a t PA
t
RFN(PA)
PA t RFN(PA)
Since [ T I ] E ~ z [TI]so+l
1
n
=n
)
. It
and h e n c e
in
s
PRA + { T I n ( ~ o ) : n < o ) , we h a v e t h e KREISEL
and LEVY r e s u l t from 1 9 6 8 : Over PA, u n i f o r m r e f l e c t i o n i s e q u i v a l e n t t o t r a n s f i n i t e i n d u c t i o n up t o
E~
(51.
Embedding and f i n e s t r u c t u r e r e l a t i o n s s i m i l a r t o t h o s e o f Peano arithmetic hold f o r theories of type as b a s i c t h e o r i e s . We s t a t e t h e corresponding f a c t s without proof:
(z)
These t h e o r i e s I T I I E ~a r e t o some e x t e n t t h e most g e n e r a l example o f o v e r PRA: I f a t h e o r y T t h e o r i e s comparable t o t h e h i e r a r c h i e s i s f o r e a c h Tl-class e q u i v a l e n t t o some ( " ) - i . e . i f for a l l n 'n T E~ ( a : ) - t h e n t h e f i n e s t r u c t u r e t h e o r e m i m p l i e s t h a t t h e s e r i e s Therefore o f t h e a n i s c o n s t a n t l y 0 o r c o n s t a n t l y some E-number E T = PRA or Vn
T =n
(Ey
) and t h u s
T = [TI]€
Y'
Y'
A s a f i n a l example we c o n s i d e r t h e " f i l l e d up" t h e o r i e s T k + P R A , where T k i s t h e s e t o f a l l t r u e I I E t l - s e n t e n c e s . I t t u r n s o u t t h a t t h e f i n e s t r u c t u r e on t h e P R A - h i e r a r c h i e s i s i n v a r i a n t u n d e r s u c h a p a r t i a l f i l l i n g up w i t h t r u e s e n t e n c e s ; i n f a c t , t h e h i e r a r c h i e s e r e c t e d o v e r t h e f i l l e d up t h e o r y T k t P R A behave f o r a l l (:)Tk+PRA n > k a s i f t h i s h i e r a r c h y had f i r s t b e e n e r e c t e d o v e r PRA and f i l l e d up a f t e r w a r d s , i . e . we have f o r a l l . n > k
-
-
t h e f i n e s t r u c t u r e r e l a t i o n s f o r PRA s t i l l h o l d . The r e a s o n i s
that
C:ktPFA(x)
Cn(x)
i s p r o v a b l e i n P R A . S o Tk d o e s n o t r e a l l y
349
REFLECTION FORMULAS OVER PRIMITIVE RECURSIVE ARITHMETIC
s t r e n g t h e n PRA c o n c e r n i n g p r o v a b i l i t y o f l ? ~ t l - f o r m u l a s i f n > k . F o r i n s t a n c e , T +PRA h a s n o t e v e n t h e s t r e n g t h t o p r o v e C k t l ( 0 ) which is k o n l y one l l - c l a s s h i g h e r t h a n T k . A l t h o u g h TktPRA i s c o m p l e t e for t h e
it i s a l r e a d y incomplete i n t h e next
p r o v a b i l i t y o f llEti-formulas, upper ll-class. The p r o o f o f Cnk t P R A ( x )
* Cn(x)
i s t e c h n i c a l a n d l e n g t h y ; s o we
drop i t .
Of c o u r s e , e x a c t l y t h e same t h i n g s h a p p e n i f Peano a r i t h m e t i c o r a t h e o r y [ T I I E ~i s p a r t i a l l y f i l l e d up w i t h Tk. We o n l y h a v e t o r e p l a c e PRA by t h e c o r r e s p d n d i n g t h e o r y and w by c 0 o r
E
~
.
4 . k - C o n s i s t e n c y and i t e r a t e d r e f l e c t i o n f o r m u l a s The s e a r c h f o r r e l a t i o n s b e t w e e n t h e Cn and t h e n o t i o n o f k-cons i s t e n c y o r w - c o n s i s t e n c y o f Peano a r i t h m e t i c l e a d s t o a n a n s w e r t o a n open p r o b l e m which was announced by SMORYNSKI i n [ g ] . He a s k e d
for " s o m e t h i n g o f t h e form
~-CON~(PA)
++
k-CON ( P A )
or
f-t
. ..El.. . . , . E ~ .. .
* ...ak...
II
a n d he c o n t i n u e s " e x a c t l y what t h e d o t s r e p r e s e n t i s n o t c l e a r
-
p e r h a p s a m i x t u r e o f r e c u r s i o n and i n d u c t i o n p r i n c i p l e s . U n t i l s u c h equivalences a r e e s t a b l i s h e d , t h e s i t u a t i o n regarding k-consistency r e m a i n s open". U s i n g i t e r a t e d r e f l e c t i o n f o r m u l a s , t h e f o l l o w i n g a n s w e r c a n be g i v e n t o SMORYNSKI's p r o b l e m :
-
Theorem * ) : The f o l l o w i n g i s p r o v a b l e i n PRA w - ~ ~ ~ ++G c 2(( E ~1 ) ~ ) 1-CON
(PA)
2-CON
(PA)
~ - C O N ~ ( P R A )c 2 ( E O )
.
C1(~O)
1-CON
(PRA)
++
C 2 ( ~ O )
2-CON
(PRA) * C 2 ( 0 )
(k+3)-CON (PA) * C 2 ( w k ( ~ d 2 ) ) , ( k + 2 ) - C O N(PRA)
l )
* C1(0)
++
F o r n o t a t i o n s s e e SMORYNSKI's a r t i c l e [ 9 1
++
C2(uk)
U.R. SCHMERL
350
Proof: This results essentially from SMORYNSKI's theorem 1.1 in [91, which reads in our notations: If T is PA or PRA, then
-
o - ~ ~ ~ G ( c2 ~ Tt{C):(O):
1-CON (T)
(k+2)-CON (T)
n<wl(o)
Ct(0) m
m
C ~ f C ~ t l ( 0(0) )
.
All the rest - except some minor technical lemmata application of the fine structure properties.
-
is a consequent
Remark: GOLDFARB and SCANLON in [31 and GOLDFARB in I 2 1 give ordinal bounds a such that functionals of k-consistency are a-recursive. These bounds coincide for 1- and 2-consistency of PA with the ordinals given above; for k-CON(PA), k23, the latter are two w powers less.
FEFERENCES : GENTZEN: f i b e r Beweisbarkeit und Unbeweisbarkeit von AnfangsfPllen der transfiniten Induktion in der reinen Zahlentheorie, Math. Pnn. 199 (1943) 140-161 GOLDFARB: Ordinal bounds for k-consistency, J. Symb. Logic 39 (197Q) 693-699 GOLDFARB and SCANLON: The w-consistency of number theory v i a Herbrand's theorem, J. Symb. Logic 39 (1974) 678-692 KLEENE: Extension of an effectively generated class of functions by enumeration, Colloq. Wath. 6 (1958) 67-78 KREISEL and LEVY: Reflection principles and their use for establishing the complexity of axiomatic systems, Zeitschr. math. Logik 14 (1968) 97-142 LBB: Solution of a problem of Leon Henkin, J. Symb. Logic 20 (1955) 115-118 YINTS: Exact estimates of the provability of transfinite induction in the initial segments of arithmetic, Zapiski 20 (1971) 134-144; translated in J. Soviet Math. 1 (1973) 85-91 SCHOTTE: Beweistheorie, Springer Verlag, Berlin 1960 SWORYNSKI: w-consistency and reflection, Proc. 1975 Logic Coll. Clermont-Ferrand, C.N.R.S. Paris 1977
LOGIC COLLOQUIUM 78 M. Boffa, D. van DaZen, K. M c A l o o n (eds.) 0 North-HoZZrmd Publishing Compmmy, 1979
L O G I C A N D THE A X I O M OF C H O I C E
H.
Schwichtenberg
(Munich)
We s h a l l p r o v e t h e f o l l o w i n g :
(1)
vx3y cp(x,y)
-3
3fVx c p ( x , f x )
is conservative over c l a s s i c a l
( f i r s t order) logic. ( 2 )
Vx3y v ( x , y )
--t
3fVx c q ( x , f x ) i s c o n s e r v a t i v e o v e r i n t u i t i o n i s t i c
l o g i c without equality. (3)
vx3y 9 ( x , y )
-b
3fVx Q(X,fX
i s conservative over i n t u i t i o n i s t i c
l o g i c with decidable equa i t y . (4)
vx3y (P(x,y)
-3
3fVX 9 ( x , f x
is not conservative over
intuitionistic logic.
i s c o n s e r v a t i v e over c l a s s i c a l and i n t u i t i o n i s t i c l o g i c .
More p r e c i s e l y :
Addition of f i n i t e l y
many i n s t a n c e s o f t h e r e s p e c -
t i v e schema w i t h a l l (number and f u n c t i o n ) p a r a m e t e r s g e n e r a l i z e d i s c o n s e r v a t i v e o v e r any f i r s t o r d e r t h e o r y i n t h e r e s p e c t i v e l o g i c . None o f t h e s e r e s u l t s i s new.
(1) i s a l r e a d y i n H i l b e r t - B e r n a y s
1939 ( p . 1 4 1 i n t h e s e c o n d e d i t i o n ) . 1974, 1966; (41,
(2) - ( 4 )
a r e d u e t o Minc 1 9 6 6 ,
n o t e t h a t ( 3 ) i s an immediate c o n s e q u e n c e of
(5).
As t o
a s i m p l e r counterexample i s i n Osswald 1975 and p r o b a b l y t h e
s i m p l e s t (which i s r e p r o d u c e d h e r e ) i n Smorynski 1978.
.
( 5 ) i s due
t o Minc a n d S m o r y n s k i ; i t was f i r s t a n n o u n c e d i n Minc 1 9 7 7 .
Proofs
a r e i n Smorynski 1978 and ( i n a g e n e r a l i z e d form d e a l i n g w i t h “ s i m u l t a n e o u s Skolem f u n c t o r s ” ) i n L u c k h a r d t R . The p r o o f s g i v e n h e r e a r e r e l a t i v e l y s i m p l e .
For (5) t h e proof
c o n s i s t s i n a p r o c e d u r e which t r a n s f o r m a d e r i v a t i o n of a f i r s t o r d e r f o r m u l a i n v o l v i n g t h e axiom of c h o i c e i n t o a d e r i v a t i o n n o t i n v o l v i n g i t . The m a i n t e c h n i c a l t o o l i s t h e u s e o f a new t y p e o f
35 1
H. SCHWICHTENBERG
352 function variables:
fz
duced, then is that
E,
. . . ,r n ,
Whenever t e r m s rl,
is a function variable.
. . . ,s
sl,
are intro-
i t s h g u l d r a n g e o v e r a l l f u n c t i o n s mapping
r
in
( r i = r . + si 1 i , j ( 3 ) a n d a l s o (1) a r e e a s y c o n s e q u e n c e s
5 determine a f i n i t e function, i . e .
As a l r e a d y n o t e d ,
(x
(provized s
of
3
)I. (51,
i n ( 5 ) i s under t h e assumption
since t h e premiss of t h e implication Vx,y
fs
The i n t e n d e d m e a n i n g o f
= y v x f y ) e q u i v a l e n t t o Vx3y q ( x , y ) .
S o we s t a r t w i t h a
p r o o f o f ( 2 ) , t h e n g i v e S m o r y n s k i ' s c o u n t e r e x a m p l e t o p r o v e (4), a n d f i n a l l y e x t e n d t h e method f o r p r o v i n g ( 2 ) t o a p r o o f of
-
(5);
only
t h i s l a s t s t e p i n v o l v e s t h e f u n c t i o n v a r i a b l e s f2.
Note:
(1) - ( 5 ) r e m a i n v a l i d - w i t h e s s e n t i a l l y t h e s a m e p r o o f s -
when a l l v a r i a b l e s x , y ,
- x, 2, ...
f,
...
a r e r e p l a c e d by f i n i t e s e q u e n c e s
...,
fix, f x. ns i m p l i c i t y we o n l y d e a l w i t h s i n g l e v a r i a b l e s h e r e .
x,
Note: -
(21,
&
of variables;
(4)
t h e n means
and ( 5 ) a l s o h o l d f o r minimal l o g i c .
However,
for
This is seen
e a s i l y from t h e p r o o f s .
Proof o f ( 2 ) : I n t h i s s e c t i o n we o n l y c o n s i d e r f i r s t o r d e r i n t u i t i o n i s t i c We s h a l l w o r k w i t h a G e n t z e n s e q u e n t
logic without equality.
c a l c u l u s as d e s c r i b e d i n Kleene 1 9 5 2 , p .
4 8 1 ( t h e r e i t i s c a l l e d G3).
F o r s i m p l i c i t y we m o d i f y i t t o i n c l u d e 1 ( f a l s u m ) a s a p r o p o s i t i o n a l c o n s t a n t a n d t r e a t -cp
as d e f i n e d b y
+
1. F i r s t n o t e t h a t ( 2 ) c a n
be reduced t o (2)'
If
1
then
Vx c p ( x , f x ) ,
1
A
+
Vx3y c p ( x , y ) ,
Proof o f ( 2 ) from ( 2 ) l : g i v e n w i t h A,P
Y w i t h A,"
Let a c u t - f r e e
r
of first o r d e r and
i n s t a n c e s t h e s c h e m a Vx3y c p ( x , y ) l e n g t h of
of first o r d e r and w i t h o u t f ,
A -, Y .
-t
d e r i v a t i o n of r , A
-*
Y be
a l i s t of g e n e r a l i z a t i o n s of 3fVx cp(x,fx).
By i n d u c t i o n on t h e
t h i s d e r i v a t i o n we c o n s t r u c t a d e r i v a t i o n o f A -, Y .
It
s u f f i c e s t o consider an inference
F i r s t t h e l e f t m o s t 3f i n t h e r i g h t h a n d s u b d e r i v a t i o n c a n b e c a n c e l l e d b y a n i n v e r s i o n lemma.
r'
T h e n t h e o c c u r e n c e s o f Vx3y -, 3 f V x ,
i n t h e a n t e c e d e n t o f b o t h s u b d e r i v a t i o n s can b e c a n c e l l e d by
induction hypothesis.
T h e n b y ( 2 ) ' Vx q ( x , f x )
i n the antecedent of
LOGIC AND THE AXIOM OF CHOICE
35 3
t h e r i g h t h a n d s u b d e r i v a t i o n c a n b e r e p l a c e d b y Vx3y c p ( x , y ) . then gives the desired derivation/of 0 Proof A,"
of
( 2 ) ' :
Let a cut-free
-D
1
d e i v a t i o n o f Vx c p ( x , f x ) ,
o f . f i r s t o r d e r and w i t h o u t f
A cut
x.
e given.
A
'Y w i t h
+
By i n d u c t i o n o n t h e l e n g t h
o f t h i s d e r i v a t i o n we c o n s t r u c t a d e r i v a t i o n o f Vx3y c p ( x , y ) ,
A
+
'4'.
It suffices t o consider
R e p l a c e a l l o c c u r r e n c e s o f f t i n t h i s d e r i v a t i o n b y a new v a r i a b l e w.
T h i s g i v e s a d e r i v a t i o n o f l P ( t , w ) , Vx C P ( x , f x ) ,
r
x.
-D
By
i n d u c t i o n h y p o t h e s i s we o b t a i n a d e r i v a t i o n o f c p ( t , w ) ,
Vx3y ~ ( x , y ) , r
x.
+
Application of
d e r i v a t i o n o f Vx3y c p ( x , y ) ,
where f t l , . . . , f t n
....f t , ) . ...,f t n i n
r(ftl,
ftl,
r
-D
x.
gives the desired
a r e a l l outermost occurrences of f-terms
in
Replace a g a i n a l l o u t e r m o s t o c c u r r e n c e s of
...,w .
t h i s d e r i v a t i o n b y new v a r i a b l e s w l ,
a d e r i v a t i o n of v(tl,wl)
a n d (V-1
(3-1
,... , c p ( t n , w n ) ,
Vx c p ( x , f x ) ,
the induction hypothesis,
r
-D
x(r(wl,.
(+3), n t i m e s ( 3 + )
.. , w n ) ) .
This gives
Then a p p l y
a n d n t i m e s (V-D).
Proof o f ( 4 ) : I n t h i s s e c t i o n we o n l y c o n s i d e r f i r s t o r d e r i n t u i t i o n i s t i c logic (with equality).
I- v x 3 y ( x f y )
-t
I t s u f f i c e s t o prove
Vx13y 1 v x 2 3 y 2 (x,"ty
A
X2#y2
c
d
A
(X1=X2
-D
y1'y2)).
C o n s i d e r t h e Kripke model
a o : a
b
T h i s i s o b v i o u s l y a model of t h e e q u a l i t y axioms.
It aoIt a.
Furthermore,
Vx3y: ( x f y )
s i n c e a i IF a # d , b # c f o r a l l i . B u t
vx13ylvx23y2
(x,
f
y1
t h i s assume t h e c o n t r a r y .
A
x2 # y2
A
(x,
Choose a f o r xl.
= x2
-D
y1
= y 2 ) ) . To s e e
T h e n y1 m u s t b e d .
Choose
H. SCHWICHTENBERG
354 b f o r x2.
Hence a
Then y 2 m u s t b e c .
0
It
= d ) . But t h i s is
(a = b + c
a = b , a2 1 1 c
a c o n t r a d i c t i o n , s i n c e a,ll-
(This counterexample
d.
i s due t o Smorynski 1978)
Proof of
(5):
We o n l y c o n s i d e r f i r s t o r d e r i n t u i t i o n i s t i c l o g i c ; s e e n t h a t t h e same p r o o f that
it i s e a s i l y
also applies t o classical logic.
F i r s t note
(5) c a n b e r e d u c e d t o
I- Vx c p ( x , f x ) , A
If
(5)'
+
Y
with A,Y
o f first o r d e r and w i t h o u t f ,
t h e n t h e r e i s an n such t h a t
1 Vx13y l . . . V x n 3 y n [ abbreviates
cp(xi,yi)
.(xi i $7
= x.
1
+
Fct(x;y)l,A
A
+
Y
,where F c t ( x , y )
= yj).
yi
(5) c a n b e p r o v e d f r o m ( 5 ) ' e x a c t l y a s we p r o v e d ( 2 ) f r o m ( 2 ) ' To p r o v e
(5)'
we c a n n o t p r o c e e d a s s i m p l y a s i n t h e p r o o f o f
above.
('2)'.
F o r , t h e r e p l a c e m e n t o f f t b y a new v a r i a b l e w i n c a s e 1 w o u l d n o t l e a d t o a d e r i v a t i o n anymore, s i n c e a n e q u a l i t y axiom t = s w o u l d b e t r a n s f o r m e d i n t o a n u n d e r i v a b l e f o r m u l a t-s
ft-fs
wzfs.
The i d e a
now i s t o r e p l a c e f b y a new v a r i a b l e f t w i t h t h e i n t e n d e d m e a n i n g t h a t i t s h o u l d r a n g e o v e r f u n c t i o n s m a p p i n g w i n t o t . To make t h i s p r e c i s e we f i r s t e x t e n d our l a n g u a g e .
V a r i a b l e s a n d t e r m s a n d now
generated simultaneously with the additional clause If r l ,
...,r
sl,
...,s
(short:
r , s ) are terms,
then fs is a
f u n c t i o n v a r i a b l e ( w h e r e f i s a n y o f t h e c o u n t a b l y many s y m b o l s reserved f o r function variables). C o r r e s p o n d i n g t o t h e i n t e n d e d m e a n i n g o f f s we a d d t h e f o l l o w i n g a x i o m s t o our l o g i c a l f o r m a l i s m : Fct(r;s)
fsr.=s r i i
f o r a l l i.
We now f o r m u l a t e a g e n e r a l i z a t i o n o f function variables, (5)''
If
t
Vx c p ( x , f s x ) ,
w i t h o u t f;,
[
empty.
Fct(r;s),
A
+
Y w i t h A,Y
tEen t h e r e is an n such t h a t
q(xi,yi)~ Fct(r,x;s,y)l,
Proof of (5)":
i n v o l v i n g t h e s e new
(5)'
which c a n t h e n b e p r o v e d by i n d u c t i o n .
A
+
of f i r s t o r d e r and
I- Vx 3 y l . . . V x
Y.
1
n
3y
n
F o r s i m p l i c i t y we o n l y w r i t e o u t t h e c a s e f o r E ,
The g e n e r a l c a s e c a n b e d e a l t w i t h i n e x a c t l y t h e same m a n n e r .
We u s e i n d u c t i o n o n t h e l e n g t h o f t h e g i v e n d e r i v a t i o n , assume t o be c u t - f r e e .
It suffices t o consider
w h i c h we may
LOGIC AND THE AXIOM OF CHOICE
We f i r s t d e s c r i b e t h e w e l l - k n o w n t e r m s from f t " .
Let f t l , .
..,ftn
355
technique of " e x t r a c t i n g f-sub-
f t be a l l f - s u b t e r m s
by i n c r e a s i n g d e p t h o f n e s t i n g o f f . L e t w l ,
. . . ,w
of f t ordered
b e new v a r i a b l e s .
F o r a n y s u b t e r m s o f f t d e n o t e b y s*
the r e s u l t of replacing a l l
outermost occurrences o f f-subterms
fti
wi
,..., wi
1
k
.
U s i n g t h e new a x i o m s o n
,... , f t i
1, f;
k
i n s by
one can p r o v e e a s i l y by
5
i n d u c t i o n on s
(*I
W
Fct(t*;w)
+
-
s(f;,)
t* d e n o t e s o f c o u r s e t l ,
5
s*.
... ,tn
; note ti
contains only w
j
with j
- Now r e p l a c e a l l o c c u r r e n c e s o f f i n t h e g i v e n d e r i v a t i o n b y f
<
i.
W
t
-
W r i t i n g t f f ) f o r t o n e o b t a i n s a d e r i v a t i o n of W
,
-
cp(t(f;,)
w w f';,t(f';,))
- -
,
W
,r
-
vxcp(x,f';*x)
+
x
Using ( * ) t h i s d e r i v a t i o n can e a s i l y be transformed i n t o a d e r i v a t i o n of cp(t*,wn)
,
W
-
VXV(X,~;,X)
o f t h e same l e n g h t
tautologics, 1.e.
,
~ c t ( t * ; w ) , r+ 5
x
f i t i s n e c e s s a r y h e r to a l l o w a s a x i o m s a l l q u a s i -
a l l t a u t o l o g i c a l consequences of t h e e q u a l i t y S
-
a x i o m s i n c l u d i n g t h e new a x i o m s o n
f';
).
By i n d u c t i o n h y p o t h e s i s we t h e n o b t a i n a d e r i v a t i o n o f
Now ( 4 ( t * , w n ) c a n b e c a n c e l l e d s i n c e i t f o l l o w s f r o m t h e s e c o n d member o f t h e a n t e c e d e n t r u l e s (3-1 vu13w
,
( w e may a s s u m e m 2 1). T h e n u s i n g t h e
( V + ) we o b t a i n
l . . . n~ 3 w ~Vx 3 y 1 . . . V x m 3 y m n l
[
i
(p(xi,yi)
A
Fct(u,x;w,x)l,T
+
x
H.
356
W
SCHWICHTENBERG
Here a g a i n w e e x t r a c t t h e f - s u b t e r m s
-
b y f+*.
from t and t h e n r e p l a c e f
T h i s g i v e s a s a b o v e a d e r i v a t i o n of
-
W
-
Vxw(x,ft*x)
Fct(t*;w)
,r
By i n d u c t i o n h y p o t h e s i s w e t h e n o b t a i n a d e r i v a t i o n o f
1
vX13y l . . . v X m 3 y m
i
(P(Xi,Yi)
A
,r
Fct(t*,x;w,x)
x(t*)
+
.
Now a p p l y (-3) a n d t h e n p r o c e e d a s i n c a s e 1 a b o v e .
References
D.
H I L B E R T a n d P.
BERNAYS
,
1939, Grundlagen d e r Mathematik
11.
Berlin21970.
S.C. H.
KLEENE,
1952, I n t r o d u c t i o n t o Metamathematics.
LUCKHARDT
G.E.
MINC
A,
,
1966
Quantifiers
C o n s e r v a t i v e Skolem F u n c t o r s .
,
S k o l e m ' s Method of E l i m i n a t i o n
in Sequential Calculi.
Amsterdam.
Unpublished,
5 1 pp.
of P o s i t i v e
S o v i e t Math.
Dokl.
7,
861-864. G.E.
MINC
,
1974
,
H e y t i n g P r e d i c a t e C a l c u l u s w i t h E p s i l o n Symbol.
J. S o v i e t Math. 8 ( 1 9 7 7 1 , 3 1 7 - 3 2 3 . G.E. H.
MINC
,
325,
02021.
OSSWALD
1977
,
1975
,
Review o f Osswald 1975.
,
{ber
Skolem-Erweiterungen
schen Logik m i t G l e i c h h e i t . K i e l 1974 ( e d .
C.
SMORYNSKI
,
1978
Annals Math.
In
J. D i l l e r , G . H .
,
Z e n t r a l b l a t t fiir M a t h .
: Proof
Miiller), Berlin,
The A x i o m a t i z a t i o n
Logic 1 4 , 193-221.
in der intuitionisti-
Theory Symposion, 264-266.
Problem f o r Fragments.
LOGIC COLLOQUIUM 78 M. Boffa, D. urn Dalen, K . McAloon ( e d s . ) 0 North-Holland Publishing Company, 1979
O N SUCCESSORS OF SINGULAR CARDINALS
Saharon Shelah I n s t i t u t e o f Mathematics, The Hebrew U n i v e r s i t y , Jerusalem, Israel.
Introduction
:
We w i l l c l a r i f y t h e s i t u a t i o n f o r t h e s u c c e s s o r o f a s t r o n g l i m i t singular cardinal A.
We f i n d a s p e c i a l s u b s e t S * ( A t ) ,
w h i c h we c a n f i n d w h i c h s t a t i o n a r y s u b s e t s o f from b e i n g s t a t i o n a r y by u-complete t h i s for successor At For A
of regular A
At
from
can be stopped
f o r c i n g (Baumgartner h a s done
=
A")).
we s u c c e e d i n c o n t i n u i n g a n i n d u c t i o n c o n s t r u c t i o n d o n e
Nwtl
for a A t - f r e e
not h
tt
(abelian) group,
and similar t h i n g s f o r trans-
v e r s a l s ; o n t h o s e p r o b l e m s s e e h i s t o r y a n d r e f e r e n c e s i n [ S h 2 ]. A s o l u t i o n o f a r e l a t e d problem
can be
- which s t a t i o n a r y s u b s e t s o f At
" k i l l e d " hy a f o r c i n g n o t adding bounded s u b s e t s of X t - w i l l
a p p e a r i n a p a p e r b y U. A v r a h a m , J . S t a v i a n d t h e a u t h o r .
Wp a l s o p r o v e a r e s u l t r e l a t e d t o t h e t i t l e b u t n o t t o t h e r e s t o f the paper,
i m p r o v i n g a result o f G r e g o r y [ G r ] : a q s u m i n g G . C . H . ,
f C J r h # N o , 0; h o l d s , w h e r e S
= 16
cf6 # cfh}; hence 0
for a n y s t a t i o n a r y S1 C S. For a r e a d e r i n t e r e s t e d o n l y w i t h t h e G C H . h i m s e l f t h e p a r t up t o s e c t i o n 1 3 . general cases than those discussed t h e end.
s1
holds
he can simplify foy
A r e a d e r i n t t r e s t e d i n more
has t o go t o
i n t h e m a i n p'art
T h e r e we a l s o s h o w t h a t t h e s p e c i a l s e t S (
*
Nwtl
1 can be
s t a t i o n a r y ( e v e n w t h t h e GCH). T h e m a i n r e s u t s w e r e a n n o u n c e d i n t h e AMS N o t i c e s [ S h 3
357
I.
358
S . SHELAH
Notation:We
s h a l l d e n o t e i n f i n i t e c a r d i n a l s by A , ~ . K , x ,
by i , j , a , B , y , < , c Let
l i m i t o r d i n a l s by 6 ,
denote a sequence
<
<
A,
N ~ < ( H ( ~ ) , E )i ; _C N ~ ,I N . I
Ni : i
i
<
n a t u r a l numbers by m , n , r , p , q . w h e r e f o r some p , x
j
ordinals
=.
Ni<
<
p,
a n d for l i m i t
Nj,
=
6,N6
U Ni. We c a l l t h i s a A - a p p r o x i m a t i n g s e q u e n c e ( f o r p ) . i<6 We d e n o t e b y d a t w o - p l a c e f u n c t i o n f r o m o n e c a r d i n a l t o a n q -
t h e r ; cf6 i s the c o f i n a l i t y of 6 ; otherwise.
cf*6
i s cf6 i f cf6
o v e r I , A C_ B mod D means I
-
-
(A
B)
(B
U
: B U (I
f i l t e r {B
-
<
K )
=
lJ
-
<
-
(A
- B)
A ) E D.
A) E
L e t c F ( 6 , ~ )= { i
CF(6,
6 and i s
i s t h e f i l t e r o v e r 6 g e n e r a t e d by t h e c l o s e d
D6
u n b o u n d e d s u b s e t s o f 6 ( s o we a s s u m e c f 6
means I
<
E
> N 1.
If D is a f i l t e r
B mod D
similarly A
D;
I f A d $ mod D ,
D t A is the
D}.
6 : c f i = K I ,s i m i l a r l y C F ( ~ , < K ) = D6,K
CF(6,p)
= D6
U CF(6,p) P
t c F ( 6 , ~ )e t c .
U
1. D e f i n i t i o n : 1) We s a y
a family a)
Py,,
K
i s g o o d f o r A i f A=A<',
K = m
or t h e r e i s -
such t h a t
lEy,Kl = A
b ) e v e r y member o f P o -A
,K
i s a s u b s e t of A of c a r d i n a l i t y
c ) every subset of A of c a r d i n a l i t y 2 ) We c a l l
and
K
K
K
c o n t a i n s a member o f Po
-A ,K
a good c o f i n a l i t y f o r A i f A
if A
such t h a t
A -A,K
i s a s u b s e t of A of c a r d i n a l i t y
c ) e v e r y s u b s e t of A of c a r d i n a l i t y
K
has a subset
such t h a t a. i s i n c r e a s i n g and f o r e v e r y j A
is
A<',K
-A,K
b ) e v e r y member o f P
d) A
=
a r e r e g u l a r and t h e r e i s a f a m i l y P 10
K
'p
2'
<
A
f o r every p
<
K
<
K,
{ai
{ai : i
< : i
<
K
<
K )
j}E P
-A
,K
ON SUCCESSORS OF SINGULAR CARDINALS
Definition
2.
=
G(A)
= {i < X
2 ) gcf(h)
= Dh
3)
Df
3.
C l a i m : 1) I f
2)
If
K
<
3 ) If
h
=
If
L
(yp
i f
for
hi,
Z
<
=
AK
h then
then
7 ) If
K
<
( n o t e t h a t w e u s e cf'
not c f )
<
< Ka,6 < ~
~
<
i s g o o d f o r .As
K
hi(i
<
i n c r e a s i n g and
11)
K
<
m
is
i s good f o r X
K
C f K ,
<
K
i s g o o d for K
a+B
1 K
i s a good c o f i n a l i t y
h
=
~a r e r e g u l a r
then
C f K
g o o d for h t h e n
K
2
#
CfKa
~ ps +u f f~~i c e
H)
are r e g u l a r ,
A, provided t h a t ,
i s g o o d f o r A}
K
i s good f o r X
K
cfu # c f K ,
W,)VK
X,K
h
i s a good c o f i n a l i t y f o r A)
K
: cf*i f G c f f h ) ]
2
6 ) If
:
{ K
i s good f o r h t h e n
i n f a c t (Vp
5)
:
{ K
t gcf(X)
i
=
Gcf(h)
: 1)
359
h then
i s a good c o f i n a l i t y f o r
K
i s a good c o f i n a l i t y t o r h t h e n
K
X
i s a good c o f i n a l i t y
for ht 8 )
If X
= z Xi, X u
increasing, 9) K
4.
If
(vp
<
and
.
<
: For
=
{c:
K
then
Ha,
K
E
Gcf(hi)
#
K,
tvp
<
~
d a two-place 5 < 6 ,
5
f o r every i
K
~
regular,
5
iK
)
= (5
:
K ,
then
I.
a l i m i t o r d i n a l such t h a t t h e r e is an d is constant)
5 < 6 , 5 a l i m i t o r d i n a l such t h a t t h e r e are s u b s e t s A.B
( v b E B)(]a : Note t h a t
'i
f u n c t i o n f r o m 6 i n t o ~ ( c f 6> H
unbounded
Remark
p,
suffice
u n b o u n d e d A C_ 5 on w h i c h So(d)
<
E Gcf(h)
cfHa
) [ in fact,
Definition Sl(d)
K
Ha)uKK
6 Gcf(Hat5tl
we let
cfp # c f K ,
<
of 5 , such t h a t
K)(Va E A ) [ a < b
d d e t e r m i n e s & ( a s Dom d ) b u t n o t
not necessarily onto
K ) ,
so i f t h e v a l u e of
K
as
is not
-+
d(a,b)
d is into clear we
s h a l l w r i t e S O ( d , ~ ) . I n t h e d e f i n i t i o n o f S l ( d ) , ~ h a s no r o l e .
S. SHELAH
360 C l a i m : For d a t w o - p l a c e
5.
1) S l ( d )
f u n c t i o n from 6 t o
:
So(d),
C_
i n t h e d e f i n i t i o n of S,(d)
2)
K
(P
=
0,l) we
can assume A , B have
o r d e r t y p e c f f , ( a n d g e n e r a l l y r e p l a c e them by unbounded s u b s e t s ) , 3 ) CF(6, 4
C
= 0,1,
If P
4)
C So(d),
K )
S,(d),
CE
c f f , > N o , t h e n t h e r e i s C E DS s u c h t h a t
C Sa(d).
6.
Definition
= (5
S2(N)
: For a A-approximating
: f,
<
7.
C l a i m : 1) I f A i s r e g u l a r , N O ,
c =
respectively,
i~'
: Let
{ a
cff, such t h a t
< A
n
:N
:
:
i
(
i
(Vi
<
order type
Proof
(see notation)
and
[ A n i E N
f,)
u p > 1,
5
1
and N
n X =
5
are 1-approximating
then
S,(F1) =
< A>
u
N.) 1
j<X
strict y between a model N and i t s u n i v e r s e ) .
B Y t r a n s i t i v i t y o f e q u a l i t y we c a n a s s u m e
i f f E, E S,(zl),
=
K
-A
to N
1
f,
,K
< 5.
The " o n l y
if" part is
Also t h e c a s e
< 6.
We h a v e j u s t a s s u m e d
E
K
Gcf(X),
so the
( a s i n D e f i n i t i o n 1 . 2 ) e x i s t s , hence belongs t o
hence w.1.o.g
i t b e l o n g s t o No, a n d h e n c e , by a s s u m p t i o n ,
. If 5 E
A C
.
We s h a l l p r o v e f, E S 2 (Vo)
thus completing t h e proof.
= cff,
appropriate P H(pl),
(
i s e a s y , s o we a s s u m e c f " ~ = c f <
00
Let
Gcf(h).
NO<
A.
1 N x a
s o w e c o n c e n t r a t e on t h e " i f " p a r t .
now t r i v i a l ,
cf*t
a n d c€*f, E
E C,
sequences
S , ( B o ) mod D gA '
I t i s e a s y t o check t h a t C i s a c l o s e d unbounded s u b s e t o f
5
f,)
1
( w e d o not d i s t i n g u i s h
Now suppose
let
5 a l i m i t such t h a t t h e r e i s a n unbounded A % 5
A,
of
f o r uoo)pl
T
sequence
S,(F1),
of order-type
then
(by d e f i n i t i o n )
there
cff,, s u c h t h a t f o r e v e r y
is an unbounded
r, <
E,,
1
A n 5 E N5.
36 1
ON SUCCESSORS OF SINGULAR CARDINALS
If A =A
-A,
lrA,Kl= A ) , and
(since hence
w e can assume P
<
5
F,
*
A n 5
E
= {B E
K
,
I B ~: i < A I
IBI
n No = P 5 -h,K
so P
:N
:
n N1
= {Bi
5
witnesses that
hence A
,€
< 51,
:
i
E
S,(mo).
Thus f i n i s h i n g .
<
So w e a r e l e f t w i t h t h e c a s e A tion 1.2,
<
(Vp
e v e r y s u b s e t of dinality
P
{ai
<
<
A of power
: i
{ai
<
: i
< jl
jj E N i .
U NO h e n c e ( a s ,€ i
1
5
.
-1
E
N:.
=
In particular,
P
-A,K
= P --x
n :N
,K
N i of c a r -
: i
by t h e c h o i c e
<
and
K } ,
and, as mentioned above,
,K
But as C)P --h
K ,
So w e can r e p l a c e A by any
w e c a n a s s u m e A = {ai
E P
E
A has order-type
is i n c l u d e d i n a s e t from
K
i t which i s unbounded i n 5 .
K,
€,,and
5
(see Definition Z ) ,
-A,K
for j
S o , a s Np n A =
A.
hence it b e l o n g s t o N
s u b s e t of of
<
~)2'
Then, by d ) of Defini-
A
So {ai
: i
6
<
EN@, 0
clearly
PA,, C_
n NE, hence f o r every w i t n e s s e s t h a t F, E S,(ro),
K ]
and t h i s f i n i s h e s t h e proof of t h e theorem.
8.
Definition
: S"(A)
-
N any A-approximating
G
X
i s d e f i n e d as (A - S 2 ( N ) ) n g c f ( h ) f o r
s e q u e n c e f o r A',
S*
i s u n i q u e l y d e f i n e d mod D X o n l y ) .
9.
Definition
to
= cfX
K
{B
<
if
for y
(Y
: For A
s i n g u l a r , a two-place
: d(B,a) Q i } has cardinality
< B <
a
<
A',
d(y,a)
Q
two-place
(Ai i n c r e a s i n g ) , then I { B : Easy
<
a
<
< ,'A
K,CL
the set
I t is called subadditive
max { d ( y , B ) , d ( B , a ) } .
t h e r e is a normal subadditive
to cfh;
f u n c t i o n d from Xt
.
( S O
f u n c t i o n d from At
is c a l l e d normal i f f o r every i
1 0 . C l a i m : For e v e r y s i n g u l a r A ,
Proof
where A i s r e g u l a r .
moreover,
: d(B,cr)<
ill
if
A
hi.
=
i<EfA
Xi
362
SHELAH
S.
x
11. C l a i m : 1) S u p p o s e
is singular,
and d i s a normal two-place X'-approximating
sequence
f u n c t i o n from A t
N
for
<
CF(Xt,
A is s i n g u l a r ,
2 ) Suppose
x ++
to
, x
5
i s r e g u l a r and i s a good
Ni
= {B < =
H e n c e P'?
<
A,
<
: d(B,cx)
a
{A
C_
: B
A*,
hence Pa C N 1
-
hence t o N 6 . E
to
K.
,
CF(Xt,
<
A',
i
N6n
: .C
<
IBI
h
such t h a t
= &}is c l o s e d
the set
K ,
and h a s c a r d i n a l i t y
and has c a r d i n a l i -
t o Nitl
x} belongs
< X.
So suppose 6 E S o ( d ) , and
A,B _C 6
are
t h e y a r e unbounded i n 6 and have o r d e r - t y p e f o r some i ( b )
Suppose f u r t h e r 6 E C ,
B,
Xt
i } belongs t o Nitl
i+l'
and f o r every b E B,
every b E
<
So f o r e v e r y a
w i t n e s s t o it ( i . e .
6
tt
f o r At'
sequence
= 16 <
Clearly
E Nitl.
and unbounded.
ty
5 for X
sequence
: 1) C h o o s e a X t - a p p r o x i m a t e
d E No,
A*
function from A t
n So(d) E S2(N).
CF(Xt,x) Proof
Then f o r some
K .
S 2 ( B ) mod D X .
c o f i n a l i t y f o r X+,and d i s a normal two-place Then f o r some A ' - a p p r o x i m a t i n g
x ) ( ~ < x< A ) ,
,
x ) n So(d)
= cfk
K
<
= c f ~ ,t V v
ic
< x.
cf6
{a : a E A,
a
<
<
6
5
Then A , B
N6 ( a s 6
b b} belongs t o p i ( b ) ,
So A w i t n e s s e s t h a t 6 E S x ) fl So(d)
(Va E A X d < b + d ( a , b )
K ,
E S2(N),
(N). 2
5 N6)
<
cf6,
i(b))).
and f o r
hence t o N
itl'
We h a v e j u s t p r o v e d
thus f i n i s h i n g the proof
of
the claim. 2 ) A similar proof.
1 2 . Claim
: Suppose A i s r e g u l a r ,
< X
K
< x,
K
-.
<
A,
n a l i t y f o r X a n d (Vu
<
x)2'
f u n c t i o n d from X t o
K
a n d f o r some A - a p p r o x i m a t e
or y =
S2(R) n CF(X,x)
Proof:
Choose
a s A-approximate
Suppose 6 E S 2 ( N )
f-
CF(X,x).
x
i s a good c o f i -
Then f o r e v e r y t w o - p l a c e sequence
for At,
C_ S l ( d ) .
sequence f o r Xt
such t h a t d E No.
We s h a l l p r o v e 6 E S l ( d ) .
The c a s e
ON SUCCESSORS OF SINGULAR CARDINALS
x
is easy,
m
As 6
E
x <
so assume
m.
S2(N), t h e r e i s a s e t { a i
< x,
s u c h t h a t for e v e r y j
x,
t i o n w i t h domain
{ai
: i
= ai.
h(i)
<
a s follows fi(j)
= d(x.,6) for j < i 3
Let h be t h e func-
j} E N g .
< x,
< x
< x =
cf6 < 6 ,
( V j
<
a n e l e m e n t xi
i ) [d(x.,xi) 1
c l e a r l y each fi
Note also t h a t xi
= f i ( j ) I.
u <
B u t now a s
is in N
depends o n l y on f i
5
=)
and function
d(xj,6)
= i
.
x
i
and x . ( j 1
* 2p <
x.
<
i )
and
6'
and { a . : j 1
i, f . So x i E N 6 for e a c h i < x. filj). 3 Now t h e r e i s a n u n b o u n d e d S x and io <
j E S
h l j E N6.
( s o Dom f . = i )
T h i s can be c a r r i e d o u t i n H ( X t ) .
<
_C 6 , u n b o u n d e d i n 6 ,
:
and is such t h a t
j
<XI
i s t h e f i r s t o r d i n a l which i s b i g g e r t h a n a
xi
I.!
: i
Clearly f o r j
Now we d e f i n e b y i n d u c t i o n o n i fi
363
K
<
i
(as for
such t h a t
I t i s e a s y t o check t h a t { x . 1
j E Sl
witnesses that 6 E S,(d).
From now o n we c o n c e n t r a t e o n s u c c e s s r ) r s o f s t r o n g l i m i t singular cardinals. 13. Conclusion
We c a n c o n c l u d e e . g .
: Suppose h
is a singular strong l i m i t .
e v e r y n o r m a l t w o p l a c e f u n c t i o n d from A t holds
t o
Then f o r
= cfh, the following
K
:
So(d)
S l ( d ) U CF(At,<
Xi
K )
- S*(Xt)
mod
.
DA+
( S o i n p a r t i c u l a r S ( d ) d o e s not d e p e n d o n d ( w h e n d i s n o r m a l )
t o e q u i v a l e n c e mod
DX+
up
).
Proof
: T r i v i a l by
Proof
: We can f i n d a A-approximating sequence
5.1,
5.3,
r y subset of N . of c a r d i n a l i t y
11 a n d 1 2 .
:<
b e l o n g s t o Nitl
N.
: i
t o
. I S 0 CF(A, < K )
At
such t h a t eve-
s S2(s).
S. SHELAH
364 15.
<
K
Clairr. : I f 6 E A - S l ( d ) ,
: If
Proof
16.
is
cf6
Definition
Define
weakly compact t h e n c f 6
For a s e t S E h
: 1)
16 <
F(S)
2)
d a two-place
f u n c t i o n from 1 t o
then c f 6 i s not weakly compact.
cf6,
2
( ~ f 6 ) ~ .
-+
let
S n 6 is a stationary subset of
A:
6 )
F n ( S ) by i n d u c t i o n on n:
Fo(S) = S , F n t l ( s ) = F ( F ~ ( s ) ) .
17.
Claim
: 1 ) F F ( S ) _C F ( S ) .
2 ) F(S*(A)) C S*(A),
h e n c e Fn
>
N~
i m p l i e s cf6
>
3 ) 6 E F~(s) i m p l i e s cfS then 6 E Fn(S) 4)
<
If a
min { c f 6 : 6 E
atn Si}
U
i
,
Si
h then
F( U S . ) = U F(Si) mod DA. iG iG Proof
1) E a s y
:
2 ) By 5 . 4 3),
4)
(and second part-by
Easy.
1 8 . Lemma
: Suppose
T h e n f o r some C be i n c r e a s i n g ,
D
E
Proof : -
i s a s i n g u l a r s t r o n g l i m i t of
A
A+
f o r every 6 E C,
'
:
CY
i
E S*(A)}
Let d b e a s i n 10. S*(A+)
A+'
flC
= So(d)
Now d e f i n e a t w o - p l a c e d*(i,j)
letting
<
cofinality
K .
ai : i < c f 6
>
continuous and converging t o 6 , t h e following holds
ti
C E D
induction)
= d(ai,a.). 1
2 S K ( c f 6 ) mod D c f 6
T h e n by 1 3 , f o r some 11 C ,
s o we n e e d o n l y d e a l w i t h S o ( d ) .
f u n c t i o n d'
from cf6 t o
K
by
:
It i s easy t o check t h a t
{ai : i E s O ( d * ) ) C_ s o ( d ) . But by 1 0 , S o ( d * ) finished.
c
cf6
- S * ( c f 6 ) (remember
K
<
c f 6 ) , s o we a r e
:
365
ON SUCCESSORS OF SINGULAR CARDINALS
x3u
19. C o n c l u s i o n : 1) S u p p o s e A i s a s i n g u l a r s t r o n g l i m i t ,
<
regular,xp
and ( v p l <
A
p
<
)!J:
IJ. T h e n F [ S * ( A t )
flCF(A+,x)] n
C F ( X +, p ) i s n o t s t a t i o n a r y . 2 ) If n
<
k'
< Kktn
w and 2
s
some s t a t i o n a r y
C_ H
some k
<
<
(VIJ <
,
n CF(Hwtl,N k ) ]
<
UP )IJ
(since 2
Np
F ~ ( S * ( H ~ + ~i ) s )n o t 3 ) Since S*(Nwtl)
by 1 9 . 1 ,
6 E Fntl(S)
il
F(S)
5
So k+n
CF(Nwtl,
>
k tn, then
h e n c e we g e t a c o n t r a d i c -
no 6 E F ~ + ' ( s ) , i . e . P
( S )
f o r some k
<
P
Q
<
<
Kk+n
<
L e t 2'' w implies
Hktn),
, a n d by
w,
=
Nktn
(n
<
(VIJ < H e ) v K k
w
since
<
Hp
;
where
Rut by 1 7 . 1 ,
CF(Nwtl,Nk).
F n + l( S ) € F ( S ) , hence
17.2
6
E
Fn+l(S) i m p l i e s
* c f 6 = H k ) , s o we g e t t h a t t h e r e i s
(since 6 E S
for s o m e P , F
Nktn),
is stationary.
i m p l i e s cf6
cf6 "ktntl
<
is stationary,
CF(Hw+l,Nk)
S*(Hwtl)
'k
stationary.
Kw i s a s t r o n g l i m i t ) .
hence,
Hence f o r
S o i n a l l c a s e s we g e t a c o n t r a d i c t i o n ; h e n c e
19.1.
(3
for
is stationary.
But i f P
t h i s contradicts 19.3.
'k
S*(Hw+l)
Then by 1 7 . 4
F ~ [ S * ( H ~ +n~ C) F ( H ~ , , , H ~ )C] F~ ( K ~ + ~ ,P H) i s s t a t i o n a r y .
k t n ,
t i o n by
=
= @
is stationary.
Fn[S*(Hwtl)
w,
some Q < w
S
w t1 then f o r
: 1) By 1 4 a n d 1 8 .
2 ) S u p p o s e Fn(S*(Hwtl))
If P
F(S)
W + l '
.
0 mod DH
w,then Fn(S*(Nwtl))
i s a s t r o n g l i m i t a n d S*(Hwtl) i s s t a t i o n a r y ,
3 ) If Hw
Proof
<
f o r every k
Fn t 1
( s )
=
+.
since
FO(S)
P
is s t a t i o n a r y but F(F ( S ) )
= s
is stationary,
FP+'(S)
is not;
P
F ( S ) i s as r e q u i r e d .
Theorem 2 0
s c
: S u p p o s e S C_
CF(A,~).
X i s s t a t i o n a r y , and
S C_ g c f ( h )
P is a p+-complete f o r c i n g t i . e .
i f
- S*(h), : i
i s a n i n c r e a s i n g s e q u e n c e o f e l e m e n t s o f P t h e n some p E P i s
for e v e r y i ) , =
S i s s t a t i o n a r y even i n t h e
universe Vp
< u> >
pi
366
S . SHELAH
Remark
: Remember
of
s C_ A .
any
B
Proof : L e t
t h a t A-complete
forcing forces the stationariness
be a A'-approximate
C
t h a t a P-name
enumerate A , We w a n t
a n d for e v e r y 5
hence
5
<
6
.
longs t o N
>
: cf6
i s d i s j o i n t f r o m S".
: pK"2
"6 E
<*
A
+
a l i m i t ,
is the
pi
For t h i s
C"
(since
of P U P x
<
cf6, p
<*
-first
N o w we d e f i n e b y i n d u c t i o n o n i
E
p'
be-
A N6.
which i s
p . f o r e v e r y j ( w h i c h e x i s t s s i n c e P i s u'-complete) 1
We l e t p i t l . B i such t h a t p' (p',B')
be such t h a t
>
pi,
C
since
was a P - n a m e
Ei
<
<
cf6.
was a P-name
P A N6, s o E i
C
So q f o r c e
of
5'
E
C.
-first
pair (p',E')
There is such
an unbounded s u b s e t of A .
Since P i s p+-complete,
cf6}.
every i
E
of
<*
is the
( pi+l,Bi)
0 ' 2 f ( i ) and p ' *
easy t o check t h a t pi, i
and A has
Let f
P such t h a t p Q q and q
E
= p , and f o r i
We l e t p o
nA
N6
6 , A n 5 f N6.
We c a n a s s u m e t h a t a w e l l - o r d e r i n g
E S).
such
c f 6 i m p l i e s fir; E N6.
t o prove t h a t not
it suffices t o f i n d q
<
=
6
such t h a t
So t r i v i a l l y t h e r e i s 6 E S , A 2 6
A,
A , a p E P , a r e i n No.
o f a c l o s e d unbounded s u b s e t o f
order type cf6,
>
s e q u e n c e for s o m e A '
there
<
It is
is q f P,
of
6.
Hence
i '
pi Q q f o r
n 6 t o b e unbounded below 6 .
a closed subset
.
Hence 6 = sup{E
6.
S".
"6 E
q
C
But So
we a r e f i n i s h e d .
21.
T h e o r e m : S u p p o s e I.!
f o r c i n g P, Proof
<
A,
such t h a t i n Vp
: First
assume A = A
and then SO
p
n N
,
so
: i
p =
< A1
is a A - a p p r o x i m a t i n g 6
= {Bi
(w.1.0.g.)
: i
<
i s a p-complete
Then t h e r e
is not stationary.
S*(X)
each B E P appearing i n {B. Clearly there
p regular.
{ B C_ A
:
A times,
< XI =
IBI
and l e t
sequence
of
{Bi
=
A',
: i
<
: i< A>.
E
No;
6 ) f o r a c l o s e d u n b o u n d e d s e t o f 6's.
s * ( A ){6~ < x
: N~
n
p =
tBi
:
i
<
6 ) ) .
A?,
367
ON SUCCESSORS OF SINGULAR CARDINALS
=
P
=
[R
= [a.
B a1t . 1
1
>,
: i Q 5
j
:
.
i}}
Q
a n i n i t i a l segment
a n i n c r e a s i n g , c o n t i n u o u s s e q u e n c e , where The o r d e r o n P i s
1,
only p
C [ G ] i s a c l o s e d unbounded s u b s e t o f : C[G
1"
S"= 0 , w h e r e S" = S'(h)'.
p w " 6 E C[ G
E P,
'v
7'
where 6
c l e a r l y f o r some l i m i t i Q 5 N
6
: i
{Bi
f'
<
: i
={Bi
A}
<
of order type c f 6 , such t h a t a n A namely { a . : j ~
it
lo If X
P = [f
<
h
,
: Dom
different
=
f
2 3 . Lemma
:
ai.
6
and Q
5 < 6 *
Now we h a v e t o p r o v e
S,
< <>,
: j
j
Since 6 E
f o r some So
s",
2 1.
i } belongs t o N6
t o A,
Note t h a t Vp
=
S"(X)
V
6
But t h e r e i s s u c h since
S o we a r e f i n i s h e d when
h
.
<X X =I .
i.e.
may h a v e a Now i n V Q d e f i -
P (the composition)is as required.
a
is regular, p< Aregular,
forcing P such t h a t i n Vp,
S
S C gcf(X).
i s not stationa-
CF(X,
Suppose h i s r e g u l a r , a E
in V,
A fl ,€ E N6.
< jo <
Range f
it
Clearly in
6 1 , and t h e r e i s no unbounded A
contradiction.
,€ < A ,
s"(x)) r-
and f o r e v e r y
A.
l e t Q be t h e c o l l a p s i n g of 2 '
There i s a p-complete
(S -
> 6).
Let p =
S".
,
i } ([aj : j
C o n c l u s i o n : Suppose
ry iff
E
5
Suppose,
g c f ( X ) , b u t S"(X)vQ n g c f ( X l v
ne P a s b e f o r e ,
22.
<
+ 1 - {jo}),
is B .
is
a n d i f G C_ P i s g e n e r i c , l e t
: i 4€,> E G ,
C[G ] = { a 6 : 6 l i m i t , a n d < a j V[G
q2 iff R1
v2.
of
is obvious t h a t P i s u-complete;
It
<
: rll
Aa
S C
X stationary, but F(S) = 4
i s an unbounded s u b s e t o f a o f o r d e r - t y p e
cfu. T h e n f o r e v e r y S' _C S w i t h
/S'I
<
A,
the family {Aa
has a t r a n s v e r s a l ( = o n e - t o - o n e c h o i c e f u n c t i o n ) . . find A'
C Aa
(a
E
S'), IA'I
<
cfa,
s u c h t h a t t h 6 s e t s Aa
S') a r e b ) a i r w i s e d i s j o i n t . H o w e v e r {A
S'}
M o r e o v e r we c a n
Ly
( a E
: as
: a E S l does n o t have a t r a n s v e r s a l .
- Ad,
,
S. SHEW
368 Proof
: See
[ Sh 11.
Lemma : S u p p o s e A i s s i n g u l a r s t r o n g l i m i t ,
24.
= 4
SQ(At)
mod D
,
A+
and l e t
= 16 <
S
: cf6
A+
T h e n we c a n d e f i n e A order-type A) { A a
#
Ho , a n d A w d i v i d e s 6 1
K,
unbounded i n a and w i t h
Aa
C a ( a E S), a -
~ ( c f a )( o r d i n a l m u l t i p l i c a t i o n ) , s u c h t h a t E S l h a s no t r a n s v e r s a l
: a
B) F o r e v e r y sal.
,
= cfA
K
S'
,
L S w i t h IS'I < A +
{Aa : a
E S ' }
has a transver-
Moreover
< At,
B') For e v e r y S ' .G S w i t h I S ' [ such t h a t
t h e r e a r e A:
L Aa(a E
S')
:
( i ) they arepairwise disjoint,
i s a b i g [ a n d even v e r y b i g ] s u b s e t of A a ,
( i i ) A:
which means
t h a t t h e r e i s a c l o s e d ( i n A ) unbounded [ r e s p . coboundedl
C
E
A"
so t h a t
(v6 E Proof
: Stage A
c ) (35 <
<
a : d(a,B)
there is A C
<
+
K
such t h a t f o r every 6
<
At,
(6 t 5
+
6 E A:).
:
There i s a normal d : At
I{6
6
6
(VC)
K )
--*
<
< ill
6, s u p A
K,
hi
h
=
,
< A,
Ai,hi
Z .
i
6 , dlA bounded,
cf6 #
K ,
and each i E A i s a suc-
cessor. Pf -
: Let
c l o s e d unbounded C L At,
A
5
G (ai,ai+l)(5<
the interval), if sup A A:
i
5
,
K )
=
such t h a t
6 E A:
0.
At
'
hence t h e r e is a
Let C
=
{ai
For each i
<
A',
= 4.
C fl So(d)
i n c r e a s i n g a n d c o n t i n u o u s , a. i
4 mod D
d be from 10, t h e n S l ( d )
i : IA51
A<,
Ai 5
<: A+],
ai find
we c a n
is closed ( i n
6 = sup(6
is a l i m i t then
: i
fl
Ail, 5
=
aitl
f o r some 5 .
i n c r e a s e s with 5 and (ai,"
i+l
)
=
U 5
A
i
5
.
Now w e define d'
by
:
ON SUCCESSORS OF SINGULAR CARDINALS if a
<
6 then d ' ( 6 , a )
= rnin { d ( B , a ) , m i n
d'(6,a)
t h a t d ' is a s r e q u i r e d .
>
if ( 3 i ) ( B
d(g,a)
>
a), a n d o t h e r w i s e
It is easy t o check
: a , D E A:}).
{<
ai
369
For s h o w i n g t h a t e v e r y i
is a s u c c e s s o r ,
E A
use subadditivity. Stage B :
<
For any a
At
the family
= {A 2 a
P
-a
<
has c a r d i n a l i t y
Pf 6
<
= {A *
the Bits,
<
dlA i s bounded,
A,
lBil
pa
E
< A,
8.
n
Bi
unbounded
K
a n d dlA bounded
: A
[ c f ( s u p A) # P
,
P'
U
-a 6 , i < K -a, i ( f o r g i v e n i,6 < A ) . L e t B: =
< So Stage C
<
2
Q
(i)
A
,
A
increasing, i n A,
and l e t , f o r i
U
1, a n d b y t h e c h o i c e
Ips . I < a1 1
B , d(B,y)
{y : y
U
-a, i
K,
dlA bounded by
i t s u f f i c e s t o prove
Bi BEBi , and A E P
<
h
6 ) .
i m p l i e s A _C B:.
s o we have proved s t a g e B.
:
If P i s a family of IAl=
Q
c f ( s u p A) # K }
A.
Bi,
i
F','i
K,
Since A E P of
U
a =
: Let
: IAl
/cI
=
h
,
s u b s e t s of A each of
then there is a set C
<
cardinality
A,
but
A such t h a t
K ,
(ii) ( V A E PI
/ A n
CI <
K
.
This is t r i v i a l . Stage D
:
We d e f i n e t h e A ' s Let
a t a.
<
yi : i
by i n d u c t i o n on a f o r a E S .
<
cfa
>
Suppose w e a r r i v e
b e i n c r e a s i n g w i t h l i m i t a , y i + A < y .1 t 1 '
For a s e t A o f o r d i n a l s , l e t a c c ( A ) = { 6 : 6 a l i m i t , sup ( A
f'
6 ) }
lpal
A
,
Q
such t h a t
( = che set of
a c c u m u l a t i o n p o i n t s o f A).
so by s t a g e C w e c a n f i n d :
ci 5
(yi,yi
t
A),
6 =
By s t a g e B , o f power
K
S . SHELAH
370
(*I f o r e v e r y
A E P
U
{u{A
its intersection with c
Y
: y
h a s power
I n f a c t we h a v e t o c h e c k t h a t
[U
pa),
(for A E cfY
i
=
K
Y
a, y E acc(A)}I
[A1
=*
lAyl
<
K
4
We l e t
IAl < A .
t A)
(K
Q
< X
ci.
Siage E
:
{Aa : a
E S ]
Because A
<
X E a c c ( A ) * cfX
:
+ / A / , hence t h e s e t h a s power <
u
Stage E
but t h i s is easy
{A : y
Pal,
E
h a s no t r a n s v e r s a l .
a , by F o d o r ' s t h e o r e m
C
a -
:
We p r o v e ( A * )
We p r o v e b y i n d u c t i o n o n a t h a t
f r o m t h e lemma.
there are big A' C A ( 8 8 6
< a,@ E
S),pairwise
disjoint.
This will
clearly suffice. C a s e 1 : For a a s u c c e s s o r o r d i n a l ,
it f o l l o w s from t h e i n d u c t i o n
h y p o t h e s i s on a - 1 . Case 2
: For
<
a such t h a t ( 3 8
a) 8 t Xw
>
a : proof a s i n t h e
first case. Case 3
an
: For
< anti,
a a l i m i t , cfa U
Ho
= 0.
an, a
, for B
5
A'
B
=
Q a,
An+l - (an 8
<
where a n
A),
I t i s e a s y t o check t h a t A ' C A
5 -
A;
are pairwise disjoint.
< a,
a n ) . pairwise disjoint.
8 E S (hence
+
Choose o r d i n a l s an
For e a c h n , b y t h e i n d u c t i o n h y p o -
t h e s i s t h e r e a r e b i g An C A ( 8 8 B Define A'
.
B
6 #
O ) ,
by
:
< B < anti
i s s t i l l b i g , and obviously t h e
Note t h a t a
E
S , s o we d o n o t h a v e t o
define A'. Case 4 : For a l i m i t ,
not case 2 ,
ded, of o r d e r type c f a (hence < A ) i n c r e a s i n g ) , such t h a t d/Ei
cfa
>
No.
There i s E
<
5
cfal
a, unboun( t h e Bi
a n d E ={L3i+l
:
i
is unbounded f o r i
<
c f a , where
ON SUCCESSORS OF SINGULAR CARDINALS Ei
={Bjt,
cfa S
<
: j
,
i }
Bitl
and each
37 I
is a successor ordinal.
(For
any unbounded A o f o r d e r t y p e c f a i s a s r e q u i r e d ) .
K,
(Remem-
b e r d i s from s t a g e A ) . We c a n d e f i n e f o r l i m i t S i n c e Bi
<
X
t
,
a
<
6
B < Bi
for
5
L e t A:
< 61.
: i
<
X
t
A B be b i g ,
Bitl
(by
pairwise
( p o s s i b l e by t h e i n d u c t i o n h y p o t h e s i s ) .
B'
A; - ( B i
B
if
We now d e f i n e A ' A' 5
Bi
we c a n a s s u m e w . 1 . o . g .
making d e l e t i o n s i f n e c e s s a r y ) . disjoint,
B 6 = s u p {Bit,
cfa,
t
$
U
where
i
A),
Bi
Bi
[Bi,
{a}
) U
,
by
:
< B
X
t
X
t
C l e a r l y , t h e A; & A B a r e b i g , p a i r w i s e d i s j o i n t a n d d i s j o i n t f r o m U [ Bi, i
Bitl
c f j # Ho,
C h e c k i n g d e f i n i t i o n s we c a n s e e t h a t for e a c h s u c h
D =d f
B,
A
n
B
5. ( i
1.
K,
t
A B i s lbjg.
D
F o r w h i c h 6 ' s h a v e we s t i l l n o t d e f i -
A).
<
Bj,
B
cf6) i.e.,
f o r which 5 E S ,
hence
So it s u f f i c e s t o f i n d p a i r w i s e d i s j o i n t
5
( j S cf6, j a limit). T h i s we d o b y i n d u c t i o n o n j . AB. 1 1 S u p p o s e we h a v e d e f i n e d t h e s e f o r e v e r y j ' < j . For j a successor
b i g A;,
among { i
<
cf6
member f o r j
: i a limit}
a successor,
note that c f j # B).
o r 6 . B S, t h e r e i s no p r o b l e m . 1
B j i s a s u c c e s s o r , h e n c e 4 S).
hence c f ( s u p ( E . ) ) #
K,
Now l o o k a t S t a g e D ,
f o r 8.. 3
continuous sequence of o r d i n a l s Since cf
E
c 5
{eE
5 < j}.
:
(yi,yi
t
A),
nedl]
..
<
: i
yi
cf 6 . 1
We t h e n d e f i n e d A
has order type
: 5 E 6 , 5 E a c c ( ~ . ) }1
so c1 6;
has power'<
It is just
n [ u { A ~ ( :~ j)( o )
J
K.
(see stage
a
>
c
Bj
=
B
j '
such t h a t 1
U
8 3. '
i
<
h a s power
{Bj(o) j ,
Bj,
cf
Bj
<
converging t o
and i n p a r t i c i l a r
K,
n ci
1
But what i s a c c ( E . ) ? 1
Otherwise,
We c h o s e t h e r e a n i n c r e a s i n g
1
[ u {A
limit}
l
5 . # No, t h e r e i s a c l o s e d unbounded C
i E C * yi where c i 'j
<
hence E . E P
K,
1
(Re-
: j(o)
<
j ,
j(o) a l i m i t ,
K.
j(o) a
A.
l(0)
defi-
S . SHELAH
372
, where
i t i s d i s j o i n t from A
'j(o) have f i n i s h e d t h e p r o o f . Stage E
a
s,
€
<
a
IA,~
<
Kcfa f o r a
<
cfa t
K
(<
<
f o r e a c h a f o r some i
and
:
IBiI
(35 <
cfa)
X 5*'s,
B
(We d e f i n e hp
< B,,
,
A+
<
a n d f o r e v e r y a.
A))
<
a
a
0'
AS such t h a t
E S ) ,
cfa F,
K ) ( V g ) ( <
We c a n
f o r any
BO, B1
<
8"
<
cIo
<
K
8
i o .< i
y ( a , ~ i + c )E A').
+
increase with 5 , a = U
B:
S
< X .
so that
tuples
Suppose
~ ( c f a ) }, w h e r e y ( a , i ) i n c r e a s e
For e v e r y a , c h o o s e B 5 G a , B 5
<
S a stationary
K ,
and f o r a n y
E
cl A a ( f o r a
there are pairwise disjoint A'
Proof
<
= {y(a,i) : i
jA*(/
with i , (hence
<
=
cfh
T h e n we c a n f i n d A* C a f o r a -
has a transversal.
a
s o t h a t A*
( V i
j
a n d e v e r y member o f S d i v i s i b l e b y Am.
At,
f u r t h e r A~ _C a , :
<
j(o)
Suppose X s i n g u l a r s t r o n g l i m i t ,
:
subset of
{Aa
, and o b v i o u s l y 'j is a l i m i t . S o we
it is a big subset of A
I t i s e a s y t o check t h a t
I
B+
[ X i ,
A,
For e a c h a
< ,
,)
d e f i n e f u n c t i o n s h o , hl,Dom
5
B
<
K ,
A
E
At,
, there are
B5
BO
= B1,
such t h a t hl(B*)
hp
h2(B")
= A.
X ( i t 1 ) ) f o r e a c h i ; t h e number o f p o s s i b l e
6, 5 , Bo > i s < X, so t h e r e i s no p r o b l e m ) . E S
choose an i n c r e a s i n g sequence
B(a,i)
( i
<
cfa)
converging t o i t . First note that B(a,i) t A<
( v a 0
B(a,itl),
<
t h
a) a .
a n d B ( a , i ) i s d i v i s i b l e by A .
Now we d e f i n e b y i n d u c t i o n o n j ordinal
y(a,j),
( i ) B(a,i)
<
a E S) h e n c e w . 1 . o . g .
= i~
t
5
(i
<
cfa,
5
<
K )
an
i n c r e a s i n g w i t h j, s u c h t h a t
y(a,j)
<
B(a,i) + A ,
The l a s t c o n d i t i o n e x c l u d e s
<
h
y'", a n d t h e c o n d i t i o n s
(ii),
( i i i )
ON SUCCESSORS OF SINGULAR CARDINALS
a r e s a t i s f i e d by A
y's,
S o we c a n d e f i n e A'
=
<
6(a,i)
y
<
: i
(y(a,i)
<
373
6(a,i) t A ~ ( c f a ) ), a n d
y(u,i)
increase
with i and converge t o a . Now we a r e g i v e n a ( o ) red.
Define A
= { y ( a , ~ it 5 )
: i
<
Note
{Aa
as
A t
< a(o)l.
a
:
requi-
c f a , f ( A a ) E Aa n B E ( a , i ) ) .
.
C l e a r l y it i s a very b i g s u b s e t of A On S n a ( o ) we d e f i n e a g r a p h
c
a n d h a v e t o f i n d Ad,
A'
there is a transversal f of
By h y p o t h e s i s , 1
<
( a l , a 2 ) i s a n e d g e i f f A1
:
n A:
al
# $.
2
:
( a ) If
=
hl(y)
= cfa2 (because y
i s an edge then c f a l
(al,a2)
E A
c f a 1. ?!
implies
"P
11) i s
( b ) T h e v a l e n c y o f a n y al ( = I{a2 : ( a l , a 2 ) i s a n e d g e
IAEI.
G
A s f i s one-to-one,
whenever A A1
al
"2
n
,
A:
nAa
then 6
1
it s u f f i c e s t o prove t h a t f(Aa
=
6(al,il)
2
61
5 Aa , a s
B8(al,il)
1
6 ( a 2 , i 2 ) ( i t is t h e b i g g e s t o r d i n a l
51
2 G y d i v i s i b l e b y A ) , s o Aa b u t f(A, ) t Aa n B562( a 2 , i 21)
E Aa 2 1 y(a2,yi2 t C2) E
y ( a l , ~ i l t 5,)
If y
# $ .
=
B6(al,il) (since y
Aa
A1 a2
E
required.
'
2 ) hence f(A,
2
52
B(a2,i2)'
1
E Aa
n 1
Now we d e a l w i t h e a c h c o m p o n e n t C o f t h e g r a p h s e p a r a t e l y . By ( a ) , a l l a ICI G
K
t
u.
E
If
C have t h e same c o f i n a l i t y , s a y
u >
n o t e t h a t e a c n A1
K
u n b o u n d e d b e l o w a , h e n c e a1 # a So l e t C
25.
=
C
*
i y ( a , ~ it
5)
:
If
5 <
u
K )
K ,
for i
and by b ) ,
u
has order type
I
A:
= { a g : 5 < u ) , a n d we c a n d e f i n e
which a r e as r e q u i r e d . each
2
u ,
A:
n 1
At,
A1
I < -
and i s
.
u u
A'
,
5 5<5 a 5 we g i v e a s i m i l a r t r e a t m e n t t o
<
P,
a E C.
Conclusion :
1) S u p p o s e H
is a strong l i m i t .
a ) T h e r e i s a f a m i l y of Hwtl
c o u n t a b l e s u b s e t s of Nutl which does
374
S.
SHELAH
n o t have a t r a n s v e r s a l , b u t e v e r y subfamily of c a r d i n a l i t y
< so t 1
has
a transversal. b ) There is an a b e l i a n group [ g r o u p ] o f power f r e e , but every subgroup of
is strong l i m i t f o r P
Suppose KwP
2)
cardinality Q
<
Kwtl
n.
which i s n o t
flu+l,
is.
Then a ) , b ) h o l d f o r
*wnt1Proof
1 a),
:
and Shelah [ MS
2 a).It
is easy t o s e e t h i s a f t e r reading Milner
1.
1 b ) , 2 b) are easy t o see.
26.
Claim
: Suppose
X i s s t r o n g l i m i t , cfX =
and : P i s p-complete
so, p <
p regular
Y ,
or among a n y p m e m b e r s o f P t h e r e a r e p w h i c h
are pairwise compatible. If i n V p
h is still a strong l i m i t
cardinal, then
S*(~+)V n c F ( X , p ) v , s*(i+)vP a r e ,equal ( i . e . , Proof
: Let
d
VP
f o r some r e p r e s e n t a t i o n t h e y a r e e q u a l ) .
At
:
n cF(.i,u)
+
K
Clearly i t i s s t i l l normal i n V
be normal.
t o prove t h a t t h e t r u t h v a l u e of
By 1 3 i t s u f f i c e s
P
.
" a E Sl(d)"
i s n o t changed, which i s q u i t e e a s y .
27.
Claim : If
x
i s supercompact,
X
> x,
cfX
< x,
t h e n S*(Xt)
is
stationary. Proof C
c
: Let
d : At
+
cfX b e n o r m a l a n d s u b a d d i t i v e , a n d s u p p o s e
At i s c l o s e d a n d u n b o u n d e d .
Suppose N(
(H(Xtt),
s u b s e t of N
n
Let 6"=
At
E),
c f h t 1 L. N ,
C l e a r l y cf6"
c a r d i n a l o f c o f i n a l i t y cfX s o c f b "
6"
E
E
N,
IINII
belongs t o N ( t h i s i s possible a s
sup(N n A').
ded, hence
C,d
C;
x
< x
and every
i s supercompact)
i s t h e s u c c e s s ~ ro f a s i n g u l a r
>
cfA.
so it s u f f i c e s t o prove
Clearly C 6"
n
N
4 So(d).
i s unboun-
375
ON SUCCESSORS OF SINGULAR CARDINALS So suppose A L 6" is unbounded, and d l A is bounded by 5 . Let A = { B .
:
i
<
B i increasing.
&*I,
for each i there is y i ,
ci =
Max { < ,d(pitl,yi),
ci =
5"
for every i.
We may assume, w.l.o.g.,
Bi < y i < Bitl , d(yi,Bi)}
Now i f i
<
<
cfh
yi E N.
<
cf6".
Let So (w.1.o.g.)
j , then by the subadditivity :
d(yi,y.) G max {d(y.,5j+l), d(5jtl,Bi+l), d(5i+l.~i)) G 5" 1 1 So dl{yi : i < cf6*> is bounded, but the set necessarily belongs
(H(A
to N , a n d , as N <
tt
1 , E ) , there is an unbounded B E h + on
which d is bounded, giving an easy contradiction to normality.
28. Remark : We in fact prove that if d is a subadditive function,
with domain a * , a
Q
a*,
and d is bounded on some unbounded A 2 a , a has an unbounded subset A"
then every unbounded A' such
A'
5a
that dlA" is bounded.
29. Conclusion : If ZFC t
"
3
a supercompact" is consistent then
the following is consistent :
ZFC
GCH
t
Proof : Suppose
x
+
11s*(Nwtl)
is stationary".
is supercompact, and also (w.1.o.g.) GCH holds.
Let h be the first singular cardinal a regular u
< x
such that S * ( h t )
S"(Xt)v
P So n o w , in V , p is N1.
17 C F ( h + , p ) " ,
Now collapse S"(A+)v
x
.
n CF(ht,u)
Levy collapsing P to collapse every conditions).
> x u' <
By 27 we can choose is stationary. to N o (hy finite
By 2 6 , in Vp,
,
+ ) vp -
and the latter obviously remains stationary.
t o N 1 by a Q which is N1-complete.
n CF(A',U)~
S*(h
We use
Again
remains stationary and i s still included in
S*(
X+)P*Q.
Oh
is not a strong requirement
30. Definition : Let A be a regular cardinal and E C h a stationary
376
SHELAH
S.
set i n i t .
<
There i s
(1) O:(E).
>
: a E E
W
such t h a t f o r every
a , Wa
lWal G l a [ , a n d f o r e v e r y X EX
i s a f a m i l y of s u b s e t s of a w i t h t h e r e i s a c l o s e d and unbounded C
s u c h t h a t X 17 a E W
h
for
all a E C n E.
L A,
every X
31.
<
There is
(2) O h ( E ) .
Theorem
x
{a :
n
=
CL
>
: a E E
Sa
such t h a t S
,
a
C
a -
and f o r
is s t a t i o n a r y i n h .
Sa}
: ( K u n e n ) : (1) For s t a t i o n a r y E
LA,
Or(E)
implies
Oh(E).
(2) For E l
3 2 . Theorem either ( i )
uK =
: Suppose
implies
Oh(E1)
= 2’ =
h
0
A
and O*(E2) i m p l i e s A
(E2)
a n d f o r some r e g u l a r
pt
a n d for e v e r y 6
# cfu
K
<
Proof
: Case : Let
( i ) is due t o Gregory [ G r
<
: a
>
h
<
{a
than
K
For a
:
E E ~ K )l
= {u Y A, l e t
Given X t h a n A,a6
5y6
C be
= [6
: 6
=
X E
K ,
1
i
<
AB
Y
-P
X &
i
<
X l w h e r e a.
:
B <
U
a, A (ai
K}.
=
2’
ut)
a,
a>. X E
Y
{AB
:
5
<
a]}).
is any s u c c e s s o r less
is the least a
f o r l i m i t 6 , and ai+l
<
=
e t Wa b e t h e s e t o f a l l u n i o n s o f n o m o r e
{a
a
s u c h t h a t f o r some y Now C ’
<
: IYI
cfa
be a l i s t o f a l l bounded s u b s e t s o f X
s u b s e t s of a b e l o n g i n g t o
(wa
A :
1.
each appearing h times ( t h e r e a r e A such s u b s e t s a s h Case t i )
p,
< v
v,16IK
O T ( E ( K ) ) where E ( K ) i s t h e s t a t i o n a r y s u b s e t
Remark
<
K
’ , or
( i i ) p is singular
Then
,
E2C_ h
C_
>
a
i
= X n a..
: ai
< 611
6 E C flE ( K ) t h e r e a r e i ( j ) a n d y .
1
<
i s c l o s e d u n b o u n d e d , a n d for 6 ( j
<
K )
such that
ON SUCCESSORS OF SINGULAR CARDINALS
Case ( i i )
<
: j
1
: For
>
p
X n ai ( j )
5
A~
<
: ( 3 j
X
:
p)
<
Let
K
33. K
=
=
6
E
W6. 6
where
V.
U
& j < v I' IVj < p.
I
p,
IQI
K j .
=
A
f ( a ) > s u p fee,.
f(a) X such t h a t f o r a E C ,
and f o r i n c r e a s i n g < 6
< a
6
: i
i
<
K
2
6
iyK
=
6i.
such t h a t
6 I V . n {f(6i) : i < 1
K}I
C o n c l u s i o n : (GCH) I f
# cfX.
Yj
a.
6 E C fl E ( K ) ,
There e x i s t s j
<
vS, 1
C_
There e x i s t s a c l o s e d unbounded C implies f ( 8 )
let
K ,
A
j
be such t h a t X n a
h
--*
Q
u
xn 6 =
.so
j
i s i n c r e a s i n g and f o r j
let f
h
=
6 such t h a t cf6 =
{,.gQ Acr
L e t W6 b e Given X
,
6
ai(j)
U
j
377
In particular OX
hence X
X
>
r'l
6 3
6 = U {X n 6 . : i <~ , f ( 6 E~ )V . }
H o , t h e n 0"
A+
holds.
( E ( K ) )
E
W
6'
h o l d s , whenever
F i n a l comments
1) T h e r e s t r i c t i o n " A s t r o n g l i m i t " a t t h e expense of complicating
i n most c a s e s c a n b e weakened
t h e r e s u l t s : assuming (Vp
and r e s t r i c t i n g o u r s e l v e s t o CF(At,
p"<X,
<x)
<
A)
x).
o r CF(Xt,,<
2 ) A m o r e s e r i o u s q u e s t i o n i s w h e t h e r we c a n , i n 7 , r e p l a c e Dg b y
X
DX.
This remains open. N o t e t h a t t h e n a t u r a l n o t i o n i s S2(F), a n d t h a t for r e g u l a r A ,
= {A 5
I+(x)
A
: f o r some A - a p p r o x i m a t i n g
i s always a normal i d e a l .
= {A
I-(?,)
5
X
: A fI
i s a normal i d e a l . {A : A
E A.
sequence
A
s,(R))
Similarly
B E $ mod D X f o r e v e r y B
E
It(X)}
The m e a n i n g o f c l a i m 7 i s t h a t
mod DX}
N,
f o r some A o ,
when g c f ( X )
It(X)
= A.
is
Another formu-
l a t i o n o f our q u e s t i o n i s w h e t h e r t h i s a l w a y s h o l d s . H o w e v e r , we c a n m e a n w h i l e i n t e r m s of
It(A)
j u s t formulate t h e l a t e r theorems
i n s t e a d o f S*(X)
(and t h e changes i n t h e proofs
378
S. SHELAH
are minor).
By t h e way i t may b e m o r e n a t u r a l t o u s e
= [6: t h e r e
S3(N)
ded s u b s e t o f 6 ,
is a function h,
(‘di
<
cf6)
Dom h = c f 6 , R a n g e h a n u n b o u n -
h / i E Ng,
and N
rl A =
6
( i n gcf(A)
6 )
it does not m a t t e r ) . 3 ) Why w e r e we i n t e r e s t e d m a i n l y
in
and n o t i n e . g .
N
w t 2 .?
The a n s w e r i s t h a t s e v e r a l i n d u c t i v e p r o o f s work f o r s u c c e s s o r s o f a n d i t was n o t c l e a r w h e t h e r t h e y f a i l a t s u c c e s -
regular cardinal,
sors o f s i n g u l a r s . 4 )
I t may b e o f
(But s e e remarks 5 and G below).
i n t e r e s t t o mention our o r i g i n a l l i n e of
which i s n o t s o t r a n s p a r e n t We w a n t t o
prove
Kwtl-approximating d : K
w t l
i s an
S,(F) i s q u i t e “ b i g ” , w h e r e
sequence f o r K
w t l
’ assuming GCH.
and u s i n g t h e ErdBs-Rado
S o we l e t
theorem
is closed of order type p r o v e t h a t i f C C_ H wtl
+ + (Nntl)Eo,
then it contains
. Ci
from t h e p r e s e n t p a p e r .
that
KO be normal,
+
thought,
C1
o f o r d e r t y p e i-4 n + l ~w i t h d c o n s t a n t
( t h e s e t of accumulation p o i n t s of C
closed subset of C of order type N
is
1
c
S,(N)
and
This proves t h a t S2(N)
ntl‘
i s i n some s e n s e b i g . 5 ) We c a n t r y t o g e n e r a l i z e 4 ) t o o t h e r c a r d i n a l s .
Let
K
<
=cf K
Definition
: Call an
if f o r every {a
<
Ma.
K
a
atn
has cardinality
(nt1)-place
...
<
:d(ao,al
<
H
<
Hatn
f u n c t i o n d from there is k
,..., a k - l , a , a k t l , . . . , a n )
:
Lemma
: Let
atn
to
normal
K
n such t h a t
= d(Oo,..
,ak ,..., a n ) }
.
Claim : There i s a normal f u n c t i o n d : K Proof
<
H
atn
--t
K .
By i n d u c t i o n o n n . -
N be an K
closed subset of K
atn
atn
-approximating
sequence f o r H
of o r d e r t y p e l n t , ( ~ t
pit,
a t n t ~ ’
a
<
A.
where p
ON SUCCESSORS OF SINGULAR CARDINALS
379
Then C h a s a c l o s e d s u b s e t of o r d e r t y p e pt which i s i n c l u d e d i n S2(m
*
Proof
: Let
d
No,
E
theorem ( l n t l ( ~
+
d : Hatn
pIt
--*
( p
on which d i s c o n s t a n t .
+
t n t l JK
If
K,
d normal.
) there
By t h e E r d i i s - R a d o
i s C1
C of
6 E C i , t h e n C1 fl6
2
order type
witnesses that
6 E S,(iS). Suppose N
6) K
<
<
p
is strong l i m i t ,
1Y ( p ) <
and
N
Nu.
If
,
cf N
K
E
is a H
E+Y
y
a successor ordinal,
-approximating
sequence
has order type 1 ( u ) ~ , then C has a U+Y Y c l o s e d s u b s e t C1 o f o r d e r t y p e u t w h i c h i s i n c l u d e d i n S 2 ( N ) .
NUtytl,
for
and C
N
Proof
: We p r o v e a s o m e w h a t
If C
N
UtB’
B
<
then there is C
P
<
n,
if a .
a successor ordinal, and C has order type>lB(p)+,
y
c
l -
< ...
(H(Hatyt1),E)
\
( T h i s i m p l e s C;
stronger statement :
C n S2(N)
v(uo
5
E
of o r d e r t y p e
s u c h t h a t for s o m e
C1 t h e n
,..., u n ) e
I{x
: v(ao,... ’ a P - l ’ X ’ ~ P
,..,an)W
Ha
S2(N)).
We p r o v e t h i s b y i n d u c t i o n on B . above,
pt,
For f i n i t e B t h i s was d o n e
and the induction s t e p is easy.
REFERENCES [ E ]
Eklof.
[ F ]
L.
Fuchs.
I n f i n i t e a b e l i a n g r o u p s , Academic P r e s s ,
London, [ G r ]
[MS
1
Gregdry, J. E.
Vol.
Symb.
Milner and S. Symp.
I 1970, Logic,
N.Y.
&
Vol. 11 1 9 7 3 . Sept.
1976.
S h e l a h , Two t h e o r e m s o n t r a n s v e r s a l s ,
Proc.
i n Honour o f ErdGs 6 0 t h B i r t h d a y , Hungary 1973.
S . SHELAH
380 [ S h 11
S.
Shelah, Notes i n p a r t i t i o n c a l c u l u s , i n Honour of Er>dZs 6 0 t h B i r t h d a y ,
[Sh 2 1
,
Proc.
Hungary
o f Symp. 1973.
A c o m p a c t n e s s t h e o r e m for s i n g u l a r c a r d i n a l s .
Free Algebras,
I s r a e l J . Math.
Whitehead problem and t r a n s v e r s a l s ,
21 ( 1 9 7 5 ) , 3 1 9 - 3 4 9 .
LOGIC COLLOQUIUM 78 M . Boffa, D. van Dakn, K. McAloon ( e d s . ) 0 North-HoZland Publishing Company, 1979
P A U L B E R N A Y S by E. Specker Eidg. Technische Hochschule Zurich Paul Bernays was born on October 17th 1885 in London; he died after a short illness on September 18th 1977 in Ziirich.
He was the son of Julius and Sara
Bernays, n6e Brecher. His father was a businessman and curriculum vitae appended to his thesis citizen of Switzerland. (1)
-
-
as he states in the
he was of jewish confession and a
Soon after the birth of Paul, the family moved to
Paris and from there to Berlin.
It is in Berlin that he attended school, from
1895 to 1907. He seems to have been quite happy at school, a gifted, well adapted child accepting the prevailing cultural values in literature as well as in music. It was indeed his musical talent that first attracted attention; he tried his hand at composing, but being never quite satisfied with what he achieved, he decided on a scientific career.
He studied engineering at the Technische Hochschule
Charlottenburg for one semester, then realizing (and convincing his parents) that pure mathematics was what he wanted to do, he transferred to the University of Berlin.
His main teachers were: Schur, Landau, Frobenius and Schottky in mathemat-
ics; Riehl, Stumpf and Cassirer in philosophy, Planck in physics. After four semesters, he moved to Gijttingen; there he attended lectures on mathematics by Hilbert, Landau, Weyl and Klein, on physics by Born, and on philosophy by Leonard Nelson. Nelson was the center of the Neu-Friessche Schule
-
Bernays was quite an active
member of the group and stayed in contact with it all his life. cation
-
in 1910
- was
His first publi-
"Das Moralprinzip be< Sidgwick und be< Kant", published in
the Abhandlungen der Friesschen Schule. (2) There were two further publications in
1913 in the same Abhandlungen, one "Ueber den transzendentalen Idealismus", the other "Ueber die Bedenklichkeiten der neueren Relativitatstheorie". ( 3 , b ) Thoughwe no longer share the difficulties discussed by Bernays, it is remarkable how calmly he takes part in otherwise rather heated controversies. There is no doubt that Bernays was deeply influenced by Nelson - by his liberal socialism as well as by his revised version of Kant's imperative demanding thepermanentreadiness to act' according to duty (Nelson lived from 1882 to 1927). In the spring of 1912 Bernays received his doctorate withadissertation (written with Landau) on analytic number theory
-
the exact title being: "Ueber
die Darstellung von positiven, ganzen Zahlen durch die primitiven, binaren quadratischen Formen einer nicht-quadratischen Diskriminante." (1) 18 1
E.
302
SPECKER
At the end of the same year he obtained his llabilitation at the University of Zurich where Zermelo was professor. His Habilitationsschrift was on function theory: "Zur elementaren Theorie der Landauschen Funktion * ( a ) " . ( 5 ) From 1912 to 1917, Bernays was Privatdozent in Zurich, There are no publications (except the one's already cited of 1913) in this period. Bernaysmusthave passed some crisis in these years.
In the short biography published in the book
"Sets and classes" (6), he states: "At the beginning of the First World War, I worked on a reply to a critique by Alfred Kastil of the Fries philosophy. 'This reply was not published
- by
the time
there was an opportunity to have it published I no longer agreed with all of it." He even considered giving up mathematics at this time - but did not see anything he felt he could do better.
Therefore, Hilbert's proposal to be his collaborator
in Gdttingen must have come as a relief to Bernays. That he found his way back to mathematics is shown by his Gdttinger Habilitationsschrift, a brilliant piece of work written in a short time.
Bernays left Zurich in 1917; his paper was sub-
mitted in 1918. The title is "Beitrage zur axiomatischen Behandlung des Logikkalkuls". (7) In retrospect it is hard to understand that the paper was not published at that time - parts of it appeared in 1926 in the Mathematische Zeitschrift. (8) Bernays explained this long delay once in the following way: "To be sure, the paper was of definite mathematical character, but investigations inspired by mathematical logic were not taken quite seriously of as amusing, half-way part of recreational mathematics.
-
they were thought
I myself had this tend-
ency, and therefore did not take pains to publish it in time. It has appeared only much later, and strictly speaking not quite complete, only certain parts.
Many
things I had in the paper have therefore not been recorded accordingly in descriptions of the devlopment of mathematical logic".
(?I
An analysis of the content of the paper will be given in some detail
- the
library at the ETH in Zurich is in possession of Bernays' copy. As to his activity in Gdttingen in the period 1917 - 1934, Bernays describes it as follows: My work with Hilbert consisted on the one hand in helping him to prepare his lectures and making notes of some of them, and on the other hand in talking over his research, which gave rise to a lot of discussions. (6) Bernays also gave lectures on various subjects of mathematics and became an extraordinary professor of mathematics in 1922.
Besides a series of papers
written as a collaborator of Hilbert (a typical title being: Die Bedeutung Hilbert's fur die Philosophie der Mathematik (10)),there are two papers where he is on his own: One with Schanfinkel (Zum Entscheidungsproblem der mathematischen
3a 3
PAUL BERNAYS
Logik (ll)),
where a case of the decision problem is treated, the other: Zur ma-
thematischen Grundlegung der kinetischen Gastheorie ( 1 2 ) (where a special case of the ergodic theorem is proved).
That his activity is only partially reflected by
his publiations is shown by references in the literature of students of Hilbert's. We learn e.g. from Ackermann in 1928 that the axiomatization of predicate calculus based on the rules
(with the well known condition on II, ) is due to Bernays. (20, p. 54) In 1933 Bernays, as a "non-aryan", lost his position at the University of GGttingen. In 1934 he moved to Ziirich - to call it a return would give a wrong impression; still in the fifties he could say: "bei uns in GGttingen" (we in Gbttingen).
In the same year 1934, the first volume of the Grundlagen der Mathe-
matik appeared. (13) It was hailed from the beginning as a masterpiece on a par with the works of Frege, Peano, Russell-Whitehead. From 1934 to his death, the home town of Bernays was Ziirich.
Being single,
he first lived with his mother and two sisters, in the last years with his sister Martha, who survives him. Twice he spent a year at the Institute for Advanced Study in Princeton (1935/36 and 1959/60), and three times he was visiting-professor at the University of Pennsylvania in Philadelphia. In Ziirich, he was first a Privatdozent, then a professor till his retirement in 1958. During his first stay in Princeton, he lectured on mathematical logic and on axiomatic set theory. His lectures on logic have been published as notes under the title: "Logical calculus"
(14) - much of the material is taken up inthe second
volume of the Grundlagenbuch (1939). (15) The consistency theorem (a central theorem of the second volume) is contained in the Princeton notes. In set theory he lectured on his own axiomatization. He had presented this system already in Gbttingen in a lecture of 1929/30, but hesitated to publish it because he felt that the axiomatization was, to a certain extent, artificial. A s
. to publish -
Bernays records,he expressed this feeling to Alonzo Church, who replied with a consoling smile:
That cannot be otherwise. (6) This perbuaded him
the work appeared in seven parts in the years 1937/1954 (16)and (almost unchanged) in the volume "Sets and classes". (6) (The same volume contains an English translation of a paper published in the anniversary volume of Fraenkel: "Schemata of infinity in axiomatic set theory". (17)
3 04
SPECKER
E.
From the first volume of the Journal of Symbolic Logic up to volume 40, he published about a hundred reviews
-
it is he who reviewed there the fundamental
papers of Gddel, Church, Gentzen, and many others. His last review is on the first volume of Schroder's algebra of logic
-
an essay of 6 pages in small print.
But it was not only through reviews that he reacted to the development correspondence was immense.
-
his
The list of his correspondents counts up to a thou-
sand persons, there are preserved up to 6000 copies of letters, many of them rather an essay than a letter. In the second part, the work of Bernays shall be discussed in more detail. H i s main works may perhaps be grouped under three headings:
( 1 ) Logic; (2) Set theory; ( 3 ) Philosophy.
There are, of course, papers which do not fall within these groups - papers on theoretical physics, calculus of variation, and especially on elementary geometry. The first (and most detailed) analysis is on his Habilitationsschrift of The paper is on propositional calculus - following Russell and Whitehead,
1918.
it is (except in studies of independence) based on the connectives of negation and disjunctions; implication is considered an abbreviation. The starting point of Bernays are the following axioms (slight variations of the axioms of PM):
(PVP) P
+
+
P
(P vs)
(PVd
+
pv(qVr) (q+r)
+
fqvp) +
(pVq)vr
((pvq)
.+
(pvr))
Rules of inference are substitution and modus ponens.
Bernays then introduces
the following two notions: Derivable formula (ableitbare Formel) and identity (allgemeingiiltige Formel), the latter being defined by the truth table method. He shows that a formula is derivable if and only if it is an identity. The non trivial part is based on the following lemma: If a non derivable formula is added to the axioms, then every formula is derivable. The lemma is proved via the normal form theorem: Every formula is equivalent to a conjunction of "simple disjunctions". Bernays goes on to remark: This consideration gives furthermore a uniform procedure to decide whether a formula is derivable or not. In the next paragraph is stated
(though not proved) what is known as "Post-
completeness": "With respect to the logical interpretation of our calculus (which was at the
385
PAUL BERNAYS
origin of this study), we obtain the result that the totality of provable formulas coincides with the totality of identical formulas. And this means that our calcul u s contains a formal systematization of those laws of logic which concern rela-
tions of truth and falsity of propositions subsisting independently of their structure and content.
Indeed, all relations between truth and falsity of propo-
sitions may be expressed with the help of conjunction, disjunction and negation, and therefore also with the help of the symbols of our calculus, and insofar these relations hold for arbitrary propositions, the corresponding formal expressions must be identical formulas in the sense defined." The next question considered is the problem which connectives form a basis. Besides giving complete answers (for the classical cases), partial systems are introduced and the following theorem is stated: If CY (say, in -,,v), either a V,A,+
or
YLY
is any formula
(and not both) is equivalent to a formula in
alone.
(It is added that there exists a finite system of axioms for the identical formulas in v , A , +
.)
In the next paragraph, the independerce of the five formulas is studied. First of all, it is shown that the formula expressing associativity is derivable from the
4 others.
Next, it is shown thatnolie of the four others are derivable from the rest.It is here (as far as I know), that "many-valued" logic occurs for the first time. Bernays describes the method as follows: "In each one of the following proofs of independence, the calculus is reduced to a finite system for the elements of which a composition ("symbolic product") and a "negation" is defined, and this reduction is carried out in such a way that the variables of the calculus are related to the elements of the system as their values.
The "correct" formulas shall be characterized by the fact that they assume
only values of a certain given subsystem." The most complicated system,introduced for these independence proofs, is one with 4 elements and a subsystem of 2 elements. The last paragraph contains a detailed study on the possibility of replacing axioms by rules. There is given e.g. a system containing the only axiom p and six rules.
-C
p
Bernays' best known contribution to mathematical logic is the work "Grundlagen der Mathematik", published under Hilbert's and his name, but written by Bernays alone.
It is mique because of the wealth of material it contains -
SPECKER
E.
386
published there for the first time (as much of what had been achieved by the Hilbert school in proof theory) or published in a more detailed form than in the original papers.
It was - for a long time - a standard reference on mathematical
logic, proof theory, arithmetization of metamathematics, recursion theory. it is unique also in another respect
-
But
"foundation" is taken quite literally, it
does not reduce mathematics to logic, or logic to mathematics - both are developed at the same time and (to some extent) the philosophy of mathematics along with it.
Bernays once was asked why he had preferred mathematics to music as a
career. One of the reasons he gave was that he had difficulty in following three tunes simultaneously - the Grundlagenbuch shows that he hadn't this difficulty in foundation. The work contains, o f course, much material original to Bernays, but as he chose not
to divide his work in definitions, theorems, proofs, it is perhaps
best not to single out results which might be given the name "theorem of Bernays". A further major contribution of Bernajw is his system of set theory, first presented in the year 1929/30 in Gijttingen, and then elaborated in a series of papers in the years 1937/54.
The basic result is, ofcourso,known to everybody -
how the fact that predicate logic can be based o n , say, disjunction, negation and existential quantification is transformed into an axiomatization of the concept
of class in such a way that a finite number of class axioms suffice to prove a general scheme of class existence. What is perhaps less well known is the careful study of basing parts of mathematics o n subsystems of the full system of axioms there are discussed e.g. three systems, sufficient to develop analysis.
It is
here that Bernays introduces his weakened forms of the axiom of choice (as the axiom of dependent choice) and shows that they are sufficient for analysis in a wide sense - including, say, Lebesgue measure theory and the theory of function spaces. If a general tendency of these studies is to be mentioned, the most distinctive feature is that Bernays tries to get along without sum axiom and power set axiom as long as possible. This is shown to be feasable in nmber theory, analysis, as well as in "general set theory".
In the book "Axiomatic set theory" (18) of 1958, a somewhat different version of the system is given - in stating the axioms, the existential form is replaced by the use of primitive syzbols. Furthermore, the succession of steps in the development of the theory is different. About half of the papers of Bernays' may be classified as philosophical. A collection offourteen,referring to mathematics, has been edited by the Wissenschaftliche Buchgesellschaft under the title "Abhandlungen zur Philosophie der
PAUL BERNAYS
387
Mathematik". (19) The first article published there dates from 1927, the last from 1971. The thinking of Bernays is characterized by his constant effort to do justice to all aspects of the problems he considered. He believed in their inherent complexity and always resisted the temptation of explaining away. A typical example is geometry - for him it had not vanished in an abstract structure and/or a part of physics, nor did he adhere to reductionism so common in foundational studies. It is clear that such an attitude excludesshortanswers to almost all problems. Nevertheless, in order to give some impression of Bernays' way of philosophical thinking, a short text is preser.ted.
(The German contains a word
-
"Sachhaltig-
keit" which no dictionary lists.It has been translated by "reality".)(The
text is
from an essay on philosophy of mathematics, presented at the International Congress of Philosophy in 1969. ( 19, p. 174,175) "It seems appropriate to attribute to mathematics a reality which, however, is different from that of material world.
That there exist other types of objectiv-
ity than that of material world is shown by objectivity in the domain of phenomena.
Mathematics is insofarphenomenologicalas it is concerned predominantly with
the study of idealized structures and is furthermore governed by the method of deduction.
In the process of idealizing,the phenomenological and the conceptual
come into contact.
(It is therefore inappropriate to oppose these two to such an
extent as is done in Kantian philosophy.)
The specific character of mathematics
as opposed to empirical science does not mean that we have in mathematics knowledge a prior?.
It seems necessary to concede that we have to learn also in the
domain of mathematics and that we have there a kind of experience sui generis. (We may call it mental experience.) This is not prejudical to the rationality of mathematics. Rather it seems a prejudice that rationality is necessarily linked to certainty. Certain knowledge in the simple and full sense is given us almost nowhere.
This is the old insight of Socrates."
For those who have not known Bernays personally, a few words on his personality may be added. As his immense correspondence, his friendliness to visitors, his acceptance
of invitations to congresses until the last years of his life. clearly show, he liked the contact with other human beings.
He was extremely benevolent, helping
many an aut.hor with his papers - from Hilbert to a high-school teacher having made some small discovery. On the other hand, he lived in an aura of detachment. He was unique in his refusal to judge other people; he never spoke badly of anybody
-
there is every reason to assume that he did not even think badly of others. When,
388
SPECKER
E.
m c e , refereme was made to a statesman almost universally recognized as one of the villains of this century, in order to induce him to a negative judgement, he replied: "My situation is so different from his, that it is not for me to pass judgement". There is no doubt that his gift of seeing everywhere the best and refraining from judgement where he could not see anything good, helped a great deal to free foundational studies from the situation where different schools are expected to fight one another.
In the name of all those who have known Bernays personal.ly, it certainly may be said: We are grateful for the privilege to have been in contact with Bernays. References (1912) "Ueber die Darstellung von positiven, ganzen Zahlen durch die primitiven, binaren quadratischen Formen einer nicht-quadratischen Diskriminante", Dissertation, GGttingen.
(1910) "Das Moralprinzip bei Sidgwick und bei Kant", Abhandlungen der Friesschen Schule, 111. Band, 3. Heft. (1913) "Ueber den transzendentalen Idealismus", Abhandlungen der Friesschen Schule, IV. Band, 2. Heft, 367-394. (1913) "Ueber die Bedenklichkeiten der neueren Relativitatstheorie", Abhandlungen der Friesschen Schule, IV. Band, 3. Heft. (1913) " Z u r elementaren Theorie der Landauschen Funktion +(a) ' I , Vierteljahresschrift der Naturforschenden Gesellschaft in Ziirich 58; Habilitationsschrift Universitat Zurich.
(1976) "Sets and classes", North-Holland, Amsterdam-New York-Oxford., Vol.8h, p. XI-XVI.
(1918) "Beitrage zur axiomatischen Behandlung des Logik-Kalkuls", Habilitationsschrift Gttingen (unpublished). (1926) "Axiomatische Untersuchung des Aussagenkalkuls der 'Principia Mathematic&"',
Mathematische Zeitschrift 25, 305-320.
Interview, unpublished. (10) (1922) "Die Bedeutung Hilberts fur die Philosophie der Mathematik", Die Naturwissenschaften 10, 93-99. (1 1 ) (With M. Schonfinkel) ( 1928)
'I
Zum Entscheidungsproblem der mathematischen
Logik", Mathematische Annalen 99, 342-372.
PAUL BERNAYS
309
( 12) ( 1922) "Zur mathematischen Grundlegung der kinetischen Gastheorie", Mathema-
tische Annalen 85, 242-255. (13) (With D. Hilbert) (1934, 2nd ed. 1968) "Grundlagen der Mathematik", Band I, Springer, Berlin.
(14) (1936) "Logical Calculus", Mimeographed lectures at the Institute for Advanced Study 1935/36, Princeton, 125 pp.
(15) (With D. Hilbert) (1939, 2nd ed. 1970) "Grundlagen der Mathematik", Band 11, Springer, Berlin.
(16) (1937) "A system of axiomatic set theory", Part I, J. of Symbolic Logic 65-77. (1941) Part 11, ibid. 6, 1-17. (1942)Part 111. ibid. 7, 65-89. (1942) Part IVY ibid. 7 , 133-145. (1943) Part V, ibid. 8, 89-106. (17) (1961)
2,
" Z u r Frage der Unendlichkeitsschemata in der axiomatischen Mengenleh-
re", in: Essays on the foundations of mathematics, dedicated to Prof. A.A. Fraenkel, Magnes Press, Jerusalem, 3-49.
(18) (1958) "Axiomatic set theory (with a historical introduction by Abraham Fraenkel), North-Holland, Amsterdam.
(19) (1976) "Abhandlungen zur Philosophie der Mathematik", Wissenschaftliche Buchgesellschaft, Darmstadt. (20) Hilbert, D. and Ackermann, W. (1928) "Grundziige der theoretischen Logik",
Springer, Berlin.
LOGIC COLLOQUIUM 78 M. Boffa, D. van Dalen, K. McAloon (eds.) 0 North-Hollmd Publishing Company, 1979
AXTRACT LOGIC AND SET THEORY. I. DEFINABILITY Jouko Vaananen Department of mathematics University o f Helsinki Hallituskatu 15 SF-00100 Helsinki 10 Finland
Definability in an abstract logic is compared with definability in set theory. This leads to set theoretical characterizations of implicit definability, LBwenheim-numbers and Hanfnumbers of various abstract logics. A new logic, sort logic, is introduced as the ultimate limit of abstract logics definable in set theory. § 0.
Introduction
The aim of this paper is to bring together, in a coherent framework, both old and new results about unbounded abstract logics (a logic i s unbounded if it is able to characterize the notion of well-ordering). Typical problems that can be asked about any logic are: (1)
Which model classes are implicitly (with extra predicates and sorts) definable?
(2) Which classes of cardinals are spectra?
( 3 ) What is the Lawenheim-number?
(4) What is the Hanf-number? In the case of unbounded logics these problems are particularly relevant as such logics fail to be axiomatizable and mostly lack workable model theory. An attempt to shed light on (1)-(4) is the main purpose of this paper. Out method is to build, right from the beginning, a close connection between abstract logic and set theory. The basic notion of the whole paper is that of symbiosis. We say that an abstract logic L*
and a predicate P
of set theory are symbiotic if, roughly
speaking, the family of A(L*)-definable model classes coincides with the family of model classes which are
A1 ( P ) .
For example, second order logic L1'
1s .
symbiotic with the power-set operation, or, what amounts to the same, A(L
I1
) = {Kithe model class
K
is A2}.
In Chapter 2 we give a new proof of the following result (essentially due to 39 1
392
J,
Oikkonen [ l o ] ) : I f An(L*)
L*
and
P
VGNLNEN
a r e symbiotic, t h e n
K
= {Klthe model c l a s s
A s a c o r o l l a r y we g e t f o r
An(€')}.
n > 1:
An(Lww) = {Klthe model c l a s s Consideration of t h e l o g i c s sort logic.
is
i s .An}.
K
An(Lww)
l e a d s very n a t u r a l l y t o what we c a l l
To g r a s p t h e i d e a o f s o r t l o g i c , l e t us c o n s i d e r a t y p i c a l many-
sorted structure
M = <M,
,. .. ,Mn;R,,.
. . ,Rm;a,,. . . ,al;>.
M c o n s i s t s of t h r e e kinds o f o b j e c t s : u n i v e r s e s v i d u a l s ai.
TO
M.,
Ri
relations
and i n d i -
q u a n t i f y over t h e i n d i v i d u a l s we have f i r s t o r d e r l o g i c ; t o
q u a n t i f y over r e l a t i o n s we have second o r d e r l o g i c ; b u t t o q u a n t i f y o v e r u n i v e r s e s ( i . e . s o r t s ) we need a new l o g i c .
Accordingly, l e t s o r t l o g i c
Ls
be t h e many-sorted l o g i c which allows q u a n t i f i c a t i o n over i n d i v i d u a l s , r e l a t i o n s and s o r t s .
I t i s c l e a r l y impossible t o d e f i n e t h e semantics of s o r t
l o g i c i n s e t t h e o r y , b u t it can be done, f o r example, i n
MKM
(Morse-
Kelley-Mostowski) t h e o r y o f c l a s s e s .
It follows r e a d i l y from t h e above a n a l y s i s of
that
is definable i n s e t theory]
Ls = (Klthe model c l a s s K (stated i n
An(Lww)
[81 p. 174).
The rest of Chapter 2 i s devoted t o an a n a l y s i s of t h e non-syntactic n a t u r e of t h e A-operation. give r i s e t o A(L
I1
We show, f o r example, t h a t t h e s e t of LII-sentences which
) - d e f i n i t i o n s , i s Il - b u t n o t
3
13- d e f i n a b l e
i n s e t theory.
r e s u l t r e f l e c t s t h e d i f f i c u l t n e s s o f f i n d i n g a simple syntax f o r Chapter 3 i s concerned with a r e s t r i c t e d A-operation, A : ,
as we may t h i n k of
LI1
as
A(LI1).
which does not
This o p e r a t i o n i s c l e a r l y r e l a t e d t o
allow t h e use o f new s o r t s ( o r u n i v e r s e s ) . L1'
This
A'( w ) ( L o w ) .
The key n o t i o n of t h i s c h a p t e r i s
t h a t of a f l a t formula of s e t t h e o r y .
We o b t a i n t h e following c h a r a c t e r i z a t i o n
of g e n e r a l i z e d second o r d e r l o g i c : I f
L*
and
P
s a t i s f y a strengthend
symbiosis assumption, t h e n
A'
(W)
(L*) = {;(Ithe model c l a s s
I(
i s d e f i n e d by a f l a t formula o f t h e
language {E,PII.
In p a r t i c u l a r L1'
= {Kithe model c l a s s
K
i s d e f i n e d by a f l a t formula of s e t t h e o r y }
ABSTRACT LOGIC AND SET THEORY
393
These results are proved in a level-by-level form.
In Chapter 4 we extend the analysis of the set theoretic nature of model theoretic definability to spectra and Lowenheim-numbers z(L*).
We characterize
the spectra of symbiotic logics and prove for L*, symbiotic with P,
I(A (L;I)) = sup {ala is n (P)-definable with parameters in A} z(An(LA)) = sup {ala is An-definable with parameters in A} (n > 1 ) .
A similar analysis of Hanf-numbers h ( L * ) non-preservation of Hanf-numbers under A
is carried out in Chapter 5. The
necessiates the introduction of a
,:I
bounded A-operation AB, and respective set theoretical notions The main result says: If L*
and P
If
and A:.
are symbiotic in a sufficiently bounded way,
then h ( L ; t ) = sup {ala is l:(P)-definable
with parameters in A)
and for n > 1 , h(An(LA)) = sup { a l a
is rn(P)-definable with parameters in A).
In the rest of Chapter 5 we consider the numbers
2 = sup {ala is n -definable} h Note that
= sup {a/a is ln-definable].
In = I(An(Lww)) and hn
= h(A(Lww))
(for n > 1 ) .
It turns out that
for n > 1 ,
I = sup {ala is A -definable} and
In
<
hn
=
In+,.
In particular, we get I(Ls) = h ( L s ) = sup { a l a
is definable in set theory].
This paper is based on Chapter 2 of the author's Ph.D. thesis and the author wishes to use this opportunity to express his gratitude to his supervisor P.H.G. Aczel for the help he provided during the preparation of the thesis. This work was financially supported by Osk. Huttunen Foundation.
VZNXNEN
J.
394 Preliminaries
§ 1.
We g i v e a t f i r s t a rough s k e t c h of t h e p r e l i m i n a r i e s , which s h o u l d he enough f o r a c a s u a l r e a d e r f a m i l i a r w i t h [21 and
[a].
More d e t a i l e d p r e l i m i n a r i e s t h e n
follow. Our a b s t r a c t l o g i c s a r e d e f i n e d r o u g h l y a s i n [2].
LQ i s l i k e
quantifier
Lwo[Q]
%
w e l l - o r d e r i n g q u a n t i f i e r and
Ls.
t h e Henkin-quantifier.
L* i s an a b s t r a c t l o g i c , c ( L * )
the
i s second o r d e r
L1'
The l o g i c which i s o b t a i n e d
i s t h e f a m i l y o f PC-classes o f
t h e sense of
181.
and
a r e o b t a i n e d by i t e r a t i o n s of t h e
Iin(L*)
c o n s i s t s of t h e complements o f PC-classes of
Il(L*)
ln(L*)
t o t h e i n t e r s e c t i o n of
Ai(L*)
i s a generalized
by adding q u a n t i f i c a t i o n o v e r s o r t s i s c a l l e d s o r t l o g i c and denoted
LI1 If
Q
If
I i s t h e HBrtig-quantifier, W
A l l l o g i c s a r e u n d e r s t o o d t o b e many-sorted.
logic.
from
[a].
in
and
Xn(L*).
1- and
Ii-operations.
The f a m i l i e s
L* L*.
cn(L*)
A (L*)
refers
In
c:(L*),
in
and
nn(L*)
a r e defined s i m i l a r l y but t h e PC-definitions a r e not allowed t o introduce
new s o r t s .
T h i s ends t h e s k e t c h .
Abstract logics
1.1.
For many-sorted l o g i c we r e f e r t o [5]. symbols and constant-symbols. type
L
then
MIK
i s denoted
If
Str(L). x
cp E
L*
(that is
an L*-sentence,
(L2) I f
M k*
cp
(that is
to
M
IMi
11.
in
A quasilogic i s a p a i r ( L l ) If
M E S t r ( L ) and
If
d e n o t e s t h e r e d u c t of
i n t e r p r e t a t i o n of
M
k* cp
and
s e t s of s o r t s , r e l a t i o n -
K.
If
i s a t y p e such t h a t
K
x
E L, then
2'
L* = < S t c * , k * > Stc*(L,cp)), t h e n
I*(M,cp)),
M Z N, t h e n
K C L,
denotes t h e
d e n o t e s t h e union of t h e u n i v e r s e s of
M.
such t h a t L
i s a t y p e and
then t h e r e i s a type
M E S t r ( L ) and cp E L*, (L3) I f
Types a r e
i s a type, t h e c l a s s of a l l s t r u c t u r e s of
L
cp
L
i s a set called
such t h a t
I* cp.
N
T h i s d e f i n i t i o n i s somewhat weaker t h a n t h e d e f i n i t i o n o f a system o f l o g i c s i n
[21, and s u b s t a n t i a l l y weaker t h a n t h e d e f i n i t i o n of a l o g i c i n The q u a s i l o g i c Lmu
mu
1 Lmu(Q ,
If
...,Qn)
Q
1
,...,Qn
[a].
are generalized
d e n o t e t h e q u a s i l o g i c which i s o b t a i n e d 1 by a d d i t i o n of t h e new q u a n t i f i e r s Q .Qn. Second o r d e r i n f i n i t a r y
q u a n t i f i e r s , we l e t from
i s d e f i n e d a s usual.
L
l o g i c , which i s denoted by
I1
..
.
Lmw, i s o b t a i n e d from
f i c a t i o n over ( f i n i t a r y ) r e l a t i o n s .
Lmw by a d d i t i o n of q u a n t i -
The f o l l o w i n g g e n e r a l i z e d q u a n t i f i e r s p l a y
a s p e c i a l r o l e i n t h i s paper: Hartig-qudntifier:
I x y A ( x ) B ( y ) ++
card(A) = c ard(Bj ,
ABSTRACT LOGIC AND SET THEORY
395
Well-ordering-quantifier: WxyA(x,y)
t-+
A
Regularity-quantifier: RxyA(x,y)
A
orders its domain in the type of
t+
-
well-ordnrs its domain,
a regular cardinal, Henkin-quantifier: 9HxyuvA(x,y,u,v)
VfVg3x3yA(x,y,f(x) ,g(y)).
Note that our Henkin-quantifier is the dual of the original one.
If L* is a quasilogic we let Lz be the quasilogic the sentences of which are those cp E L*
for which
L E A
and cp E A, and the semantics of which
L*. For example, (Lmw)H(K) will be
follows that of
LKW if the syntax of
is defined in the usual set theoretical way (see e.g. [ 3 ] ) . We denote L;F Lmw 1 I1 by w:L and in general by LZw. LWw(Q ,..., Qn) and Lww are shortend L(Q1 ,..., $)
to use
and
LAh to denote
L".
As usual, LA
(LmW)A. For w < h E A, we
denotes
(LmA)A. Similarly LAG
denotes
(LmG)*.
If
K <
A , then
LKK added with the weak LKX does not make much sense, but we redefine it as second order quantifiers 3X( 1x1 5 a A . . . ) for a < A . The obvious set theoretic definition gives LKX c H ( K + lal) whenever
X =
K,.
A class of structures of the same type is called a model class if it is closed under isomorphisms.
If K C Str(L)
is a model class, then the model
class Str(L) - K is denoted by K. A model class K are
L
and cp E L*
tM
such that K = Mod(cp) =
usual, that a quasilogic L*
- L+,
We say, as
L*
and L+
L + , L* 5 L+,
are equivalent,
if they are soblogics of each other.
An abstract logic is a quasilogic L* (Lb) If L
M E
c* cp}.
is a sublogic of another quasilogic
if every L*-definable model class is L+-definable. L*
is L*-definable if there
E Str(L)IM
and L'
are types such that
such that
L C L', then L* 5 L'*
and for cp E L*,
Str(L'),
M
k*
cp
if and only if
MI,
I* cp.
(L5) For every rudimentary set A, type L E A and cp v $
in
and cp,
$ E
Li, there are cp
Li such that Mod(qx$) = Mod(cp) flMod($)
A
and Mod(cpAjr) =
Mod(cp) fl Mod($). (L6) For every rudimentary set A, types and Vccp in L i then
L,L' E A
such that if L' - L
and
(0
E
L'* there are 3ccp
consists of the constant-symbol c,
VZN~EN
J.
396 I t i s obvious t h a t L*
,..., Qn ) ,
I1 L,,
and t h e i r f r a p e n t s a r e
( L 6 ) h o l d s f o r t h e n e g a t i o n , we
If t h e analogue o f ( L 5 ) and
abstract logics. call
1
Lmw, L m u ( Q
a Boolean l o g i c .
When we a r e o n l y i n t e r e s t e d i n t h e d e f i n a b l e model c l a s s e s o f an a b s t r a c t logic
L*, we sometimes w r i t e L* = {KI t h e model c l a s s
i s . . .}
K
meaning t h a t a n a r b i t r a r y model c l a s s
K
i s L*-definable
if and o n l y i f
K
is...
S o r t 1oDic
1.2.
i s o b t a i n e d if t h e I1 f o l l o w i n g f o r m a t i o n r u l e i s added t o t h e r e c u r s i v e d e f i n i t i o n o f L -formulae: Lu:
The c l a s s o f formulae of i n f i n i t a r y s o r t l o g i c
mu
If
cp
i s a formula and
formulae
.
To d e f i n e t h e s e m a n t i c s o f
Ls
mu
is a s o r t , then
s
and
3scp
we have t o work i n t h e
MKM
Vscp
are
theory of classes
o r i n any o t h e r t h e o r y i n which s a t i s f a c t i o n f o r f o r m u l a e o f set t h e o r y i s d e f i If
nable.
i s a type, s
L
a s o r t , s f? L
and
L ' = L U { s ] , t h e n f o r any
M E S t r ( L ) we d e f i n e
M
k
M This defines
3sq1
i f and o n l y i f
3N E S t r ( L ' ) ( N = M & N IL
Vscp
i f and o n l y i f
VN E S t r ( L ' ) ( N I L
Lf,
a s an a b s t r a c t l o g i c .
We d e n o t e
= M
-+
k
N
L ; ~ by
cp) cp).
L'.
It a p p e a r s
t h a t s o r t l o g i c has n o t been s i n g l e d o u t as a l o g i c b e f o r e , a l t h o u g h it has been s t u d i e d i n a s e m a n t i c a l form i n [ l o ] . Let
lI1(Lw),
n 2 1, be t h e s u b l o g i c of
,L:
t h e f o r m u l a e o f which have t h e
form 3SlVS2..
where
sl,
...,s
.3(V)Sn'p
a r e s o r t s and
cp E .:L :
IIn(Lmw) b e t h e s u b l o g i c
Let
Of
,L:
c o n s i s t i n g of formulae of t h e form VS13S2...v(3)sncp
where
sl,
..., s
ln(Lmw) and
nn(L
a r e s o r t s and mW
)
cp
I1
E L,,.
are definable i n
For each
ZF.
n <
Note t h a t
m,
the abstract logics
&(Lmu)
and
Iln(Lmw)
a r e c l o s e d undpr second o r d e r q u a n t i f i e r s . Let
ln(Lmw)
A ( L ) b e t h e s u b l o g i c of t h e formulae of which a r e equivan mw l e n t t o ll ( L )-formula. An(LW) i s an a b s t r a c t l o g i c b u t it does n o t seem t o n mw
397
ABSTRACT LOGIC AND SET THEORY
ln(Lmw)
have such a simple syntax as
nn(Lmw). I n f a c t t h e c l a s s of
and
)-formulae r e f l e c t s t o a c e r t a i n e x t e n t t h e p r o p e r t i e s o f t h e underlying
A (L
n mu model of s e t t h e o r y and changes when t h e model i s changed. i s just
1.3.
[a].
i n t h e sense of
A(Lmw)
An+l (Lmw). The fragments
&(LA),
Note t h a t
U nn( L mu) )
More g e n e r a l l y , A ( & ( L m u )
nn(LA)
and
-
A,(LWw)
An(LA) a r e d e f i n e d s i m i l a r l y .
Extension-ogerations
We review t h e d e f i n i t i o n of t h e A-operation from 181 because t h e d e f i n i t i o n n a t u r a l l y l e a d s t o both more g e n e r a l and more r e s t r i c t e d o p e r a t i o n s . cp E L*, L '
If
and
IN E
Str(L)INlL' = M &
~(M,cp) = A model c l a s s
K
i s n-defined
K
of t y p e
by
M E Str(L'), l e t
C L
I*cpl.
i s 1-defined by
L'
L - L'
is
tp
N
L - L'
i s f i n i t e and
i s f i n i t e and
K = [M E Str(L')IVN E S t r ( L ) ( N I L , = M
+
I* @ ) I .
N
i s I(L*)-definuble ( n ( L * ) - d e f i w b l e ) i f it i s 1-defined (n-defined) by some
K cp
if
cp
E L*.
A(L*)
Finally, K
i s A(L*)-defhzble i f it i s both
l(L*)-
and I[(L*)-definable.
g i v e s r i s e t o t h e semantics o f an a b s t r a c t l o g i c , b u t t o f i n d a syntax f o r
t h a t l o g i c seems a s d i f f i c u l t a s f i n d i n g a syntax f o r
A(L*)
t h a t i n s p e c i a l cases
s p e c i f i c l e t us agree t h a t a r e &-tuples c l a s s as
Ji
U,
Ji E L'*
E L*,
and
cp
To be
1-defines t h e same model
One of our r e s u l t s w i l l imply t h a t
a l e s s a r t i f i c i a l syntax.
[81 $ 4 ) .
i s t h e a b s t r a c t l o g i c t h e sentences o f which
A(L*) where
Note however,
A,(Lmw).
has a b e a u t i f u l syntax ( s e e e.g.
l(L*)
The f a m i l i e s
and
Il(L*)
A(L*)
h a r d l y has
can a l s o be made i n t o
a b s t r a c t l o g i c s i f , f o r example, a model c l a s s which i s 1-defined by
cp E L*
a s s o c i a t e d an a r t i f i c i a l sentence
r(L*)
pends only on t h e syntax of that
L* i s unbounded i f By i n d u c t i o n on
and
An+,(L*)
=
zn(Lmw), e i t h e r
L* W
n < w
Note t h a t t h i s syntax f o r
and not on t h e u n d e r l y i n g s e t t h e o r y .
is de-
We say
i s A(L*)-definable. we d e f i n e
A(ln(L*) U nn(L*)). as a 7 -extension of
In+l(L*) =
l(nn(L*)),
nn+l(L*)
Lmw o r as a fragment of
obvious t h a t t h e two i n t e r p r e t a t i o n s a r e e s s e n t i a l l y e q u i v a l e n t .
b u t it i s
L:u,
A l l standard
l o g i c s a r e s u b l o g i c s o f A 3 ( Lmw ) and t h e r e f o r e t h e o p e r a t i o n s A n , r e l a t i v e l y u n i n t e r e s t i n g , a p a r t from t h e i r r e l a t i o n t o s o r t l o g i c . If t h e above d e f i n i t i o n o f
= n(In(L*))
Now we have two i n t e r p r e t a t i o n s f o r
l(L*)
and
iT(L*)
n > 2, a r e
i s modified by r e q u i r i n g t h a t
398 L of
J. V & h N
has no new s o r t s o v e r and above t h o s e of L', t h e e s s e n t i a l l y weaker n o t i o n s 1 n ; ( L * ) - and l l ( L * ) - d e f i n a b i l i t y a r e o b t a i n e d . Let A i ( L * ) b e d e f i n e d a s
A(L*)
above. More g e n e r a l l y we d e f i n e and A 1~ + , ( L * ) = A1, ( z 1~ ( L * ) u
1.4.
1
f.;+,(L*)
ni(l:(~*))
nn(~*)).
f o r t h e a ' t h l e v e l of t h e ramified hierarchy.
i s a c a r d i n a l number ( i n i t i a l o r d i n a l ) " , R g ( x ) c a r d i n a l " and operation.
Fw(x,y)
[41.
is t h e predicate
"x E
fin", of
i s the predicate
course.
i f it i s d e f i n a b l e w i t h a An(P) is
w.p.i.
ln(P)
5
2.
PK(y) is the set
"y = P K ( x ) " .
A
ln(P)
i s t h e power-set
P
{x _C yI 1x1 <
( n n ( P ) ) w.p.i.
( nn ( P I ) - f o r m u l a
In( P )
i f it i s b o t h
yn(P)
and
( n n ( P ) , A n ( P ) ) - d e f i n a b l e w.p.i.
HC(x) =
X.
and
K}
pW,(x,y)
n n ( P ) - f o r m u l a e a r e de-
( = with parameters i n )
w.p.i.
A
A predicate i s
A.
nn(P) w.p.i.
A
"x
i s a regular
as a p r e d i c a t e , meaning t h e p r e d i -
The s e t s of z n ( P ) - and
A predicate is
fined as usual.
i s t h e p r e d i c a t e "x
x = P ( y ) , where
sn
However, we w r i t e
is t h e predicate
Cd(x)
i s t h e l e a s t o r d i n a l which h a s t h e same power a s
Card(x)
m a x ( c a r d ( T C ( x ) ) , s o ) . We sometimes u s e cate
=
Set theory
Our s e t t h e o r e t i c a l n o t a t i o n f o l l o w s m o s t l y t h a t o f
Ra
1 nn+,(L*)
= I;(nn(L*)),
An o r d i n a l
A.
i f the predicate
"x E a"
a
is.
The b a s i c r e p r e s e n t a t i o n s
I n t h i s c h a p t e r we d e f i n e t h e symbiosis o f a l o g i c and a p r e d i c a t e o f s e t t h e o r y , and prove t h e main r e s u l t a b o u t s y m b i o s i s (Theorem 2 . 4 ) .
The c h a p t e r
ends w i t h some remarks on t h e non-absolute n a t u r e of t h e A-operation. By i t s v e r y d e f i n i t i o n an a b s t r a c t l o g i c d e t e r m i n e s two p r e d i c a t e s o f s e t theory: Stc
and
b.
I t i s c o n v e n i e n t f o r o u r purposes t o e s t a b l i s h a c o n v e r s e
r e l a t i o n , t h a t i s , a s s o c i a t e every p r e d ic a te with a general i zed q u a n t i f i e r . Suppose
P = P ( x l,....x )
K[P] = { M I M P(al Let
%
E
is a p r e d i c a t e of s e t t h e o r y .
<M,E,al,
,. .. .a
... , a
>
such t h a t
M
Let
i s t r a n s i t i v e and
) }.
be t h e generalized q u a n t i f i e r a sso c ia te d with
K[P]
and
ABSTRACT LOGIC AND SET THEORY An e l a b o r a t i o n of t h e proof o f t h e well-known
Proof. A,
[ 3 1 p. 83) g i v e s ( 1 ) .
( s e e e.g.
definable.
( 2 ) i s t r i v i a l as
fact that
kLmu
is
i s even L [ P I W W -
KEPI
0
The above lemma shows t h a t other.
399
and
L[P]
a r e i n a s e n s e d e f i n a b l e from each
P
We t a k e a s l i g h t weakening o f t h i s p r o p e r t y a s t h e d e f i n i t i o n of symbiosis.
D e f i n i t i o n 2.2.
theory and
-on _ A
A
Suppose
L*
i s an abstract logic, P
a transitive class.
a predicate o f s e t
L* and
We say t h a t
are symbiotic
P
i f the following two conditions hold:
(Sl) If
Q
K[P]
(5’2)
The logic
L* and
E L*, then Mod(cj?) i s is A(Li)-definabZe. is symbiotic
L*
on
are symbiotic on
P
Note t h a t s y m b i o s i s on
A
1
(P) w . p . i .
{Q,LI
if there i s a predicate
A.
# 0 such t h a t
i m p l i e s s y m b i o s i s on any t r a n s i t i v e
A
The following p a i r s are symbiotic on
Examples 2 . 3 .
P
A = HF, we omit the clause “on A”.
If
A’ ? A .
HF f o r any rudimentary
A:
and
(1)
LDIA
1.2)
LQA
and
Q, if
131
LWA
and
On,
(4)
LIA
and
Cd,
(5)
LRA
and
Rg,
16)
Lil
and
Fw,
171 ,L
P,
and
n
pww
n
Q
i s a generalized q u a n t i f i e r and
K
= h’,
t
H(K)
f o r any rundsmentary
LAK
and
pwKJ
19)
LAG
and
On.
2
H(K)
The proof o f ( S l ) i s s i m i l a r t o t h e proof of 2 . 1 ( 1 ) i n any o f ( 1 ) - ( 9 ) .
As a t y p i c a l example o f t h e proof of ( 5 2 ) , l e t u s c o n s i d e r
is
A
w:
(8)
Proof.
is unbounded,
.
The folZowing p a i r s are symbiotic on and
LQ
~(LI). NOW
*
<M,E,a>
E
<M,E,a>
1 Wxy(xEy)
K[d]
A
Vxy(yEx
--
A
Vxyz( zEy
A
Vz(zEa
A
Vxy(Vz(zEx
+
A
yEx
A
A
xEa
-+
yEa)
xEa + zEx)
A
Ixy(xEz)(yEa))A ZEy)
-+
x=Y) .
o
A
(4).
Recall that
W
400
VL~NLNEN
J. Note t h a t , i f
in fact, as
is s$mbiotic with
L*
P
0)
(#
on
A
then
Lz
i s unbounded,
is A(L*)-definable, it s u f f i c e s t o o b s e r v e t h a t
K[P]
A
Wxy(xAy)
c+
there are
and
M, E
a,,
...,a
such t h a t
,. .. . a > E K[P] and < M , E , ~,..., ~ a n > 1 VXY(XAY + X E Y ) .
<M,E,al
Conversely, it i s by n o means t h e c a s e t h a t e v e r y unbounded l o g i c i s
A.
s y m b i o t i c on some
L(W,Q l , . . . , Q n , . . .
We s h a l l i n d i c a t e l a t e r why t h e l o g i c
)n<w
i s n o t symbiotic. The n e x t theorem i s t h e b a s i c r e s u l t about symbiosis and about r e l a t i o n
It was proved i n t h e a u t h o r ' s Ph.D.
between l o g i c and s e t t h e o r y i n g e n e r a l .
t h e s i s El31 and appeared l a t e r , b u t i n d e p e n d e n t l y i n [ l o ] . h e r e f o r completeness. Theorem
L*
and
2.4.
A
Suppose
and
t
A
are t r a n s i t i v e classes, P
A
LA
an abstract logic extending
K
12)
K
Proof. cp E LE.
is is
l(Li)-definabZe,
(1)
+
Z,(P)
is
K
+
( 2 ) : Suppose
1-defines
cp E L i
I,(P )
w.p.i.
K.
Let
such t h a t
Lo E A
cp(x,y)
,...,x
is
f-f
cp(x,a))
ll(P)
and
)
L
i s s y m b i o t i c , t h e r e i s an
i.
a ' = TC(Ia>)
i
and
[$(x
tcu
v a r i a b l e s t o v a r i a b l e s of s o r t
Let
a E A.
of s e t theory l e t
by r e p l a c i n g atomic formulae
tEu
Let
L 1 3 Lo
Lo
and
rl
and
cl,
E
... , c
such t h a t
q
L
K[P].
,,...,
Cn)lE
-
d c l
,...,c
)
$(xl
,...,x n )
and changing a l l bound
such t h a t
<M.,E>.
be
E
For any f o r m u l a
E
extended w i t h
a r e t h e c o n s t a n t symbols i n t h e t y p e of
L* - s e n t e n c e o b t a i n e d from 2A "c
E(t,u))
'8 E L t A
1-defines t h e c l a s s
L.
be o b t a i n e d from
(i.e. be
Let
(E A).
a sort not i n
,,... , x n ) I E
by
c l a s s o f well-founded e x t e n s i o n a l s t r u c t u r e s and
I* cp)}.
N
&
A.
a b i n a r y p r e d i c a t e symbol n o t i n
L*
L E A:
K o f type
( 1 ) : Suppose Vx(x E K
$(xl
Then
Ao.
Now
(2)
where
a predicate,
on
A.
w.p.i.
K = { M E Str(L)13W E S t r ( L o ) ( N I L = M Hence
P
and symbiotic with
the j o l l o d n g are equivaZent for any model c l a s s (1)
We r e p e a t t h e proof
The proof c l e a r l y owes a g r e a t d e a l to [ 2 ] .
'8
and
By ( 5 2 ) t h e r e i s an Suppose K[P].
P = P ( x l ,. Let
i.
As
1-defines t h e
5
L
3
2 -
.. ,xn)
be t h e
L
1
40 1
ABSTRACT LOGIC AND SET THEORY
by u n i v e r s a l l y quantifying over %(x) Let
be t h e formula
Vy(yEx
...,c
cl,
a s t r u c t u r e of type
a1
,...,am
i n which any atomic
L
E, E
Now
K.
and
is true.
cp(M,a)
P.
i
N Y M = <... ,M. ,€,. ..>
Then
P.
Let
Clearly
elements. c,
Finally, l e t
is be
5
Suppose a t f i r s t t h a t
K.
5
by l e t t i n g
is a transitive set.
is interpreted as
A E S t r ( L ) and V ( A , a ) , whence
a'. x
be a t r a n s i t i v e set which r e f l e c t s
N
Mi
Let
of
that
E,
N
For t h e converse, suppose
such t h a t
i. b
i s s a t i s f i e d by ele-
)
is true.
M can be expanded t o a model of
t h e universe of s o r t reflects
R(x l , . . . , x
YA and w e prove t h a t it 1-defines
Hence
cp(M,a)
i n s t e a d of
E
,...,am )
i f and only if R ( a l
t h e conjunction of
I
f o r every element
$ ( x ) be t h e LE-sentence which s a y s , using
ments
M E
using v a r i a b l e s of s o r t
t+\s(/cEbcpc(x))
A E K.
a
in
As
K
M.
Let
k M
As
serve as
N
Let
5.
k
A E Mi
5 , Mi
such t h a t
i s closed under isomorphism,
MLEK. Corollarv 2 . 5 , (1)
For any rudimentary c l a s s
A(LWA) = {Klthe model c l a s s
A: A},
w.p.i.
K
is
A1
is
Al(Cd) w.p.i.
is
A2
= A ( L ~ ~ )
(2)
A(LIA) = N t h e mode2 c2ass
K
(3)
I1 A(LA ) = {Klthe mode2 c l a s s
K
(4)
A(Lw
If
(51
) = {Klthe mode2 c l a s s
A 2 H(:),
K
= A+,
X 2
w,
In [ l l ] a predicate
P(x)
is
w.p.i.
K
is called
A}.
Al(Pww )>,
then:
A ( L A K ) = {Klthe mode2 c l a s s
f o r some formula
K
A),
n
i s A l ( P w ) n.p.i.
local i f it i s
A}.
of t h e form
3a(Ra
k
cp(x))
~ ( x )of s e t t h e o r y , and a proof i s sketched t o t h e e f f e c t t h a t
a p r e d i c a t e i s l o c a l i f and only i f it i s equivalent t o a t h i s with Theorem 2.4 y i e l d s : Corollarv 2.6.
A model c l a s s
i s I(L")-definabZe A
f i n e d by a local property w . p . i .
12-predicate.
Combining
i f and only i f it i s de-
A.
Theorem 2.4 can be immediately i t e r a t e d t o y i e l d a r e s u l t about ln-defina-
402
VGNLNEN
J.
bility. A form of the following corollary was first proved by J. Oikkonen in [ l o ] with a different proof.
Corollarv 2.7.
cate, and
Suppose
.A 5 A
A and
are t r a n s i t i v e classes, P a predi-
Li a Boolean logic extending
L E A and for any
n < w:
is ~ , + ~ ( ~ i ) - d e f i n a b L e ,
(I) K
is
K
(2)
Zn+l(P) W.p.i.
A.
Proof. We use induction on Suppose then K
If n = 0, the claim follows from 2.4.
n.
is ln+l(Li)-definable and
By induction hypothesis Mod(cp) is
n
and therefore K
> 0.
nn(P) w.p.i.
K = { M E Str(L)13N E Mod(cp)(NIL = is ln+l(P) w.p.i.
M
N
A
Let cp E n n ( L z )
A.
\
1-define K.
Now
cp)},
A.
For the converse, suppose (2) holds.
K is
P on K of type
LA and symbiotic with
Then the following are equivalent f o r any model class
Ao.
Let
S be a Il (P)-predicate such that
A. By 2.4 K is 1-defined by some cp E L[SIA. By 2.1 Mod(cp) is A1(S) w.p.i. A; and therefore An+l(Li)-definable. Hence K is zl(S)
w.p.i.
In+,(Li)-definable. Corollarv 2.8. For
n > 0
and
any rudimentary c l a s s
fGP
(1)
An(LWA) = {Klthe model c l a s s
K
(2)
An(Lfil) = {Kithe model c l a s s
K is
Note that K[Pw]
II(L
is
I1 fore A(LA ) 5 A2(LA).
ww
is
An An+,
w.p.i. W.p.i.
A:
A}, A}.
)-definable, whence L[Pw] 5 A (L ) 2 wo
and there-
A 2(LA ) 5 A2(LWA) 5 A ( L i l ) . Hence in I1 A 2 (LA ) , and therefore An(LA ) An+l(LA) for all n > 0. If this is combined with 2.8, the following obtains: I1 fact A(LA )
-
Corollarv 2.9. For
On the other hand
n
-
> 1,
An(LA) = {Kithe model c l a s s
K
is
An
w.p.i.
A}.
Therefore in MKM:
Li = {Kithe model class
K zk definable in s e t theory w . p . i .
A}.
The second part of the above corollary was stated on page 174 of [ E l . As A(L1'),
A 1 (Lww ) is just the usual first order logic Lww, and A2(Lu,) is that is, essentially second order logic, it would be tempting to conjec-
ture that A (L ) is essentially third order logic. This is not the case, 3 ww however. By familiar methods (see e.g. [91) one can prove that for any analytical
ABSTRACT LOGIC AND SET THEORY ( t h i s can be improved, s e e [ 9 1 ) o r d i n a l
a
t h e a ' t h o r d e r l o g i c i n A-equivalent
It seems p l a u s i b l e t o p u t
t o second o r d e r l o g i c .
403
As(LA)
above t h e whole notion
of h i g h e r o r d e r l o g i c and c o n s i d e r it r a t h e r as a f r a g n e n t of a q u i t e new power-
1 a second o r d e r 1 l e a d s t o some-
S i m i l a r l y it seems i m p l a u s i b l e t o c a l l
f u l logic, sort logic.
q u a n t i f i e r , o r even a g e n e r a l i z e d second o r d e r q u a n t i f i e r , as
t h i n g f a r beyond second and h i g h e r o r d e r l o g i c , v i z . s e t t h e o r y .
We r e t u r n t o
t h e problematics of second o r d e r l o g i c i n t h e next c h a p t e r . The
A-operation can be used t o g i v e a v e r y n e a r c h a r a c t e r i z a t i o n
of symbiosis: Proaosition 2.10. ( ~ 1 ) ~If :
Proof.
Suppose
Then A f L i )
a predicate. $
E L;,
-
Li
i s a Boolean l o g i c extending
A(LIPIA) i f and onZg i f
t h e n Mod($) is
Suppose at f i r s t t h a t
A1(P) w . p . i .
( S 1 ) A and ( 5 2 ) hold.
LA
and
P
and
ISZ)
A. By Theorem 2.1
every L[P] - d e f i n a b l e model c l a s s i s A l ( P ) w.p.i. A, whence by 2 . 4 , A L[PIA 5 A ( L i ) . Therefore A(L[PIA) 5 A ( L i ) . On t h e o t h e r hand, i f K
i s Li-
-
d e f i n a b l e , t h e n by ( S l ) K i s A l f P ) w.p.i. A, whence by 2.4 K i s A(LIPIA)A A(L[PIA). The converse i s immediate i n view of 2.4. d e f i n a b l e . Hence A ( L 2 )
We can use 2 . 1 0 t o show t h a t t h e l o g i c symbiotic on any
L i = L&, that
Indeed, suppose
A.
we may assume
K[P]
A = HF.
i s A-definable i n
L* = L(W,Q1,
-
...,Qn, ...) n < w
i s symbiotic w i t h
L;I
By 2.10, A(L*)
L+ = L(W,Q1,
P
on
A(L[PIww). Let
... , Q n ) .
Now
-
A(L*)
a contradiction.
i s not As
A.
n < w
-
such
A(L[PIww)
A(L+),
The e x i s t e n c e on non-symbiotic unbounded l o g i c s may seem t o l i m i t t h e a p p l i c a b i l i t y of Theorem 2.4. the logics A = U A
L+"
(n < w),
However, i f where
L+"
i s t h e union ( i n t h e obvious s e n s e ) o f
L*
i s symbiotic w i t h
Pn
An, t h e n f o r
on
n'
A ( L 2 ) = {Klthe model c l a s s
K
is
A1(Pn) w.p.i.
f o r some
An
Thus t h e range o f Theorem 2.4 extends t o many non-symbiotic l o g i c s .
,..., Qn ,...
d(L(W,Q1
= {Klthe model c l a s s f o r some
Note however, t h a t
Lw w ( W , Q l ,
1
,...,yn,-..u.
,.. .,Qn ,... )n<w
For example:
K
is
Al(H
K
is
A1
,,..., 8,)
n < w}
= {Klthe model c l a s s {&l
n < u}.
i s symbiotic on
w.p.i.
HC.
I n t h e next r e s u l t s we i n v e s t i g a t e t h e a b s o l u t e n e s s of symbiotic l o g i c s .
us c o n s i d e r t h e following t h r e e p r o p e r t i e s of an a b s t r a c t l o g i c
L*
Let
and a predi-
J. V & k N
404
cate P:
Stc* is I , ( P )
(Al)
found with
and cp v j i , cp
l1(P)-functions.
S such that if cp E L*, then
There is a A,(P)-predicate
(A2)
in ( L 5 ) and ( L 6 ) can be
Ji, 3ccp and V e v
A
The conditions ( A l ) and ( A 2 ) together form a natural notion of P-absoluteness of L* generalizing the notion of an absolute logic in
[ e l . Note that
( A 2 ) -+ ( S l ) .
The following lemma is obvious: Lemma 2 . 1 1 . (11
If
and
L*
are symbiotic on A
P
and
L+ 5 A ( L * ) , then
satisfy (SIJA. (21
If
and
L;
P
s a t i s f y ( A 2 ) and
L+
A
then
L+ 5 A(L;),
L+ and
and
P
satisfy
P
(sl)A. The next result generalizes a theorem by Burgess (Theorem 2.2 in [81) which says that no unbounded absolute logic is A-closed.
The proof remains almost the
same.
Suppose
Theorem 2 . 1 2 .
L+
N
Proof.
Suppose
Wl E Str(L)(M
1'
1
on
A
and
Let
K is clearly A,(P).
M
P
s a t i s f y ( S I I A but not ( A 2 1 .
P
S is a A (P)-predicate such that if cp E L+, then
cp ++ S(M,cp)).
IMlM P
K =
Let
i s a Boolean logic symbiotic with
L*
Then L+ and
A(L;).
such that
By 2.4 there is a cp E L+
=
c-f
M
1'
cp
c-f 1
1
S(M,a)l.
such that K = Mod(cp).
S(M,cp) *
M $ K,
a contradic-
tion.
Suppose
Corollarv 2 . 1 3 .
on
A
L* i s a Boolean logic, L*
and they s a t i s f y fA2).
Then
L1; i s not A-closed.
where
i s a predicate of s e t theory,
(1)
LIPIA
(2)
LQA
(3)
LWA3 L I A . LRA. L Y , L W I W l , L W I G .
where
Lt\
such t h a t
-
P are symbiotic
The following logics are not A-closed:
Corollarv 2 . 1 4 .
Hence, i f
and
P
Q i s a generalized q u a n t i f i e r such t h a t
i s a symbiotic logic extending
A(L~;) LQ1
. ..:Q.
LQ
i s unbounded.
LA, there are no
Q
1
,.. . ,Qn
ABSTRACT LOGIC AND SET THEORY
405
The f o l l o w i n g theorem g i v e s a n o t h e r a s p e c t o f t h e f a i l u r e of s y n t a c t i c a l methods i n c o n s t r u c t i n g A-extensions. Theorem 2 . 1 5 .
c a t e and LA Stc
A(L;)
and
i n $1.
A(L*)
are t r a n s i t i v e classes, P
5 A
A
Stc
a predi-
an abstract Zogic such t h a t there i s a l,(P)-functionembedding
L;
into
A
Suppose
Recall t h e definition o f
;,I
and
is
n2(P)
L;
P on
i s symbiotic with but not
1,(P).
Ao.
Thex the predicate
A ( L ~ ) and
Therefore
P
do not s a t i s f y
(A1I .
Proof.
By d e f i n i t i o n StCA(L;)(L,X) (stc*(Lo,cp)
++
39$L0L1 E T C ( X )
& stc*(L1,q) &
= < Q , L ~ . ~ , L , >&
VM E S t r ( L ) ( 3 N E S t r ( L o ) ( N I L = M
&
N
I*c p ) *
M
+
N
k* $ I ) ) ) )
(VN E S t r ( L l ) ( N I L = This proves t h a t let
R(x,y)
tions
K[Tl fines
Let where
f
is
Stc
is
A(L;)
n2(P).
To prove t h a t
b e a II ( P ) - p r e d i c a t e which i s n o t 2 g such t h a t
Stc
12(P).
We c o n s t r u c t l l ( P ) - f u n c -
whence by 2.4 t h e r e i s an L*-sentence ( P ( C ~ , C ~ , C ~ which ) 1-deA For any x l e t q x ( y % E ) be t h e L -formula ( s e e t h e proof o f 2 . 4 ) -W
9(E) b e a n L*-sentence which 1 - d e f i n e s t h e c l a s s o f models A i s well-founded and e x t e n s i o n a l .
(9(E) & q X ( c , 3 ) & q y ( c 2 , E ) ) By ( A l ) we may assume t h e r e i s a t y p e z = I-
Xy
a s s o c i a t e d with We d e f i n e
I,(P),
ll(P)
K[Tl.
E
i s not
and
For any
x
sentence
cates
A(L;)
and
z
= L'
c3,c2,c,
XY
and
are E.
-+
y
<dom(E),E,c > 3 be t h e
let
XY
d c , ,c2,c3).
L'
XY
[,(P).
Let
and
6
such t h a t Let
L
XY
IE L'* and t h e p r e d i XY xyA be t h e subtype o f L '
xy
b e an a r b i t r a r y v a l i d L*-sentence.
J. V%&NEN
406
f(X,Y d X , Y
Now ( * ) h o l d s a s is n o t t o o d i f f i c u l t t o s e e .
It f o l l o w s , f o r example, t h a t t h e r e i s no
a given
mw
13-formula
which d e c i d e s whether
1 - d e f i n e s t h e same model c l a s s as a n o t h e r g i v e n L’I-sen-
L’I-formula
WW
h a s a p r i m i t i v e r e c u r s i v e s y n t a x , it seems I1 u n l i k e l y t h a t any similar s y n t a x can be found f o r i t s fragment A ( L ) o r f o r t e n c e n-defines.
So, a l t h o u g h
LEw
MW
A~(L*). We end t h i s c h a p t e r w i t h some remarks on d e c i s i o n problems o f s y m b i o t i c For s i m p l i c i t y we o n l y c o n s i d e r l o g i c s o f t h e form
logics.
L:u.
The d e c i s i o n
Val(L* ) = {cp E HFlcp E Lw : and cp i s v a l i d } . I t WW ww I1 i s known ( s e e [13] and [ 1 2 ] ) t h a t V a l ( L W w ) i s t h e complete Ti - s u b s e t o f HF. 2 More g e n e r a l l y , i f L* i s s y m b i o t i c w i t h F and L* i s s u f f i c i e n t l y s y n t a c t i c
problem o f
L*
L* = LQ
(e.g.
is the set
f o r some
Q ) , then
A proof of t h i s can b e found
Val(Liu)
in [ l 3 ] .
i s t h e complete Il ( P ) - s u b s e t o f HF. 1 V a l ( L I u w ) and V a l ( L R w w )
For r e s u l t s about
s e e [141.
§ 3.
F l a t d e f i n a b i l i t v and second o r d e r l o g i c
I n t h i s c h a p t e r we c o n s t r u c t t h e p a r t o f s e t t h e o r y which c o i n c i d e s w i t h second o r d e r l o g i c i n t h e same way a s t h e whole s e t t h e o r y c o i n c i d e s w i t h s o r t logic. Definition 1.1.
Q u a n t i f i e r s of the form
(1)
3x(HC(x) 5 HC(yl U...U
(2)
Vx(HC(x) 5 HC(yl U. ..U y n )
are called
,...,y,)) ,. .. , y n ) ) of fZat f o m l a e
y n ) & ‘.o(x,yl
fZat q u a n t i f i e r s .
+
cp(x,yl
The s e t
smallest s e t containing ~o-formulae and closed under
of s e t theory i s the
and f l a t
&,v,-
quantifiication. b
b
The l E ( P ) - and n n ( P ) - f o r m u l a e a r e d e f i n e d by i n d u c t i o n on
b
I o ( F ) - and Ro(F)-formulae are j u s t t h e l o ( P ) - f o r m u l a e . formulae o f t h e form ( 1 ) where
cp(x,y l , . . . , y
formulae of t h e form ( 2 ) where
cp(x,yl
formulae a r e d e f i n e d a s u s u a l .
It i s e a s y t o s e e t h a t t h e s e t of & , v , ~ ,3xEy,VxEy
ll-formula i s
and ( 1 ) above.
)
,. .., y n )
is is
b nn(P). b
ln(P).
1b (P)ZFC-formulae
n
as follows:
(;)-formulae
are
IIn+l(P)-formulae a r e In(
b ZFCP)ZFC- and n n ( P )
i s c l o s e d under
Note t h a t by Levy‘s theorem “41
p . 104) every
ABSTRACT LOGIC AND SET THEORY
407
The whole p o i n t o f f l a t formulae i s t h e f o l l o w i n g r e f l e c t i o n p r i n c i p l e : Lemma 3 . 2 .
and
$ ( yl , . . . , y m ) f = f
N
such t h a t
(b
..,bm)
2”
N
M
N
such t h a t
b2,
...,bm
E N
& IL(yl,b2
,...,b,))
-+
f
$(Y1,.-.,Ym)’
.,xn).
where
$ ( y l , ...,ym)
let
$(f,b2
a
a transitive
For any s u b f o r n u l a
).
HC(f) 5 HC(b2 U...U
t o be t h e smallest t r a n s i t i v e s e t containing
functions 3..
( [ 4 1 p. 99) t h e r e i s cp(xl , . . . , x
reflects
and f o r any
)
E
an arbitrary
a E M , H C ( M ) = HC(a)
such t h a t
M
1.
3 y 1 ( H C ( y 1 ) 5 HC(b2 , . . . , b m )
Choose
d X 1
a
cp(xl.....x
of
... ,Ym)
*(Y,,
,. . .,x
a
is a f l a t f o m l a and
By t h e u s u a l r e f l e c t i o n p r i n c i p l e
containing
N
cp(x,
reflects
M
Proof. set
cp(xl, ...,x )
Suppose
Then there i s a t p a n s i t i v e s e t
set.
bm)
and
,...,b m ) .
and c l o s e d under t h e
r u n s t h r o u g h t h e subformulae of
0
The above lemma shows, among o t h e r t h i n g s , t h a t e v e r y f l a t formula i s ( u s i n g Theorem 3.7.2 o f D e f i n i t i o n 3.3.
and
-on _ A
Suppose
L* is an abstract logic, A
a predicate of s e t theory.
P
L*
and
P
a transitiue class
are strongly symbiotic
if the following two conditions are s a t i s f i e d
(SSl)
If cp E L*,
(SS2)
K[P]
then
Mod(cp)
is Abl ( P ) w . p . i .
{Q,L}.
is A i ( L f i ) .
S t r o n g symbiosis i s h a r d e r t o come by t h a n s y m b i o s i s . not
AEFc
[41).
1
For exmple,W
A , ( L I ) - d e f i n a b l e ( e s s e n t i a l l y b e c a u s e i n c o u n t a b l e domains
and Theorem 7.3 of
131 can b e u s e d ) , whence L I
I
is
i s redudent
i s not s t r o n g l y symbiotic.
T h i s f a i l u r e can b e r e g a r d e d a s an i n d i c a t i o n of t h e i n c o m p l e t e n e s s of t h e d e f i n i t i o n of
L I , r a t h e r t h a n as a c h a r a c t e r i s t i c p r o p e r t y o f
LI.
The
s i t u a t i o n i s d i f f e r e n t w i t h second o r d e r l o g i c which seems t o r e s i s t s t r o n g symbiosis i n an e s s e n t i a l way, a s we s h a l l prove i n a moment. Examules 3 . 4 .
The following pairs are strongly symbiotic on
(1)
LIPIA and
(2)
LA(W,Q) and
13)
L ~ ( w ) and
P, Q, if
LA(W,I)
and
Cd,
(5)
L ~ ( w , R ) and
Rg.
Theorem 7 . 5 .
Li
Q
i s any generalized q u a n t i f i e r ,
On,
(4)
cate, and
HF f o r my
A:
rudimentary
Suppose A 5 HC and A E A are t r a n s i t i v e s e t s , P a predian abstract logic extending LA and strongly symbiotic with
408
J. V & ~ N E N
Then the following are equivalent f o r any model class
P on Ao.
K o f type
L E A: K i s li(Li)-definable,
(1)
b Z,(P)
(2) K is
w.p.i.
A. The implication (1) -+ (2) is obvious.
Proof. We follow the proof of 2.4.
For (2) + (l), suppose cp(x,y) Vx(x E K
is
l:(P),
a E A
and
cp(x,a)).
t+
Let a' = TC((a)).
Let
v be the conjunction of 5
(as it is defined in the
proof of 2.4) and the first order sentence which says that there is a bijection which maps all elements of the sorts in
v
L. Using Lemma 3.2 one can prove that of
p
by
L3
L3
one-one to elements of the sorts in still 1-defines K.
has the same power as its L-reduct.
But every model
Hence the new universes introduced
can be dispensed with in favour o f new predicates, and therefore v 1
be converted into a A (L*)-definition of 1 A
K.
can
0
The proof of Corollary 2.7 carries over immediately and we have:
Corollarv 3.6. Suppose
A S HC
A S A are t r a n s i t i v e s e t s , P
and
i c a t e , and
Lz a Boolean logic extending
P on
Then the following are equivalent f o r any model cZass
type
Ao.
L E A
n
and f o r any
a pred-
LA and strongly symbiotic with
K of
< w:
i s li+,(LI)-definable,
(1)
K
(2)
b K is ln+l(P) w . p . i .
Corollary 1 . 7 .
For
A.
A 5 H(ol) :
(1)
1 An(LA(W)) = i K [ t k e model c l a s s
(2)
1 An(LA(W,I)) = {Kithe mode2 c l a s s
b
is An
K
K
A},
w.p.i.
is Ab(Cd) w.p.i.
A}.
The following corollary is proved mutatis mutandis as Proposition 2.10: Corollarv 3.8.
predicate.
\SSZiA: I f
Suppose
Then A:(Li) $
Li i s a Boolean logic extending
-
i f and only if
A;(L[PIA)
E Lz, then Mod($) i s A1(P) b w.p.i.
-
LA and
(SSZl
P a
and
A.
It f o l l o w s that second order logic is not strongly symbiotic, because there 1 I1 1 is no Q such that L1' (- Al(L ) ) A1(LQ). Second order logic is rather the closure of first order logic under the define f o r any abstract logic L*:
(L*) =
{KI the
111 -operation.
model class K
More exactly, let us
is AA(L*)-definable for some n < to}.
409
ABSTRACT LOGIC AND SET THEORY 1
1
= A:u)(Lwu) = A(w)(LW) = A ( u ) ( L I ) .
C l e a r l y , L1'
f i e s t h e s i n g l e - s o r t e d i n t e r p o l a t i o n theorem.
1
A(u)(L*)
Note t h a t
satis-
The f o l l o w i n g c h a r a c t e r i z a t i o n o f
and second o r d e r l o g i c f o l l o w s from 3.6:
A;u)(L*)
C o r o l l a r v 1.9.
Lz
Suppose
a predicate such t h a t
Li
i s a Boolean logic extending P
and
LA
P
and
are strongly symbiotic on A 5 H ( u l ) .
Then the following hold:
1 A(w)(LI) = {Klthe model c l a s s
111
Lil
(2)
K
i s definable w i t h a f l a t f o m l a o f
{E,P} w . p . i .
the language
A].
K i s definable with a f l a t f o m l a w . p . i .
= {Kithe model class
A}.
To sum up, second o r d e r d e f i n a b i l i t y c o r r e s p o n d s t o f l a t d e f l n a b i l i t y i n s e t I1 )-) d e f i n a b i l i t y corresponds t o A -
t h e o r y , i m p l i c i t second o r d e r ( t h a t i s A ( L
2
d e f i n a b i l i t y i n s e t t h e o r y , and f i n a l l y , d e f i n a b i l i t y i n sort l o g i c c o r r e s p o n d s R e c a l l t h a t by Theorem 3.7 of [ 2 ] ,
t o d e f i n a b i l i t y i n s e t theory.
f i r s t order
d e f i n a b i l i t y corresponds t o A y - d e f i n a b i l i t y .
1 1 a l r e a d y t h e whole i m p l i c i t s t r e n g t h o f second o r d e r l o g i c .
It i s well-known
t h e same i s
( s e e e.g.
L r r 5 A(LQ,),
191) t h a t t h e II - p a r t o f second o r d e r l o g i c has
%
because
1
i s II,-definable
Another way o f s a y i n g
(see
[TI).
T h i s f a c t has
t h e f o l l o w i n g more g e n e r a l a n a l o g u e : P r o o o s i t i o n 3.1C. L*
Suppose
A
5
A;~,cL;)
Proof.
c
L*
Suppose
i s symbiotic with
formula i n t h e l a n g u a g e
{E,P}
and
there is a strong l i m i t
LY
R(a)
A (P) 1
A, t h e n
K
A,(P)
and
1 cP(a,b).
w.p.i.
Similarly
If
are t r a n s i t i v z classes and
I I i ( L u w ) I A ( L i ) , then
on
P
.
A.
Suppose
Then
cp(a,b)
w.p.i.
a E A.
such t h a t
A s t h e assumption
R(a)
1
reflects
Il ( L
In view of 3.9 it s u f f i c e s
is
Z,(P)
w.p.i.
cp(a,y)
A.
is a flat
cp(x,y)
h o l d s i f and o n l y i f
P, a
) 5 A(LZ)
1 ww A , t h e above e q u i v a l e n c e shows t h a t
icp(a.y)
A
i s d e f i n a b l e by a f l a t formula of s e t t h e o r y i n t h e language
K
is
w.p.i.
5A
A.
Ao.
A(L;).
t o prove t h a t i f {E,PI
and
HC
i s a Boolean logic symbiotic on
and
b
are i n
implies t h a t is
I,(P )
R(a),
Pw i s w.p.i.
A.
P r o p o s i t i o n 3.10 can b e improved by c o n s i d e r i n g s u i t a b l y d e f i n e d Aa ( w)-Operat i o n s , where a i s an o r d i n a l d e f i n a b l e i n f i n i t e o r d e r l o g i c o r LY E A ( s e e
t91). The r e s u l t s about a b s o l u t e n e s s o f s y m b i o t i c l o g i c s i n t h e p r e v i o u s c h a p t e r c a r r y over t o s t r o n g l y symbiotic l o g i c s a s follows: I,et us consider t h e following properties:
(SAl)
Stc*
is
found w i t h
$bp)
and
Qv$, W$, 3ccp
11 ( P ) - f u n c t i o n s .
and
Vccp
i n (L5) and ( L 6 ) can b e
J. V G h N
410
b
(SA2) There is a A (P)-predicate S 1
k*
t/M E Str(L)(M
cp
++
such that if cp E L*, then
S(M,cp)).
The condition (SA1) and (SA2) together from a notion of strong P-absoluteness of L*.
Note that if P
is omitted, strong absoluteness coincides with the
notion of absoluteness, because every I1-predicate is comes only when some non-trivial predicates P second order logic is &-absolute
1: .
So the difference
are considered. For example,
but not strongly Pw-absolute (see the remarks
after 3.8). The following theorem is proved as 2.12:
and
L*
Suppose Boolean
Theorem 3.11.
Then L+
L+ 5 Ai(L2).
Corollary 1.12.
and
and
P are strongly symbiotic on
A
P s a t i s f y ( S S I I A but not ( S A 2 l .
L* and P are strongly symbiotic on A 1 L2; i s not Al-closed.
Suppose Boolean
and s a t i s f y ( S A 2 1 .
Then
1 Corollarv 1.13. The following logics are not A -closed: 1 (1)
LIPIA, where
12)
LA(W,Q), where
P i s a predicate of s e t theory,
Q i s a generalized q u a n t i f i e r .
Li i s a strongly symbiotic logic on A extending no generalized q u a n t i f i e r s Q' . Qn such t h a t
Hence, i f
A~(L);)
-
LA, there are
..
L ~ ( Q,..., Q ~ ) . 1
Also the proof of Theorem 2.15 carries over:
Suppose
Theorem 1.14.
b
L* and
P
are strongly symbiotic on
V, s a t i s f y
( S A l l , and there i s a I 1 ( P ) - f u n c t i o n which embedds Lmw i n t o b b i s I$,(€') but not 12(P). the predicate Stc
(L*)
L*.
Tken
1 1
This theorem shows how difficult it is to find a syntax for A -extensions, 1
whereas the full A(w)-extension has a simple primitive recursive syntax. The situation is hence -irnilar as in the case of A-extension.
5 4.
LGwenheim numbers
The purpose of this chapter is to transfer the definability results of
5
2
from the level of model classes to the level of spectra and in particular minima of spectra, that is Lowenheim numbers. Definition
4.1. Suppose
L* i s an abstract logic and
u)
E L*.
The s p e c t m
411
ABSTRACT LOGIC AND SET THEORY
of
cp, S p ( q ) , i s the c l a s s {card()!) ( M
k* cpl.
The indexed family
i s called the f a m i l y of L*-spectra.
4.2.
Examples
(1)
The c l a s s of successor cardinals and the c l a s s of l i m i t cardinals are LI-spec t r a .
x
5 p K ) > i s an L I L s p e c t m .
(2)
{x~~IK(K+
(3)
The cZass of r e g u b r cardinals and the class of weakly inaccessible cardinals are LR-spectra. i s an L I ' - s p e c t m .
(4)
{2KI K
a cardinal1
(5)
{zK(K
i s measurable} i s an LII-spectrum.
For other examples of spectra see [131 and [141.
E: I s
The following problem i s c a l l e d t h e spectrum problem for ment of an a r h i t r a r y L*-spectrum a g a i n an L*-spectrum?
LW, f o r example, has a n e g a t i v e s o l u t i o n because but
{K/K
No} i s n o t .
>
{ K ~ K5
The spectrum problem f o r
answer - t h i s w i l l be d i s c u s s e d l a t e r . f o r some i d e n t i t y - s e n t e n c e
C = Sp(cp)
S p ( 7 cp).
LI
fi
1 i s an
Q
L1'
has a
i s an L'I-spectrum,
C
and t h e complement of
This f a c t has a more g e n e r a l analogue.
LW-spectrum
can have a n e g a t i v e
The spectrum problem f o r
p o s i t i v e s o l u t i o n f o r a r a t h e r t r i v i a l reason: i f
t h e comple-
The spectrum problem f o r
C
then
is just
A t f i r s t we n o t e t h e following
t r i v i a l lemma: Lemma
4.3. Suppose
C
i s a class of cardinals and
structures , where card(A) E C . only i f C ' i s I,(L*)-definable. 1
Then
C
C'
i s the c l a s s of
i s an L*-spectrum Cf and
If t h i s i s combined w i t h Theorem 3.5 and C o r o l l a r y 3.9, t h e following charact e r i z a t i o n of s p e c t r a y i e l d s : Theorem
4.4. Suppose
i c a t e , and on
Ao.
Lz
A 5 HC and A. 5 A are t r a n s i t i v e classes, P a preda Boolean logic extending LA and strongly symbiotic with P
Then
Sp(Lt\) = {C 5 CdlC 1
is
b
&(P)
SP(A(~)(L~ = ) I)C S CdlC
w.p.i.
A},
i s definable by a f l a t formula i n the language
{E,P] w . p . i .
A},
J. V&hII?,N
412
SP(L;')
{c E CdlC i s definable with a
=
w.p.i.
f k t formula o f s e t theory
A).
Definition 4.5. Suppose L* i s an abstract logic. The L8wenkeiwnumber Z(L*) of L* i s the l e a s t cardinal K such t h a t min(c) 5 K f o r every C E sp(L*), i f any such K e r i s t . Equivalently, l(L*) i s the l e a s t cardinal K such t h a t i f cp E I,* has a model, then cp has
a model power 5
K.
It is well-known that
l(L*) A
exists if A
is a set.
Theorem 4.6. Suppose A and A0 -c A are t r a n s i t i v e classes, P a predi c a t e , and i$ a Boolean l o g i c extending LA and symbiotic w i t h P on Ao. Then for any n < w: 2(An(Li)) = sup
If Z(AJL$)) moreover
{ K ~ K
A}.
i s IIn(P)-definable w . p . i .
i s a limit cardinal (e.9.
n > 1
LI 5 A(Li)j, then
or
l(An(LA+)) = sup {ala i s Kn(P)-definable w . p . i .
m. Suppose at first that a is a 1 (P)-formula and a E A
cp(x,y)
VB(B 2 a
++
A 1.
is II (P)-definable w.p.i. such that
A.
Suppose
cp(5,a)).
Let K be the class of linearly ordered structures the ordertype of which is an ordinal 2 a. K
is clearly ln(P) w.p.i. A, whence K
But every model 'of K has power 2 card(a). Z(An(Li)). For the converse, suppose K X = min(Sp(cp)) < ~(A~CL;~)). Now VB(B > X
whence X
-
3y
Hence a
2 83M( IM] = y & M 'F* cp))
is II (P)-definable w.p.i. A.
0
Corollam 4.7, For my rudimentary s e t
A and
n > 1:
(I)
l(LIA) = sup { a l a i s n,(Cd)-definable w . p . i .
(2)
l(LA
(3)
l(An(LA)) = sup {ala i s II -definable w . p . i .
(4)
l(Li) = sup {a(a i s d e f i n a b k i n s e t theory w . p . i .
II
= sup {ala i s
n 2-definable.w.p.i.
<
Let cp E Lz such that
l(An(Li)).
K <
is ln(Li)-definable.
min {card(M)lM E K}
5
A),
A}, A), A)
( i n MKM).
ABSTRACT LOGIC AND SET THEORY
413
Part (2) of the above corollary was proved earlier but independently in 161. LGwenheim-numbers can also be characterized in terms of a notion of describability. This notion is related to the notion of indiscribability (see [ 4 ] p. 268) but differs mainly in that less parameters are allowed. Definition 4.8. Let D be a s e t o f f o m l a e of s e t theory. An ordinal a i s D-describable w . p . i . A i f there are a ~ ( x ) E D and an a E Ra r l A such t h a t
and R6
rk(a)
4 . L Suppose
Then there i s a
such t h a t
6 > a
V6(B 1 a
R6
k
$(y)
reflects
P.
a
11 (P)-formula and
a E A
A.
such that
* d6,a)).
be the ll(P)-formula 3xcp(x,y)
$(a).
Ra
is ll,(P)-defimbZe w . p . i . 6 i s ll(P)-describubZe w . p . i . A .
P i s R-absolute and
Proof. Suppose cp(x,y) is
Let
B < a.
P i s R-absolute if euery
The predicate Lemma
for
q(a)
Then y
clearly implies 6 >
->
f3
+
R
a.
Y
k
$(a).
and B
the least 6
such that
Hence $ ( a ) describes B .
RE
1 $(a)
Suppose P i s R-absolute and a is ll(P)-describabZe w . p . i . Then a + 1 is n,(P)-definable w.p.i. A.
Lemma 4 . 1 0 . A.
Proof. Suppose d x ) is a ll(P)-formula and a E A such that R B k @(a) if and only if 6 ? a . Let $ ( y , x ) be the ll(P)-fonuula which says that cp(x) is true in a transitive set which reflects P and the ordinal of which is < y. If $(@,a), then (because P reflectslfor some y < 6 Ry 1 cp(a), whence 8 > a . On the other hand, if 6 > a , then Q(6,a) as one can choose Ra as the required transitive set.
Suppose A and A E A are t r a n s i t i v e s e t s , P an R-absol u t e predicate, and Li an abstract logic extending LA and symbiotic with P on A o , and l ( L z ) i s a limit cardinal. Then Corollarv 4 . 1 1 .
:
l ( L * ) = sup Iala A
i s ll-describable w . p . i .
A}.
Proof. The claim follows immediately from 4 . 9, 4.10 and 4.6. Lemma 4.12.
Suppose
0
a is f i r s t order describable ( t h a t is described by
414
VEGNEN
J.
A. Then a i s II2-definable w . p . i .
some formula of s e t theory) w . p . i . Suppose
Proof.
1 cp(a)
Rg Let
$(y,x)
cp(x)
i s a formula and
i f and only i f
be t h e 12-formula
a € A
B 1 rk(a)
B 1 a.
1 cp(a)".
"Ry
such t h a t for
A.
Then
JI(B,a)
i f and only i f
B 2 a. C o r o l l a r v 4.11.
A:
For any s e t
I1
l ( L A ) = sup {ala is f i r s t order describable w . p . i .
A}.
Another way of f o r m u l a t i n g C o r o l l a r y 4 . 1 1 i s t h e f o l l o w i n g : P r o o o s i t i o n 4.14.
A
Suppose
and
C
A,
are rudimentary s e t s , P an R-
A
absolute predicate, and Li an abstract logic extending LA and symbiotic and l ( L i ) is a l i m i t cardinal. Then with P on A. l ( L z ) = the l e a s t
a
=l1( p ) < V , E , a > aEA'
such t h a t
I n particular,
-_ l ( L k l ) = the l e a s t
such t h a t
a
The above r e s u l t suggest t h e study of o r d i n a l s
Let us denote t h e p r e d i c a t e ( * ) of
a by
a
aEA'
such t h a t
The following lemma w i l l be
Dn(a).
most u s e f u l :
The predicate
Dn(a)
Proof. Let
S(x,y)
1 -predicate
with one f r e e v a r i a b l e "z
is IIn, f o r
Lemma 4 . 1 5 .
is a
1 -formula
y
be t h e
( s e e e.g.
which i s u n i v e r s a l f o r
[41 p. 2 7 2 ) .
w i t h one f r e e v a r i a b l e
n > 1.
y".
Let If
F(z)
1 -formulae
by t h e A,-predicate
F(z), l e t
f(z,a)
be t h e
Ra. f i s c l e a r l y A2. Let S o ( x , y ) be t h e A1-predic a t e which is u n i v e r s a l for 1 -formulae w i t h one f r e e v a r i a b l e y. Now we have:
r e l a t i v i z a t i o n of
Dn(a)
z
to
VY E RaVz E w ( F ( z )
++
and t h e r e f o r e
Dn(a)
is
TIn.
+
(So(f(z,a),y) v
1
S(z,y))),
0
If o is nn-definable w . p . i . A, then there i s a i s An-definable w . p . i . A (n > 1 ) .
ProrJosition 4.16.
such t h a t
Proof.
Let
6
cp(x,y)
be a n,-formula
and
a
E
A
such t h a t
8 2 o
ABSTRACT LOGIC AND SET THEORY
Let
Dn-l(~) & Ru 1 3x - cp(x,v)". 4.15 it i s A 2 . Let
$(u,v) be t h e A - p r e d i c a t e
D,(a)
may not be
We claim t h a t
By r e f l e c t i o n t h e r e i s an o r d i n a l
6.
such
If
1 5 then
y
6 5 y , whence
f o r some
Corollarv
6
B(y,a).
1
e(w,a)
such t h a t
be t h e
1 a.
be t h e l e a s t
6
e ( y , a ) , then
$(&,a)
0
4.17. For any rudimentary s e t
A
l ( LA ) = sup I a / a is A2-definabZe w . p . i .
and
n > 1:
A).
Z ( A ~ ( L ~ ) =) sup {ala i s A -definabZe w . p . i .
Dn(a )
Let
On t h e o t h e r hand, i f
6 5 6 5 y.
e(w,v)
defines an ordinal
$(6,a).
I1
The p r e d i c a t e
Note t h a t
"
b u t by t h e proof of
111
3u 5 w$(u,v).
An-predicate
415
A}.
i s a c t u a l l y e q u i v a l e n t t o a LBwenheim-Skolem-theorem,
a s t h e following theorem shows: Theorem
4.18. The following are equivalent for any n
(11
Z ( A ~ ( L ~ ~=) K) ,
(21
Proof.
In
Note t h a t b o t h
(1) and
cp E Ln(LKw) has a model, t h e n
of power
<
7,
=
RK
( 2 ) imply
1 "cp
because i f
K
=
a E RK
such t h a t
cp(a)
If ( 2 ) h o l d s and whence
n = 2 , it follows from
holds.
Let
cp has a model
Suppose t h e n ( 1 ) h o l d s .
it follows from a s u i t a b l e i n d u c t i o n hypothesis. and
3,.
has a model",
(1).
So ( 2 ) i m p l i e s
K.
Dn-,(~) holds
thet
> 1:
K
K
Suppose
=
We may assume
IK, and i f n
> 2,
c p ( x ) i s a &-formula
be t h e c l a s s of o r d i n a l s
such
a
Dn-l(~) and Ra cp(a). By Theorem 2.4 and ( 1 1 , t h e r e i s a 6 E K 6 E K . A s Dn-,(~), we have R K I. cp(a), as r e q u i r e d . !J
that that
Corollary
4.19.
If
extendible, then
Proof.
If
K
K
§
5.
2
Z ( A ~ ( L ~ ~=) )K .
i s supercompact, t h e n
These f a c t s a r e proved i n [ l l ] .
D7(K).
l ( A (L ) ) =
is supercompact, then D (K); 2
if
K
KW
K.
If
K
such
is
i s extendible, then
0
Hanf-numbers
Hanf-numbers can be c h a r a c t e r i z e d i n t h e same way as LGwenheb-numbers. has t o b e a r i n mind, however, t h a t
A
One
does n o t preserve(Hanf-numbers ( s e e 1151).
Therefore we i n t r o d u c e a new n o t i o n of d e f i n a b i l i t y , bounded d e f i n a b i l i t y , which
i s n e a t enough t o p r e s e r v e Hanf-numbers b u t s t i l l almost as powerful as A- or A,-
416
J. V * h N
definability.
This notion was f i r s t s t u d i e d i n [ 1 5 1 .
chapter i s Theorem 5.6.
The main r e s u l t of t h i s
The chapter ends with a discussion on d e f i n a b l e o r d i n a l s
and s o r t l o g i c .
Let P be a predicate of s e t theory. A predicate of s e t theory i s 1B, ( P ) w.p.i. A i f there are a I o ( P ) f o m k (P(X l , . . . , ~ n , ~ , ~and ) a E A such t h a t
D e f i n i t i o n 5.1.
,.. .,x
S(xl
)
vxl.. .vxn(s(x,,.
..,xn)
f-t
.
3xcp(xl,. , ,xn.x.a))
and vx l...~x s(xl A.
,...,x
) B
is n A l
S
(IX~(P(X,
,...,x
,x,a)I
is a s e t ) .
if - s ( x l ,...,x n ) is B i s both l l ( P ) and II:(P)
i s I I ~ ( P ) w.p.i. w.p.i. A if S
An example of a A (Cd)-predicate which is not (provably)
i n [151.
1 Note t h a t every I1-predicate i s ~ X ( P ( X ) ++
Z:(P)
A
1;
B A,(Cd)
by Levy's theorem.
3 x ( d x ) & W&rk(y) c rk(x)
-t
w.p.i.
w.p.i.
A.
i s given
From t h e f a c t
- cp(y))
B
it follows t h a t every I , ( P ) - p r e d i c a t e is I,(P,Pw). Therefore t h e r e i s no need B t o define 1 (P)-predicates f o r n > 1 - they would coincide with t h e l n ( P ) predicates. Now w e define t h e model t h e o r e t i c analogues of t h e above n o t i o n s . w i t i o n 5. 2 .
Let
be an abstract logic.
L*
definable i f it is 1-defined by an L*-sentence VA ~ K V BE E(A,cp)(card(B) 2
i? is
K is IIB(L*)-definable if i f it i s both 'A
lB(L*)-
A model c l a s s cp
K
is
lB(L*)-
such t h a t
K).
lB(L*)-definabZe. K is AB(L*)-definable
and IIB(L*)-definabZe.
i s a n a t u r a l operation on l o g i c s and resembles A-operation so much t h a t
it i s i n f a c t not a t a l l obvious t h a t t h e r e i s any d i f f e r e n c e between them. a treatment of
AB
see 1151.
For
We pick up some of t h e r e s u l t s o f [ 1 5 1 t o t h e
following lemma ( n o t e t h a t ( 1 ) below f a i l s f o r
A):
Lemma 5.1.
(I) (2)
A'
-
przserves Lthdenheim- and Hanfnumbers.
l B ( ~ * Z(L*) )
i f
-
L"
5
A~(L*) or
L* is one of the following
logics ( o r a fragment of onel L , ~ , Lmw(w), L , w ( ~ a ) , Hence AB(L*) A(L*) f o r such L*. (3) v = L < F l i e s A ~ ( L I ) A(LI).
-
L,~-
ABSTRACT LOGIC AND SET THEORY Con(2FC + A B ( L I )
(4) If Con(ZF), then
417
+ A(L1)).
Related t o t h e bounded notions of d e f i n a b i l i t y i s a new notion of symbiosis as w e l l :
Suppose L* is an abstract logic, P a predicate of s e t theory and A a t r a n s i t i v e c l a s s . L* and P are bowdedly symbiotic a A if the following two conditions are s a t i s f i e d : B (BSZ1 If cp E L*, then Mod(cp) is A,(P) w.p.i. fcp,L} ( B S 2 ) K [ P ] i s AB(LI)-definable. D e f i n i t i o n 5.4.
The pairs of example 2.3 are a l l boundedly symbiotic. We omit t h e proof of t h e following theorem because t h e proof would be mutatis mutandis a s t h a t of 2 . 4 . Theorem 5.5.
“2
cate, and with
Suppose A and A. E A are t r a n s i t i v e classes, P a predian abstract Zogic extending LA and bowdedly symbiotic
on Ao.
P
Then the following are equivalent:
(2)
K is lB(Li)-defiwlbZe,
(2)
K
is I B, ( P ) w . p . i .
Theorem 5.6, Li
cate, and P
on Ao.
A.
Suppose A and A _C A are t r a n s i t i v e classes, P a predia Boolean logic extending LA and boundedly symbiotic with
Then
h ( L i ) = sup Iala
is l:(P)-definable w . p . i .
n > 1:
and for
h ( A n ( L i ) ) = sup Iala
l:(P)
A}
Proof.
is l n ( P ) - definable w . p . i .
A}.
In o r d e r t o prove t h e two claims simultaneously, l e t u s agree t h a t B means l n ( P ) . Now, l e t n > 0. Suppose t h a t a i s l n (P)-
for n > 1
d e f i n a b l e w.p.i.
A.
Let
o r d e r type of which i s
K
< a.
be t h e c l a s s of l i n e a r l y ordered s t r u c t u r e s t h e K
is
B d e f i n a b l e ( u s i n g 2.4 and 5.4, l n ( L i ) B l(An(LA))-definable.
1B ( ABn ( L i ) )-definable.
B J, E A n ( L i )
there is a
If
n > 1 , then
If
li(P)
w.p.i.
for n > 1 means L1’
B 5 An(Lf;)
K.
As
does not have a r b i t r a r y l a r g e models. B f o r every K < a . Hence c a r d ( a ) < h ( A n ( L i ) ) . such t h a t
K
ln(Li)).
l:(Li)-
Hence
whence by 5.3 ( 2 )
K
n = 1 , t h e same conclusion follows t r i v i a l l y .
which lB-defines
o n l y , J,
a < h(A:(Li)).
and t h e r e f o r e
A
For t h e converse, suppose 5 X = sup Sp(cp).
Now
K
K But
has models of power 11,
is
K
is Hence 5 card(a)
has a model of power
It follows e a s i l y t h a t
< h(Ai(L2)).
Let
cp
be i n
A:(L,)
Z
K
418
J.
Hence
B
i s In(P)-definable w.p.i.
X
VGNXNEN A.
n
For any rudimentary s e t A: I1 (1) h(LA ) = sup { a l a is 12-definabZe w . p . i .
Corollarv 5&
is Z,-definable
A}.
(21
h(An(LA)) = sup Iala
w.p.i.
(3)
In MKM: h ( L i ) = sup Iala is definable i n s e t theory w . p . i . = Z(Li).
A}
(n > 1 ) . A}
P a r t ( 1 ) of t h e above c o r o l l a r y was proved e a r l i e r , but independently, i n
[61 ( s e e a l s o 111). Let us w r i t e
In f o r
sup { a l a
i s Rn-definable),
hn
sup { a l a
i s In-definable}.
for
By what we have a l r e a d y proved: ( f o r
n > 1)
In t h e next few lemmas we s h a l l e s t a b l i s h t h e mutual r e l a t i o n s of t h e o r d i n a l s
In ,h n .n
It t u r n s out t h a t t h e following n o t a t i o n i s h e l p f u l :
< w.
t
= the least
a
such t h a t
Dn(a)
= the l e a s t
a
such t h a t
= the least
a
such t h a t
z(An(Law)]=
a.
In 5 t 5 hn f o r n > 1 .
Trivially
FOP n > 0 , In < tn.
Lemma 5.8.
Proof.
Let
be t h e In-formula which i s u n i v e r s a l f o r
S(x,y)
with t h e free v a r i a b l e
y.
a = {cp(y)Icp(y)
1 -formulae
Let
i s a In-formula such t h a t
-Id
y)
d e f i n e s an ordinal}. a E Rw+, Vu
and t h e r e f o r e
E xS(u,y).
y E Rt
. n
Therefore
This
ln _<
Lemma 5.9.
Now
y y <
a E Rt
$(a,y)
.
L e t $(x,y) be a In-formula equivalent t o n i s t r u e f o r some y , whence Jl(a,y) i s t r u e f o r some
i s an o r d i n a l which i s g r e a t e r than any TI -definable o r d i n a l .
tn.
If n
> 1, then
tn is &-definabZe,
and hence
tn
< hn.
ABSTRACT LOGIC AND SET THEORY
R e c a l l from 4.15 t h a t
Proof.
Va(a < t n t f - D
Hence t h e claim follows from
(a) &V6 < aiDn(6)).
0
If n > 1 , then h = l n + l .
Lemma 5.10.
Proof.
nn.
is
Dn
419
Suppose
cp(x,y)
i s nncl-definable and
a
1 -formula
is a
such
that
Let
$(x) be a ln-formula saying t h a t
all
y
E
6
there are then
6 < x.
and
Rx
- $(6).
such t h a t
$(x)
Let
6.
ln-defines
t h e contrary, t h a t is
6 < a.
holds, a contradiction.
2,
Corollarv 5 . 1 1 .
i s an o r d i n a l and
x
be t h e l e a s t of them.
6
*(B)
and
holds f o r
6 < 6
# 6.
whence
y
Hence it s u f f i c e s t o prove t h a t
a 5 6.
Suppose
6 < 6 , t h e n by ( * )
and
Therefore
a 5 6.
l3
<
h3 = l,,
<
then
Therefore
Hence i f
$(y),
y 5 6
If y E R 6
< h2 =
cp(y,6)
VxQ(x,a), a c o n t r a d i c t i o n .
Vx$(x), t h e n
( ~ ( 6 ) . On t h e o t h e r hand, i f
Therefore $(6)
If
~ p ( y , B ) . Hence
0
h,, = l 5 <...
are c a r r i e d o u t w i t h parameters, t h e following
If t h e p r o o f s of 5.8-5.10 theorem y i e l d s : Theorem 5.12.
is a rudimentary s e t and
A
Suppose
n > 1.
Then
z ( A n ( L A ) ) < h(An(LA)) = Z ( A n + , ( L A ) ) . C o r o l l a r v 5.13.
(MKMI
Proof.
dxl,
sets in dxl
Suppose
Ra
...,x
$(al
1m
V.
,...,a
1.
k <
Hence
W.
Ra
We may assume
,...,a
@(al
The p r e d i c a t e " a
is Il - d e f i n a b l e and t h e r e f o r e 2 [ l l ] p . 86), and hence
Z2
<
Hence t h e 1st supercompact i s tendible, then
D
3
(K)
m
< w
1.
Dm-l(a).
...,a
al,
such t h a t
R
NOW
,...,a
+(al
Therefore
< a.
Therefore
a 2 Z(LS ) HF
*
I f the required cardinals e x i s t , then
1st measurable < 2, < 1st supercompact
Proof.
Let
,...,x
V.
( a R
For t h e converse, suppose
).
Then every d e f i n a b l e o r d i n a l must be
C o r o l l a r y 5.14.
such t h a t
a
i s a formula of s e t t h e o r y and
)
Ra, a = l(Ls ) , such t h a t Q(al ,..., a ). HF i s equivalent t o a -formula d x l
,...,x
for a sufficiently large
Ra
= h ( L i F ) = the l e a s t
2(L;F)
<
h2 = Z3 < 1st extendible < h g .
i s measurable" i s < 1,.
t2 5
K.
If
K
1,.
H&ce t h e 1 s t measurable
i s supercompact, t h e n
The p r e d i c a t e " a
12- d e f i n a b l e
l3
(see
i s supercompact" i s
and t h e r e f o r e
( s e e [ l l ] p. 1 0 3 ) , and hence
D,(K)
< h,. <
t3 5
If K.
K
n2'
i s ex-
The p r e d i c a t e
)
420
3 . V&bEN
"a is extendible" is 113
and therefore the 1st extendible is
Hence the 1st extendible < h j .
13-definable.
0
So we see that the Lowenheim- and Hanf-numbers of even the lowest levels of sort logic exhaust a wide range of large cardinals. This would seem to suggest that the logics A (L ) are rather strong indeed. In connection with 5.14, note n A that the ordinals ln,hn,n < w exist even if there are no large cardinals; they exist in L, for example. It seems to be a rather common phenomenon that the Lawenheim-number of
R
logic is smaller (often substantially) than the Hanf-number (see e.g. 5.12). However, in the second part of this paper we shall construct a model of set theory where the Hanf-number of LI
is smaller than the Lowenheim-number of LI.
In that model the spectrum problem for LI has a negative solution, because there is a cardinal a spectrum, but
between l(L1)
K
{hlh <
K}
and h(L1)
such that
{h(h
K}
is
is (obviously) not.
I1
We end this chapter with a remark on another way of characterizing h(LA ) . Definition 5.15. An o r d i m 2
a i s weakly f i r s t order describable w . p . i . A if there are a formula ~ ( x ) of s e t theory and an a E Ra n A such t h a t
R6
1d a )
for
62 a
and R6
Q(a) f o r arbitrary large
Theorem 5.16. Suppose
A
6 < a , 6 2 rk(a).
i s a rudimentary s e t .
I1 h(LA ) = sup {ala i s weakly f i r s t order describable w . p . i .
A}.
BiblioaraDhv [l]
J. Barwise, The Hanf number of second order logic, J.S.L., 37 (1972),
pp. 500-594. [2] J. Barwise, Absolute logics and
LmU, Annals of Math. Logic, 4 (1972),
pp. 309-340. [31 J. Barwise, Admissible sets and structures, Springer-Verlag, Berlin and New York, 1975. [4] F. Drake, Set theory, North-Holland, Amsterdam-Oxford-New York, 1974.
[51
S.
Feferman, Many-sorted interpolation theorems and applications,
Proceedings of the Tarski Symposium, A.M.S. Proc. Symp. in Pure Math., 25, pp. 205-223, A.M.S. 1974.
42 I
ABSTRACT LOGIC AND SET THEORY
A. Krawczyk and W. Marek, On the rules of proof generated by hierarchies, Lecture Notes in Mathematics, 619, Springer-Verlag, Berlin and New York,
1977, pp. 227-240. M. Krynicki and A. Lachlan, On the semantics of the Henkin quantifier,
J. Symbolic Logic, to appear. J. Makowsky, S. Shelah and J. Stavi, A-logics and generalized quantifiers, Annals of Math. Logic 10 (1976),pp. 155-192. R . Montague, Reduction of higher-order logic, The theory of models (Addison,
Henkin, Tarski, editors) North-Holland, Amsterdam, 1965, pp. 251-264. ~
J. Oikkonen, Second order definability, game quantifiers and related
expressions, Cornmentationes Physico-Mathematicae, 48 (1978). R. Solovay, W. Reinhardt and A. Kanamori, Strong axioms of infinity and elementary embeddings, Annals of Math. Logic 13 (1978), pp. 73-116. L. Tharp, The characterization of monadic logic, J.S.L.
38 (1973), pp.
481-488. J . Vaananen, Applications of set theory to generalized quantifiers, Ph.D.
thesis, University of Manchester, October 1977.
J. Vaananen, Boolean valued models and generalized quantifiers. Reports of the Department of Mathematics, University of Helsinki, 1978.
J. Vaiinanen, A-operation and Hanf-numbers. Preprint, Oslo, 1978.
IOGIC COLLOQUIUN 78 M . Boffa, D . van Dalen, K . McAloon l e d s . ) 0 North-Holland Publishing Company, 1979
LATTICE PRODUCTS VOLKER WEISPFENNING University o f Heidelberg
A b a s i c technique i n a l g e b r a i s t h e r e p r e s e n t a t i o n o f an a l g e b r a i c s t r u c t u r e A as a 'compound' o f ' s i m p l e r ' s t r u c t u r e s . One way o f d o i n g t h i s i s t o r e p r e s e n t t h e elements o f A by g l o b a l o r p a r t i a l f u n c t i o n s ; more s p e c i f i c a l l y , one c o n s i d e r s r e p r e s e n t a t i o n s o f A as a s u b d i r e c t o r ' p a r t i a l ' s u b d i r e c t p r o d u c t o f s t r u c t u r e s {Ai}ieI.
The u s e f u l n e s s o f t h i s procedure depends on two p o i n t s :
1.
How much i n f o r m a t i o n i s a v a i l a b l e on t h e f a c t o r s Ai
2.
How much i n f o r m a t i o n i s t r a n s f e r a b l e f r o m t h e Ai
?
to A ?
S i n c e a r b i t r a r y s u b d i r e c t p r o d u c t r e p r e s e n t a t i o n s y i e l d v e r y l i t t l e on p o i n t 2, a number o f concepts has been proposed t o improve t h i s s i t u a t i o n by s p e c i f y i n g more p r e c i s e l y , how ' t h i c k ' o r how ' t h i n ' A i s i n t h e ( g l o b a l o r p a r t i a l ) d i r e c t p r o d u c t o f t h e Ai.
I n t h i s connection, r e p r e s e n t a t i o n s o f f i r s t - o r d e r s t r u c t u r e s by
s e c t i o n s i n sheaves have r e c e i v e d growing a t t e n t i o n i n l a s t 15 y e a r s
-
f i r s t from
a l g e b r a i s t s and then a l s o f r o m model t h e o r i s t s . W h i l e t h e a l g e b r a i c papers c e n t r e on r e p r e s e n t a b i l i t y and c h a r a c t e r i z a t i o n o f f a c t o r s ( p o i n t l), t h e model t h e o r e t i c a l ones deal m o s t l y w i t h t h e t r a n s f e r o f p r o p e r t i e s f r o m f a c t o r s t o f u n c t i o n s ( p o i n t 2 ) . The concept o f a ( g l o b a l o r p a r t i a l ) l a t t i c e p r o d u c t i s s u i t e d f o r b o t h purposes. I t reduces r e p r e s e n t a t i o n s t o two s t e p s : 1.
Expansion o f a g i v e n s t r u c t u r e by a d i s t r i b u t i v e l a t t i c e and c e r t a i n ' t r u t h v a l u a t i o n s ' t o an a b s t r a c t l a t t i c e p r o d u c t .
2.
Representations o f d i s t r i b u t i v e l a t t i c e s by r i n g s o f s e t s .
Most o f t h e model t h e o r e t i c a l i n f o r m a t i o n c o n t a i n e d i n a r e p r e s e n t a t i o n i s a l r e a d y c a p t u r e d i n t h e f i r s t s t e p . As a consequence, model t h e o r e t i c a l t r a n s f e r p r i n c i p l e s ( p o i n t 2 ) can be f o r m u l a t e d and proved i n a ( l a r g e l y e f f e c t i v e ) s y n t a c t i c a l manner f o r a b s t r a c t l a t t i c e p r o d u c t s . The f o l l o w i n g s k e t c h i s i n t e n d e d t o o u t l i n e some concepts and r e s u l t s i n t h e theor y o f l a t t i c e p r o d u c t s . A d e t a i l e d t r e a t m e n t w i t h complete p r o o f s w i l l appear
elsewhere. F o r s i m p l i c i t y , we r e s t r i c t o u r s e l v e s h e r e t o g l o b a l f u n c t i o n s ; s i m i l a r results hold f o r p a r t i a l functions. A l a t t i c e space (LS) i s a p a i r (IJ),
where I i s a non-empty s e t , r i s a s u b l a t t i -
ce o f P(1) w i t h I ~ X , f l x = f i .(1,s) i s c a l l e d boolean i f Z is a boolean a l g e b r a . T o p o l o g i c a l n o t i o n s such as compactness, t h e s e p a r a t i o n axioms To,T1,T2, ous maps, homeomorphisms a r e d e f i n e d as i n a t o p o l o g i c a l space (I,U),
423
continu-
where d i s
V. WEISPFENNING
424
t h e l a t t i c e o f open s e t s . L e t
Ac,G
Ai be a s u b d i r e c t p r o d u c t of s t r u c t u r e s Ai
for
a f i r s t - o r d e r language L, and suppose (1,Z) i s a LS. Then t h e t r u t h - v a l u e o f an L-formula cp(Z) a t %An
[@(a)] =
i s d e f i n e d by
.A
{ i € I I Ail=cp(3(i))} f o r a l l atomic
i s a global
and &An.
So l a t t i c e p r o d u c t (GLP) o v e r ( I & i f [ c p ( t l ) l E t h i s i s a c o n d i t i o n r e s t r i c t i n g t h e ' t h i c k n e s s ' o f A i n n A i f r o m above. Correscp
ponding c o n d i t i o n s bounding t h i s t h i c k n e s s f r o m below a r e g i v e n b y t h r e e types o f 'patchwork p r i n c i p l e s ' f o r A: T h e n - d i s j o i n t g l o b a l patchwork p r i n c i p l e (x-DGPW),the w-global patchwork p r i n c i p l e (x-GPW), and t h e K - s t r o n g g l o b a l patchwork p r i n c i p l e (K-SGPW), where x i s a c a r d i n a l o r
-.
Each o f these p r i n c i p l e s says t h a t c e r t a i n
f a m i l i e s o f f u n c t i o n s {ak}kEK i n A can be patched up t o a f u n c t i o n a i n A w i t h r e s p e c t t o a c o r r e s p o n d i n g f a m i l y {ak}kEK l a t t i c e o f % ) , provided card(K) < K
.
of elements o f X ( o r o f a s p e c i f i e d sub-
Instead o f giving the exact definitions,
we
i11us t r a t e t h e (3-DGPW) (3-GPW) (3-SGPW) Then t h e f o l l o w i n g r e l a t i o n s h o l d : l.(i) (ii)
x-SGPW If
j
K-GPW
j
K-DGPW.
(1,X) i s compact, then
a-GPW
+
and s i m i l a r f o r o-DGPW,
--GPW
( i i i ) I f (1,Z) i s compact and boolean, then 3-DGPW
0-SGPW.
--SGPW.
The s i g n i f i c a n c e o f these p r i n c i p l e s i s t h e f o l l o w i n g : w-DGPW and w-GPW a r e most o f t e n encountered i n ' n a t u r a l ' a1 g e b r a i c r e p r e s e n t a t i o n s . --GPW
characterizes the
s t r u c t u r e of g l o b a l s e c t i o n s o f a f u l l sheaf o f L - s t r u c t u r e s (comp. M a c i n t y r e [19731). --SGPW
i s t h e most i m p o r t a n t p r i n c i p l e f o r model t h e o r e t i c t r a n s f e r theo-
rems, s i n c e i t a l l o w s t o i n f e r g l o b a l ' e x i s t e n c e f r o m
local e x i s t e n c e
i n the f o l l o -
.
( I n the Z ( [ c p ( Z , a ) ] 2 a) f o r &An, a E x case of --GPW t h i s does n o t h o l d i n g e n e r a l , unless t h e e x i s t e n c e 3Zcp(Z,Z(i)) i s unique i n each f a c t o r Ai.)
wing sense: [32cp(Z,%)] 2
a
As i n d i c a t e d above, t h e r e p r e s e n t a t i o n o f an L - s t r u c t u r e A by a GLP proceeds v i a t h e i n t e r m e d i a t e concept o f an a b s t r a c t g l o b a l l a t t i c e p r o d u c t (AGLP). L e t 3 3 be t h e language o f l a t t i c e s w i t h 1, and l e t L* be t h e t w o - s o r t e d language ( L , B , { [cp(Z)] I ~ ( 2 atomic ) L - f o r m u l a } ) , where [cp(Z)1 i s a function-symbol h a v i n g arguments of s o r t L and values o f s o r t B then t h e n a t u r a l L*-expansion
A*= ( A , Z ,
.
I f now A happens t o be a GLP o v e r (I,Z), ([cp(Z)]}
t h e f o l l o w i n g axioms f o r e v e r y a t o m i c L - f o r m u l a 2 . ( i ) ~ ( 2 )H
[ C P ( Z ) I = 1 ; ( i i ) ni!!l[xi=yil
)
o f A apparently s a t i s f i e s
( ~ ( 2 :)
n [cp(~)l s [4p)1
.
LATTICE PRODUCTS
425
Using t h i s f a c t , we now d e f i n e an AGLP as an L * - s t r u c t u r e B ( w i t h L - p a r t BL,B-part
B3 ) s a t i s f y i n g Z ( i ) , ( i i )
and t h e axioms f o r d i s t r i b u t i v e l a t t i c e s w i t h 1. Then a
s t r o n g converse t o t h e above can be proved: 3.REPRESENTATION THEOREM. L e t B be an AGLP. Then any r e p r e s e n t a t i o n o f Baby a l a t t i c e space (1,Z) extends i n an e s s e n t i a l l y unique way t o a r e p r e s e n t a t i o n o f B by A* ,where A i s a GLP o v e r (I ,Z). I n principle, every representation o f a f i r s t - o r d e r structure A by global sections
i n a s h e a f can be o b t a i n e d f r o m t h i s theorem by s u i t a b l e c h o i c e o f an expansion o f
A t o an AGLP B and a r e p r e s e n t a t i o n o f
%.
F o r t h e known r e p r e s e n t a t i o n s o f r i n g s
and l a t t i c e - o r d e r e d r i n g s and groups i n Hofmann [1972] and Keimel [1971] t h i s method i s i n most cases s i m p l e r than t h e o r i g i n a l one. A c r u c i a l p o i n t h e r e i s , t h a t t h e AGLP B t h a t comes up ' n a t u r a l l y ' i s sometimes n o t t h e one t h a t y i e l d s t h e s h e a f r e p r e s e n t a t i o n . I n s t e a d one passes t o a s u i t a b l e meet-homomorphic image o f BB t o o b t a i n a new AGLP 8 ' . Another method c o v e r i n g these r e p r e s e n t a t i o n s which o v e r l a p s w i t h s p e c i a l cases o f 3. has been found i n d e p e n d e n t l y b y Krauss and C l a r k [19791. C a l l t h e r e p r e s e n t a t i o n o f an AGLP B induced by t h e Stone r e p r e s e n t a t i o n o f Ba t h e c a n o n i c a l r e p r e s e n t a t i o n o f B, and c a l l a LS (1,X) a Stone LS i f i t i s To and i f
nD 6 uE i m p l i e s t h a t t h e r e e x i s t f i n i t e s e t s D ' c D , E ' c E w i t h nD' i uE'. Observe moreover t h a t wDGPW, a-GPW, W-SGPW make sense f o r AGLP's, too. Then 1. and 3. y i e l d : f o r a l l D,EcZ,
4.COROLLARY. L e t B be an AGLP s a t i s f y i n g W-DGPW, a-GPW, W-SGPW, r e s p e c t i v e l y , and l e t B be c a n o n i c a l l y r e p r e s e n t e d by A*,where f i e s --DGPW,
--GPW,
--SGPW,
A i s a GLP o v e r
(IJ).
Then A s a t i s -
respectively.
W i t h n a t u r a l l y d e f i n e d concepts o f morphisms f o r GLP's and AGLP's t h i s c a n o n i c a l r e p r e s e n t a t i o n becomes an e q u i v a l e n c e between t h e c a t e g o r y o f AGLP's and t h e c a t e g o r y o f GLP's o v e r Stone l a t t i c e spaces. I f B i s an AGLP c a n o n i c a l l y r e p r e s e n t e d by A*
, we
denote A by
B,
i t s factors by
Bi,
and t h e u n d e r l y i n g space by ( S p B , 5 ) .
C a l l an AGLP A an a b s t r a c t boolean p r o d u c t (ABP), i f A a i s
a boolean a l g e b r a . The
b e s t r e s u l t s c o n c e r n i n g t h e t r a n s f e r o f model t h e o r e t i c p r o p e r t i e s f r o m t h e canon i c a l factors
Ai
o f A t o A a r e o b t a i n e d f o r ABP's A and p r o p e r t i e s d e f i n e d i n
terms of u n i v e r s a i and e x i s t e n t i a l formulas. O t h e r p r o p e r t i e s r e q u i r e c e r t a i n 'maximum p r i n c i p l e s ' f o r A. The method o f p r o o f c o n s i s t s i n an e f f e c t i v e r e d u c t i o n of e x i s t e n t i p l L*-formulas
t o c e r t a i n normal forms. I t i n v o l v e s besides Feferman-
Vaught-type *arguments (see V o l g e r [19761 ) a c o m b i n a t o r i a l theorem r e l a t e d t o P. H a l l ' s theorem on d i s t i n c t r e p r e s e n t a t i v e s . One o b t a i n s i n t h i s way e.g. a characterization
of e x i s t e n t i a l l y
complete ABP's:
L e t K be an v 3 - t h e o r y i n L, and l e t
c
be t h e c l a s s o f a l l ABP's f o r t h e language
V. WEISPFENNING
426
L* such t h a t A s a t i s f i e s 3-DGPW and a l l c a n o n i c a l f a c t o r s Ai a r e models o f K. Then E i s an i n d u c t i v e , elementary c l a s s . Moreover,v+axioms f o r c can be e f f e c t i v e l y c o n s t r u c t e d f r o m K. Denote t h e c l a s s o f e x i s t e n t i a l l y complete s t r u c t u r e s i n a
m ( Z ) an e x i s t e n t i a l L - f o r m u l a , BE
c l a s s A by E ( A ) . L e t A=,
t h e p o t e n t i a l t r u t h - v a l u e o f ~ ( 3 i) n
.
w i t h BiP v ( a ( i ) ) > 5.THEOREM. L e t A a
c
by
-
[(p(3)] =
An.
Then we d e f i n e
{ i d p A I ex. A i c B i t K
I n terms o f t h i s concept, E(E) can be c h a r a c t e r i z e d as f o l l o w s :
.
Then AE E(E) i f f
L-formulas q ( X ) and a l l 36
An,
Ag i s atomless and f o r a l l e x i s t e n t i d l
Bq(Z)]rint[cpol
and
cll~(ti)]trcp(a)l
(where
i n t , c l a r e taken i n t h e t o p o l o g i c a l space SpA w i t h z A as a b a s i s o f open s e t s ) .
-
A t f i r s t glance, t h e second c o n d i t i o n i n t h e theorem appears t o be j u s t s l i g h t l y weaker than t h e c o r r e s p o n d i n g c o n d i t i o n E(Mod K) f o r
all
[ m ( 3 ) 1 d [ q ( a ) ] which means t h a t
Ai
E
iESpA. I n case K has a model companion, t h e two a r e i n f a c t e q u i -
valent: 6.COROLLARY. Suppose E(Mod K ) i s elementary. Then E ( L ) = {AEC I A A.EE(Mod K ) f o r a l l i d p A }
-1
atomless,
i s a l s o elementary.
T h i s can be used t o r e p r o v e t h e e x i s t e n c e o f a model c o m p l e t i o n f o r commutative r e g u l a r r i n g s , commutative r e g u l a r f - r i n g s ( comp. M a c i n t y r e [ 19731, Weispfenning [ 1 9 7 5 ] ) , and t o prove t h i s f a c t f o r l a t t i c e - o r d e r e d a b e l i a n groups w i t h p r o j e c t o r and weak u n i t . The s i t u a t i o n changes r a d i c a l l y , i f K has no model companion: There e x i s t s such a
K and AEE(E) w i t h "0 c a n o n i c a l f a c t o r Ai€E(Mod K ) . More gene-
r a l l y , f o r any c o u n t a b l e u n i v e r s a l t h e o r y K such t h a t Mod K p r o p e r t y , and any n<w
has t h e amalgamation
t h e r e e x i s t s AE E(E) w i t h a t l e a s t n f a c t o r s
Ai 4
E(Mod K ) .
T h i s a p p l i e s i n p a r t i c u l a r t o (non-commutative) s t r o n g l y r e g u l a r r i n g s , where K i s t h e t h e o r y o f s k e w f i e l d s . The b e s t general i n f o r m a t i o n we have on t h e f a c t o r s
Ai
i s the following: 7.COROLLARY. Suppose AEE(E). Then Aik ThV3(E(Mod K ) ) f o r a l l iESpA, and A.EE(Mod K) i f
-1
(i)E5.
A c h a r a c t e r i z a t i o n s i m i l a r t o theorem 5 can be g i v e n f o r 1-extensions i n E. O t h e r p r o p e r t i e s t r a n f e r a b l e by o u r methods i n c l u d e e l i m i n a t i o n o f q u a n t i f i e r s ,
Xo-cate-
g o r i c a l model companions, model t h e o r e t i c r e s u l t a n t s , n-completeness, n - d e c i d a b i -
lity, N o - c a t e g o r i c i ty, prime model e x t e n s i o n s . K.H. Hofmann [ 19721 , Representations o f a l g e b r a s b y continuous s e c t i o n s , B u l l . AMS 78, 291-373. K. Keimel [1971] , The r e p r e s e n t a t i o n o f l a t t i c e - o r d e r e d groups and r i n g s by sect i o n s i n sheaves, L e c t . Notes Math. 248, S p r i n g e r , Heidelberg. P.H. Krauss & D.M. C l a r k [1979] , Global s u b d i r e c t p r o d u c t s , AMS Memoirs 210, Providence, R. I. A.MacintyreC19731 ,Model-completeness f o r sheaves ..., Fund.Math.87, 73-89. H. Volger [1376] ,The Feferman-Vaught theorem r e v i s i t e d , C o l l .Math. 36, 1-11. V.Weispfenning [ 19751 ,Model-completeness and e l i m i n a t i o n o f q u a n t i f i e r s f o r subd i r e c t p r o d u c t s o f s t r u c t u r e s , J . o f Algebra 36, 252-277. REFERENCES.
LOGIC COLLOQUIUM 78 M. Boffa, 4. van Dalen, K . McAloon ( e d s . ) 0 North-Holland Publishing Company, 1979
SOME
0-FIELDS
OF
SUBSETS OF
REALS
BOGDAN WFGLORZ University
of
WrocZaw
The origin of the present considerations lies in two following problems of Ulam: (I) L e t B b e a a - f i e Z d of s u b s e t s o f t h e r e a l l i n e R, w h i c h c o n t a i n s a l l Lebesgue measurable s e t s . Suppose t h a t , f o r e v e r y un-
c o u n t a b l e p a r t i t i o n V of R s u c h t h a t V 5 8 and e a c h member of V is u n c o u n t a b l e , t h e r e i s a s e l e c t o r of V i n B . Does B = P [ R ) ?
(See [lo], Problem 3 4 ) . (11) L e t B b e a o - f i e l d o f s u b s e t s of t h e r e a l l i n e
R,
which
c o n t a i n s a l l B o r e 1 s e t s . S u p p o s e t h a t , for e v e r y p a r t i t i o n V of i n t o two-elements
s e t s t h e r e i s a s e l e c t o r of V i n 8 . Does B
=
R
P(R)?
(See [ll], p.15 and [12]). In 1975, E. Grzegorek and I solved completely both problems by showing the following theorem. THEOREM. There exists a orcomplete field B of subsets of the real line R such that: (a) all Lebesgue measurable sets are in 8 ; (b) all subsets of R of the cardinality less than 2 w are in 8 ; ( c ) for every family of pairwise disjoint two-elements subsets of R there exists a selector of V in B; and (d) B # P ( R 1 , i.e. B is proper. See [ 4 1 and [ 5 1 . Since the formulation of problems (I) and (11) suggested a positive answer rather than the negative one, and our Theorem above gives a negative answer to both (I) and (11), we were trying to "save" anything from Ulam problems by adding some extra assumptions on the field B in the above formulation of (I) and (11). The first attempt in this direction has been formulated in our paper 1 5 1 . We were asking there the following problem. 421
B. Q G L O R Z
428
(111) L e t B b e a o - f i e l d o f s u b s e t s of t h e r e a l l i n e E, w h i c h c o n t a i n s a l l L e b e s g u e m e a s u r a b l e s e t s . S u p p o s e t h a t , for e v e r y p a r t i t i o n V 5 B of R t h e r e e x i s t s a s e l e c t o r of V i n B . Does B = P ( R ) ?
W e c o n j e c t u r e d t h a t t h e a n s w e r i s NO, a t l e a s t i n ZFC
(Notice t h a t , i f e.g.
Z W i s s i n g u l a r and e a c h subset o f
R
+
CH.
of t h e
c a r d i n a l i t y l e s s t h a n 2 W i s L e b e s g u e m e a s u r a b l e , t h e n t h e answer f o r (111) i s Y E S ) . A n o t h e r a p p r o a c h i n g i v i n g s o m e e x t r a a s s u m p t i o n s t o ( I ) and (11) had b e e n s u g g e s t e d t o u s b y P r o f e s s o r S . G l a d y s z . Namely, h e drove our a t t e n t i o n on t o t h e f a c t t h a t r e a l l i n e
R
i s a g r o u p , con-
s e q u e n t l y it seems t o b e n a t u r a l t o a s k i n a l l q u e s t i o n s ( I ) , ( 1 1 ) and (111) a b o u t f i e l d s i n v a r i a n t u n d e r t r a n s l a t i o n s .
Unfortunately,
i t i s q u i t e e a s y t o show t h e f o l l o w i n g p r o p o s i t i o n . PROPOSITION. L e t B b e a p r o p e r A-complete t h e real l i n e
R,
f i e l d of subsets of
which c o n t a i n s a l l one-element
s u b s e t s of
B i s i n v a r i a n t u n d e r t r a n s l a t i o n s . Then f o r e a c h a p a r t i t i o n V of i n 8.
R
2
5
6 < A
R.
Suppose
there is
i n t o a t l e a s t 6 e l e m e n t s s e t s w i t h o u t any s e l e c t o r
( F o r more d e t a i l e d d i s c u s s i o n see [ 1 5 1 ) . T h i s P r o p o s i t i o n shows t h a t t h e r e o n l y r e m a i n s t h e f o l l o w i n g
q u e s t i o n ( b e i n g a n " i n v a r i a n t " v e r s i o n o f ( I ))
.
(IV) Let B be a n i n v a r i a n t under t r a n s l a t i o n o - f i e l d
of s u b s e t s
o f t h e r e a l l i n e R, w h i c h c o n t a i n s a l l L e b e s g u e m e a s u r a b l e s e t s . S u p p ose t h a t , f o r e v e r y p a r t i t i o n V of R i n t o u n c o u n t a b l e s e t s t h e r e i s a s e l e c t o r o f V i n B . Does B = P ( R ) ? I t t u r n e d o u t t h a t assumming s o m e t h i n g l i k e CH i n b o t h q u e s t i o n s (111) and
( I V ) t h e answers a r e a g a i n NO.
The l a s t q u e s t i o n d i s c u s s e d h e r e , w h i c h i s a l s o c l o s e l y r e l a t e d w i t h U l a m p r o b l e m s , arose when G r z e g o r e k and I w e r e t r y i n g t o s o l v e ( I ) and (11). Namely, o u r way o f c o n s t r u c t i n g f i e l d s r e q u i r e d i n ( I )
and (111, w a s t h e f o l l o w i n g . W e w e r e s e a r c h i n g f o r B t o b e a f i e l d g e n e r a t e d by t h e f i e l d o f a l l Bore1 s u b s e t s o f
R
a n d by a o - i d e a l
w h i c h h a s some c o m b i n a t o r i a l p r o p e r t i e s and e x t e n d s t h e i d e a l o f a l l
s e t s h a v i n g t h e Lebesgue measure z e r o . T h i s h a s l e d u s t o t h e f o l l o wing c l a s s o f i d e a l s . DEFINITION. An i d e a l 1 o n
t i t i o n U of of U i n I .
K
K
is an U l a m i d e a l i f f f o r every par-
i n t o a t l e a s t two-elements sets, t h e r e i s a s e l e c t o r
SOME U-FIELDS
OF SUBSETS OF REALS
429
I t has turned o u t t h a t t h e c l a s s of U l a m i d e a l s p l a y s q u i t e a n
rsle
important
i n i n v e s t i g a t i o n s of s t r u c t u r a l p r o p e r t i e s of i d e a l s .
( F o r more i n f o r m a t i o n s a b o u t Ulam i d e a l s see [ 5 1 , a l s o [ 1 3 1 , [ l ] and [ 2 ] ) N e v e r t h e l e s s , u n t i l Autumn
1 9 7 7 , t h e f o l l o w i n g problem had
been open.
( V ) L e t J b e a A-complete i d e a l o n K . K such t h a t 1 5 I ?
Does t h e r e e x i s t a X-com-
p l e t e Ulam i d e a l I o n
Extending some p a r t i a l r e s u l t s of Grzegorek and m e [ 5 ] , and T a y l o r [ 9 1 , and, i n f a c t , u s i n g T a y l o r ’ s t e c h n i q u e , w e p r o v e t h a t t h e answer f o r ( V ) i s YES.
( F o r more i n f o r m a t i o n s s e e 1 9 1 and [ 1 4 1 ) .
The a i m o f t h i s p a p e r i s t o g i v e l a r g e r o r s h o r t e r o u t l i n e s of t h e p r o o f s of t h r e e theorems a n s w e r i n g t h e q u e s t i o n s (111), ( I V ) and
(V). L e t b e g i n from t h e end, i . e .
b e g i n w i t h t h e problem (V). To
simplify t h e formulations, r e c a l l t h e following notion introduced by A . T a y l o r ( s e e e.9.
[9]).
DEFINITION. An i d e a l 1 i s f r i e n d l y w i t h r e s p e c t t o a c l a s s of i d e a l s on
K ,
i f f o r each K E
t h a t t h e i d e a l s K and by K and
TI
*
T
K
t h e r e i s a permutation
I a r e compatible, i.e.
t.
I I
of
K
K
such
t h e i d e a l generated
I i s proper.
Notice t h a t , i f I i s f r i e n d l y with r e s p e c t t o t h e n each i d e a l from
K
K
and I i s U l a m ,
c a n b e e x t e n d e d t o a n U l a m i d e a l . Thus, t o
s o l v e ( V ) i t i s s u f f i c i e n t t o f i n d a Arcomplete U l a m i d e a l which i s f r i e n d l y with respect t o t h e c l a s s
K
o f a l l A-complete i d e a l s on
A ( A s a m a t t e r o f f a c t , t h i s p o i n t may be q u i t e d e l i c a t e , b e c a u s e no
K.
p r o p e r e x t e n s i o n of an Ulam i d e a l i s f r i e n d l y w i t h r e s p e c t t o t h a t class
-
s e e [ l ] and [13]). I n f a c t a l l p a r t i a l r e s u l t s i n s o l v i n g
( V ) w e r e o b t a i n e d u s i n g t h a t way. The f i r s t
one h a s been o b t a i n e d by Grzegorek and m e ( s e e [ 5 ] ) ,
and i t was j u s t s u f f i c i e n t t o s o l v e problems ( I ) and (11). Namely,
w e had proved t h e f o l l o w i n g theorem. THEOREM. The i d e a l N S ,
l a r uncountable c a r d i n a l
Kp
K
o f a l l n o n s t a t i o n a r y s u b s e t s of a reguis friendly with respect t o the c l a s s
of a l l i d e a l s s a t i s f y i n g t h e following combinatorial property:
J E
Kp
i f f t h e r e e x i s t s a p a r t i t i o n U of
of which h a s t h e c a r d i n a l i t y
K,
K
into
K
sets, each
such t h a t no s e l e c t o r of U i s i n J .
Then, a much b e t t e r r e s u l t , f o r K-complete i d e a l s on o b t a i n e d by A.
Taylor 1 9 1 .
K,
was
B. Q G L O R Z
430 THEOREM.
Let
K
b e a s u c c e s s o r c a r d i n a l . Then t h e i d e a l NSK of
a l l n o n s t a t i o n a r y subsets o f of a l l K-complete i d e a l s on
K
is friendly with respect t o the c l a s s
K.
The proof u s e s t h e f o l l o w i n g t e c h n i q u e . L e t J be a g i v e n i d e a l on
K.
Consider t h e s e t S o f a l l f u n c t i o n s g:
5
K,
w e have
q-l({cl)l < f <J g
iff
I t i s easy t o s e e t h a t < J
K +
K
such t h a t f o r e a c h
I n t r o d u c e t h e r e l a t i o n < J on S by
K .
1 5 : g ( ~ s) f ( c ; ) } E J . i s well-founded whenever J i s wl-complete.
Moreover, i f f i s a cJ-minimal
element of S then f
c o m p a t i b l e . Thus u s i n g t h e K-completeness
*
J and NSK a r e
o f J and t h e f a c t t h a t
K
i s a s u c c e s s o r c a r d i n a l , w e c a n e x t e n d J by a d d i n g a new s e t A such t h a t f i s one-one on A , and t h e i d e a l s f * J ( A ) and NSK a r e compatib l e . ( F o r more i n f o r m a t i o n s and d e t a i l s see [ 2 ] o r [ 9 1 ) . U n f o r t u n a t e l y t h i s proof d o e s n o t work when More p r e c i s e l y , one c a n show ( s e e e . 9 . with respect t o
KK
K
is inaccessible.
[ l ] ) t h a t NSK i s f r i e n d l y
( t h e c l a s s o f a l l K-complete
i d e a l s on
K )
iff
K
i s a s u c c e s s o r c a r d i n a l . Another d e f e c t of t h i s p r o o f l i e s i n t h e f a c t t h a t i t i s n o t t o o e a s y t o see what c a n w e d o whenever J i s n o t a K-complete i d e a l on K T o o m i t a l l t h o s e problems l e t i n t r o d u c e t h e f o l l o w i n g i d e a l s on K . L e t X 5 K b e r e g u l a r .
x
E
ri
iff
e
here i s a cardinal
< h
and a r e g r e s s i v e
f u n c t i o n f : X + K s u c h t h a t , for e a c h 5 we h a v e
If-l({cl)I <
I t i s n o t t o o d i f f i c u l t t o check t h a t i f A
X c e s s o r c a r d i n a l t h e n 1 ; = NSK and t h a t l K
9
=:K
and
K
K
i s a suc-
N S K o t h e r w i s e . Now, re-
p e a t T a y l o r ' s p r o o f . Namely, l e t J be a g i v e n A-complete Consider t h e s e t S A of a l l f u n c t i o n s g:
K
e
+ K
i d e a l on
K.
such t h a t t h e r e i s a
c a r d i n a l 0 < X t h a t f o r e a c h 5 < K , /q-l({c})l < e g . A s b e f o r e i n t r o 9 duce t h e r e l a t i o n < J on S,,. Again < J i s well-founded whenever h 2 w 1 '
* J
Moreover, i f f i s a iJ-minimal element of S,, t h e n f c o m p a t i b l e . Thus, u s i n g t h e f a c t t h a t , f o r some e f ( V c < K ) l f - l ( { c > ) I< e f and J i s h-complete,
<
and 1 ; a r e
h we have
w e can extend our i d e a l J
by a d d i n g a new s e t A such t h a t f i s one-one on A and t h e i d e a l s
* J ( A ) and 1: a r e c o m p a t i b l e . ( F o r more i n f o r m a t i o n s and d e t a i l s see [141). T h i s c o n s t r u c t i o n y i e l d s t h e f o l l o w i n g theorem which s o l v e s t h e problem ( V ) . f
SOME 0-FIELDS OF SUBSETS OF REALS
THEOREM.
A.
Let
J
b e a A-complete i d e a l on
a A-complete Ulam i d e a l 1 on
431
K.
Then t h e r e e x i s t s
such t h a t J 5 I .
K
To s o l v e (111) and ( I V ) we u s e a n o l d n o t i o n i n t r o d u c e d by L u s i n and S i e r p i f i s k i ( s e e p r o p e r t i e s L and S i n 1 7 1 , pages 3 6 , 8 0 , 8 1 and
82). DEFINITION. ___-
A subset X 5
R
i s a Lusin s e t i f f o r each set N of
Lebesgue measure z e r o , w e have I X n N I
s
w
A key t o s o l v e ( I V ) i s t h e f o l l o w i n g theorem. THEOREM.
(Assume C H ) .
There e x i s t s a n i n v a r i a n t under t r a n s l a -
t i o n s a-complete i d e a l 1 on
R
which c o n t a i n s a l l s e t s of Lebesgue
measure z e r o s u c h t h a t f o r e v e r y p a r t i t i o n V o f
R
i n t o uncountable
sets t h e r e i s a s e l e c t o r of V i n 1. T o prove it, w e c o n s t r u c t ( w i t h a s m a l l m o d i f i c a t i o n of
Sierpidski's construction
-
[a]) a
s e e e.g.
Hamel b a s i s E f o r
which i s a L u s i n s e t . L e t E = i e : a < ~
~ and 1 l ,e t , f o r C( < w l , Ea 5 < e l . Put 5' U{Et;: 5 < a ] , and d e f i n e a f u n c t i o n r : R + w ~ , by r ( x ) = a
b e t h e l i n e a r subspace of Fa = Ea
-
R
R
spaned by t h e set { e
-
i f f x E Fa. The c r u c i a l p o i n t of t h e p r o o f i s t h e f o l l o w i n g f a c t . FACT. -
I f N h a s Lebesgue measure z e r o t h e n r ( N ) i s a n o n s t a t i o -
n a r y s u b s e t of wl. Indeed,
suppose t h a t f o r some N o f Lebesgue measure z e r o , we have
that r ( N ) is stationary. F i r s t remark t h a t w i t h o u t l o s s of g e n e r a l i t y w e c a n assume t h a t r i s one-one on N .
Since
R
t h e c o u n t a b l e f i e l d of r a t i o n a l s
i s t r e a t e d a s a l i n e a r space over
0,
we c a n assume w i t h o u t l o s s of
g e n e r a l i t y t h a t t h e r e i s some n < w and non-zero r a t i o n a l s s o ,
...,sn
s u c h t h a t each x t N h a s t h e f o l l o w i n g form x = s e o f f0
+
... +
snea
n
,
where
a
0
>
...
> a n'
Then, u s i n g n t i m e s Fodor Theorem, w e c a n assume, a g a i n w i t h o u t loss of g e n e r a l i t y , t h a t t h e r e a r e c o u n t a b l e o r d i n a l s p1 > t h a t f o r each x € N , w e have x = s e
O f f
+ s e 1'
+...+
s e
'n
...
> 93, s u c h
B. Q G L O R Z
432 1
But then the set C = - ( N - ( s e + . . . + s e ) ) has Lebesgue meat 1 81 n Pn consequently r(C) is uncountable, but sure zero and r(C) = :?N), C 5 E and E is G Lusin set which is impossible. This contradiction proves our Fact. To prove the theorem, define the required ideal I on X E 1
iff
R
by:
r(X) E N S w 1
It is easy to see that I is a a-complete ideal on R containing all sets of Lebesgue measure zero. To see that 1 is invariant under translations it suffices to notice that if x,y E R are such that r(x) < r(y), then r(x+y) = r(y). Consequently, for each a E R and each X c R we have r(X) A r(X+a) 5 5 r(a). Finally, let V = {Va: a < w l } be a partition of R into uncountable sets. Since, for each 5 < u l r the set r-l({t)) is countable, the family {r(Va): a < w l } consists of uncountable sets. By Sierpidski Rifining Theorem (see C71), there is a family IU,: a < w l } of pairwise disjoint uncountable sets such that, for each a < w l , Uasr(Va). It is easy to see that there is a selector G of U,: a < w l } in NS,,. But then Y 1 ( G ) E 1 and r-’(G) n Va # 0, for all c1 < w l . Consequently, there is a selector of V in I . This finishes the proof of our Theorem. Using this Theorem and Corollary 3 from [ 5 1 we can see that the field generated by our ideal 1 and the field of all Bore1 subsets of R is a proper field which gives the answer NO for the problem (IV). Thus we have the following theorem. THEOREM. B. Assume CH. There exists a proper a-field B of subsets of R such that (a) B contains all Lebesgue measurable sets; (b) B is invariant under translations of R; (c) for every partition V of R into uncountable sets there is a selector of V in 8 . We do not know if the assumption of CH in Theorem B is redundant. To solve (111), we need a stronger concept than just Lusin sets. R a strongly Call, after T.G. McLaughlin (see C 6 1 ) , a subset X Lusin set, if for each Lebesgue measurable set N we have TX n N I 5 w iff N has Lebesgue measure zero.
SOME U-FIELDS OF SUBSETS OF REALS
473
Then n o t i c e t h a t i f X i s a s t r o n g l y L u s i n s e t t h e n
Assume CH.
w e c a n o r d e r X i n t h e t y p e w 1 s u c h t h a t f o r e a c h s e t N of p o s i t i v e Lebesgue measure, t h e s e t N n X i s a s t a t i o n a r y s u b s e t o f X . T h i s f a c t y i e l d s some c o n s e q u e n c e s . F i r s t o f a l l , r e p e a t i n g t h e arguments used i n t h e proof of Theorem B , w i t h a Hamel b a s i s b e i n g a s t r o n g L u s i n
s e t , w e c a n g e t t h e f o l l o w i n g s t r e n g t h e n i n g o f Theorem B . THEOREM.
measure m on
R
B'.
A s s u m e CH. Then t h e r e e x i s t s a a - a d d i t i v e i n v a r i a n t
s u c h t h a t , f o r e a c h Lebesgue m e a s u r a b l e s e t X I i t s
Lebesgue measure i s j u s t m ( X ) , and t h e
a-field of
m-measurable
s e t s s a t i s f i e s t h e t h e s i s of Theorem B. Another a p p l i c a t i o n o f s t r o n g L u s i n s e t s i s t h e f o l l o w i n g t h e orem. THEOREM. C .
a u-field
(Brzuchowski
-
C i c h o d ) . A s s u m e CH.
B o f subsets of t h e r e a l l i n e
t i o n V 5 B of
R
R
Then t h e r e e x i s t s
s u c h t h a t f o r each p a r t i -
t h e r e e x i s t s a s e l e c t o r of W i n 8.
I n d e e d , l e t X b e a s t r o n g L u s i n s e t . L e t I b e t h e i d e a l of a l l n o n s t a t i o n a r y subsets o f X.
L e t B b e t h e a - f i e l d g e n e r a t e d by 1 and
t h e f i e l d of a l l Lebesgue m e a s u r a b l e sets. Then B s a t i s f h e s a l l r e q u i r e m e n t s of Theorem C . T h i s theorem s o l v e s (111). I n f a c t , t o p r o v e Theorem C , much
less t h a n CH i s n e c c e s s a r y . On t h e o t h e r hand, J. Cichod h a s i n f o r med m e , t h a t t h e r e g u l a r i t y o f 2 w i s n o t enough t o p r o v e Theorem C . For more d e t a i l e d d i s c u s s i o n and o t h e r a p p l i c a t i o n s of s t r o n g L u s i n
s e t s see [ 3 1 .
REFERENCES
[I1
James E. Baumgartner, Alan T a y l o r and S t a n l e y Wagon, I d e a l s on uncountable c a r d i n a l s , ( t y p e w r i t t e n pages!.
[21
, S t r u c t u r a l p r o p e r t i e s of i d e a l s , t a t i o n e s Mathematicae.
[31
J a n Brzuchowski, J . Cichod and B. Wgglorz, Some a p p l i c a t i o n s of s t r o n g l y L u s i n s e t s , ( t o a p p e a r ) .
[41
E . Grzegorek and B. Weglorz, On e x t e n s i o n s of f i l t e r s ( a b s t r a c t ) , J o u r n a l of Symbolic L o g i c , 42, ( 1 9 7 7 ) p.113.
( i n p r i n t ) Disser-
[51
, E x t e n s i o n s o f f i l t e r s and f i e l d s of s e t s ( I ) ( i n 25 ( S e r i e s A ) , (1978), 275-290. p r i n t ) J . Austral.Math.Soc.
161
T.G. McLauqhlin, M a r t i n ' s Axiom and some c l a s s i c a l c o n s t r u c t i o n s , B u l ' i e t i n . of t h e A u s t r a l i a n M a t h e m a t i c a l S o c i e t y , 1 2 , ( 1 9 7 5 ) p.351-362.
434
141 151
B. WFGLORZ
W. Sierpidski, Hypothbse du Continuum, Monografie Matematyc ne, Vo1.4, Warszawa-Lw6w, (1934). , La base de M. Hamel et la propriGt6 de Baire, Puhl.Math.Univ. Belgrade, 4, (1935), p.220-224. A. Taylor, On saturated sets of ideals and Ulam's problem, Fund. Math. (in print). S.M. Ulam, Scottish Book, typewritten pages, L w ~ w ,1935-194 , A collection of mathematical problems, Interscience Publishers, Inc., New York, (1960). , SIAM Review, Combinatorial Analysis in Infinite Sets and some physical theories, 6, (1964), p.343-355. B. Weglorz, Some properties of filters,in Set Theory and Hierarchy Theory V, Proceeding of the Conference held at Bierutowice, Poland, September 17-24, 1976; in Lecture Note in Mathematics ~01.619, SpringerTVerlag, Berlin, Heidelber New York, (1977). p.311-328. , On extensions of filters to Ulam filters, Bull. Polon. Acad. Sci. (1978), (in print). ~ -, Large-invariant ideals on abstract algebras, (handwritten pages).