ISILC - Proof Theory Symposion: Dedicated to Kurt Schütte on the Occasion of His 65th Birthday. Proceedings of the International Summer Institute and ... in Mathematics)

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

500 ~ ISILC Proof Theory Symposion Dedicated to Kurt SchLitte on the Occasion of His 65th Birthday Proceedings of the International Summer Institute and Logic Colloquium, Kie11974

Edited by J. Diller and G. H. MiJller

Springer-Verlag Berlin.Heidelberg-NewYork 1975

Editors Prof. Justus Diller Westf~.lische Wilhelms-Universit~t

Institut fQr mathematische Logik und Grundlagenforschung Roxeler Stra6e 64 44 M6nster/IBRD Prof. Gert H. MLiller Mathematisches Institut der Universit~t Heidelberg Im Neuenheimer Feld 288 69 Heidelberg 1/BRD

Library of Congress Cataloging in Publication Data

ISILC Proof Theory Symposium, University of Kiel, 1974. !S_TLC Proof 'Y'aeory Symposium. (Lectures notes in mathematics ; 500) Text in English oF German. i. Proof theory--Congresses. 2. Sch~tte, Kurt --Bibliography. I. Sch~tte, Kurt. II. Diller, Justus. IIl. M~Zler, Gert Heinz, 1923IV. International Summer Institute and Logic Colloquium, University of Xiel, 1974. V. Series: Lecture notes in mathematics (Berlin) ; 500. QA3.L28 no.500 [ Q A g . 5 L ] 510'.8s [511'.3] 75-40482

AMS Subject Classifications (1970): 02D05, 02D99, 02E05, 02F29, 02F40

ISBN 3-540-07533-X Springer-Verlag Berlin 9 Heidelberg 9 New York ISBN 0-387-07533-X Springer-Verlag New York 9 Heidelberg 9 Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin 9 Heidelberg 1975 Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

Kurt Sch~tte

VORWORT Der v o r l i e g e n d e der P r o c e e d i n g s q u i u m Kiel

Band

" S y m p o s i o n on P r o o f T h e o r y "

des I n t e r n a t i o n a l

Summer

ist ein Tell

I n s t i t u t e and Logic C o l l o -

1974 - ISILC -, yon d e n e n der a n d e r e T e i l in e i n e m e i g e n e n

Band der L e c t u r e Notes

erscheint.

Die h i e r g e s a m m e l t e n A r b e i t e n b e h a n d e l n T h e m e n aus dem w e i t e r e n B e r e i c h der B e w e i s t h e o r i e M~nchen,

anli~lich

seines

65.

und sind P r o f e s s o r Dr. Kurt Sch~tte,

Geburtstages

gewidmet.

A r b e i t e n w u r d e n auf dem S y m p o s i o n ~ b e r B e w e i s t h e o r i e Kurt

Sch~tte

V i e r yon d i e s e n zu E h r e n yon

im R a h m e n des ISILC am 2 . 8 . 1 9 7 4 v o r g e t r a g e n ,

zehn A r b e i t e n w u r d e n in den S e k t i o n e n des ISILC v e r l e s e n auf der T a g u n g als A b s t r a c t elm s c h r i f t l i c h e s Kurt

Sch~tte

vor.

S y m p o s i o n an,

e i n g e l a d e n wurden,

weitere oder lagen

Der K i e l e r V e r a n s t a l t u n g

s c h l o ~ sich

zu dem F r e u n d e und n a h e K o l l e g e n yon die in K i e l nicht

anwesend

sein k o n n -

ten. Kurt S c h ~ t t e hat wicklung Salzwedel war dort

in der Altmark, 1933 David

beigetragen. studierte

Hilberts

es jedoch Paul Bernays, durch

in den l e t z t e n 30 J a h r e n w e s e n t l i c h

der B e w e i s t h e o r i e

nicht bei der m a t h e m a t i s c h e n

letzter Promovend.

Logik,

dienst und legte

1948 das A s s e s s o r e x a m e n

in GSttingen,

Logikern,

wurde

Mit

der H i l b e r t - S c h u l e ,

Seholz,

1936-1945

Noch w i h -

am M a t h e m a t i s c h e n

In-

er zu dem k l e i n e n K r e i s yon

die G r u n d l a g e n f o r s c h u n g

in D e u t s c h -

ihm A c k e r m a n n und A r n o l d S c h m i d t Behmann,

aus

H e r m e s und S c h r S t e r aus

Schule.

1950 f o l g t e nach Marburg~

Heinrich

Bedingt zunichst

in den S c h u l -

in H a n n o v e r ab.

er H i l f s k r a f t

die in der N a c h k r i e g s z e i t

der M ~ n s t e r a n e r

1945 g i n g e r

und s e i t d e m g e h S r t e

land w i e d e r a u f g e b a u t haben:

Sch~tte

s o n d e r n a r b e i t e t e yon

Nach dem Z u s a m m e n b r u c h

stitut

In e r s t e r Linie war [3] b e t r e u t e .

Lage b l i e b Kurt

als N e t e o r o l o g e .

rend s e i n e r S c h u l t i t i g k e i t

in

er in B e r l i n und G ~ t t i n g e n und

der seine D o k t o r a r b e i t

die s c h l e c h t e w i r t s c h a f t l i c h e

zur E n t -

G e b o r e n am 1 4 . 1 0 . 1 9 0 9

er als w i s s e n s c h a f t l i c h e r

wo er sich

1952 h a b i l i t i e r t e .

Assistent

Wihrend

A r n o l d Schmidt

er in den fri~hen

50er J a h r e n auch ~ b e r die G r u n d l a g e n der G e o m e t r i e und ~ b e r L a g e r u n g s probleme arbeitete,

konzentrieren

sich seine V e r ~ f f e n t l i c h u n g e n

seit

den s p i t e n 50er J a h r e n m e h r und m e h r auf die L o g i k und B e w e i s t h e o r i e . 1959 e r s c h i e n in der " G e l b e n S e r i e " Wiss.

S p r i n g e r Verlag)

der er s e i n e von G e n t z e n b e e i n f l u ~ t e gramms k l a r und u m f a s s e n d Neufassung

( G r u n d l e h r e n der Nathem.

seine gro~e N o n o g r a p h i e

in

A u f f a s s u n g des H i l b e r t s c h e n P r o -

f o r m u l i e r t hat.

in V o r b e r e i t u n g ,

"Beweistheorie",

Von d i e s e m Buch ist

eine

die yon C r o s s l e y ins E n g l i s c h e ~ b e r s e t z t

Vl

wird.

Im Jahr 1959/60 war Sch~tte als Castprofessor am Institute for

Advanced

Study in Princeton,

schen Hochschule versity. beiten,

1961/62 an der Eidgen~ssischen

in ZUrich und 1962/63 an der Pennsylvania

In diesen Jahren publizierte nimlich

die Arbeit

er drei besonders

TechniState Uni-

wichtige Ar-

[24] zur einfachen Typentheorie,

rer zu den ersten, n i c h t k o n s t r u k t i v e n

die spa-

Beweisen yon Takeutis Fundamen-

talvermutung

durch Takahashi

ten

[31], in denen er g l e i c h z e i t i g mit Feferman die genaue

[30] und

beweistheoretische stark kritische

und Prawitz gefUhrt hat, und die Arbei-

Stirke der verzweigten Typenlogik durch die erste

Ordinalzahl

charakterisiert

hat.

1963 nahm er einen

Ruf auf den Lehrstuhl fur Logik und Crundlagenforschung phischen Seminar der Universit/t matische

Institut

der Universitit

Sein Ergebnisbericht seher Logik" wurde 1952,

Kiel an,

"Vollstindige

1966 g i n g e r

M~nchen,

am Philosoan das Mathe-

an dem er seither t/rig ist.

Systeme modaler und intuitionisti-

1968 abgeschlossen.

Dazwischen,

beginnend

schon

arbeitete Kurt Sch~tte immer wieder Uber Systeme zur konstruk-

riven Bezeichnung yon Ordinalzahlen, Ver~ffentliehungen

[14],

[37],

was seinen Niederschlag

[38], [41] und [42],

schiedenen Arbeiten seiner SchUler gefunden hat. glied der Bayerischen Akademie Kurt SchUtte heiratete

Haus gef~hrt,

Seit

1973 ist er Mit-

der Wissenschaften. im Jahr 1937 Friulein Hanna Lechte.

Das Ehepaar Sch~tte hat zwei TSchter, sela MSncke.

in den

aber auch in ver-

Frau Sigrid Dreyer und Frau Gi-

Herr und Frau Sch~tte haben stets ein gastfreundliches das auslindischen

Kollegen

ein StUtzpunkt

in Deutsch-

land wurde und an das die zahlreichen Freunde und Ciste der Familie gem

denken. Die Herausgeber dieses Bandes danken zuerst den Kieler Kolle-

gen Arnold Oberschelp und Klaus Potthoff fur deren organisatorische Leistung und den erfolgreichen Verlauf des ISILC und darin des Symposions Uber Beweistheorie.

Ferner geht unser herzlicher Dank an Frau

HeBling und Frau Schaefer fur die Erledigung u m f a n g r e i c h e r arbeiten,

Schreib-

an Frau Ernst fur die Erstellung des Schriftenverzeichnisses

und an die Herren Dipl.-~athematiker

Vogel und Rath fur die Durchsicht

zahlreicher Manuskripte.

sagen wir dem Springer-Verlag

aufrichtigen

SchlieBlich

Dank fur seine entgegenkommende

J. Diller

(M~nster)

Mitwirkung.

G.H. M~ller

(Heidelberg)

INHALTSVERZEICHNIS

Verzeichnis Buchholz,

der P u b l i k a t i o n e n

Wilfried,

von Kurt

Curry,

J o h n N.,

in c o m b i n a t o r y

Feferman,

Solomon,

partial Felscher,

der A n a l y s i s

Walter,

type-free

Yoshito,

Calculability

functionals

of finite

(a r e v i s e d v e r s i o n ) Georg,

Leivant,

Daniel,

Lopez-Escobar,

on a t h e m e

E.G.K.,

pleteness Luckhardt,

of the p r i m i t i v e

type

Horst,

generalization

for a r i t h m e t i c

of P r a w i t z )

elements

logic

in a c o n s i s t e n c y

Wolfgang,

Church-Rosser-Theorem

endlich langen Termen 0sswald,

Horst,

198

proof 233

ffir X-Kalkiile mit u n -

.............................

Uber Skolemerweiterungen

s t i s c h e n L o g i k mit G l e i c h h e i t

182

com-

......

for s i m p l e type t h e o r y I . . . . . . . . . . . . . . . . . . . . . . . . . . Xaa2,

164

(va-

Intuitionistic

second-order

152

of

...................

and Wim V e l d m a n ,

The real

119

recursive

due to Schfitte ..............

of a r e s t r i c t e d

73

over the n a t u r a l n u m b e r s

on a recent

theorems

56

of

I .........

........................

Strong normalization

riations

theories

...............................

Observations

completeness

44

K o n s t r u k t i o n e n mit B e w e i -

sen und S c h n i t t e l i m i n a t i o n Hanatani,

26

Funktionalin-

and c l a s s i f i c a t i o n s ,

Kombinatorische

4

standardization

.........................

Non-extensional

operations

........

..............................

Justus und Helmut V o 6 e l , I n t e n s i o n a l e terpretation

Kreisel,

Sound functors

of g e n e r a l i z e d

logic

Sy-

...........................

and Anil Nerode,

H a s k e l l B., A study

Diller,

............

N o r m a l f u n k t i o n e n und k o n s t r u k t i v e

steme von 0 r d i n a l z a h l e n CrossleN,

Sch~tte

257

in der i n t u i t i o n i -

.....................

264

Vlll

Pfeiffer,

Helmut,

W(X) Pohlers~

Eine V a r i a n t e

f~r O r d i n a l z a h l e n

Wolfram,

Prawitz,

Dag,

inductive Comments

Bruno,

Schwichtenberg,

for the p r o v a b i l i t y

definitions

of truth

Bemerkungen

Helmut

Wainer,

and r e c u r s i o n in h i g h e r types Takeuti~

Gaisi,

Troelstra,

Consistency

Anne S., N a r k o v ' s

for t h e o r i e s

proofs

Infinite

.......

290 320

terms

...........

and M a r k o v ' s

sequences

271

and the

.....................

and o r d i n a l s

principle

of choice

procedures

.........................

zu Regel und S c h e m a

and S t a n S.

ite-

.......................

on G e n t z e n - t y p e

267

of

i n d u c t i o n in s y s t e m s w i t h n - t i m e s

classical notion Scar2ellini,

............................

An u p p e r b o u n d

transfinite rated

des B e z e i c h n u n g s s y s t e m s

341 365

rule

..................

370

VERZEICHNIS DER PUBLIKATIONEN VON KURT SCHUTTE Stand M~rz 1975 Monographien LI]

Beweistheorie. Springer Berlin-G~ttingen-Heidelberg 1960 (Bd. 103 der "Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen").

[2]

Vollst~ndige Systeme modaler und intuitionistischer Logik. Springer Berlin-Heidelberg-New York 1968 (Bd. 42 der "Ergebnisse der Mathematik und ihrer Grenzgebiete"). Wissenschaftliche

Aufs~tze

[3]

Untersuchungen zum Entscheidungsproblem der mathematischen Logik. Math. Ann. I09 (1934), 572-603.

[4]

Uber die ErfGllbarkeit einer Klasse von logischen Formeln. Math. Ann. 110 (1934), 161-194.

[5]

Uber einen Teilbereich des AussagenkalkGls. Comptes Rendus des S@ances de la Soci6t6 des Sciences et des Lettres de Varsovie XXVI 1933, Classe III, I-3.

[6]

SchluBweisen-Kalkiile der Pr~dikatenlogik. (1950), 47-65.

[7]

Beweistheoretische Erfassung der unendlichen Induktion in der Zahlentheorie. Math. Ann. 122 (1951), 369-389.

[8]

Mit B.L. van der Waerden: Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz? Math. Ann. 123

Math. Ann.

122

(1951), 96-124. [9]

Die Eliminierbarkeit des bestimmten Artikels in Kodifikaten der Analysis. Math. Ann. 123 (1951), 166-186.

[10]

Eine Bemerkung ~ber quasirekursive Funktionen. Logik u. Grundlagenforschung I (1951), 63-64.

Arch. f. math.

~I]

Beweistheoretische Untersuchung der verzweigten Analysis. Math. Ann. 124 (1952), 123-147.

~2]

Mit B.L. van der Waerden: Das Problem der dreizehn Kugeln. Math.

Ann. 125 (1953), 325-334. ~31

Zur Widerspruchsfreiheit (1953), 394-400.

~4]

Kennzeichnung von 0rdnungszahlen durch rekursiv erkl~rte Funktionen. Math. Ann. 127 (1954), 15-32.

~5 ]

Ein widerspruchsloses System der Analysis auf typenfreier Grundlage. Math. Zeitschrift 61 (1954),~160-179.

~6]

Uberdec~angen der Kugel mit h~chstens acht Kreisen. Math. Ann.

129 (1955), 181-186.

einer typenfreien Logik. Math. Ann.

125

2

[17]

Ein Schlies

[18]

Die Winkelmetrik in der affin-orthogonalen Ebene. Math. Ann.

[19J

Gruppentheoretisches Axiomensystem einer verallgemeinerten euklidischen Geometrie. Math. Ann. 132 (1956), 43-62.

~.oj

Schlie~ungss~tze fur orthogonale Abbildungen euklidischer Ebenen. Math. Ann. 132 (1956), 106-120.

[21]

Ein System des verknGpfenden Schlies u. Grundlagenforschung 2 (1956), 55-67.

[22]

Der projektiv erweiterte Gruppenraum der ebenen Bewegungen. Math. Ann. 134 (1957), 62-92.

[23]

Aussagenlogische Grundeigenschaften formaler Systeme. Dialectica 12 (1958), 422-442.

[24]

Syntactical and semantical properties of simple type theory. J. of Symbolic Logic 25 (1960), 305-326.

[25]

Ein formales System der klassischen Aussagenlogik mit einer einzigen GrundverknGpfung. Arch. f. math. Logik u. Grundlagenforschung 5 (1961), 113-118.

[26]

Logische .Abgrenzungen des Transfiniten. Freiburg/~ttuchen 1962, 105-I 14.

[27]

Der Interpolationssatz der intuitionistischen Pr~dikatenlogik. Math. Ann. 148 (1962), 192-200.

[28]

Lecture Notes in Mathematical Logic (On Metamathematics). The Pennsylvania State University 1962/63.

[29]

Minimale Durchmesser endlicher Punktmengen mit vorgeschriebenem Mindestabstand. Math. Ann. 150 (1963), 91-98.

[30]

Eine Grenze fur die Beweisbarkeit der transfiniten Induktion in der verzweigten Typenlogik. Arch. f. math. Logik u. Grundlagenforsehung 7 (1964), 45-60.

~1]

Predicative well-orderings. Formal systems and recursive functions, Proc. of the 8th logic coll. Oxford 1963, S. 280-303, Studies in Logic, Amsterdam 1965.

~2]

Probleme und Methoden der Beweistheorie. (1965), 562-567.

~3]

Neuere Ergebnisse der Beweistheorie. (einstGndiger Rahmenvortrag). Proc. of the International Congress of Mathematicians at Moscow 1966.

~4]

Nit J.N. Crossley: Non uniqueness at ~2 in Kleene's 0 . Arch. f. math. Logik u. Grundlagenforschung 9 (1967), 95-101.

~5]

Zur Semantik der intuitionistischen Aussagenlogik. Contributions to Mathematical Logic, North-Holland Publ. Company, Amsterdam 1968.

fur Inzidenz und 0rthogonalit~t.

Math. Ann.

129 (1955), 424-430. 130 (1955), 183-195.

Arch. f. math. Logik

Logik und Logikkalk~l,

Studium Generale 18

[36]

On simple type theory with extensionality. Logic, Methodology and Philosophy of Sc. III, North-Holland Publ. Company, Amsterdam 1968.

[37]

Ein konstruktives System von Ordinalzahlen I u n d II. Arch. f. math. Logik u. Grundlagenforschung 11 (1969), 126-137, und 12 (1969), 3-11.

[38]

Nit H. Levitz: A characterization of Takeuti's ordinal diagrams of finite order. Arch. f. math. Logik u. Grundlagenforschung 14 (1970), 75-97.

[ 39]

Mit J. Diller: Simultane Rekursionen in der Theorie der Funktionale endlicher Typen. Arch. f. math. Logik u. Grundlagenforschung 14 (1971), 69-74.

[4O]

Artikel zu den StichwSrtern "Beweistheorie","Finit", "Formales System", "Hilbertsches Programm", "Metamathematik", "Widerspruchsfreiheit" in: J. Ritter (Hrsg.), Historisches WSrterbuch der Philosophie, Schwabe & Co. Verlag Basel/Stuttgart, ab 1971.

[41]

Einftthrung der Normalfunktionen ~ ohne Auswahlaxiom und ohne Regularit~tsbedingung. Erscheint in: Arch. f. math. Logik u. Grundlagenforschung.

[42]

Mit W. Buchholz: Die Beziehungen zwischen den Ordinalzahlsystemen 2 und ~(~). E rscheint in: Arch. f. math. Logik u. Grundlagenforschung.

NORMALFUNKTIONEN

UND KONSTRUKTIVE

SYSTEME

VON 0RDINALZAHLEN Herrn Professor Dr. Kurt Schdtte 65~ Geburtstag

zum

gewidmet

W. Buchholz

Als Hilfsmittel

fGr beweistheoretische

verschiedenen Autoren konstruktlve zahlen elngefGhrt, lassen.

~(~) entwickelt

Im folgenden

werden

klassischen

0rdinalzahltheorle

sollen konstruktive

her

Bezeichnungssysteme

, die ihrem formalen Aufbau nach eng verwandt

slnd mit den Systemen E(N) , 2, W(X) yon SchGtte [12], die aber im Gegensatz llche Erkl~rung

wurden von

die zwar auf sehr einfache Welse induktlv definiert

slnd, sich aber nur schwer v o n d e r verstehen

Untersuchungen

Bezeichnungssysteme I)" fGr Ordinal-

[17] und Pfeiffer

zu Jenen eine relatlv einfache

in der klassischen 0rdlnalzahltheorie

[11]

und natGr-

besitzen.

Wir

werden dabei von gewissen von FEFERMAN und ACZEL [I] stammenden - und yon BRIDGE in [3] n~her untersuchten

- Normalfunktlonen

die sich sehr elnfach und ohne Bezugnahme sischen Theorie definieren

Arbeit:

Entwicklung gewisser

nisse Gber die Funktionen |

Dabel enth~lt

schon bei Bridge

Wohlordnungsbeweis

In w

behandelt In w

fGr die Systeme ~(T), ~({g~)

und mlt wird eln gegeben.

der Wohlordnungsbowelse

[19]

yon

, Od(1) yon Kino [8] , Z(N)

, 2 und W(X) yon Pfeiffer

nur

werden

[ 11],[ 12] .

I) z.B. die Systeme O(n) von Takeutl [17]

im wesentlichen

(ohne Beweise).

Dieser Bewels ist eine Verallgemeinerung

yon SchHtte

w

8(~) im Rahmen

[3] bewlesener Ergeb-

8(T) und ~({g})

den Systemen anderer Autoren verglichen

SCHU-TTE [ 17] und PFEIFFER

I- 3 beinhalten

mit zum Tell neuen Beweiseno

die speziellen Bezeichnungssysteme konstruktiver

Die Paragraphen

der Bezelchnungssysteme

der klassischen 0rdinalzahltheorie. eine Zusammenstellung

ausgehen, in der klas-

lassen.

Zum Inhalt der vorliegenden die nichtkonstruktlve

|

auf Hauptfolgen

[11],[12]

W. Buchh olz w

D i e

5

N o r m a 1 f u n k t i o n e n

Wir legen hier (in w

- w

8a

die axlomatlsche Mengenlehre yon Zermelo-

Fraenkel mlt Auswahlaxlom zugrunde. Die Klasse On der 0rdinalzahlen sei in fibllcher Weise so definlert, da6 ~ = {~ e On J ~ < a] fGr Jedes a ~ On ist. Ksei

die Klasse der Kardinalzahlen

~

, d.ho die Klasse derjenigen 0r-

dlnalzahlen > ~ ,

die sich nicht blJektlv auf eine kleinere 0rdinalzahl

abbilden lassen,

k~

sei die 0rdnungsfunktion der Klasse KU{0) , d.h.

die (elndeutig bestimmte) Funktion, die On ordnungstreu und bljektlv auf KU{0] abbildet. Es ist also_~ o =0 und ~ = M ~

fflr ~ > 0 .

Zu B ~ 0 n

gibt es

genau ein ~ e 0 n mlt ~ < B<~+1 ; wir definieren S ~ : = ~ und ~ + : = ~ + i . S6 hei6t die Stufe yon ~ . Aus dem Auswahlaxiom folgt, da6 8+ regul~r ist, d.h. ffir jedes ~:~--~B + mlt a < ~ + ist auch sup~(q)J q ~ ]

<~+

o

Eine 0rdlnalzahl V hei6t additive Hauptzahl, wenn fflr alle ~ , q < auch ~+q < ~ ist. Die Klasse der addltlven Hauptzahlen bezeichnen wlr mit H.

H ist eine echte Klasse, und k ~ ~ ist die 0rdnungsfunktion yon H.

Zu 0 ~

c 0 n glbt es elndeutig n>/ I u n d

additive Hauptzahlen 71/>..~

mit ~ = ~ I + ' ' ' + ~ n ; wir definieren H[~] := {~I' .... ~n }_ und H[0] := ~ o Es gilt: K C H , Vv~0n(7~ H~-*H[v] = [~]) , H[q] C H [ ~ + q ] C H [ ~ ] U H[q] . Ist M eine wohlgeordnete Menge,

so bezeichnen wir mit IJMII ihren 0rd-

nungstyp, d.h. dlejenlge 0rdlnalzahl, die ordnungsisomorph zu M i s t . Wir verwenden folgende Mitteilungszeichen: , ~ , ~ , 6 , ~ , q , ~

(auch mit Indizes) f~dr 0rdinalzahlen,

, ~

ffir Elemente von K U [0] ,

i , k , m , n

f~r nat~rliche Zahlen,

A\B

ffir die Mengendifferenz

[xJ xr A A X @

B} .

Definition der Funktionen | Im folgenden bezeichne ~ eine abz~hlbare Menge von Funktionen f:D(f)--~K mit ~ @ D ( f ) ~ 0 n m , 1~<m. ~ fl,f2e~A

xI~D(fl)A

erfGlle die folgende "Unit~tsbedingung":

x~eD(f2) A fl(xl)=f2(x2)~

f1=f2A

x I =x 2 9

Wir deflnieren :

j~ J

:= {f(x)J f ~

=f(~1,~176

~

) mlt f ~

Unter Bezugnahme auf ~

xcD(f)]

~

~C~]

:= (~1,.o~

und (~I .... '~m ) ~D(f)

] , wenn

.

werden nun 0rdinalzahlenmengen C (~, G) und Funk-

tionen ~a 1) durch transfinite Rekursion nach ~ definiert:

I) W o e s

erforderlich Ist, werden wlr die Abh~nglgkeit von ~

Schreibwelse C~(~, B) 9 @ ~

zum Ausdruck brlngen~

durch die

6

W. Buchholz

Co(~,8 ) := {0}u

Cn+l(C~,8)

:=

{

@~,

{~I H[~] ~ Cn(~,8) } u

{~1~1"~1^~[~]~Cn(~,8)}u

C(ct,~) := ~

{s~nl ~ < ~ ~ ~ c ( ~ , s ~ )

~

[~,~} ~ Cn(a,~)}

i)

Cn(Ct,~ )

:= {r

Kr(~)

8

~$c(~,~)} UK

: O n --9 Kr(a)

sei die Ordnungsfunktion

der Klasse Kr(~ ) .

Die Elemente von Kr(~) heiSen a-krltlsche Zahlen. Folgerungen (I)

@e8 e H

(2)

7 r

(3) (4)

Cn(e,~) c Cn+1(a,G ) c C(~,G) C(e,~) Ist die kleinste gegendber Addition und allen Funktlonen a~s ~ abgeschlossne e Obermenge yon {0} U 8 mlt der Eigenschaft:

(c)

8)

~<~^

>

HI7] c Cn(a, 8 )

~c(~,s~q) ^ ~,Q~c(~,8)

~

s~c(~,~)

9

Anmerkung In Aczel [I] und Bridge [3] wird C(e,~) als die kleinste gegen~ber Addition und allen Funktionen a u s ~ abgeschiossene Obermenge von [0} U definiert, welche (C') ~< ~ A ~,~ eC(~,8) ~ @ ~ cC(~,8) erfdllt. In [4] wird gezeigt, da5 fdr spezielle Mengen ~ (wie z.B. fdr die in w behandelten ~ T und {g] ) die oben definlerten Mengen C~(~, ~) ebenfalls die Bedlngung (C') erf~llen. Zumindest in jenen F~llen ist also die bier gegebene Definition der @ ~qulvalent zu der ursprHngllchen Definition yon Feferman und Aczel [I] . - Dadurch, da~ man die Bedingung ~ e C(~,e~q) in die Definition der Mengen C(a, 8) einfdgt, k a n n m a n Corollar 1.16 und Theorem 1.17 aus Bridge [3] (bzw. Folgerung (12) und Lemma 5 der vorliegenden Arbeit) unmlttelbar beweisen, ohne vorher den sehr das Theorem 1.15 aus [3] fuhren zu m~ssen. Diese M~glichkeit wurde yon SCHUTTE entdeekt und dem Autor w~hrend der Abfas-

kompliziertBe~W~

sung seiner Dissertation mitgeteilt. Lemma I

a)

~i~< ~e ^

b)

8 Lime s zahl

~i~< ~2

>

C(cb,8~) = c(a2,~,2)

>

c(~,~) = ~

I) Wir schreiben O~Q fir @~(Q) .

c(~,~)

2) Dies folgt aueh aus [3] 1.16 .

W. Buchh olz

c) d)

~ < %^v~(~4~<%~&c(~,~)) ~ < % > ~r(%) c Kr(c,~)

e) f)

c~1 < c~2 #1 < 152

> ;"

7

>

c (c h , ~) = c (c%, 15)

8a 115~< 8cu28 1~1 ",.< 8cqB1 < 8c~62

Lemma 2 Ist

~1 = min{~ I ~ ~ c ( a , ~ ) } ~-.< 01 < ~+

Beweis. Da ~

9

c(~,~)

, so g i l t : = c(~,01)

,

~1 c K r ( a )

abz~hlbar ist, mu~ die Menge C(e, 0) eine M~chtigkeit < 15+

haben. Also ist ~< 01<15 + . Aus 15a01cC(~,15) Folgerungen:

.

(5)

C(~,@aO) = C(a,0)

(6)

c(,~,e,~(~,-1))

folgt 01 ~ C(~,61)=C(c, 15).

= c(~,~r3+1)

Satz I Es gilt

S@a15 = $6

und

@c~

= ~

Beweis. Nach Lemma 2 ist sup[~ I ~

fHr ~ >0 Kr(e) ^ ~ < ~+] = 6+. Da 15+ regular

ist, mu6 also [ql | < 0+] = 6+ sein. Folglich ist ~ < S| = $6 9 - Ist 0 < ~ , so ist (wegen K C K r ( ~ ) ) ~ =| Mit S| folgt ~ ~< ~ < | = ~ , d.h. ~ = ~ c ~ Folgerungen:

(7)

6~K

(8)

~..< 7~C(~,~)~K

(9)

7 e Cn(C~,6 )

Satz 2 @ ist Normalfunktion,

z

d.h. e

~_

8~6 e K

,,." >

~ ~1-~1

,,',

~[~] c

c(~,~)

S~ e Cn(C~,15)

ist eine streng monotone Funktion von

On in On mit | = sup{| I q c O} fur Jede Limeszahl Ferner gilt: (i) Kr(0) = H (ii) Kr(~+1) = [~a151 a ~ C ( ~ , S a S ) v @aS = ,6 :} (iii) Kr(a) = ~ a Kr(~) , wenn a Limeszahl ist . Beweis. Wir beweisen zun~chst

@s0 < B+, d.h. ffir ein ~ .

(i) - (lii).

(i): Nach

6 9

(I) ist K r ( 0 ) C H .

Aus 0 r folgt 8~ Cn(0,15) durch Induktion nach n, also ist H C K r ( 0 ) . (ii): 1. Ist | 1 6 2 so gilt | und @ a O = O 9 Ist ~ a ~ K , s o gilt: ~9~0 c Kr(~+1) < )| @ C(a+1,~aS) 2. Aus @ ~ 5 ~ K und ~ ~C(a,@a15) folgt mit Lemma Ic ea8 ~ C(a+1,ea15) . 3. Aus | ~ K und e a B = 6 folgt | ~ Cn(a+1,| dutch Induktion nach n. 4. Aus ~ ~ C ( ~ , ~ ) und 6 < @ ~ 8 folgt mit Lemma la @~8 e C(a+I,@~15) Aus I.-4~ folgt unmittelbar die Behauptung

(ii).

8

W. Buchholz

(iii): Fdr Limeszahlen

= {~lv~(~<~ Aus

-~

e gilt

6~C(~,~))}UK=

(i) - (iii) und Lemma

supM r K r ( a )

Kr(e) = [81 ~ [C(~,~)} U K =

~0~{~1 ~ C ( ~ , ~ ) } U K =

la,b folgt durch transfinite

.ich~teere f d r jede Teilmenge M yon K r ( ~ )

f~r jede Limeszahl

~Ojr(~) Induktion nach

. Daraus f o l g t

|

~u~ @~q

6 9

.Folgerungen

(Io)

eo6 = ~

,

e1~ =

(11)

@~IB 1 = 8o282 ai e C(~i'e~iSi)

Wegen der Bedingung

sB

und ' 8i < | ~

~ f d r i = 1,2

C(~,|

e l = ~2 ' 61 =

J

in der Definition von C(~,6)

82

folgt

nun aus Satz I , (I) , (7) und (11) unmittelbar: eC(~,@~q)

, q<@~q

>

~ < ~ und

~,q e Cn(e, 8 )

Satz

b)

~(~1% ~ < c ~ ^ ~ ~ C ( ~ l , e ~ ) )

Beweis.

a) Aus der Voraussetzung

~

e~(e%#)

folgt mlt Lemma

= e%~

la,c C(~1,q) = C(~2,~)

f~r alle ~ | . Also ist Kr(~1) O (| +I) = Kr(~2) Q (| ~ +I). Daraus folgt e~1~ = ~ 2 8 . - b) Aus der Voraussetzung folgt: Es gibt ein

c(~,e~28) und | ~(ec~B) = e~26 , also |

mit ~ 1 4 ~ r folgt

Kr(~+1)

(Lemma la,d,e).

Mit Satz 2(ii)

~ @~i(8~26 ) ~ 8~26 .

Aus Satz 3a und Lemma la,c folgt: V~(~I~< ~ - ~C(~i,e~16)) <--# V~(~14 ~<% und mit Satz 3b folgt daraus Lemma 3 9

~

Lemma 3 ~1 < c~2 •

el e C(~2,@~262)

A

BI < |

~

@~161 < 8~2~ 2

Lemma 4

e~

~ c(~,e~)

Beweis.

>

c(~,e~B) n ~+ = e ~

Durch Xnduktion nach n zeigt man Cn(~,e~6 ) N 6+ C

@G~ : Ist 7 < 6+

u_nd H[~] ~ Cn_i(~,8), so folgt mlt I.V. (Induktionsvoraussetzung) H[V] c ea6 , also ~ < @~8, denn @~8 ist Ja elne additive Hauptzahl. - Ist 7 E I~ I fl C(G,@~6) N 6+ , so folgt |

>~ 7 r C(G,@~6) , also nach Vorausssetzung

7 < 8~6" - Ist 7 = @~q ~ Cn(~,e~6) N 6+ mit ~ < a

und ~,q e C n _ 1 ( ~ , e ~ B ),

W. Buchh olz so folgt nach I.V. < @aB

(wegen ~..<7)

folgt nach Lemma 3

~ < |

7 = ~D

9 ;

aus ~ < ~

, ~eC(e,@aB)

,

< 8cB -

Lemma 5 ~ Cn(~,~)

A

7 r C(~1,@~1(B1+l))"C(e1,~eiB I)

Beweis dutch Induktion nach n.

B I ~ Cn(~,B)

Es sel B~< 81#Y~dasonst

trivial ist. Wegen 7 e Cn(e,~)--C(eI,|

o
~

Aus ~ ~ C ( ~ I , e ~ I ( ~ + ~ ) ) - C ( ~ , S ~ )

C(~I,Se181) fdr mindestens @in ~ r dazu die Folgerungen (2), (3),(8)).

I) gilt ~un

die Behauptung ~<|

,<7 und

folgt ~ ~ C ( ~ , S ~ ( ~ + ~ ) ) \

bzw. ~ ~ ~[~] ,wen/% 7 ~ K (siehe Aus der I.V. folgt daher ~

Cn_1(a,B ) , falls HI7] c Cn_1(e,B ) oder ~ [ F ] c Cn_1(e,B ) (F gK) ist. Es bleibt nun nur noch der folgende Fall zu behandeln: (~) ~ : e~h A ~ < e ~ ~ e C ( ~ , @ ~ ) A [~,~} c I. Ist ~ = 7 , so ist ~ r Cn_1(e,B), und mit der I.V. 2. Sel n <~ =@eIB1" ~ Mit ~ ~ C ( ~ , S ~ ) folgt nach Satz also ~i~<~ . Aus ~I~< ~ folgt mit Satz 3 @~i ~ = e ~

Cn_i(~,8 ) . folgt ~i g Cn_1(e, 8). 2(ii) @~i~1 ~ Kr(~+1), oder e ~ I ( 8 ~ ) = e~q,

also ~I r {q,7} = Cn(e,B). 3. Sei ~ I ~ I < ~ und ~ < 7 9 Wegen 7 e C(a1,@e1(B1+1)) folgt daraus mit

(@),(6) ~nd D:~=I ~ < ~

folgt auSerdem

BI e Cn_1(e,~)

9

w

Aus ~< ~

un~ {~,~} ~ C ( ~ I , ~ ( ~ + ~ ) ) .

7~C(~I,~IBI)

D i e

{~,~] ~ C(~I,|

und

Also ist nach I.V.

F u n k t i o n e n

~G

| ist die 0rdnungsfunktion der Klasse Kr(a) der a-kritischen Zahlen. Wir betrachten nun im folgenden gewisse Hilfsfunktionen @ , deren Wertebereich jeweils die Klasse K r ( a ) \ K r ( ~ + 1 ) der maximal e-kritischen Zahlen ist. Diese Funktionen stehen naturgem~5 in enger Beziehung zu den Normalfunktionen 8 (vergl. Satz 4a,b) , sind aber besser als jene zur rekursiven Darstellung yon 0rdinalzahlen Anmerkung: 0ffenbar ist {@~(~+I)I

geeignet.

a 4 B} c K r ( ~ ) ~ Kr(c~1)

. Die maximal

~-krltischen Zahlen bilden also eine echte Klasse. De finitionen K-r(~) :=

Kr(~)-. Kr(a+1 )

~(~)

mlnC~l ~

~-a

:=

~k~(~)~

sel der 0rdnungsisomorphismus

Bemerkung: Die 0rdnungsfunktion

von {BI S~(a)~< ~] auf

von K-~r(a) ist

k~(S~(~)

K-r(c). +G)

9

10

W. Buchholz

Wit schreiben wieder ~aG statt % ( 6 )

9

Aus Satz 2(ii) folgt:

(~3)

e~

~ K~(~)

Lemma 6 2(a) = min[q I a ~ Beweis.

C(~,eaq) }

Sei qo :=min{ql

keine Limeszahl

und

~(a) ist keine Limeszahl

~eC(~,e~q)]

Wegen Lemma

.

Ib und Satz 2 kann qo

sein, also ist ~o < @aqo und somit wegen

(13)

qo = ~(e) "

Lemma 7 a I e Cn(a, 8) Beweis.

~

p,(C~I) e C((~,8)

A

S~(~I)

s Cn((~,6)

Sei ~(al) + 0 .Nach Lemma 6 ist also ~(al) = 6 1 +

I mit al e C(a I,

ea I ( 6 1 + I ) ) \ C ( a 1,ea 161) . Aus al e C(a, 6) kC(a1,| folgt mit Lemma la a I < a oder eaiS1< 6, also I r 6). Mit (9) und Lemma 5 folgt die Beh. Definitlonen I 0

, ,

L ~6

:=

Fdr

e~< 6

Satz

4

Ffir

S~(~) ~< 6

sei

a)

~6

b)

sea6 = $6

c)

e~181 = e % 6 2

wenn 6 = 6o+n sonst -e+

bzw.

8

mit

~(~) < 6o = |

die eindeutig

S~(a i) ~< 6i

bestimmte

(i = 1,2)

und

Zahl ~ mlt

n < co

~ + 7 = ~ 9

gilt:

= e ~ ( ~ ( ~ ) + (- s~(~) + ~) + ~ 6 )

>

ch = %

A

61 =

m2

Beweis. a) I. Set S#(~) ~< 6 und 60 := #(a) + (-S#(~)+ 6) 9 Ist 8o < e ~ 8 o, so ist auch 80+ ~ 6 < e ~ ( 6 o + L ~ 6 ) 9 Ist 6o=e(ZSo , so ist 6o eine ~-Zahl, und nach Lemma 6 muB ~ ( ~ ) < 6o sein ; daraus folgt 6 = 6 o ; also i s t s I und daher ebenfalls

13o +Z~6 < @~(~o + s

9 Nach Lemma 6 !st m e C(G,

ea~(a)) c C(a, ea(6 o + ~ ) ) . Mit (13) folgt ea(~(a)+(-S~(e)+6)+~a6) e ~ ( ~ ) . 2. Sei 7 eK-r(a). Aus (13) und Lemma 6 folgt 7 = eaq mit ~(a) ~<~ < e a q . Es gibt nun genau ein ql mit p ( a ) + ql + c a b = q 9 Sei 6 := S ~ ( e ) + ql ' also ql = - S ~ ( ~ ) + 6 " Wie man leicht nachprfift, ist ~a6 = ~aq , denn aus L a S = 1

oder Lab= I folgt q = 6 + ~ a q . E s

f o l g t T = ea(~(a)+(-S~(a)+G)+/aS).

3. Aus 61 < 6 2 folgt 61 +6~61 < 62 +La62 . Nach I. ,2. ,3- liefert die Zuordnung 6~-~ea(~(a) + ( - S # ( a ) + 6) + ~ 6 ) eine ordnungstreue und bijektive Abbildung yon [ql S~(a) x< q] auf ~ ( a ) . Daraus folgt die Behauptung a).

W. Buchholz b) folgt aus a); c) folgt mit Lemma

11

Id aus der Definition

yon

Satz 5 Ist

S~(~) ~< ~o '

so gilt

a)

~ < ~ ~ ~'~o ~ c(~,~)

b)

~ ~ ~o

Beweis.

:

>

~ c(c~,~)

Sei 7

e~o

~ < ~ ^ ~,~o ~ c ( ~ , ~ )

~>

:= 8--~h~

und

~ c(~,~)

~ := V ( ~ ) + (-S~(~)+ q o ) + t ~ o 9 Dann gilt:

(i)

7 = @~q A ~ cC(~,@~q)

(il)

H[~ o] ~ [s~(~)] u H[U]

^

q < |

(siehe

(13) und Satz4),

a) Sel ~ < ~ und ~ , q o e C ( a , 8) . Mit Lemma 7 folgt ~ e C(a,8) ~,~ e C(a, 8) und (i) foigt 7 s C(~,B) 9 b) Sei 8 x < 7 r

. Wegen

~,q ~ Cn(~,8 )

w

Mit

Wir betrachten

und

nun die Menge

(i l) folgt welter H[~o] ~ Cn(~,8 ).

K o e f f i z i e n t e n m e n g e n T

derjenigen

durch elnen aus den Zelchen 0 , + , 8 , f ( f 6 ~ ) darstellen K 7

lassen.

(ffir ~ T )

Nach Einffihrung

Induktive

gebildeten

sogenannter

Charakterisierung

- vorausgesetzt,

Definition

0 eT

(T2)

[71 .... ,Vn] c T N H

(T3)

a,~

(T4)

(~1 . . . . .

A

%

A

S~(a) 4

die sich

0rdlnalterm

Koefflzlentenmengen

V1>/...>/Vn

A

K V,

erh&Iten,

der Menge T und der <-Relation

erffillt gewisse

elner 0rdlnalzahlenmenge

(T~)

c T

0rdlnalzahlen,

werden wlr dutch die S~tze 8 und 9 Hilfsmlttel

die eine rekursive T ermSgllchen

auf

Rekurslvit~tsbedlngungen.

T

n >I

~>

6

71+...+7 n E T

>

%.] ~ T A f ~ ' 9 A ( ~ 1 ' " " % , ) ~ D ( f )

~(~ ~ T

--=~

f(%,...,%) ~ T

.De f inition

d.h. A ist die klelnste Kardinalzahl Folgerungen:

gegenGber

allen Funktionen

> ~ .

(14)

T=A C(c~,~) ~ A

(16)

6g

A.<

,

(1) folgt daraus mit _/1__2) ~ < a und

(2),Lemma 7 , u n d

0 r d i n a I t e r ~ e

9 Aus ~ < a

aus

~

abgeschlossene

12

W.

Buchholz

Satz 6 Es ist

T = C(A,O)

Beweis.

I. Mit Satz 5 und T c A folgt T c C(A,O)

dutch Induktion

thud T fl ~I = ~AO

nach den Definitionsregeln

2. Dutch Induktion triviale mit

S~(~) ~ [~] U H [ ~ o ]

folgt

beweist

man

folgender:

no . Nach dem Beweis

7 = |

e Cn+I(A,O)

= C(A,@A0)

Definition

Die Definition so abgefa6t

mit

folgt aus

und

(T2)

~'~o e T . Mit

folgt mit Lemma #

der Koeffizientenmengen

erfolgt

dureh T-Induktion.

, de6 die Definition

K~7 und K~7 fflr 7 r T

Die Regeln

elndeutlg

wird.

(TI),...,(T4)

Ffir M c T

K~7

:= K H[7]

,

wenn 7 e T \ H

2.

K~7

:= K:~[7 ]

und

K 7

:= K ~ [ 7 ]

,

wens ~ c T N K und ~ + ~

3.

K;7

:=

und

K 7

:=

,

wenn ~ eT O H

4.

K~@aO 9 --

:= [ ~ U K * a U K ~ B

und

wenn

a,O E T , S~(a) ~

~

~

[7} K eeO usa

wind K 7" und K 7 endliche

K ~ wind additive F~r M c On sei:

Hauptzahlen

<

:= K K a U K ~ O

~+ ~

Teilmengen

.

und 7 < ~ +

~ ~ < e)

~

:<

3~(~

^ a~

,

~)

<--4

gilt:

* mit

KS~7<

(2) , (8), Lemma 4 und Satz 5 9

Satz 8 FHr

a e T

Beweis.

gilt

:

Aus Lemma

s~(a) 4 ~ <--~

S~(a) ~

B

~

>

KS~a <

a

6 und Satz 7 folgt:

J~(sB = sn A

von

~ ( ~ < ~)

Satz 7

dutch T-Induktion

9

(7+~) N a + V~(~ ~ M

8~Kr(a) kK

7.

von T . Die Elemente

: <~ } ~.

acc(~,e~))

:=

~ .

M < ~

M

sind

sei K ~ * ) M

und

Beweis

(T3)

T 0 ~I = 8AO

:= K H[7]

und

7 = ~o

yon Satz 5b ist dann H[~o] c Cn(A,0 ) .

Kg7

Fflr 7 e T

nicht

~
(~3)

I.

Fdr 7 c T

(d.h.

.

~ T .

T = C(A,O)

Induktive

T-Induktion

Cn(A,0 ) c T . Der einzige

c Cn(A,O ) folgt sit I.V.

7 = ~o

dutch

(TI),...,(T4))

und ~,~ eCn(A,O ) . In diesem Falle

Aus

3. Aus

nachn

Fall ist dabei

~C(~,|

.

<

>

K S0a < a

W. Buchholz

13

Satz 9 Fflr a,a I e T und 8,81 mlt

S~(a)

~< 8 , S~(~ 1) ~< 131

a)

KSS~ U {8] <

b)

a I < a A KS#Ia I U {81] < @a8

gilt:

8~8 ),

@'o,181 <

~{5

Beweis. a) Aus Satz 4a folgt 8a# = @a~ fflr ein ~>/8. Nach

(13) ist also

a ~ C(a, OaB) und B~< B < @a~ . Mit Satz 7 folgt daraus K s s a U [8) < 8~8 9 b) Ist $81 $ $8 , so folgt die Behauptung aus Satz 4b. - Sei $81 = $8 Aus al< a , S#(el)~< 81 und KSBIa I < 8e8 folgt nach den S~tzen 7 und 8 a I e C(a,~a~) . Mit 81 < ~aB , Satz 4b, Satz 5a und Lemma 4 folgt daraus

8a181 ~

c ( c ~ , ~ c ~ ) n ~+ = ~c~8 9

Nichtkonstruktive

Definition yon

~(~)

Durch die induktive Definition yon T wird zugleich Jedem 7 e T in eindeutlger Weise ein aus den Zeichen O , +, @, f (f ~ ~ ) gebildeter Ordinalterm ~)[7] zugeordnet, der in der klassischen 0rdinalzahltheorie die Zahl 7 darstellt. ~[o]

= o

, ~[1]

Wit nennen @[7] die "~)-Darstellung yon ~".

= ~oo

, ~[~+2]

= eo8oo

+

~oe

+ ~oo

(Beispiele:

)

Das Bezeichnun~ssystem 8(~ sei die Termmenge {~)[~] I ~ r T) zusammen mit der durch ~[~] < ~[ 6] : <==> 7 < 6 deflnierten <-Relation auf dieser Menge. Mittels der S~tze 4b, 8, 9 lASt sich fdr Terme 8aib i die Entscheidung dardber, ob ~aib i e ~9(~ ) bzw. 8alb I < 8a2b 2 ist, auf die Entscheidung entsprechender Fragen ffir kfirzere Terme zurHckffihren. Gilt ~hnliches fdr die Terme der Gestalt fal...a m ( f ~ ) , so kann das Bezeichnungssystem ~(~) rekursiv definiert werden; dies trifft z.B. (wie wir in w

zeigen werden) fdr die in w

eingefdhrten Systeme ~(T) und ~({g]) zu.

Von besonderem Interesse ist stets die Teilmenge_8o(_~) := {@[7] I e T~IL von e ( ~ ) ; denn fflr a = ~)[7] r 8o ( ~ ) ist nach Satz 6 = l l { x g ~ ( ~ ) I x < a}ll , d.h. ein Term aus @ o ( ~ ) repr~sentiert in dem wohlgeordneten er ist.

-

System ~( ~ ) genau die 0rdinalzahl,

Der Ordnungstyp yon @ o ~

Eine Erweiterungselgenschaft

deren 8-Darstellung

) Ist 8A0 .

von @ ( ~ )

Nehmen wit an, man habe mlt Hilfe elnes Systems 8(~o) elne Reihe bewelstheoretlscher Ergebnisse erhalten, aber zur Behandlung gewisser weiterer Probleme reiche der durch ~(9o) dargestellte Ordlnalzahlenabschnitt nicht mehr aus. Man wird versuchen, durch Erweiterung der Menge ~ ein fi~r den gewitnschten Zweck gendgend starkes Bezeichnungssystem

14

W. Buchholz

e( ~ ) herzustellen. Dabel wird man Jedoch mSglichst so verfahren wollen, daS ~ ( ~ ) in kanonischer Weise elnem Teilsystem yon ~( ~ ) entsprlcht, d.h. da~ ~ T $~ T 9 ist und fdr jedes ~ ~T #~ die ~-Darstellung yon ~ bezdglich ~o mit der bezdglich ~ Gbereinstimmt. Hierzu liefert der folgende Satz 10 ein im allgemeinen ausreichendes Kriterium. Definition

f1~

:: {(x,f(x)) I x~D(f) A f(x)< v}

91,,

---

of1,, I f ~ .

(f~9)

A fl,,+~}

Satz 10 Set

v eK ,

FGr

~o := "~.~v

~,8~< v

u n d gelte

gilt dann:

Daraus folgt speziell

V~(V~ I~ 1fl~

|176 = @9~

und

T{~=C~(v,0)Ov

~ "~[7]

< v )

.

C ~ ( c , 8) = C ~ ( ~ , 8 ) O v 9

und

O~

= O#

fGr c< v

.

Beweis durch transfinite Induktion nach a (mit Satz 2) . w

D i e

S y s t e m e

~-(T)

U n d

~([g}

)

und ihre Beziehungen zu anderen Bezelchnungssystemen T , ~ m~gen (rekursive) 0rdinalzahlen < gl bezelchnen. De finitionen

1.

~..,. :: [{(0,n1+~)} I ~ ~','} ~(~) :: ~(~) , e ~ 8 ,: e ~ 8 ,

c~(~,8) ,: c~(~,~)

2. go := {(~'a1+~) I ~ E On} ge := 0rdnungsfunktion yon [h I V ~ ( ~ < ~ ge8 g

[ g~(8+I) g~8)

:= :=

, ,

usw , fGr ~> 0

-- g ~ ( h ) = h ) ]

wenn 8 = 6o + n sonst

mit

g~(6 o)._. = 80

{((~,~),g~) I (~,~)~ onxon}

Folge rungen (17)

A T = ~I+T

'

(18)

~T =

(19)

gel61 < gc~282 gilt genau dann, wenn elner der folgenden drei

~TIQI+T

F~lle vorliegt:

A[g} = min{~ I ~ = g~0 } , wenn

(17a)

K n Q I + T c CT(~,8)

w~<

(i)

e I < c~2

B I < gc~262

(ii) (iii)

~I = ~2 c~2 < ~I

81 < 82 g~181 ~< 82

{g} erfdllt also die Unlt~tsbedingung yon Selte 2 .

w. Buchholz

15

Lemma 8

a)

T < ~ und ~, ~ 4 ~I+T

~">

b)

~ < I1~o(~)1] = e ~ + ~ o

und

e~ = @~ und

I1~(~)11

<

Beweis. a) folgt aus Satz 10 und (18)

CT(e,~) = C~(a,#) fl ~I+ T

@~nl+T+lO

b): Sei T < ~

ffir

~

<

9 Nach Satz 6, (17)

und a) ist ll~o(r)II = @T~I+T0 und ~(T) = CT(~I+T,0) ~ ~I+T " Nach (17a) u. Lemma 3 liefert die Zuordnung ~b-~@T~1+~0 bZW. ~b--geT(~1+T+~)0 eine ordnungstreue Abbildung yon T in eT~I+T0 bzw. yon CT(~I+T,0)O ~I+T in

e~(n1+ +n1+~)0.

~ 4 ]I@o(~)II und

A l s o ist

I18(~)[]< e~n1+~+~0 9

Vergleich der Funktlonen @T und e {g} ~~ungl

von Jetzt an sei ee~ := @{g}eG , c(e,~) := c{g}(a,~)

usw~

Lemma 9

Beweis o Sel ~ := ~D

. 4 a) Sel ~ e ~ x< ~ x< ~ < ~ " Dann ist S~ = ~ o mit @ ~ ~< ~o naeh Lemma 4 ~ o ~ C ( ~ , | und somit auch ~ C(~,ez~) . Mit e ~ = e~G . - b) Aus e < ~@zG folgt Se = ~ mit ~ < @ ~ . folgt welter e < ~ + I < ~ und D~+I e C ( ~ , @ ~ ) 9 Mit ~ < ma 3 folgt daraus Definition:

eaS~< 6 ~ + 1 G :=

,

< ~I * also Satz 3 a folgt Wegen ~ < DI | und Lem-

< 8~5 9 %(-I)

:=

~n+1(~ ) :: ~q~n(~)+1(B+1)

ffir

~ e 0 n U (-I]

Lemma 10

Sei a)

8 g ~I U (-I} ~n < ~ n

und

~n "= ~n (8) ;

(~+I) x< ~n+1

b) ~o4 ~ < ~%(~+I) c) ~ < eenB A ~ n ~

,

und

~> ~>

darn% gilt:

sup(~ n I n r ~} = ~ 2 ( ~ + I )

~ < ~(~I) ~ < en~

Bewels. a) Durch Induktion nach n zelgt man: ~n ( ~ n

(~+I) < ~ G [ ~ + I )

und C n ( ~ , ~ o + 1 ) N ~ 4 c i<~co(C(~i,6~i(6+1))O~i) 9 - b) Aus der Pr~misse folgt mit a) ~n x< ~ < ~n+1 ffir ein nc(o. Ist ~n = ~ ' so gilt die Behauptung nach a) . Ist ~n < ~ ' so folgt ~ < ~Aq~n+1(~+1 ) ~< ~q~(~1). c) folgt aus b) .

16

W.

Buchholz

Definition

Satz 11

FGr

a)

er~

b)

c~(~,e~#) n ~

~ < ~

= ea(~.+

4+T

gilt

:

e)

= c(~,e~)

n

wenn

Beweis durch transfinite

Induktlon nach 1. [e~B I ~. ~< ~) c K r ~ ( ~ ) c K r ( ~ ) Beweis : 1. 1.

(z = 0

:

1.2. a = ~ + I

Krl"(0)

= H = Kr(0)

(glelchzeitig

.

: Nach Satz 2(ii) ist KrT(a)

ql = 8T~ql } " Nach I.V. ist 8T~ql = | f01~{ 0 < ql und somit ~ < ~ @ ~ ( ~ * + ql) Aus

~ ~CT(~,eT~ql )

folgt

fdr a) und b)).

~$K

= [@T~ql I ~ ~ CT(~,ST~ql) V ) . Aus ql = @T~ql (es sei wieder ~ := ~ 4 ) "

, also ebenfalls

~ < s

+qq)

"+--_

Andererseits folgt aus ~ < s und der I.V.: ~. ~< q , C T ( ~ * e ~ q ) 0 C(~,| ~ + und (e~q= q < ~ - ~ . + q = 8~q) . Zusammengefa6t ergibt slch: KrT(~) = [e~ql ~ < ne~q ~ (~ [c(~,e~q) v q = e~q)) ~ Kr(~) . Aus ~. ~< 13 folgt @a~ = 8~q fGr ein q mit ~ < a ~ < ~ | ~@gq , 1.3, ~ Limeszahl : Nach I.V. gilt [8[q I [. ~< q] = KrT([) fHr [ < ~ . Daraus folgt KrT(~) = ~@~ KrT(~) c [<~a Kr([) = Kr(~) . Ist a. ~< und [ < a , so ist 2.

eo, IB r

[. ~< ~

Kr'r((~)

===~

und

8~

= 8~q

mit

B~< q ,also @ a # ~ K r T ( [ ) .

o,. ~< I~ 9

Beweis indirekt. Sei ~ < ~. , d.h. ~ :-- ~ 8 ~ < ~ und ~ = ~.. Aus Lemma 9a und der I.V. folgt 8~B = 8[~ = eT[o . Ferner gilt [ r CT([, eta0) , also nach Satz 2(ii) 8 ~ 0 ~ KrT(~+I) . Es folgt @~8 ~ KrT(~) 9 Aus I. und 2. folgt 3.

@T~8 = 8 ~ ( c . + 8) 9

Durch Nebeninduktion

C n~ ( ~ , ~ s )

n ~+ ~ c ( ~ , e ~ ) f~r e < ~ 8 ~ " Man beachte dazu die a < ~8~6 ist nach I. 8a~ r KrT(~) .

nach n beweist man

und Cn(~,| 0 e § c C~(~,8~6) folgenden Hinweise: Wegen

Ist 7 r K n ~ + , so ist 7 = ~ ~< ~ mit ~ < ~I R S g # ; daraus folgt nach Lemma 10c ~ < ~ q ~ ~< @ ~ , also ~ ~ C ( ~ , @ ~ ) . ~ ( 4 7 a ) M K D a + c C ~ ( ~ , 8 ~ ) . Fdr

~ < e

und

n~ ~< n

Folgerung:

Ist

T ~< 8 D ~ 0

(2o)

e~

= e~B

ist nach I.V.

und

und

fund wegen ~. < n I)

~ < ~I+T

'

so gilt

C~(a, 13) n a+ = c ( a , ~ ) n a+

@~n = 8Y~n

17

W. Buchholz

Mit Hilfe von Lemma 10 und Satz 11 k8nnen wir nun noch eine interessante Eigenschaft der Zahlen 6 ~ 8

(~ < ~ ) beweisen.

Definition einer Normalfunktion

~: ~I

~ ~I

~(~+~) := il~(m~)ll ~

:=

sup[~

I ~< 6 }

(~ao~ Lemma 8~ ist

, wenn ~ Limeszahl Ist.

~4

II~o(~)II , a~so

~

< II~(~)II = ~(~+~)

- )

Satz 12

~)

~,~ =

Bewels.

~(~)~

( ~' := Ableitung von ~ )

FGr a = ~ ( I + 8 )

+ n mit ~r ~I U {-I}, n r ~ sei

Lemma 10 ist ~ : ~i--~ ~I Normalfunktion. (i)

~=

~

, wenn

~=~
~

:= ~n(8 ) . Nach

Ferner gilt: ,

(ll) ~e < ~a < ~(~+I) , wenn a = ai+I Bewels durch transfinite Induktlon nach ~ . I. gO = ~ = ~ o (-I) = ~0 9 2. Ist e Limeszahl, so folgt ~a = ~ aus der I.V. und der Stetigkeit yon ~ und ~ . 3. Sel a = a 1 + 1

und @~I ~< %~I < ~e . sel a I = ~(I+~) + n

(mlt 8 ~ n I U [-I]) und ~ :=6~qn(~+1 ) . Es gilt ~a I= ~n(8) , ~ a = ~n+1(G). Aus Lemma 10 folgt ~ n ( G ) < ~ und ( ~ n ( G ) + 1 ) . = ~+1 , also nach Satz 11 @ T ~ n ( 8 ) + 1 0 = ~n_+i(8)<$ . Aus ~a1~< ~ a 1 = ~ n ( 8 ) folgt mit Lemma 8 ~ = H~(~a)ll < @T~ - ( A ~ 0 , also ~ < @a Aus ~ ( 8 ) + I ~ < ~ a < ~ folgt

mlt Lem~a 8",/,~= e'~n~:n ()'~, +10 ..< 11~7o(~,~)11 < 118(~,=)11 = ,~(c,..,-'~) . Aus

(i) folgt die Behauptung a). Zu b): Offenbar kommen als Fixpunkte

yon ~ nur Limeszahlen ~=~

in Betracht;

< > ~(~.(I+~))=~

und

also ist ~'= ~' . Es gilt aber:

%(e'(1+6))=~qD8(fur

6e~I)

9

Und wegen

Vergleiehe mit anderen Bezelchnungssystemen I. Die SchGtte-Fefermansche Sei k(G) Es gilt:

Grenzzahl F ~

(nach Aczel [I]

) :

:= 8~+~ , e+ := min{q I a 4 k(q)] , ~ := @ e ( e + + 8 ) . ~08 = to# , und fdr a > O ist k ~ ( ~ q ) die Ordnungsfunktion von {q I ~ ( ~ <

~ --~q

=q)}

9 k~k(~) ist die Ordnungsfunk-

tlon yon {~ I ~ : ~ 0 } Daraus folgt F ~ = ~ql 0 und Kr(e+1 ) = [q I 8~q = q} 2. Die SehGtte-Pfelffer-Systeme In [5] wlrd gezelgt

Z ~ ~(~)

fdr e < Fo

7(N) , 7 : und

Z AZ(N) = 8(N)

.

"

18

W. Buchholz

3. Die Systeme

W(X) yon Pfeiffer:

In [6] wlrd gezeigt mit Satz

W(X) ~ ~(T)

(mit T := -I +IIXII ) 9

12, dab die yon Pfeiffer

in [12],w

Daraus

definierte

folgt

Grenzzahl

e*

gleich e ( ~ Q + I)0 ist . I

4. Die Systeme

0d(I)

yon Kino:

In [6] wird gezeigt wenn

5. Die Funktlonen In Bridge

w

FP(a,-)

yon @

ea{g}o

K o n s t r u k t i v e

u n d

auf die klassische

gleichzeltig:

1. Termmengen

X , ~ , R

2. Eine Abbildung

0rdinalzahltheorie

S :~

identlsch

sehen kann, (vergl.

gekennzeiehneten

bzw. e([g})

, die dbrigen Regeln

Wohlordnung Abkdrzungen:

I. Induktive 1.0.

Rc~-%

von ~(T)

rekursiv

ohne

definieren

Systeme

angehen.

auf deflnlerte

10,11 definierten

Definitlonsregeln beziehen

II(X ,<X)ll = T

bezlehen

slch auf belde

a.< b

: <'..'.... ~--

-'l(b < a)

M<

a

:~

;

Vx(xeM--~

x < a)

a~< M

:~

>

~x(x~M A

a~< x)

Ro

R u {o}

a und b slnd identisch

% , Q , R

(%, <)

(19))

9

Die

slch auf e(T) Systeme.

Bel

da6 elne rekursive

sei .

:<

yon

'.-

gegeben

Paar

Bezelchnungssystem

und Folgerung

a = b

Definition

~(T) und ~({g})

setzen wlr auSerdem voraus,

(X , <X) mlt

:=

S y s t e m e

Kua

die S~tze 4b,8,9

mlt ' b z w . "

der Definition

der

fGr diese

ist das Im Anschlu6

mit dem auf den Seiten

bzw. e({g})

da6 die Grenz-

) R U {0}

Ca und 3. Koefflzlentenmengen 4. Eine zweistelllge Relation <

e(T)

insbesondere

~({g} )

Wohlordnungsbeweis

Wir definleren

mit der Be-

.

werden wir die Bezeichnungssysteme

und einen konstruktiven

Wie man lelcht

ist

D a r s t e 1 1 u n g

8(T)

Bezugnahme

% 6~(T+I),

dab FP(~, -) im wesentlichen dberelnstimmt,

gZeleh

Im folgenden

II0d(I),<J

von Isles:

auf ~p+1

/(G(I,I),I)

und

ist .

[3] wird gezeigt,

schr~nkung

zahl

~0d(I),
r := -I +IIlll < eA{g}o

19

W. B u c h h o l z

1.1.

0 e %

1.2.

al,...,a n r Q

1.3.

a , b e % A

I.#'.

~

1.#J~

a,

=

A

1
A

>

an%...% a I

a1+...+a n e

>

Ksba < a

X

b e ~

2. I n d u k t l v e :=

gab ~

Definition

2.1.

Sa

2.2. 2.3.

S(a1+...+an) S~ab := Sb

3. I n d u k t i v e

a

,

yon

wenn

:=

Sae

a

r

ffir

eo

a

e

O

Sa 1

Definition

von

Kua

und

Kua

ffir

a e~

3. I.

K*O := Ku0 :=

3.2. 3.3-

K~(al+...+an) := K~/~lU...UI~u)an KuSab : : {a} UKua UKub und KuSab := Kua. UKub

3.4-

Kua

:= ~

3.5:

K*v

:= K v := ~

3.5'.'

KA~u ~ab :=UK(u~U K(u~~

4. I n d u k t i v e

und

Kua

Definition

yon

a < b

4.1.

oCa

4.2.

u e ~ A

u~< b

>

4.3.

u e ~ A b < u

>

4.4.

a 1 + . . . + a m < b1 + . . . + b n den

>

:= {a}

u r ~o--

, wenn

u < Sb

,

wenn

a r ~

mit

Sa ~< u

,

wenn

v e~

nit

u < v

,

wenn

u < gab

ffir

a ,b e

o
|

< u gilt

(1~< m , I % n , 2
in g e n a u

folgen-

zwei F~llen:

(i)

m < n

und

(ii)

Es glbt ein k mit ai = bi

4. 5-

und

ealb I < Ga2b 2

ai = bi

ffir

1 ~< i ~< m

I ~< k % m i n { m , n }

fdr

I ~< i < k .

gilt

in g e n a u

folgenden

a I < a2

und

KSblal U {b I] < ~a2b 2

(li)

a I = a2

und

b I < b2

4.6'.' galb I < ga2b 2 (i)

a I < a2

gilt

ak < bk

und

drel F~llen:

(i)

(iii) a 2 < a I und FUr u , v e ~ gilt:

,

~alb I ~< K s b 2 a 2 U {b2} u < v : ~ =~ u <X v in g e n a u

folgenden

und

b I < ga2b 2

(li)

a I = a2

und

b I < b2

(iii)

a2 < a I

und

galb 1 %

drei F~llen:

b2

Definition Als den Grad tretenden

Ga

Zeichen

eines Terms + , 8,

g, 0~

a deflnieren u ~X

wlr die A n z a h l

der in a auf-

20

W. Buchholz

Zur R e k u r s i v i t ~ t

yon

(%, <)

Wir nennen 2 Ga die K e n n z l f f e r der Aussage

a < b

duktion nach der K e n n z i f f e r I.-4.

die Entscheidbarkeit

Die E n t s c h e l d b a r k e i t ergibt

der Aussage a ~ 9

und 2Ga+2Gu+1 ~ie K e n n z i f f e r

2Ga+2 Gb die K e n n z l f f e r

der Menge ~

folgt nun unmittelbar bzw.

Berechenbarkeit

bzw. B e r e c h e n b a r k e i t

9 Durch In-

aus den D e f i n i t i o n e n

yon a e % , a < b , ~JA 9

yon a c ~ , a e ~

, Sa

fdr a e ~

sieh auf trivlale Welse.

Mitteilungszelchen a, b ~ e , d , e , x , y

fur Elemente yon

u ,v , w

fGr Elemente

Satz

yon

o

13

(~, <) ist eln linear geordnetes

a)

~ ( a < a)

b)

a ~ b

c)

a < b

,. >

und

a < b

b < c

System

oder ~

, d.h.

f~r a,b,c e %

gilt:

b < a a < c

Man beweist a) dutch Induktlon nach Ga , b) dutch I n d u k t l o n nach C a + Gb, c) durch Induktion nach C a + Gb + G c

9

Folgerungen (21)

a ~ b

(22)

sa % a

(23)

sa<

Satz

~

~

Sb

a < b

>

oder

a = b

a < b

14

a)

~ab e %

~

b)"

{a, b} < gab

KsbaU{b}

< ~ab

Beweis.

4 KSb a U { b } ==>

a) Man zeigt durch Induktlon nach Gc :

c

b/IMan zeigt durch Induktion nach Gc :

c 4 {a.,b}

Lemma

~

c < ~ab c < gab

11

a)

c ~ Kua

....>

c ~ r

b)

C = C1+...+c n (mit n ~ 1

A

c 4

a

^

, c I .....c n ~ Q)

Sc4

A

u

Sc ~ u

>

KuC = {c I ..... C n} c)

v < Sc

d)

v,
).

,,>

KvSc c KvC

K~K~O=KvO

e)

u..
~-

K~o ~ K~o

W. Buchholz Lemma

21

12

KuC < a Beweis

A

u < v

A

~)av e 9

durch Induktion

>

KvC < ~)av

nach Gc . Wir behandeln nur den Fall v < c =~clc 2 .

AUS { C l } U K u C I U K u C 2 = K~C < a folgt mit der I.V. < ~av . Ist Sc = v , so folgt(mit KvC I U {c2} < ~av

, also

Sc > v , so folgt

De

Lemma

KvC < ~3av

cI < a

und

11b) KvC = { e C l C e }

nach der Definition

KvC I U K v C 2

, cI < a

und

yon < . Ist

KvC = KvC I U KvC 2 < ~av .

r

W o h l o r d n u n g s b e w e i

s

De finitionen Q ' QI ' Q2

bezeichnen

Teilmengen

{ce Q

I

Q n u + :=

{ccQ

I Sc~< u}

W[Q]

:=

{aeQ

I QAa

:=

{a ] V V ( V r Q O u

Q0a

:=

WQu

yon

c < a }

ist wohlgeordnet

}

> Kv a c Q)}

:: wt nu +]

Folge rungen (24)

Qlr

,., = Q 2 n u

(25)

W Q ist der gr~Ste wohlgeordnete wohlgeordnet, a ~ wQ

;.

=

A

w uQI = wQ2

Abschnitt

yon ~ O u

+ , d.h. W Q Ist

und es gilt: <

"

a e ~

0u+

A

~

N a c wQu

De finiti on Eine Teilmenge

0ffenbar

(AI)

a r Q

~

(A2)

u c Q

--~

ausgezeichnet Sac

von % , U l q d e s

Q U

sei Q irgendeine sei

, wenn gilt:

Qnu + = wQ

"

ist Jede ausgezeichnete

Im folgenden

Lemma

Q yon X heiSe

M u := ~

Teilmenge

fest gew~hlte ,

WU

:= W Q

von % auch wohlgeordnet. ausgezeichnete

Teilmenge

.Dies~e[f,ebi&[emma~emsch[ie~Lie}l.

II

II

13

a r Q

und

u c Q

~.

Kua c Q

Beweis.

I. u < Sa : Aus a e Q folgt mit

(AI),(A2)

a e Wsa, Mit u c Q 0 ( S a )

22

W.

folgt daraus

Buchholz

K u a c Q . 2. Sa ~ u

: Aus a, u

Aus a cW u folgt mit Lemma

11d

g i l t K u a c Mu . Mit K u a ~

a e W u und

Lemma

E Q folgt a e Q N u + = W u.

K v ( K u a ) = Kv a c Q

fGr v r Q n u . A l s o

(25) folgt d a r a u s Kua c Wu c Q .

14

u r Mu

und

u~Q

~.

Beweis.

Sei u e M u , u ~ Q

und

ucQ v := m i n { w ~ Q

und u ~ v r Q n v + = W v , a l s o n a c h Satz

(25)

I u ~< w}

. D a n n gilt U e M u = M v

u r WvCQ.

15

a)

a +be

X

b)

~ab c %

und

a ,b ~ Q

und

a, b e Q

....).

a+b

>

e Q

eab

~ Q

Beweis. a) Sei a+b ~ % ist a,b e Q O u

und a,b e Q

9 Wir

setzen u

+ -- w u ; a l s o ist a + b e M u , und

wohlgeordnet.

Wir

folgt d a r a u s

a + b ~ W u c Q . - Sei

man

z e i g e n nun,

sich l e i c h t H b e r z e u g t ,

treue und bijektive ist,

:= S(a+b)

folgt daraus

= Sa , a+O

Muna

und M u D b

dab M u O ( a + b ) w o h l g e o r d n e t ~

llefert

Abbildung

yon ~

die W o h l o r d n u n g

:= {d

I dr

die Z u o r d n u n g

yon ~P(a+b)

sind

ist; mit a+b r u

A

a~
c~-+a+c

eine

9 Da ~ N ( a + b )

Ob auf C

:= a . Es

=

. Wie

ordnungs* (~Oa) UM u

.

b) A b k G r zung: I a, b e Q S[a,b]

: <~---->

A

eab c 9

YccQYdeQ( c < a V d e Q ( Sb~< d < b

Hilfssatz Beweis

:

~[a,b]

A

(c,d~O)

2. e ~< K u a U

{b}

B[a,b]

: ecQ

< e = ecd

und e ~ M u O@)ab . D a n n

I.V.

folgt

KuCU{d}

und B[a,b]

ist

Durch

KuC U { d } c

c Q . -

. - Sei n u n transfinite

Sei n u n

Ist c = a

c < a . Dann Induktion

W e g e n K u C C M u ist n a c h L e m m a schon bewiesen.-

ist u e Q O u + = W u.

folgt m i t a) aus der I.V. 13

(25)

: Aus ee Mu folgt mit Lemma

fur v r Q0u . A l s o

) e e Q

: Aus a,b,u ~Q folgt mit Lemma

c Q

ecd r Q )

n a c h Ge :

M i t e E M u und e ~< K u a U {b} f o l g t d a r a u s n a c h 3. K u a U { b }

Q

~.

durch Induktion

und g e l t e

I. e = c + d

>

~. ~ a d ~

e s MSb N ~ a b

des H i l f s s a t z e s

Sei u := Sb

A

A eta E ~

11d

{b} c Q 0 u + =

11d K v ( K u C U [ d } ) = Kve

(siehe Satz

und d < b

, so folgt e g Q

ist

c e Q

11d f o l g t

B[a,v]

W e g e n @ c d = e ~ ~ u. Sd = u

und

fur v g Q n u Kw c c Q

A u s B[a,b]

14a)

. Mit aus d ~ Q

zu beweisen.

n a c h v z e l g e n w~ z~c~s~

Mit L e m m a

Wu -

e c Wu c Q .

Mu o @ab

KvC c Q

u < v ~ Q

KvC c Mv .

Kua~

Kv c CQ 9

fdr v c Q :

K u c c Q Ist

fur alle w e Q O v

und S b < v

.

folgt mit L e m m a ~ e

ist KuC < c . M i t c

< a , u < v ,

W. Buchholz

~av c % und Lemma Kvc c M v N ~ a v folgt.

, woraus nach der I.V.

Wegen a ~ Q Lemma

12 folgt daraus KvC < ~av . Es gilt also B[a,v]

Damlt ist

Mit Lemma

Kv c c Q

(wegen V x r KvC ( G x < Ge)

ffir a l l e v

ist Sac Q und somlt

11b und Satz

14 O ~ Q , also

rQ

bewiesen

KsaC c Q

15a folgt nun

Wegen c < a

c c Q U{O}

) KvC c Q

ist

9 Wegen Q r

ist der Beweis des Hilfssatzes

folgt mit Lemma

und

.

c E Q 9 Aus ~[a,b] , c ~ Q, d e Q, c < a

e = ecd r Q . - Damit Aus ~[a,b]

23

Sc ~ Sa. gilt nach

folgt

beendet.

13 und dem eben bewiesenen Hilfssatz

~ab e

MSb und M s b O ~ a b c QO(Sb) + = WSb , also nach (25) ~ab ~ WSb ~ Q . Folglich gilt : ~[a,b] > Gab r Q Daraus schlie~t man durch "geschachtelte" Lemma

transfinite

Induktion fiber Q auf die B e h a u p t u n g

15

u r Mu

und

Beweis.

Aus u E M u und Muff u c Q folgt

~nu

c Q

~

w u ist a u s g e z e i c h n e t

und W u o u = Muff u . N a c h Lemma Daraus ffir v r u~a

b)

folgt nach

(24)

9 so ist S a = u E W

Definition

(da Q w o h l g e o r d n e t

13 ist Q c N u ; also ist

W~ u = W u n U +

u n u . W u erffillt also

(A2)

und

:=

~{Q

u ~ Wu ist) u r W u WuDu

= Q0u

.

W~ u = W v = Q O v + = W u 0 v +

(Seite

u . Ist a c W u n u

:

und

18)

Zu (At): Ist a~ W u und

9 so folgt S a c W u wegen WuflU=Qnu.

J Q Ist a u s g e z e i c h n e t

)

Anmerkung: Sel

vy(y

Pru[ Q~ Q~] :

nu § ^

dann gilt:

3QIV%( a ~ % A vx(x~ QI -* Sx~Q1) A A vn~Q I (Pr [ % % ] ^ ( Pru[%%l -~ Q1 n ~+ ~ %))) Satz 16 Ist ausgezeichnete

Teilmenge

von % .

Beweis. Hilfssatz:

Sind QI ' Q2

u r QI U Q 2 Bewels des Hilfssatzes

ausgezeichnet,

so gilt:

A u~< QI A u~< Q2 )Q10u+ = Q2nu+ durch transfinite Induktlon nach u ~ QIUQ2 :

Sel u ~ QI und u~< Q2 " Aus der I.V.

folgt

u c QI n u + = w QI = w Q2 . Mit Lemma

14 und ux
Somit gilt

= Q2nu

~ also gilt

Q I 0 u + = WuQ2 = Q 2 n u + .

Aus der D e f i n i t i o n satz folgt,

QI n u

.

yon ~/ folgt Va(a ~ - - ~

dab Jede ausgezeichnete

FGr u g ~ g i l t

also

:

~/~u+

= U~Q

Sac ~ )

, und aus dem Hilfs-

Nenge eln A b s e h n i t t n u+J

ur Q

^

yon

~

Ist.

Q Ist ausgez.} =

24

=

W. Buchholz

~J { WQ

Satz

I u e Q

A

Q ausgez.}

= Wu~

17

Es gilt

~0~ = ~

,

und folgllch

Ist

9

wohlgeordnet.

Be~els. sel ~u := ~u~ u n d Wu := Wu~ 9 Aus Lemma 15 und satz 16 folgt:

(1) U ~ M u ~ Daraus

MunU~

folgt 0 r ZJ

>

u ~ ~/

9 Nach den S~tzen

15, 16 ist

~

abgeschlossen

ge-

genGber + und ~ . Wir mHssen also noch

(H)'

x~

beweisen. erfolgt l.V.:

~

bzw.

(H)" a , b

~ZJ

durch "geschaehtelte"

transfinite

a, b r ~ A V c c ~ V d e ~ Y ( c <

a

v

(c= a A d < b)

folgt mit Lemma

fdr alle v E ~ N u

u e ~

; also ist

durch N e b e n i n d u k t i o n

b
< gab

und

Se ~ _ r

. Mit Se eMse und Lemma ist also

Ksee c ~

folgt.

-- Wir haben nun

daraus

gab = u r ~J~ .

)

~/

gcd r ~ r

u ; mlt N.I.V. zur Annahme ~ <

14 folgt daraus

) .

Ve(e g MuNU-->

u . Mit Lemma

~d~< Se . Dann w~re

, woraus nach Lemma

; dieser :

KvU = K v a U Kvb c

9 - Wir beweisen nun

c, d r ~ N

Se E ~d'folgen, was im W i d e r s p r u c h e r Mu

13

nach Ge: Sel e r

folgt S e ~ MSe . - N e h m e n wlr an, es w~re mlt

(II)UausfHhren

Induktion Gber

Sei u := gab . Aus a , b e ~ --9 e r ~ )

gab ~ ~/

....>.

Wir wollen hier nur den Beweis yon

und I.V. wdrde Se steht.

Se g ~ N u

daraus

- Also ist . Wegen

11b und Satz

u e M u und M u N u c ~f bewlesen

11c

Se = g c d

. Mit

15a e r ~f (I) folgt

Anmerkung Beim Beweis der W o h l o r d n u n g hler v e r w e n d e t e n R e k u r s i o n dber ---, Kva c W[

kann man mlt schw~cheren

auskom~en:

(X,<x) Mengen % ])}

c @(T)

Man deflnlere

, so dab

%

~J{W[%Du+]

I ~ ~ X} . da6 ~

Wie man lelcht nachprHft,

elne ausgezelchnete

slch ~ =

@(r)

als den

durch transfinite

= {a I V v c X ( v <

fdr alle u ~ X Ist . A u 6 e r d e m deflniere man

woraus folgt, Satz 15 ergibt

von~(Tl

Beweismltteln

gilt dann

Menge mlt X c ~

u--*

?z#:= Mu~= %

,

ist. Mit

9

LITERATUR [I]

Aczel, P.: A new approach large countable

[2]

Bachmann,H.:

to the Bachmann method

ordinals.

Die N o r m a l f u n k t i o n e n

und das Problem der ausgezeichn~e~

Folgen von 0rdnungszahlen.

ViertelJahresschrlft

den Gesellschaft

(1950).

ZHrlch 95

for d e s c r i b i n g

(unver~ffentlicht)

der N a t u r f o r s c h e n -

W. Buchholz

[ 3]

Bridge,J.:

Some problems in mathematical logic. Systems of ordinal

functions and ordinal notations.

[ 41

25

Buchholz,W.:

Ph.D.Thesls, Oxford 1972 .

Rekursive Bezeichnungssysteme

fGr Ordinalzahlen auf

der Grundlage der Feferman-Aczelschen Normalfunktionen e Dissertation Mthachen 1974 .

[ 51

Buchholz,W.,SchGtte,K.:

.

Die Beziehungen zwlschen den Ordlnalzahl-

systemen Z und ~(~) . Erschelnt in: Arch. Math. Log.u. Grundl. [ 6]

Buchholz,W.: b-bet die Beziehungen zwischen den Systemen 8({g]) , W(X) und Od(I) . (unverSffentlicht)

[ 7]

Isles,D.: Regular ordinals and normal forms. Intuitlonism and Proof Theory,

Proc.conference Buffalo 1968, North-Holland,Amsterdam(1970)

[ 8]

Kino,A.: On ordinal diagrams.

[ 9]

Levltz,H. und Sch~tte,K.: A Characterization of Takeuti's Ordinal

Jour. Math. Soc. Jap. 1 3 ,S. 346 (1961)

Diagrams of Finite Orders. Arch. Math. Log. u. Grundl. 1_~, S.75 (1971)

[1o]

Pfelffer,H.: Ausgezeichnete Folgen fGr gewlsse Abschnitte der zwelten und weiterer Zahlenklassen.

[11]

Dissertation Hannover 1964 .

Pfeiffer,H.: Ein Bezeichnungssystem fGr Ordinalzahlen.

Arch. Math.

Log.u. Grundl. 1 2 , S. 12 (1969)

[12]

Pfelffer,H.:

Ein Bezeichnungssystem f~r Ordinalzahlen.

Arch. Math.

Log. u. Grundl. 1 3 , S.74 (1970)

[13]

Pfelffer,H.: Vergleich zweler Bezeichnungssysteme len. Arch. Math. Log. u. Grundl.

[I~]

(1972)

Pfelffer,H.: t~ber zwei Bezelchnungssysteme Arch.Math.Log.u. Grundl.

[15]

15 ,S.41

fdr Ordinalzah-

fGr Ordinalzahlen.

16 , S.X3 (1973)

Pfelffer,H.: Bezelchnungssysteme

fdr Ordinalzahlen.

Communic.

of

the Math. Inst. RiJksunlversitelt Utrecht 1973 - I

[16]

SchUtte,K.: Kennzeichnung von Ordnungszahlen durch rekursiv erkl~rte Funktionen.

[17]

SchUtte,K.: Ein konstruktives System yon Ordinalzahlen. Log. u. Grundl.

[18]

Math.Annalen 127 , S.15 (1954) Arch. Math.

11 , S. 126 (1968) und 12 , S.3 (1969)

Schdtte,K.: EinfGhrung der Normalfunktlonen 8G ohne Auswahlaxlom u n d o h n e Regularit~tsbedlngung.

Erscheint in: Arch.Math.Log.u. Grundl.

[19]

Takeuti,G.: Ordinal diagrams I, II . Journ. Math. Soc. Jap. ~ , S.386

[20]

(1957) und 12 , S.385 (1960) Weyhrauch,R.W.: Relations between some hierarchies of ordinal functions and functlonals. Ph.D.Thesis stanford 1972 9

SOUND FUNCTORS Dedicated

to Kurt Sch~tte on the occasion of his 65 th birthday

J N Crossley

and Anil Nerode ~

We answer some questions Crossley-Nerode

[1974]

~hereafter

we consider

recursive

an infinite

field and related

systems

equivalence

are not necessarily

what we call

soundly based

representative independent n B

~

set

~ sound

shows that,

The

referred types

basis

for

generated

important

is a recursively

O~ n B

RETs are

enumerable

is a basis for

Our main

sense,

original

spaces over

These are types with a

~).

in a very strong

In particular

of vector

where finitely

The most

types.

such that

to as CN).

(RETs)

situations

finite.

for which there B

same as that of Dekker's

I.

left open in our m o n o g r a p h

theorem

07~(we also call

(Theorem 3.6)

the theory of sound types

RETs

is the

(of sets).

categories

In CN we considered use slightly

simpler

be a fixed recursively model.

We further assume

for

~

,

and

(an explicit

(2)

~3)

that

if

for

x, that

that

presented (i)

that

is, a formula

set

We now

this paper we let

c o u n t a b ~ infinite x = x

x c cl(al,...,an~

x c cl~a I .... ,an~

of models.

Thus throughout

code of) the finite

of T h ( ~ , a l , . . . , a n ) where k is minimal. [to appear]

certain categories

categories.

saturated

is a minimal

formula

is uniformly

recursive

~al,...,an~

and

then we can find an irreducible

~(x,al,...,an)

such that

~

in

formula

is an atom

and ( ~ , a I .... ,an) , ~!k v ~ v , a I ..... an) (A result of C.J. Ash, see Crossley-Nerode

shows that we only need parameters

This work was supported by the A u s t r a l i a n in 1974.

from

Research

cl(a I ..... an~.)

Grants

x

Committee

J.N. Unlike

CN we do not require Condition ~

Condition

(2) gives an

has a dimension

and Shepherdson ence algorithm

that every algebraically (the cardinality

(2) is the analogue

spaces

irreducible

formula.

sense for fields.

conditions

that there

as Ash and Nerode

However,

not true for the original

of

allow us to show as we did in Crossley-

equivalence

dimension which

in the categories

categories

CI(TFg)

[1975] have

s

of CN as

is

shown this cl

is

is not a

for those categories. ~i(Fs

has as objects

and as morphisms

the algebraically

(which have of course one-one partial

~i(Yfg)

is contained

monomorphisms

in the category

with one-one partial

s

C(Fs

of

define

CI(FFD)-RZTs to be the equivalence of

under

$ = s infinite

functor between the usual

9Zt 0

sends

$

set of

above.

categories

sense which

of 9 the

elementary

where

the

Thus we can of objects

of

in

CI(~ )

~(gF~.

is the model of equality with an

number of individuals. as specified

s

classes

the isomorphisms

where

identify the underlying ~

subsets

extensions).

extensions

are the same as those of s

recursive

of arbitrary

recursive

objects

( = objects

closed

simply and solely the inclusion maps between

objects

model

to find the analogue we call this the

is an effective

invariant under recursive

may

of a

It also easily yields

for field extensions;

defined below. functor

sub-

and an algebraic depend-

(3) allows us effectively

polynomial

three

closed

of any basis).

~.

the minimal

These

27

N0-categorical.

for vector

in the usual

basis for

[to appear]

to be

The condition

algorithm

Condition

A. Nerode

algebraic dependence algorithm (cf. Fr6hlich

[1956]).

linear dependence a recursive

~

(i) ensures

system of

Nerode

Crossley,

is referred ~7~0

We write

CI(F{/~

we mean

inclusions

to as

with a basis $I

for

"sets"

We

for any other

~i(~0).

By a

(as in CN) a functor

to inclusions.

in

28

J.N.

Crossley,

A. Nerode

LEMN_A 1.1 The map recursive

cl

: $I § r (9"s

equivalence.

cl A = cl B

in

functor from

~(~).

$

to

Proof. [to appear],

In fact, HoweVer,

$

with

if, and only if,

cannot be extended

The first statement

the second follows

to a

is proved

from Ash-Nerode

shows

in Crossley-Nerode [1975].

that some restriction

on the

of CN is necessary. We write

by

cl

in

~(~dT~) in general.

Lemma I.i therefore categories

is a functor which commutes

A ~ B

V

for the map

induced on

$-RETs

to

r

cl.

THEOREM

i. 2 A

r

the range of

X is soundly

Proof. @71 = cl B

If

X = V(Y)

for some set

ire set of all natural Conversely, some sound basis infinite Theorem

function

3.2,

Y = $-RET

But then

X

so

B 9 A

enumerable f. so

if,

~

3.8

is in

is soundly based.

and

set which

= cl B.

But

with

in the recurs-

A

~7. 9 X

has

is an independent

is the range of the one-one [to appear],

cl f-l(B) 9 V(Y)

where

X = V(Y).

[to appear]

we have the stronger result

of that ps4oer ).

THEOREM

1.3

$-RETs

such that

There

~7. c X

is contained

Then by Crossley-Nerode

From Crossley-Nerode (Theorem

~$

exists

B

is soundly based then some

Suppose

cl f-l(B)

(f-l(B~)

then there

B.

numbers, if

B.

recursively

recursive

based if, and only

V.

is a function

dim on soundly

based

r

to

9 .N. Crossley, V(X)

= X

for every

S-RET

V dim(Y)

= Y

for every

s

dim

and

A. Nerode

29

X Y

which is soundly

and

only

based.

COROLLARY

1.4

For

all

$-RETs

X,

So far we see that based

~(a~-types.

Dedekind

is no one-one

effective

analogue finite

mapping

A

S-types

is a b i j e c t i o n

definition

of a set

A

being

map of

onto

a proper

this

systems

DEFINITION

A

subset

treated

to soundly between

is that

of

a set

partial

A.

of

the natural

The

is r e c u r s i v e l y

recursive

where

that

A.

function

For an algebraic

the s i t u a t i o n

infinite

finite was

subset

on the u n d e r l y i n g

However,

may be

a proper

is no one-one

definition

are finite.

generated

set

A

of

finitely

definition

structure O~

in CN

generated

when

finitely

is as follows:

1.5

A soundly based

C(~)-RET

A = V(B)

for some Dedekind

COROLLARY

1.6

V

2.

on

V

(due to Dekker)

there we always

Dedekind

is a b i j e c t i o n show

X = Y.

that

if there

onto

we used

systems

V

if,

types.

Dedekind

but

VX = VY

We next

Dedekind's there

if,

Y,

S-RET

A

is said to be D e d e k i n d

if

B.

S-RETs

is a bijection from Dedekind

to soundly based

C(~)-types.

The categories In CN

category

~

w

we d e f i n e d

a category

to be appropriate

C

with

if it satisfies

a distinguished the f o l l o w i n g

sub-

eleven

30

J.N.

axioms.

(i.i)

We shall

Crossley,

A. Nerode

show that the categories

s

are appropriate.

Concreteness. The objects

maps between

objects

of of

C

are sets.

~.

The morphisms

Composition

of

~

in the category

are

is compos-

ition of maps. This (1.2)

is true by definition

for

~i(~?~).

Injections. All morphisms

of

Again

for

trivial

~

are one-one maps. ~i(9~)

since all morphisms

are

inclusions. (1.3)

Inverses. If

p

-i

,

then

p

p -i

is a morphism

of

is a morphism

The only morphisms

and

of

in

p, as a map,

has an inverse

s fi(~?Z)

with

inverses

and

@L

as maps are

identities. (1.4)

Inclusions. If

0/,~

then the inclusion This (1.5)

are objects map

~

is true for

CI(~Yl~

If a set of morphisms

of

then that intersection The intersection

Pi

r

is a morphism

is a subset of

of

~.

by definition.

Intersections.

of maps)

is

of

§

to

has an intersection

of inclusion maps

with domain restricted

restricted

C

n(codomain Pi

to

(as a set

is in {Pi

n{A i : i E I}

: i c I~.

: Ai9

§

B .i

:

i

c

I}

and codomain

So (1.5) holds

for

CI(F~5).

J.N.

Crossley,

A. Nerode

31

Directed unions.

(1.6)

The union of any set of morphisms inclusion,

is a morphism For

dual reasons

CI(9~) to

of

of

C,

directed under

~.

this reduces

to arbitrary unions

and holds

for

(1.5).

Subcategory.

(1.7)

~

is a full subcategory

We define are the o b j ~ t s o f

~

of

~.

to be the full subcategory

~i(YgL)

which are finitely

generated,

whose objects or equivalent-

ly, are of finite dimension.

Restriction.

(1.8)

If O/

p

9 dom p

is a morphism

of

then the bijection

~,

r

q c p

is an object of with domain

0%1

C

with

is a m o r p h i s m

i

of

r

If

@D 9 ~

Since

and

~i(~)

p 9 ~

then

q 9 ~

has all and only

inclusions

this holds

for

cz(~). Approximation.

(1.9)

If o~

with

p

is any m o r p h i s m

p' 9 p

of

form a directed

C,

then the morphisms

set with union

p'

of

p.

I

For subsystems, (i.i0)

0~L 9 ~ i ( ~ ,

so

(1.9) holds

~ for

is the union of its finitely generated ~i(Yrt).

Compatibility. If

p, q

there is (under and moreover,

~)

are morphisms

of

a least morphism

~

compatible r

in

~

in

such that

~,

then p, q ~ r,

r 9 ~

For ~ , ~ 9 ~162 n{ ~ : ~r~ ~ 4 & ~ ~ } : cl(~U~) is the r e q u i r e d o b j e c t . Since Og,Y~ are f i n i t e d i m e n s i o n a l so is c1(~ u ~). The same argument works f o r morphisms so (1.10) holds for

r (Z/V) .

J.N.

32

(i.ii)

Finite

p 9 ~163 then there

that

q ~ ~162 and Let

@g

So

(1.4)

since U

form

functor

(i.I),

Let ~

.

A'

such

be a basis for

Since both A'

~, A',

is the required

which

categories

is one-one.

(1.2) hold since

U(O~),

U(p -I)

and

U

U(~)

Then

f.

: ~ ~ ~' satisfies

is a one-one

U(~L) n U ( ~ ) ~t

c

hence if

U(r

in

~

is combinatorial

= U(~)

since

just

3.12,

use

a

If

U

is a o n e - o n e

U(r

U(C)

U(q)

n ~)

=

that is,

is of the

where

U(Ct) q

in

is the

C. U

1emma.

2.2 functor

~ U(f~)

is one-one,

the inclusion map of

of functor)

By the proof of CN Theorem

we

U

But any object of

in the category

Now

(and is

U(r

if, and only if,

= ~

c U(~)

is (by our definition Ot

U

n f2-

~.~ .

U(p)

functor.

since

~

U(~) ~ ~'

is an inverse of

sections.

LEMMA

C

Then the map with domain

appropriate

then

if, and only if,

inclusion of (1.5)

~

exists.

are

By CN Theorem 3.12,

if, and only if, U(~)

in

(i.ii).

U(p) 9 U(~)

if, and only if, U(~)

[That is,

(1.10).

If

unique)

be in B'

generated.

not necessarily

p ~ q.)

be a basis for

~, ~'

Proof. (1.3)

implies

f

2.1

is a c o m b i n a t o r i a l -

is finitely

which is the identity on

satisfies

Suppose

(1.1)

or

~

such a A' u B'

CI(97L)

THEOREM

f ~ q

A' u B'

ff~ 9 ~

in

is a finite map

p : ~t

and let

codomain

A. Nerode

generation.

Every morphism if

Crossley,

then

preserves

(arbitrary)

inter-

J.N.

Crossley,

A. Nerode

i'

commutes

33

r T

i

ol

uI

if, and only

if,

r

U(r')

u(%~)

> u(~') commutes

U(i)

u,

o(i')

u,

u(~)

> u(~)

U(r)

Proof. shown U(r'i) if,

Remember

(by our definition = U(i'r)

U(r')U(i)

c U(r')

Suppose

inclusion.

P

Then

P

under

so is

inclusion, Since

(x,y) 9 U(Up) ~,

is one-one. since

U

are the inclusions if, and only if,

Hence

r'i = i'r

if, and only

is a functor.

U~(x,y)

if, and only if,

is of the form

of

So if Since

U(i')

r'i = i'r

is a set of morphisms

set of morphisms

for

U

= U(i')U(r)

U(r)

if

U(i),

2.3

COROLLARY

(1.6)

since

that

CN, p.9).

U

C.

Moreover,

of

U(C)

{U(p)

: p 9 P0 }

by Lemma

inclusions

U~(x,y)

is finitely

r up

2.2,

generated

UU(p)

where by,

(x,y) 9 U(up) then there exist Pi {p : p 9 P} is a directed set there

Pl u ...UPn c__ p. Finally therefore

directed if

where P

under P0

is a

is directed

P0"

preserves then

r c r'.

c__ U(Up).

U~(x,y) say,

Conversely,

9 ~163

By

(i.ii)

(al,bl),...,(an,bn).

such that (ai'bi) 9 Pi" exists p e P such that

But then U~(x,y) c__ p hence (x,y) 9 U(p). U(Up) c__ UU(p) and we see that U preserves

directed

unions.

(1.7) p'

Let

: U(O~)

~ c_ U ( ~ )

= U(~ implies

If

U(O~),

p' = U(p)

U(~)

E U(~162

for some

p 9 ~

then since

U maps

34 onto

J.N.

U(~162

(1.8)

and

follows

(1.9)

~

= m{q

: q c U(p)

= ~

& q 9

and

Therefore

r 9 ~ U(~

p' e ~

o~,}.

U(p),

by ( 1 . 7 )

But t h e n ,

U(q) ! U ( r )

for

r

p,

a g a i n by t h e

such that

U(p),

(i.ii) will be satisfied

Indeed we shall have

q

then

P,q ! r

are contained

lemma,

U(r)

U(q) c U ( r ) .

is

in the

This

f

in the cases we are

generates

p

implies

f

2.4

and

U

U(~

U(~

is a c o m b i n a t o r i a l

: C § ~' 9

then

~

By Lemma c o~,

then

p 9 U(q)

where

is m o r p h i s m

P0

2.2, U

U

functor

is m o r p h i s m

p ~ q q

which

is o n e - o n e

combinatorial.

if, and only

is continuous

for some

is the unique

if,

U(p) ~ U(q).

on morphisms

then

p c U(r)

element

such that

and if

if, and only p = U(P0).

combinatorial.

This corollary be onto

U(p)

for some

U(p).

P0 ~ r U

~162 b y ( 1 . 6 )

9

& q 9 U(~

p 9 o~, and also Thus

& p,

: q i

Proof.

if,

p

= m{q

2.3.

If

if

c

the proof.

generates

Hence

: p,

& q = U(p')

Condition

and onto

C.

~162

9

: q c_ U ( p )

concerned with.

COROLLARY

: p,

= U{q

morphism of

completes

c p

u{u(p,]

U(q) 9 u ( o r

some s m a l l e s t smallest

of

similarly.

=

by C o r o l l a r y

A. Nerode

is a full subcategory

U(p) - - u ( u { p ,

(1.10) If U(p),

Crossley,

in CN, Example

shows that

3.1.

it was essential

for

V

not to

J.N. COROLLARY

A. Nerode

35

2.5

If

is a combinatorial

U : r ~ r

of categories

then

U

Proof. U(p)

Crossley,

9 oe,

If

Hence

is morphism p c u-l(q)

functor

c U(r)

and an isomorphism

combinatorial.

for some

q 9 C'

as in the proof of Corollary

U(p) Hence, by Lemma

-I

if, and only

if,

and

p 9 ~

then

2.4,

p 9 r.

2.2, p c r

if, and only if,

u-l(p)

c U -I (r) .

Hence

s a u-l(r) So

(u-l)~(s)

= u-l(s)

3.

The main

theorems

Now suppose exists

since

identify subsets

N of

and

N

easy

We now write

(In

support

see

that

Crossley-Nerode

It follows

LEMMA

that

A,

[to V

is

numbers

for

appear]

.

Such a basis We may

and so regard objects

to be and

we use

~

recursive.

and an arbitrary

= cl A

c r.

combinatorial.

for the functor

supp(A), A)

U(s)

induced by A ~0~

(cl A)

supp

for than

as

cl.

Then

= B

rather

$I

9 V($I)

n N.

V(B)

N-supp

of

we

it B 9

is $I"

supp.)

one-one.

3.1 V($I)

being

of V(supp

basis

is u n i f o r m l y

V

N

if,

is m o r p h i s m

is a recursive

with the natural N.

the to

U -I

a 9 cl{a0,...,a n)

Given the basis

define

if, and only

is an appropriate

the set of finitely Proof.

category

with

generated morphisms

By Theorem

2.1 it suffices

of

~

= V(~

I)

V($I).

to show

V

is one-one

36

J.N.

and combinatorial on objects functor

V

preserves

from

V(CL)

It follows degree

1 assumption)

is combinatorial into

~ I V(f) e V(~ follows for

consists

This

combinatorial

so this follows

(without

too.

the

that

V

cl dom f

f. Since

g e ~

f e ~ I

generates

is of this form, so (i.ii)

it holds

the proof.

and

supp

: V($I) ~ $I

V

are one-one partial

functors.

and supp

is uniformly

3.1 and Corollary

are partial

recursive

recursive.

and the basis

2.5

But N

is recursive

at once.

3.3 (i)

Every

~ V($I)

(partial recursive)

combinatorial

~s induced by a (partial recursive)

G : $I ~$I" (ii)

strictly

7.4

inclusion map from

From the proof of Theorem

x ~ cl(a0,...,a n}

functor

on morphisms,

inclusion maps,

is finitely generated

: $I ~ ~I (~'~)

Proof.

F : V($1)

is one-one

of finite maps.

completes

we only need to show

COROLLARY

V

the inclusion

g e ~

is a

3.2

V

recursive

is a unique

and every

is one-one V

But there is at most one so

from the proof of CN, Theorem

extending

that every

COROLLARY

V(~)

V

We have also shown

as we are only treating

= ~

V($I).

We have just shown

supp.

inclusions.

into

for there

cl codom f

A. Nerode

(i. Ii) holds.

as ~ has an inverse

so

inclusion

and

Crossley,

indeed,

G = supp FV

Every combinatorial

functor

functor combinatorial

will do. G : V($1) ~ V($I)

is

combinatorial.

Proof.

Since

closed under composition

(partial recursive) part

(i) is obvious.

combinatorial

functors

are

J.N.Crossley,

(ii)

V

therefore

on

is strictly $i)

is.

A. Nerode

combinatorial But

37

since any functor on

F = VG supp

so by CN Theorem

(a n d

$ 7.9

F

is

strict. Because

COROLLARY

of Corollary

3.3 we write

F = V(G)

if

F = VG supp.

3.4 If

F,G

: $I ~ $I

are combinatorial

functors

then

V(GF)

=

V(G)V(F). Proof.

Obvious.

We note that the preceding equally

to functions

COROLLARY

of several

and following

arguments

corollaries

apply

(cf. CN p.8).

3.5 Let

f : E ~ E

be a recursive

there is a partial recursive

combinatorial function.

combinatorial

function

~

Then

from sound

types to sound types such that dim

for finite dimensional

Proof. present

Corollary Finally

"combinatorial recursive

types

Immediate

F(X)

X

from CN, Theorems

8.3 and 8.4 and the

1.3.

section we leave the reader

functor" may be replaced

combinatorial

dim

X.

3.3 and Theorem in this

= f

functor"

Summing up our results

everywhere

to check that

by "partial

in the above. we have:

THEOREM 3.6 The map V Moreover,

from

$-RETs

to soundly based types is a bijection.

it is a bijection from Dedekind

$-RETs

to soundly based

38

J.N.

Dedekind ire)

types.

Further

combinatorial

combinatorial

induces a b i j e c t i o n from

All we need

V(F)

= V(G)

FX = GX

N.B.

$-RETs

by Corollary

recursive V($I)

from

$I

to

from

$I

into

V($I)

VFX = VGX

functions into

on sound

~(~)

For example,

V

~(~)-types,

but not into

V($1)

is morphism

combinatorial

3.1 is not morphism

combinatorial

in general.

Hamilton and CN extensions

4.

In his on natural

[1969,1974]

numbers

Hamilton

to sound types

extension

to sound types

hA

showed how to extend relations

in the case of vector

(Hamilton called his sound types

V-RETs.)

of a partial

function

was defined by first extending

(r

on D e d e k i n d types and then by restricting

values

one obtained

consider

a function

restricting

C(h)

which

One of these

Since finitary, to

V($I)

are special

is an appropriate V($I)(R )

for

0-chain

types

R

We now write

r

of elements

of

for V($I)

to a relation to sound

One can further of course.

We now extension

to the notion of frame. [1972].

category,

even if

a relation

we can immediately

(that is, finite

to natural

r

3.1 the

of natural

cases of a general

due to Ellentuck

as we have dimension,

by taking dimensions,

r

to sound types.

two modifications

we can define

But in fact, selves

involves

is essentially

h

as a relation

to sound arguments,

show that these two extensions procedure

h

spaces.

In CN Section

numbers

of

But

out that this theorem does not cover

but by CN Example

~i(F/v)

F = G.

1.4

combinatorial

~ ~(9~/U).

composition.

to show is part of the last sentence,

since a functor may be combinatorial even though

(partial recurs-

to (partial recursive)

if, and only if,

It must be pointed

all partial

A. Nerode

functions on sound types which p r e s e r v e s

is that

implies

V

functions on

Proof. that

Crossley,

isomorphism

V

is not

on chain types. restrict types)

our-

and then,

numbers. r

Let where

~($i)

V : $I ~ r

denote

the set

is the closure

J.N. Crossley, functor. extension

For the present assume V h V of a partial function

A. Nerode

39

is finitary recursive, then the h on natural numbers to natural

numbers is defined as follows. Let0• V~ : $I ~ ~I0 and V 1 : $I ~ CII be closure functors. Let r = r r Then Ci(h), the extension of h to ~I' may be defined as the set of Dedekind r (X,Y) such that there exist ~ 9 ~ and ~ 9 ~ such that (~ ,~) is attainable from some recursive h-frame. An h-frame is a frame K ~ ~162 x or such that (~,~) 9 K implies h(dim 07~) = dim (cf. CN Section 25). We set

hV = r Thus h V is just and values. Since

n (V_o($I) • V l ( $ i ) ) .

r

restricted to sound types as its arguments

Vi($1)

is appropriate we can extend

by taking s = V0($I) • VI($I) for this extension.

V_0($I) x V_I($I) write V($i)(h)

h

to

above.

Let us

THEOREM 4.1 (i) function

x Vl($i))(h)

and

hV

coincide

h

(for

a partial

on the natural numbers).

(ii) partial

(r

(V0($I)

function

x Vl($i))(h )

on the natural

and

hV

coincide

(for

h

a

numbers).

Proof. First we must correct the proof of CN, Theorem 31.1 (CN, p.107). Suppose (X,Z), (Y,Z) 9 r Let ~ i 9 X, Og2 9 ~, 6 3 9 ~ so that ~2, 0713 are in the range of V I. By Theorem 25.5 there are a recursive h-frame H I such that ( ~ i , ~2) 9 ~ ( H I) and a recursive h-frame H 2 such that (#g l, #63) 9 ~(H2). Further ~ 0 x V~(~ is a recursive frame and trivially (~l-l, ~ 2) and ( ~ i , Og~) 9 ~(~ 0 x VI(~ So by CN Lemma 27.1, since ~r~I, ~2, ~63 are Dedekind we have (#L I, ~2) and (0/I, ~3) are attainable from the recursive

h-frames

K 1 = H I n (~162 x Vl(~

and

K 2 = H 2 n (~162 • Vl(~

40

J.N. Crossley,

Now continue as in CN. 0 ~.

This proves

A. Nerode (i)

if we replace

C0' r

by

~I' Next we can restrict for

i = I, 2

Dedekind. exist ~i

so that

K i _c V0(~

x VI(~

as for ensuring

~&•

is

Thus if (X,Y) ~ Ki(h), X, Y are saund types then there ~ V0($1) ' 0L2 ~ VI($1) such that ( ~ l l , ~ 2) is attainable

from a recursive

h-frame

contained in V0($I ) x VI(~ i Since Vi($I) ! r we trivially have This completes the proof of (ii).

(X,Y) ~ V($i)(h ). V($i)(h ) ~ r Recall combinatorial number

K I, K 2

by exactly the same technique

if

(Nerode

[1961])

h(x+n)

that

h(x)

Thus

is eventually recursive

is recursive combinatorial

for some natural

n.

COROLLARY 4.2 For any binary

relation

~I(R) n (V0($1) Proof.

If

Since

c X,

W

specifying

natural numbers

is eventually

Proof.

such that

as parameters

recursive

are sound Dedekind

cl{al,...,an}

by

at some but not all arguments

combinatorial.

This follows at once from Nerode

[1961], Theorem

corollary.

Now we turn to the case that e

X, Y

is total if, and only i~ every function obtained

8.1 and the preceding

a

and e ~

4.3 hV

h

numbers

e ~ (H n V0(~ I) • VI(~ for any frame H such that c ~ (H). So the left-hand side is contained in the right. Vo($i) x VI($I) ~ r x r 1 = s I the other inclusion follows.

COROLLARY

of

~

on the natural

• VI($I) ) = (V0($1) • Vl(~i))(R).

(X,Y) e r

types then there exist (~,7) (~ ,~)

R

V

is uniformly recursive.

is not finitary but Of course the Dekker-

J.N. Hamilton

Crossley,

case comes under this.

A. Nerode

Since

V(~

does not consist

only finite objects we modify the definition

D~FINITION

~$I(R)

of a relation

to sound t y p ~ is the set of all those V~

9 X

recursive

with

R-frame

Hamilton R'

~

~$I(R)

[1969,

where

A = VY

~ ~9

R'

Y

74]

5.

Examples

defined

the extension {A 9 ~($)

R~

of a relation

: dim ~ 9 R'}.

%~

subspaces

of a space

B = {bj

effective

operation

way of making contained therefore,

9

in fixed

: j ~ J}

~ for %~

in terms of

A

and

into a functor

$-RETs.

dim A 9 R' l.S)

if, and only

if, and only

by Definition

if,

R-frame 4.2.

direct

sum of two

is to choose bases ~,?~

and then to define

: i ~ I} u {u2j+l This B.

: j 9 J}

is easily made

However,

does not work unless

enumerable)

our attention

numbers

to

(internal)

{u2i

for

we restrict

a set of natural

the

is a basis

(recursively

R

coincides with

from some recursive

~ = V_~Y e ~$I(R)

to be the space spanned by {u i : i < w}

of

(by Corollary

way of defining

%P,Ir

A = {a i : i c I},

$(R)

is attainable

if, and only if,

R V'

extension

if, and only if,

Y 9 $(R)

One obvious

where

such that there

is attainable from some

V

types to be

types Hamilton's

A 9 R V'

such that

~

i~ @ i ~

such that

is the extension

for some

(in

vector

X 9 ~($i)

on natural numbers

4.3

Proof. if,

R

~ ~ I.

For Dedekind

A

9 $

on RETs to soundly based

COROLLARY

of

of extension.

4.2 The extension

exists

41

independent

an

the obvious A, B

are

sets.

to spaces of the form

then we can readily construct

V(A) a

If, for

42

J.N.

(combinatorial, V($1)

Crossley,

and indeed morphism

A. Nerode

combinatorial)

functor

@

on

by setting V(A)

where

A + B = {u2i

Then chasing

9 V(B)

: i e A}

in the following

= V(A+B)

u {u2i+l diagram

: i e B}. shows

induces

the required

functor. A

A'

V (A)

V (A')

V(A) 9 V(B)

,->

V(A') 9 V(B') t

V(B)

V(B')

B

B'

In fact, we can fill

in all the horizontal

lines too by our earlier

results. Tensor product

of vector

We simply use the fact that if {bj

: j ~ J}

basis

for

functor

on

recursive

Department

then the vectors

i~

|

V($1)

~

Again

spaces

3~, Id~

{a i | bj

example.

{a i : i ~ I}

: i ~ I, j e J}

this readily

(which is m o r p h i s m

if suitably

is another

have bases extends

combinatorial

to a combinatorial and partial

encoded).

of Mathematics,

Monash University, Australia

and

form a

Clayton,

Victoria,

J.N.

Crossley,

43

A. Nerode

REFERENCES

CROSSLEY,

J N and A NERODE

[1974]

Combinatorial Grenzgebiete,

[to appear] DEKKER,

Effective

functors. Ergebnisse Vol. 81, 1974.

der Math.

u.

ihrer

dimensions.

J C E

[1969]

Countable vector spaces with recursive Part i, J.S.L. 34, 1969, 363-387.

[1971]

Two notes on vector spaces with recursive operations, Notre Dame J. Formal Logic 12, 1971, 329-334.

FROHLICH,

A and J C SHEPHERDSON

HAMILTON,

A G

operations,

[1969]

D.Phil.

[1970]

Bases and a-dimensions of countable vector spaces with recursive operations, J.S.L. 35, 1970, 85-96.

[1974]

Recursive equivalence types of vector J. Aust. Math. Soc. (to appear).

Thesis,

Oxford,

1969.

spaces,

A STUDY OF GENERALIZED STANDARDIZATION IN COMBINATORY LOGIC by Haskell B. Curry

In [2] Theorem 13A7 it was shown that if function and

ml,m2,...,m n

is a partial recursive numerical

are natural numbers such that

defined, then the combinatory

ob

ob

f(ml,m2,...,mn)

which we associated with

that its normal reduction does not terminate. clude from this that that

f

f(ml,...,m n)

We were unable at that time to con-

has no normal form; for we did not have a proof, for

the kind of generalized reduction there considered, that if an form

XN,

is unis such

then the normal reduction of

X

must terminate in

ob X XN.

has a normal This question

we purposely left open in accordance with the policy announced in the preface of [2]. However we have recently received an inquiry which has brought this matter again to our attention; this inquiry has led to the present paper. In this paper I shall derive a result which includes this situation as a special case.

The precise hypotheses will be stated later.

But in a preliminary way I can

say that I will consider reductions which are stepwise generated quasi-orderings ([I],w

in which each step is a replacement of a certain component called a r e-

de_~x. The redexes are of two kinds, which I shall call primary and secondary. assumptions will be satisfied if the primary redexes are the ~-redexes of version ([I],w

or the redexes of weak reduction ([2],w

redexes are the 8-redexes of [i], w

The

X~-con-

and the secondary

All reductions considered here are generated

by redexes of one or both of these kinds.

The principal theorem of this paper is

that it follows from the assumptions that any such reduction can be standardized; an incidental result (Lemma 2) is that if a redex is not used in a standard reduction, then there will be a redex at the end of the reduction. w

Since the Z-redexes of [2],

are special cases of 8-redexes, the question of the epening paragraph is answered

by the corollary of the principal theorem. The problem of the opening paragraph was partly answered by J.H. Morris, Jr. in [4]. Morris, however, simply remarks that since Property (E) was proved for ~8-conversion in [I],w

the standardization theorem can be proved for it by a proof sim-

ilar to that in [1],w

This is true, but there are a lot of fussy details about

H.B.

Curry

45

it; and it is not yet certain that the case where reductions by primary redexes are weak reductions are included in it.

The present proof proceeds by a different method

Property (E) is not even mentioned - and has an abstract approach which may have other applications than those mentioned.

Weak reduction is certainly included.

In the preparation of this paper I have profited by discussion with Hindley, both at the recent conference in Swansea and in correspondence. that some of my preliminary guesses would not work.

He has pointed out

He is considering making some

further generalizations going far beyond that made here; these he may publish later. But the responsibility for any errors that may be found here is mine alone. Before stating the formal assL~aptions, it will be necessary to explain certain matters relating to terminology and notation.

In principle the usage here is the

same as that in [I] and [2] (which is approximately the same as that in [3]); but there will be certain deviations and some matters which require emphasis.

The speci-

fications in regard to usage of letters apply to those letters with affixes, except in the case of 'RC' , where the contrary is explicitly stated. The term 'ob' will be used here as in [I] and [2]. The reader may, if he likes, substitute the word 'term'

The letters, 'U'

'V', 'X'

'Y'

and 'Z' will be used

for unspecified obs; these are usually the stages in a reduction (see below). The terms 'component' and 'replacement' will be used here as in [2],w

ex-

cept that here a component is always an occurrence of an ob in a particular position in some reference ob, say

X.

The letters 'M', 'N' , 'P', 'Q', 'R', 'S'

for components, the last four for components of special kinds. M

and

N

of the same

ponent

M

shall be said to be senior to another component

to

just when the beginning of

M)

either

N

X

is a part of

will be used

Note that components

can overlap only if one is a part of the other9

M

M

N

(and

N

to be junior

lies to the left of the beginning of

or lies wholly to the right of

M.

A comN;

then

The ob of which a

component is an instance will be called the form of the component, and will be denoted by enclosing the symbol for the component in square brackets (cf. [I]w ents of the same form will be said to be equiform. in

X

does not disturb any part of

X

wholly

Note that a replacement of

of

M

by

M"

trace of

N

and M

N

by

N',

M

to the left of the beginning of

nor more generally any component which does not overlap with placement in

Compon-

M.

M,

Two replacements,

will be said to be interchangeable if the first re-

followed by the replacement corresponding to the second one in the

gives the same result as if the roles of

M

and

N

were interchanged.

This requirement is unambiguous if the traces are unique and equiform to the originals. If

M

and

N

do not overlap the two replacements are always interchangeable9

A redex is a special kind of component. stand for redexes; of these and

R

P

The letters 'P', 'Q', 'R', and 'S' will

will always be primary and

S

secondary, whereas

may be either unless otherwise specified in the context. Each redex

associated with a component

R C,

called the contractum of

R;

R

Q

is

and the replacement

46

of

H.B.

R

by

RC

Curry

is called the contraction of

traction rules; the rule to be applied to

R. R

The association is by a set of con-

is uniquely determined by [R] and

specifies [R~]; but the rules may involve intuitive variables so as to apply to more than one form.

Thus the rule for ~-redexes is that if

R

is of form

(Xx.X)Y

RC

is of form

[Y/x]X,

R

is of form

KXY,

RC

is of form

X,

that for K-redexes is that if

and so on.

Redexes to which the same contraction rule applies

will be said to be of the same type. type.)

then

then

(Thus equiform redexes are always of the same

It will be asst~aed that a primary redex is never of the same type as a sec-

ondary redex. If Q,

R

and

Q

in symbols

are redexes in the same

R]Q,

X, the residuals of

are defined as in [I],w

or

[2],w

R

with respect to

According to the

asst~ptions below, a reduction is a chain of contractions; then residuals with respect to a reduction, including a null reduction, are defined as in [i] or [2], ~.c. In connection with reduction I shall use the terms 'step' and 'stage' as in [I]w

Note that if

preceding

The letter and D1

D2

Xk

is the k'th stage, the k'th step is the step i~aediately

Xk. 'D'

will be used for reductions.

begins where

followed by

contraction of

D1

D 2. R

If

leaves off, then R

If

DI+D 2

D1

and

D2

are reductions

is the reduction consisting of

is a redex, the one-step reduction consisting of the

will be called

(R).

The terms 'primary' and 'secondary' will be extended from redexes to contractions of them and to reductions.

Thus a primary contraction will be contraction of a pri-

mary redex, and a primary reduction is one consisting entirely of primary contractions.

Similarly for secondary contractions and reductions.

All reductions in this

paper are chains of contractions of one or the other of these two kinds. The term 'constant' is here undefined; its meaning will depend on the application. Thus in B-conversion the constant components are those without free variables; in weak reduction every component is constant. If

R

is a redex in a stage of a reduction

in the next step, then R

is senior to

Q.

R

D,

and

Q

is the redex contracted

is said to be bypassed (by that step or by

Then a reduction

D

Q) just when

is said to be standard just when no redex

is contracted in

D

The reduction

is said to be normal just when no redex is bypassed by any step.

If

M

D

which is residual of a redex bypassed by the next preceding step.

is a component in a stage

Xk,

then the beginning of

in any later stage

Xk+ j

intermediate step.

Thus it makes sense to speak of component

the same beginning as In a reduction

M D

M

can be recognized

if there is no change to the left of that beginning in any N

in

Xk+ j

as having

or beginning to the left or right of the beginning of

M.

the redex contracted in the first step will be called the

starting redex; that contracted in the last step the final redex.

H.B.

Curry

47

The assL~nptions of this paper will now be listed. They are called EI-EI2. that

E1

and

discussion.

E2

Note

have already been mentioned, and in fact used, in the preceding

They will not be explicitly mentioned hereafter.

El.

A reduction is a stepwise generated quasi-ordering in the sense of [i],

E2.

Every step in a reduction is the contraction of a redex.

E3.

Two distinct redexes in the same

E4.

If

~4A3.

R

and

Q

If

R

E6.

If

R1

the same

X,

then

E7.

and

If R

a single redex then

R" E8.

E9.

R2

R"

If

P

Q

R2[ Q

R]Q

Q

Q

is any third redex, all in

X

and

R

is senior to

by

QC;

R.

If Q

otherwise

which is senior to

pC

then

Q

~,

then

Let

P

be a primary redex in

X"

and

P

and that contraction of

R,

is equiform to X

to

Y~

and

R. Q

is

is residual of a redex by-

to

X,

P" Mi

P

X,

and

~...,M

be nonoverlapping equi-

all of which are junior to by replacing each P"

M i by

X

to

Then if where

each of these is contracted, the result, in

X

is trans-

NI,...~N

are

will be a redex of the same type as P.

are secondary redexes and the

reduces

P.

Ni,

Y.

Ni. are their contracta,

Then the residuals of the

be a nonoverlapping set of components each of which is equiform to the

El0.

is

P.

Suppose further that the

P"

RIQ

is a part of

R"

is a primary redex whose contraction reduces

also equiform constant components, then

ing

is a re-

is void.

which has the same beginning as

Y

RIQ

are disjoint sets.

are redexes in

form constant components in formed to

every member of

are distinct redexes~ and and

is the result of replacing

a primary redex in passed by

are the same redex,

RII Q

and

X~

R.

Q

and

never begin at the same point.

are redexes in the same

dex of the same type as ES.

X

Y',

Mi;

M i will and if

is the same as the result of contract-

X'. If

D

is a primary reduction from

standard reduction

D"

from

is the seniormost redex in

X

X

to

Y

X

If

S

is a secondary redex, then both

El2.

If

S

is a secondary redex, then

S

Y~

then there is a primary

of which a residual is contracted in

Eli.

component of

to

such that the first redex contracted in

S

S

and

SC

D"

D.

are constant components.

is not a primary redex~ and no proper

is a redex of any kind.

We now embark on the proof.

It consists mainly of nine lemnas.

first three are preliminary results which are frequently referred to.

Of these the After them it

may be well to look ahead and observe that the theorem follows easily from Lemmas 6

48

H.B.

and 9.

Curry

The proof of Lemma 6 requires the apparently much more complex Lem~as 4 and

5, and similarly Lepta 9 requires Lemmas 7 and 8.

There are strong analogies between

Lenmms 4 and 7, 5 and 8, and 6 and 9.

Lemma I. P

Let

R

be a redex in

be a primary redex in

dex bypassed by

R.

Proof.

R

by Eli

R

If and

Y

X

whose contraction reduces

which is senior to

R C.

Then

is primary, this follows by E8.

RC

are constant componerts.

P

Suppose

If we replace

R RC

by

X.

By E9 this replacement will convert

Q

in

X,

R

and gives

P

to

Y.

Let

is secondary.

shall be brought back to which is senior to

X

is residual of a re-

R P

as residual when

in

Y

Then we

into a redex

R

is contracted,

q.e.d.

Lemma 2.

Let

R

be a redex in

starting redex is junior to stage of

D,

R.

X

and let

D

be a standard reduction whose

Then there is a unique residual of

R

in every

and, if the stage is not t~e last, this residual is senior to the redex

contracted in the next step.

Proof.

Let

next step be

Xk

be the k'th stage in

Qk+l"

k = 0;

Rk

of

R

for

R0

is

Hence it suffices to show that, if sis that it is true for

in

X k,

and

R

and

R

I ~ m ~ n,

and

Rm_ 1

and hence of

R,

not senior to to

Q~.

If

is senior to

Qm"

in

Rm

Qm+l

be standard.

Xm,

Qm+l"

such that

0 ~ k ~ n

is senior to

is senior to

Q1

it is true for

Qk+l"

This is

by hypothesis.

k = m

on the hypothe-

Then

and

Then there is a unique residual

By E7 there is a unique residual is senior to

Qm+l

is senior to

~. Rm

Suppose by

E3,

m<

n

Rm_ 1 Rm and

of Rm

On the other hand, if

Qm+l

is secondary, it cannot contain

an equiform and similarly positioned redex in D

be standard.

Thus

Rm

in Rm_ I, is

and hence it is senior

is primary this contradicts, by Lepta I, the hypothesis that

El2), and hence lies entirely to the left of it.

hypothesis that

k

Rk

k = m-l.

Assume this inductive hypothesis. Xm_ I,

and let the redex contracted in the

Then we have to show that for every

there is a unique residual trivially true if

D,

Rm

D (by

It must therefore be residual of

Xm_l,

which again contradicts the

must be senior to

Qm+l"

This completes

the induction and the proof of the lemma.

Lemma____3. Let seniormost redex in of

Q

D

be a primary reduction from X

D

to

Y~

of which a residual is contracted in

which is contracted in

tracted in

X

D

begins at the beginning of

begins to the left of the beginning of

Q.

and let

Q

be the

D.

Then the residual

Q;

and no redex con-

H.B. Proof.

Let

Xk

(0 ~ k ~ n)

tracted in the k'th step be Pk+l

for

such that

residual of

Rk

Pk+l

for

Pk

k

k < m,

such that

This is trivial for Now since

It follows by Lemma 1 that Xk_ 1 which is senior to

Rk

k = m.

Rk

has

Thus

Pm+l R.

By E7, Rk

R

the other hand Q contracted.

Rk

k,

R

Lemma 4. of some

S i.

D,

Let Let

and let

P~,

is the unique

R R~

X

which is a replacement for Pk

of an Pm+l"

in

X

Rk_ 1

in

Rk_ 1

begins

Thus by de-

and begins exactly at the beginQ,

R,

by the hypothesis on

for if it did every

Q,

and no residual of Q.

Q.

Pk+l Q

On

in

D

would be

The len~aa follows by defini-

be equifom secondary redexes in to

to and including

U,

reduce in

U

Y

to

X~

and let

as sole residual in

(ii)

reduces X

Then there is a stage

R

to

U

V;

U

Y

in

D

such that:

and senior to the re-

if

D~

consists of the stages of

to

V,

and

D1

is

D ~ + {R~}~

D,

then

(iii) there is a standard reduction

such that every redex contracted is an equiform residual of some

its starting redex is not residual of a redex bypassed by Proof.

We can suppose the

Si

Let

Xk

k > j, X k will be obtained from by an instance of [sC]. Let

Xr

be the first stage of

a residual of r < n. and, for

R~

and

R~

Xj

X~

is

D2

from

Si~

and

Since they are

this

Rn

D

is senior to must be

and

X

is

Y.

by replacing each residual of

which contains a redex Sr+ I.

Let this

By Lemna 2 there will be a unique residual k ~ n

X

R.

Rk

of

This proves (i).

is

R~.

are numbered in standard order.

be the k'th stage, so that

down D1

nonoverlapping (El2), the redex contracted in the k'th step is a residual of S~.

D

in which each redex contracted is residual

Z.

having

{R~

a standard reduction from Z

Y

be a redex which is residual of one bypassed by the last step

dex contracted in the next step;

it

k ~ m.

0 ~ k ~ m.

must be the same as

$I,$2,...~S n

{R~

there is an

to

in

Pm+l"

be a standard reduction from

V

Pm+l

is the unique residual, and

cannot begin to the left of R

and

Rk

for residual it is a primary redex.

cannot begin to the left of

Hence by E3,

Pm+l'

and hence at the beginning of

is residual of a redex

This

Rk,

would begin to the right of the beginning of

(i)

which

Suppose it is true for a given

Pm+l

scending induction my claim is true for all

of

Pm+l

there is a redex

is residual with respect to

Pk-

exactly at the beginning of

tion of

and let the redex con-

and at least as far left as any

0 ~ k ~ m

does not begin as far left as

does not either.

ning of

D,

Then there will be a

begins exactly at the beginning of

R k.

Then since Pk'

(0 ~ k < n).

k > m.

I now claim that for every Xk

49

be the k'th stage in

Pk+l

begins further left than any

Curry

Xr

R~ be

Si,

Sk; call

Then for j < i ~ k

such that U.

P~ in every

R

is

By hypothesis Xk,

r_
50

H.B. Again,

R~

it were, then stated for

Curry

is not residual of a redex bypassed by the preceding step. r > 0

X r.

and

Xr_ 1

This proves (ii).

It remains only to prove (iii). The residuals of secondary redexes equiform to the El2).

For if

would be an earlier stage having the properties

They will be carried by

Si,

{R~)

Sr+l,...,Sn

are a set of

and hence constant and nonoverlapping (Eli, into another set of secondary redexes

S~,...,

S ~ all of wb_ich are equifom to the S.. If R, and hence R~ is primary, this m i ' follows by E9 and Eli; if R is secondary, it will follow since R~ cannot overlap the residual in

U

of any

S i.

Now let

D2

be the standard reduction formed by

contracting these residuals in standard order, and let its terminus be is the result of replacing simultaneously every

S~

make the analogous replacements in U we are led to R. R

By E9 if

R

Lemma 5. D2

~R}

Let

will reduce

D1

Y

to

R,

Z9

V

to

Y~

to

Z

R,

If

D3

m

Let D~ ing

steps. Q

X

to

q.e.d.

V

V.

Let

D3

with final redex

R.

D2

is residual of

be a standard reduction

X

to

Z.

D3

contains

m-i

D1 + D2

is standard.

be the starting redex of D3;

then

steps, moreover if

p

D3

D3,

and let

{Q}

m.

reduce

is a standard reduction from

m > i,

the starting redex of

begin to the left of the beginning of

Q

Assuming

steps, I shall prove that it holds when

This will prove the lemma by induction on

be the rest of m-i

Z 9 ~ Z,

whose starting redex is not residual

is void the lemma holds since

that the le~na holds if has

is transformed to

and if secondary, does not lie wholly to the left of it.

Then there is a standard reduction from

D3

R~

Z9

such that the starting redex, if primary, does not begin to the left

of the beginning of

Proof.

and

and such that every redex contracted in

one of a set of equiform secondary redexes in Y

Then

be a standard reduction from

be a standard reduction from

of one bypassed by from

Y,

Then

But if we

is primary, and from properties of nonoverlapping replacements of

is secondary,

Let

Z'.

by its contractum.

D3,

Y

to

Y'.

Y"

to

Z

Let contain-

if primary, cannot

(by Lemma I), and, if secondary, it cannot

lie entirely to the left of that beginning. If

Q

D1 + D2 + D3 the

is not the residual of a redex bypassed by the last step of is the standard reduction sought.

D, X, Y, Z,

and interpret the

R

Otherwise we invoke Lermna 4.

of that lemma to be the present U,

R~, DI, D 2

D2,

of that len~aa as

D2, U 9 ~

V, Y, Y9 + Q~, DI, D +2

Q

then We take

respectively;

here.

§

If the starting redex of and so is not bypassed by from

X

to

V"

R.

D1 Let

is not D1

be

Q~,

then it is the starting redex of

D I + D E.

Then

D1

D2,

is a standard reduction

in which the final redex is Q~. On the other hand, suppose Q~ is + D E. Then D 1 is (Q~}. If Q, is primary, then since it

the starting redex of

has the same beginning as

Q,

it does not begin to the left of the beginning of

R,

H.B.

Curry

51

and hence, by E7, is not residual of a redex bypassed by secondary, then

Q,

tirely to the left of

R;

but in that case, since

of the redexes contracted in to the hypothesis about and

D I + DE

R.

If

can be residual of a redex bypassed by

Q.

D2

after

Thus

Q,

is again a standard

We thus have a

D~

U',

Q

Q,

R

Q

and

Q,

are

only if it lies en-

cannot overlap (by El2) any

would have to do so also, contrary

cannot be residual of a redex bypassed by

D{

reducing

X

to

{Q,}.

V" and ending in

satisfying the hypotheses of this len~na for

DI,

with

X,

9

V

Y

and

D 2,

R

interpreted as

wise

D~

+

X, V', Y , and

it satisfies the hypotheses for

D2

satisfies the hypotheses for

Q,

respectively.

Le~na 6. D

from

Let

X

to

D3,

and

be a secondary redex in

R

Y

Let

R

is not residual of a redex in {R0}

reduce

The reduction preted as

to

D3

{S}.

has

m-i

duction from

X

Lemma 7. (0 _< k _< m)

X

to

X'.

{S} + {R} X',

Y

that:

X

U V;

if

to

D

be

{S}

and

Z, D

and let S,

By the hy-

Z,

q.e.d.~

reduce

Z

X

to

Y.

U

will be the R0,

D

will be the present Y'.

Then

be a standard primary reduction from D,

inter-

X,

V

DI,D2,D3,X,X~,Y',Z

Y

Y.

Let

Xk

Pm

and let

U -: Xr_ and a redex

S~

in

then

in

Y,

and

S,

is

D ~ + {S,},

if

D2

is formed from the final segment of

D1

U

is senior to

down to and including

D1

S,

to

which is bypassed by

and

contracting the residual of

X

and let the redex contracted in the k'th step

D

U,

Pr+l;

D r starting at

in each stage and replacing D2

{S} such

{S,}

is a standard reduction from

equiform) residual with respect to that contraction, then

Xr

Y:

q.e.d.

S,

Let

to

{R~ + D 3.

Then there exists a stage

to

{S} + D

Y

By that lenlna there will be a standard re-

is the sole residual of

V

reduce

is residual of such an

will be the present

consists of the stages in

reduction from

{R}

then

R

(iii)

Proof.

to

will satisfy the hypotheses of Lemma 4 with

be a secondary redex in Z.

S D~

V,

and

S

to

D,

senior to

Let

be the k'th stage in

(i)

(ii)

to Let

Let

reduce

and let

Assume, therefore, that

will satisfy the hypotheses of Le,~na 5.

Pk"

steps. X

as

Like-

Then there is a standard reduction

In the conclusion of that lemma,

will be the present

be

X~

Z.

be the starting redex of

standard reduction sought. and let

D2

Z.

Proof. If

S

be a standard reduction from

If we take

of this lemma with the same changes.

pothesis of the induction there is a standard reduction from

Let

R,

reduces X

to U,

by

PI. by its (unique and is a standard primary

Z.

be the first stage in

as residual and is senior to

Pr+l"

D

in which there is an

By hypothesis

r < m.

S,

which has

By Lemma 2 there is

52

H.B.

a unique residual be

S.

Sk

of

S~

Curry

in every

Xk,

r ~ k ~ m;

Furthermore Le~aa 2 shows that every

Sk

for

By descending induction it follows from E4 that every by El2 it does not overlap have the same beginning. Again

S,

Pk+l

and so, by E7, all

D2

Sk

Len~aa 8.

Y

Let

D1

Pk+l"

are equiform to

S

and

Xr,

for otherwise

Xr

If we make these interchanges

S.

Let

D2

X

to V

whose final step is

be a standard primary reduction from

Y

to

Z

whose starting rede• S,

Let

m

m > 0,

let Y~, If

X

to

R

Let

D3

be a

Z.

be the n~nber of secondary contractions in

involve an induction on

occur in a stage

SC.

if primary, does not begin

and, if secondary, does not lie wholly to the

Then there is a standard reduction from

D3o + {R~ + D y

must

Pk+l"

is secondary, and hence,

Sk

whose starting redex lies entirely to the right of

Proof.

Sm

This proves (ii).

be a standard reduction from

to the left of the beginning of

If

S,.

is interchangeable with

standard reduction from

left ot it.

and

satisfying (iii). This completes the proof.

contraction of a secondary redex to

Sk

S,

is senior to

is not bypassed in the step, if any, before

Again each

V

is

This proves (i).

would not be the first stage with such an

we have a

Sr

k < m

D 3.

The proof will

m. be the first secondary redex contracted in and let its contraction reduce

m = 0,

D o3

let

be the whole of

Y*

to

Y'.

D3,

and

Y~

D;

let it

Let

D3

be

Z.

be Then

D o3

is a standard primary reduction, and, by E8 and an induction, no redex contracted in D o3 begins to the left of the beginning of S. Further D 3 is a standard reduction with only m-i secondary contractions; and the starting redex of D3, if primary, does not begin to the left of the beginning of

R

(Lemma i),

and if secondary does

not lie wholly to the left of it. By El0 there is a standard primary reduction D 2 from V to Y . No redex o contracted in D I + D 3 can begin to the left of the beginning of S. Therefore, by Lemma 3, the starting redex of of a redex bypassed by

S.

D2

does not either, and hence it cannot be residual

Thus the reduction

The last paragraph proves the lemma if that the lem~a holds if there are If

R

m-i

D1 + D2

m = 0.

is standard.

Henceforth we assume

secondary contractions in

will be the standard reduction sought.

By that lepta there is a stage

U"

in

D2

R;

D2,

then

D1 +

Otherwise we apply Lemma 7.

such that there is an

is senior to the redex contracted in the next step, has and has the same beginning as

and

D 3.

is not residual of a redex bypassed by the last step of

D 2 + {R} + D 3

m > 0

R

P~

in

U"

which

as sole residual in

Y ,

also its residuals, all of which are equiform,

are interchangeable with all the redexes contracted later in

D2 .

If we make these

H.B.

Curry

53

interchanges we have a standard primary reduction tracting before

R,

in

U',

If

9

*

is not the first stage in

D2,

redex, which is the starting redex of S.

Then

DI, 9

viz.

V 9,

the result of con-

D2,

O

then

D2

U"

is not residual of a redex bypassed by

is the first stage in

residual of a bypassed redex in

U.

D2, i.e.

V,

For suppose it were.

wholly to the left of the beginning of

D2

is not void and its starting

D1 + D2O + {R,}, will be a standard, reduction from

On the other hand, if

D3

from

U , to Y'. Further if D 2 consists of the steps, if any, of o D 2 + {P~} will be a standard reduction from V to V'.

then

U"

D2

O

S;

if

R

then

X

R,

to

V9

is not the

Then it would have to lie

should be the starting redex of

this would contradict the hypothesis about that starting redex; otherwise, since o D 3 begins to the left of the beginning of S, it would con-

no redex contracted in

tradict the hypothesis that reduction from

X

to

D3

be standard.

Then we get a

D1

which is a standard

V'.

Thus we have a situation similar to that at the close of Lemma 5. of this lemma with satisfied by a and

D3

DI,

defined as above, in the place of

in the place of

D3;

and

D3

has

m-I

DI, D~

Len~aa 9.

Let

D

from X

to

Z.

Proof.

If

P

be a primary redex in

be a standard reduction from

D

Y

X

in the place of

secondary contractions.

inductive hypothesis there is a standard reduction from

and let

The hypotheses

X, V, Y, Z, S interpreted as X, V', Y', Z, R~, respectively are

X

to

Z,

By the

q.e.d.

whose contraction reduces

to

Z.

D2,

X

to

Y~

Then there is a standard reduction

is a primary reduction this follows by El0.

be

the redex of the first secondary contraction in

D.

Let

D3

is a standard reduction whose

is a standard primary reduction, and

be

S

where

D~

D

If not, let

D~ + {S} + D3,

starting step, if primary, does not begin to the left of the beginning of otherwise does not lie wholly to the left of the beginning of Y

to

Y*,

and

{S}

reduce

Y*

to

Y',

the standard primary reduction from from

{P} + D O .

then

D

If

S

X

to

then

D3

Y

reduces

S. Y"

there is a standard reduction from

Proof.

X

to

If there is a reduction

Let

X

to

Xk

reduce Let

D

D

Z,

be

which is obtained according to El0

+ {S} + D 3 will be the standard reduction sought.

Theorem.

D~ Z.

and

is not residual of a redex bypassed by the last step of

D ,

Otherwise we can apply

Lepta 7 to obtain a situation which is a special case of Lemma 8.

reduction from

Let to

S,

By that lemma

q.e.d.

from

X

to

Y,

then there is a standard

Y.

(0 _< k _< n)

standard reduction from

Xn

to

be the k'th stage in Y,

and also from

Xn_ 1

D. to

Then there is a trivial Y.

Suppose that there

54

H.B.

is a standard reduction from reduces

Xm

tion from

to

Xm+ I. Y

If

Rm+ 1

Y.

Let

Rm+ 1

be the redex whose contraction

is a primary redex, then there is a standard reducif

Rm+ 1

Hence, by descending induction on

m,

there is a standard reduction from

m

to

to

by Ler~a 9;

for every

Xm

Xm+ 1

Curry

such that

Corollary.

0 ~m

~ n.

For

is secondary, this follows by Lemma 6.

m = 0

If there is a reduction

D

from

form, then there is a normal reduction from

Proof.

we have the theorem,

X

to

X

to

by Len~aa 2 there would be a residual of

Department of Mathematics Pennsylvania State University University Park, Pennsylvania

16802

R

in

R Y,

and

Y

to

Y

q.e.d.

is in normal

Y.

By the theorem there is a standard reduction

pose that at some stage the seniormost redex

Y

Xm

Y.

Sup-

were not the one contracted.

D

from

X

Then

which is impossible.

to

H.B. Curry Bibliography [i] Curry, Haskell B. and Feys, Robert. Combinatory logic, vol. I, Amsterdam, North Holland Publishing Co., 1958. Third printing 1974. [2] Curry, Haskell B., Hindley, J. Roger, and Seldin, Jonathan P. Combinatory logic, vol. II. Amsterdam, North Holland Publishing Co., 1972. [3] Hindley, J. Roger, Lercher, Bruce, and Seldin, Jonathan P.

Introduction

to combinatory logic. Cambridge, at the University Press, 1972. [4] Morris, James Hiram Jr. Lambda calculus models of progranming languages. Thesis, Massachusetts Institute of Technology, 1968.

INTENSIONALE Kurt

FUNKTIONALINTERPRETATION

SchGtte Justus

K. G~del hat Theorie

zum 65.

D i l l e r und Helmut

wurde

der B a r - F u n k t i o n a l e

und der i n t u i t i o n i s t i s c h e n sind die F o r m e l n

in

kutiert

wird,

HA

l~Bt

nete Theorie)

Deshalb

G e g e b e n sei EinfGhrung voraus, HA

von

Allein Vw~

B^:= 3vVwB

C A , abh~ngig in A b h ~ n g i g k e i t VzU

aus

und vonder

Annahme

Dieses

der I m p l i k a t i o n

a l l e i n in A b h ~ n g i g k e i t

fGr die ~ aus ~ folgt.

Da sich

wird m a n im a l l g e m e i n e n viele v e r s c h i e d e n e

~

Vereinigung

HA w (das

der T h e o r i e n

^ der D i a l e c t i c a - U b e r zu begriinden ist.

Dureh

C, in der

Implikations-

B ~ C . Wir setzen

, aber aueh B ^ , bereits vonder

eine H e r l e i t u n g definiert Wahrheit

ein y a u s r e c h n e n

Argument

yon

bezeich-

H einer F o r m e l

yon

sind. von

lassen,

wird auch bei der Dia-

angewandt.

Ist ein solches

durch B e t r a c h t u n g

yen

y be-

H A yon u n t e n

von z die Objekte w a u s z u r e c h n e n , nach

oben b i n ~ r v e r z w e i g e n kann, A s t e n yon H A endlich

Objekte w erhalten.

so da~ ~ aus

yon dem n o - c o u n t e r e x a m p l e - A r g u m e n t

Aw~W~ ab, das

Zusammenfassung

Es mu~ also

folgt.

[8] a n g e b e n und das auf der E n t s c h e i d b a r k e i t Als s c h e m a t i s c h e

Formu-

in den v e r s c h i e d e n e n

solche

liche M e n g e W geben,

N-HAw

auftritt.

C A := 3yVz~

man u m g e k e h r t

so versucht

Typen

yon v, also u n a b h ~ n g i g folgt.

T,

schon nicht

wie folgt

Herleitung

lectica-Ubersetzung oben,

SehGttes

und mit

Herleitung

frei

stimmt, nach

(vgl.

eine V a r i a n t e

H A yon oben nach u n t e n Vw~

Z a h l e n auf-

der T h e o r i e

endlicher

genannte natGrliche

eine n a t G r l i c h e

[6] u n t e r

hat und in [2], Ende yon w 2, dis-

die fGr I m p l i k a t i o n e n

B eventuell mehrfach

, mu~ sich aus

fGr das

die die

Diese

In d i e s e n S y s t e m e n

Fassung

Heyting-Arithmetik

erh~It m a n aus H e i n e

dab

fortgesetzt.

Typs a u f t r e t e n

wird in [2]

in der

der k l a s s i s c h e n

vom Typ der n a t G r l i c h e n

[13] "neutral"

D eingefGhrt,

eine A n n a h m e

Systeme

sich die D i a l e c t i c a - I n t e r p r e t a t i o n

fortsetzen,

und T ist.

HA

interpretiert. [11] und Howard

intensionaler

hGheren

auf die i n t e n s i o n a l e

setzung

von S p e c t o r

[3 ]). Wie Howard bemerkt

ist die in T r o e l s t r a HA

Funktionale

Analysis

zu G~dels

Gleiehungen

lierung

Vogel

auf e x t e n s i o n a l e

aus G l e i c h u n g e n

im Gegensatz

in der aueh

gewidmet

1958 in [4] die H e y t i n g - A r i t h m e t i k

Dialectica-lnterpretation

gebaut,

Geburtstag

T der p r i m i t i v - r e k u r s i v e n

Verwendung

DER A N A L Y S I S

Dieses

G~del in

sich

endweicht

[4] und K r e i s e l

von B beruht.

ergibt

eine

Argument

in

57

J. Diller, H. Vogel

",@

xB

(--,T)

3vVw~

xB

c B-~C

HA:

I

x

3vVw~

x

) (~I)

^

.

.

.

.

x

Vv3yVz 3 endl. Menge W:(Aw~W~ ~ U),

und aus der letzten Formel erh~lt man mit Auswahlaxiomen die ^-Ubersetzung der Implikation (B ~ C ) ^ := 3X,W,Y Vv,z(Ax < Xvz ~[v,Wxvz] ~ ~[Yv,z]) GSdels Theorie T wird also zun~chst durch einen beschr~nkten Allquantor

Ax
, der als Junktor variabler Stellenzahl,

liger Junktor aufzufassen ist, zur Theorie sich

HA~ nach

[2],

T^ erweitert.

als t-stelDann l~Bt

Satz 3 bez~glich der ^-Ubersetzung in

T^ inter-

pretieren. Die ^ -Interpretation "verf~lscht" demnach Herleitungen in HA w weniger als die Dialectica-Interpretation. Ziel der vorliegenden Arbeit ist es, die A-Interpretation auf intensionale Systeme der intuitionistischen und der klassischen Anasysis fortzusetzen,

die unmittelbare

Erweiterungen von

HA w sind.

Dazu wird in w I die Theorie T^ um ein Schema SimBR~ der simultanen Bar-Rekursion vom Typ ~ erweitert, und in dieser (quantorenfreien) Erweiterung gel bewiesen

T^+ SimBR~ wird eine verallgemeinerte

Fan-Re-

(Satz I). Mit dieser Fan-Regel wird in w 2 die Bar-In-

duktion vom Typ ~ in Anlehnung an Howard

[5], w 3, in

H~+{ A ~

} +

SimBR o hergeleitet. Daraus ergibt sich mit [2], Satz 3, Korollar I, die Interpretation der intensionalen intuitionistischen Analysis HA+m { A ~ A ^} +BI~

bezGglich der ^-Ubersetzung in

~ + SimBR~ . Zur In-

terpretation der klassischen Analysis wird in w 3 ein neuer Beweis fGr die negative Version klassischer Auswahlaxiome in der intuitionistischen Analysis gefGhrt, der Theorem 4B in Howard

[5] versch~rft,

insbesondere von Folgen-Kedierungen unabh~Lugig macht, und den Bowels vereinfacht.

w I. Simultane Bar-Rekursion und Fan-Re~el Wir verwenden die Ergebnisse und die Terminologie von [2] und [3]. T^ bezeichnet wie in [2] die mit den Junktoren-~ und ^ und dem beschr~hqkten Allquantor (A ( ) formulierte Theorie der mit den Kombinatoren K und S und dem Iterator J eingefGhrten primitiv-rekursiven Funktionale endlicher

(linearer) Typen. Wit schreiben

s
fGr

Ns~t=0

58

J. D~ller,

und

s-
genschreibweise.

. Wie in

[4 ],

Auch Typtupel

H. Vogel [11 ] und

werden

L2 ] v e r w e n d e n

wie einzelne

wit die F o l -

Typen mit ~,~,...

bezeichnet. Definition vom Typ

(o~)oo~

I. In

T^ sei zu jedem Typ

definiert

durch

([3],

, := ~o[RO~ F~r

(1.1) (1.2) (1.3) (1.4) (1.5)

IT

9 die Beschr~Lukun~

Definition

I

8)

(x+y) ]) j

ist in

T^ h e r l e i t b a r

(wir s c h r e i b e n

clt

ffir l~ct ):

clO = 0 ~ clNt = ~y[P((c It)y)(ct)(t~y)] s
Definition vom Typ

2. Zu jedem Typ T ist das V e r k e t t u n g s f u n k t i o n a l

(o~)o~o~ 9

= cts

cx = dx ~ clt = d It

:= T

definiert

~c,x,uw[P((clx)y)u(x~y)]

Wir s c h r e i b e n

c*tu

ffir

* ctu

(2.1)

s
= cs

(2.2)

t~s ~

(c.tu)s

= u

(2.3)

smt ~

(c*tu) IS = els

(2.4) (2.5)

(c*tu)INt = c*tu = (clt)*tu c*t(ct) = clNt (1.1) bis

Gleichungen len k ~ n n e n Folgen

(2.5) g e l t e n

gehen dabei

funktionalen

eines

Zu jedem Typtupel

T^ die B a r - F u n k t i o n a l e

Y,G,H,c,t,u

Bar-Rekursion

~;

die a n g e g e b e n e n

fiber. Mit diesen

und B a r - I n d u k t i o n

((od)o)((oo)o~)((~)(o~)o~)(o~)o~ le Terme

ffir Typtupel

T^ h e r l e i t b a r :

ohne Typen

Funktiona-

ffir endliche

Typs formulieren.

~ ein Typtupel. von

ebenso

Dann ist in

in G l e i c h u n g s t u p e l

wit B a r - R e k u r s i o n

von Objekten Sei

*

durch

9 ffigen wit zu den Grund-

B~

vom T y p t u p e l

und zu den A x i o m e n

passender

Typen folgende

von

Axiome

T^ ffir al-

der s i m u l t a n e n

vom Typ ~ hinzu:

Y(c*tu)
~ Bo YGHctu

= G(c*tu)(Nt )

Y(c*tu)~Nt

~ B ~ YGHotu

= H(Bo~ YGH(c*tu) (Nt)) (c*tu) (Nt)

SimBR~

Die so entstehende

Theorie

Das Schema

SimBR~

aus gen (3)

2n

Implikationen.

(2.4) Fctu = F ( c l t ) t u

bezeichnen besteht

Setzt

wir mit

~+ SimBR~

ffir ein Typtupel

man abkiirzend

F

.

T= ~1,...,~n

:= B ~ Y G H

, so gilt we-

J. Diller, Also ist

es keine

ma vereinfacht

Einschr~nkung,

sich dann w e g e n

H. Vogel

speziell (2.5)

89

u=ct

zu setzen.

Das Sche-

zu

Y(o,Nt)
~ Fct(ct)

= G(clNt)(Nt)

Y(oINt)~Nt

~ Fct(ct)

= H(F(clNt)(Nt))(clNt)(Nt)

SimBR o Wir b e n u t z e n Folge

demnach

das Paar

(c0,...,c(t-1))

bezeichnet,

konstante)

herauszulesen

(3.~) (3.2) (3.3)

3. Ist

felgende

F = B0 YGH

einfache

, se n e n n e n

aus der der

wir das T e r m t u p e l

F =BYGH

und der V e r k G r z u n g

F' yen F

Zusammenh~Luge :

Y(clt)
-+ F'ct

= G(c It)t

Y(c,t)~t

~ F'ct

= H(F(clt)t)(clt)t

Fctu = F'(c*tu)(Nt) Bemerkun~

F'

I. Die h i e r a n g e g e b e n e

ist zu der in Howard

schw~cher

verwendete mit

o~

endliche

eine

:= ~ c , x [ P ( G ( c , x ) x ) ( H ( F ( c l x ) x ) ( c l x ) x ) ( x ~ Y ( c l x ) ) ]

Bar-Rekursien nend

vem Typ

allein

ist.

die Verkfirzun~ ven F. Zwischen bestehen

fiir die

Funktion

(schlieglich

F'

als K o d i e r u n g clt

Term t nicht

Definition

c lt,t

der L~Luge t, w ~ h r e n d

als die ven Specter

folgende

.'= B ~ Y G H

Fassung: felgende

Y(c It)
kombinaterische

[ 6 ] ~quivalent. [11], H o w a r d

Zu jedem Typtupel Axieme

Fassung

der

Sie ist anschei-

[5 ] und Luckhardt [10] , das es B ~

9 gibt

erfGllt-

= G(c It)t

~ F'ct = H ( ~ u [ F ' ( c . t u ) ( N t ) ] ) ( c l t ) t

.

Es gilt T A + ~ - B i m B R ~ ~ SimBR o , denn ist Schema

F' = B ~ Y G H SimBR~

, so erfiillt das durch

. Umgekehrt

T + S i m B ~ + Ext werin

Ext

~u[F'(c*t u)(Nt)] l~gt

definierte

F das

,

die E x t e n s i o n a l i t ~ t s r e g e l

Schema ~ - S i m B R o , w e n n aus

(3.3)

abet nur

~ ~-SimBR~

Denn die Verkiirzung F' yen

Anscheinend

gilt

G6 aus Specter

F = BYGH

erfGllt

wegen

[11 ] bezeichnet. (3.1),

(3.2) alas

(3.3) felgt

= Xu[F(ctt)tu]

sich die Aquivalenz

= F(clt)t

beider

Schemata

nicht

ohne Ex-

t ensi onalit ~t n a c h w e i s en. Bemerkung

2. Bei H i n z u n a h m e

in

T^ aus der e i n f a c h e n

und

9 einelementig,

yen P a a r f u n k t i e n a l e n

Bar-Rekursien

also T y p e n

sind.

BR

felgt

SimBR~

, beider

die T y p t u p e l

Paarfunktienale

sind aber in

o T^

60

J. Diller,

H. Vogel

nach Barendregt [I ] nicht definierbar. T r o t z d e m ist die simultane primitive R e k u r s i o n nach [3] in T^ herleitbar. Die Frage, ob sich SimBR entsprechend

in

T^+BR

Beschr~kte bar,

herleiten

Quantoren

trachten

wir

Vx~t

Ax
s=O

beschr~nkten

s=r

Definition (o~)o

und T e r m t u p e l n F

G,H

.'= B~ YGH I H1wct

definier-

Insbesondere

s%0

ist.

Ax~t

be-

eder fiir steht

in einer Fassung,

dabei

fiir

die diese

verwendet"

4, monotone

tupel

quantorenfrei

~Ax~t

T^ ~quivalent

man die B a r - R e k u r s i o n

Quantoren

offen.

vom Typ o stehen.

als Abkiirzung fur

, was dazu in

. Oft benutzt

bleibt

sind b e k a n n t l i c h

w e n n sie vor Gleichungen

minIslx~t} = 0

l~t,

Bar-Rekursion.

passender

Typen

Zu jedem Term Y vom Typ ist in

TA+ G i m B R

ein Term-

mit

= P(Got)(Hwot)(maxIx~Y(clx)ix
de fini ert. Hierfiir ist in

T^+ SimBR~

(4.1)

Vx~Nt

Y(clx)<x

(4.2)

Ax~Nt

Y(cix)->x ~ Fct(ct)

Denn well

Daraus

~ Fct(ct)

max{x-'Y(ctx)Ix
= Hwct

, falls

H1wct

= Gct

sonst.

folgt

sofort

Im Baum

6~

den w l~uft ~o

durch

kodieren.

die endlichen

Fo

hat dann in kSnnen

zu

Fo

gehSrigen

in

T^

ausrechnen. Definition Hbfwt

definierte MA~

Folgen

w]k,k

die Breite

gilt

.

von n a t ~ r l i c h e n

die F u n k t i o n e n b:~ o ~ ~

ist,

ist, und

SimBR~

vom Typ

Zahlen

schneidet

aus

b(wlk)k

der festen

60

f~r die stets

einen Fan wk< b(wtk)k

. Zu jeder F u n k t i o n

fiber alle F u n k t i o n s w e r t e

Folgen

sind

oo . Der Fa-

(wO,... ,w(k-1 )) , die wir durch

die F~den w verlaufen,

endlichen

L~nge

yon f in den

k folgenderma2en

5. Es sei H das durch

= maxIf(w.tY)(Nt)ly
Funktional

vom Typ

Y(clx)->x

Folgen

wir alas M a x i m u m

,

Y(clx)_>x

(4.2) mit

Jede F u n k t i o n

in dem genau

ist.

~ Ax
gerade

und

= H(F(ciNt)(Nt))(clNt)(Nt)

der endlichen

aus,

f:B ~ -~ ~

= G(oINt)(Nt)

Ax
(4.1) und

die F~den o d e r W a h l f o l g e n wlk,k

=0

(3) h e r l e i t b a r

aus

T^

. Dann d e f i n i e r e n

((oo)oo)((oo)oo)oo

wir das F a n - M a x i m u m

durch

M A X bft = J ( H b ) f t O ~ 1 7 6 . Fiir

MAX bft

schreiben

wir auch

maxIf(wlt)tiAx
wx
.

J. Diller, Dann ist in

(5.~) (5.2)

H. Vogel

T^ h e r l e i t b a r

M A X bfO = fO~176

MAX b f ( N t )

= N[&X b ( H b f ) t

= maxImaxIf(w*tY)(Nt)ly
(5.3)

ix
Dies folgt

wxWb(WlX)X

mit I n d u k t i o n

Demnach h a t

aus

Es s e i ~o

jeder

(*)

~o zuordnet.

einen Fan

b:B o ~ ~

aus

Fan ~ in

B o , den von

(6)

Funktionen

vorauszusetzen. wann

eine Folge

(oo)oo

g einen T e i l b a u m

in

so erh~lt

m a n als Bild

gw

Fan.

ein Faden Wir k~nnen

ungesichert

seien Terme

^Y(gwlx)~ (6) in

(6) folgt

gwlx = g(wlx)Ix

einen

Fiir die auf

in in

~o

~o

' der genau

T^ ausdriicken,

Fan liegt und durch

ist: g,b,Y

der Typen

(6). Dann sei

FAN[w,t]

(oo)oo

,

die Formel

x)

T^ h e r l e i t b a r

^ Y(gwlt) _~ t ~ A y < b w t Aus

ein

Einschr~nkung,

ist

(.) l~uft.

es gelte

~o auf.

wird,

~o a u f ~ e z ~ h l t e n

(o6)o

in

oben durch

Fall

, und

Beweis.

in

Folge

' der wie

bwx = b ( w l x ) x

6. Gegeben

FAN[w,t]

~o

~o

und

und

(oo)o

vom T y p -

von

die endliche

(.) in dem von g, b aufgez~khlten Y vom Typ

Dann ist aus

Also

in

Definition

Ax
Bo

g und b ist es keine

Folgen

Ax
(6.1)

g,b

In diesem

die endlichen

ein F u n k t i o n a l

von Objekten

eine A b b i l d u n g

in

z~hlt Fo

herausgeschnitten

gwlx = g(wlx) Ix

durch

wlk,k

Eigenschaften.

Folgen

definiert

Auf diese Weise

Bo

erwiinschten

endlichen

Folge

-< J ( H b ) f ( t - ' x ) ( w l x ) x .

, g ( w Ik) ( k - ] ) >

man g auf

definierten

die

g:~o ~ o

endlichen

(g(w I1)0,...

Beschr~ukt

MAX

B~ d e r Baum d e r

Eine F u n k t i o n

, die

J(Hb)f(t~Nx)(wlNx)(Nx)

das Funktional

tupelo.

} I Ax
~ f(wlt)t-~ NAX bft

x
in

61

wegen

FAN[w.tY,Nt]

(2.3)

= g(w*tY)Ix

A bwx = b(wlx)x

= b(w*tY)X

gilt FAN[w,t]

^ Y(gwlt)-~ t -~ A x < N t

FAN[w,t] ~ A x < N t Aus d i e s e n b e i d e n

Formeln

k a t i o n iiberlegt man sich Das folgende

Ay
Y(g(w*tY)IX)- ~ x ,

(w*tY)X < b ( w * t Y ) X die Behauptung.

.

Die u m g e k e h r t e

Impli-

selbst.

Lemma iibertr~gt

Kreisels

Gedanken

[9], den Howard

62

J. Diller,

H. Vogel

in [ 5], Lemma 3C auf einzelne F~den im Baum

6~ anwendet,

auf Fans in

diesem Baum. Lemma I, A b s c h ~ t z u n ~ der Fan-Tiefe. tion 6 mit

(6) ist in

T^+ SimBRc

Zu

g,b,Y

wie in Defini-

ein Term L d e f i n i e r b a r mit

(7.1)

Vx~t Y(clx)<x - Lct = 0

(7.2)

A x ~ t Y(clx)_>x-~ Lct = ] + m a x { L ( c * t g w t ) ( N t ) I A x < N t

Ftir

to:= L0~

(7.3)

ist in

wx
T + SimBR~ h e r l e i t b a r

F A N [ w , t o] ~ Y ( g w l t o ) < t o"

Der y o n fe t

g,b

aufgez~hlte,

durch Y fundierte

Fan h a t

maximal die Tie-

o

Beweis. T^+ SimBR~

Nach

(4.1),

einen Term F m i t

Vx-
Fct(ct)

Ax-x -~ Fct(ct) L sei die V e r M i r z u n g

(4.2) und

(~3) aus [2] gibt

es in

(3) und = 0 = I +N/O(b(kw,x[F(clNt)(Nt)(gw(Sx))])(NNt).

yon F (vgl. D e f i n i t i o n

3). Dann gilt

(7.1) und

Ax-
wx
= ] +max{L(c.tg(wlNt)t)(Nt)IAx
(7.2) wegen

sionalit~t, weil hier nach l~enge betrachtet wird. Aus

(7.]) und

(6). Die letzte (5.2),

Ax
Ax
gwlt

(5.3)

8(Lct)

ftir c, so erh~lt man

to

mal, so ergibt sich wegen

Ax
~

einer endlichen

wx
Iteriert man diesen Schritt

Hieraus

,

gilt ohne Exten-

(5.3) nur das M a x i m u m

(7.2) felgt wegen

Setzt man hierin speziell

Gleichung

wx
durch

(7.2) die B e h a u p t u n g Der Term g,b

t o aus

(7.3)

to=LO~

.

9

(7.3) ist i.a. gr~2er als die Tie-

aufgez~hlten,

dutch Y f u n d i e r t e n Fans

Einen genauen A u s d r u c k ftir die Tiefe von

~o

erh~lt man wie folgt

F~. in

T^+ SimBR~ + BR ~ . Man berechnet

zuerst

ten Anfangsstticke aller F~den,

die in dem von g aufgez~thlten T e i l b a u m

von

B~ liegen,

indem man mittels

die L6_uge der durch Y u n g e s i c h e r -

SimBR~

einen Term L definiert

mit

J. Diller,

LoW

Vx~t

Y(clx)<x

Ax~t

Y(clx)>-x ~ Lwr

:= LwO~

dee Fadens

-~ Lwct

in

B~,

net man das M a x i m u m

= 1+Lw(c*tgwt)(Nt

dieser

L~ngen

Induktion

LoW

B o herausgeschnittenen

man mittels

BR o einen Term M mit -- Mwt

)

des durch Y u n g e s i c h e r t e n

wie m a n durch

durch b aus

Lo(wlt)
63

= 0

ist die L/nge gw

H. Vogel

, wobei

Fan

AnfangsstGeks

erkennt.

Dann b e r e c h -

die W a h l f o l g e n

Fo verlaufen.

w in dem

Dazu definiert

= 0

L o(wlt)_~t -. Mwt = I + m a x I M ( w * t Y ) ( N t ) l y < b w t } to:= M O ~ 1 7 6 sendere dung von kursive

regel

-

BR ~

und

Die in Satz

I aus

TA

herleitbar.

Schlusses

"von

folgende

Nx

a I und z F o l g e n

sind,

y nicht

in

Beweis. a:= alx[t~Nx ]

t~Nx setzt.

B

:=

Satz

~

die V e r w e n -

das p r i m i t i v - r e -

verallgemeinerte

Induktions-

der S u b s t i t u t i o n s r e g e l wir sie in f o l g e n d e r

in

T^

Fassung

T^ ist mit der S u b s t i t u t i o n s -

A ~ Ax[0~

bzw.

und x nicht

,

Variablen

entsprechender

Typen

in t auftritt.

die I n d u k t i o n s r i c h t u n g und

Aus den Pr~Lmissen folgt

tun, indem man in den s:= Slx [t~Nx]

,

dann

und ~

auf

Ax[O]

aus Lemma

(Nxst ~ Ax[t~Nx] ) .

folgende

und schlieSt

mit

Satz

I aus

[2]

.

I und diesem K o r o l l a r

I r verall6emeinerte

Substitutionsregel

In

x-
W i r folgern

erfiillt insbe-

herleitbar:

Axz[t~x,a])

und damit

Fall durch

f~r x substituiert

O-
B x[t]

121.

yon T e r m e n

M a n kehrt

Ay<s(x-
auf

I aus

s I ,A,t

stets

und

vermeidet

auf x".

, A y < s I Axz[Nx,a1~

wenn

Nun setzt m a n

verwenden

~

5.

[2] f o r m u l i e r t e und mit Hilfe

Regel direkt

Ax[t]

Pr~missen

sie in diesem

Unten

zu Satz

von

Beweis

aus D e f i n i t i o n

zul~ssig

Korollar regel

ersetzt MAX

ist in

Fan-Tiefe

Der oben a u s g e f G h r t e

Funktional

auch direkt eines

ist dann die genaue

(7.3).

Fan-Regel.

Regel

direkt

In

T^+ SimBR~

ist mit der

herleitbar:

Y(clx)<x + Oc[clx] ,

Ay<X(clx)XZDcx z[c*xU(clx)xyz,Nx,V(clx)xyz] wenn

Y,X,U,V

e,x,y,z tritt.

Terme

der Typen

passender o~,o,o,~

Typen nicht

sind,

Dox [o o ]

-" Dc[ClX]

in denen die V a r i a b l e n

auftreten,

und y in D nicht

auf-

,

J. Diller, Beweis.

Nach dem Satz G b e r

E3] d e f i n i e r e n

wir T e r m t u p e l

H. Vogel simultane

g,h

Wertverlaufsrekursion

aus

durch

gwt = U ( g w It)t (wt) (hwt) hwO = z hw(Nt)

= V(gwlt)t(wt)(hwt)

und d e f i n i e r e n

einen Term b durch

bwt = X(gwlt)t(hwt) Durch

Induktion

nach x folgt

ieren

in den P r ~ m i s s e n , , ]

(6) und

des Satzes

hwx = h ( w l x ) x

gw

Dcxz~

(8.1)

Y(gwlx)<x

(8.2)

Ay
S e t z e n wir in te

to,

(8.3)

die Indizes

fGr c und

wir in

(8.1)

fGr x speziell

FAN[w,to]

* DEgWlto,to,hwto]

Aus

folgt ~ (Ay
ist wegen

GberflGssig.

(8.4)

g,b,Y

in Lemma

I definier-

~ DEgwlx,x,hwx])

Also folgt hieraus

(8.1)

die V o r a u s s e t z u n g

und aus der zweiten

Y(gwlx)~x

Pr~misse

(8.2)

Ay
Wenden

wit auf

erhalten

woraus

~ DEgwlx,x,hwx]

(8.3) und

(8.4) das K o r o l l a r

sofort

~ D[gwlO,O,hwO]

I aus

[2] an,

so

DEoO~,O,z]

folgt.

, Damit

w 2. F u n k t i o n a l i n t e r p r e t a t i o n HA

bezeichnet

E131 "neutral" Sei ~

die Menge

eine Menge

der B a r - I n d u k t i o n

Heyting-Arithmetik

von

T^

(um Q u a n t o r e n

yon nicht

der B a r - F u n k t i o n a l e

ordnet

ist der Satz bewiesen.

hier wie in E2~ die intensionale,

genannte

eine E r w e i t e r u n g

setzung

zu Satz

wir FAN[w,O]

pen).

dann

~ D[gwlx,x,hwx]

der ersten Pr~misse

9 FAN[w,x]

dann

fGr z. Wenn folgt

~ DEgwlx,x,~x]

das zu

(7.3)

(6.1)

hwx

fortlassen,

~ D[gwlx,x,hwx]

so ergibt

Y(gwlx)~x

Hierin

e,x,z

. Wir s u b s t i t u -

jeder Formel

zu

endlicher V, 3

T^ g e h ~ r i g e n

von T r o e l s t r a

Typen.

aller

Funktionalen

aus w I). Die in E2~ angegebene Avon

HA W + ~

HA

eine Formel

ist

endlicher

Ty-

(z.B. ^-Uber-

A ^ ~ 3vvw~

zu;

65

J. Diller, H. Vogel darin sind

v,w

disjunkte Tupel von in A nicht frei auftretenden Va-

riablen, und ~ ist eine Formel aus In

HAw ist nach

~+ r

[2], Satz 2, das Formelschema

A~

~quiva-

lent zu der Vereinigung folgender Axiomschemata: ACo~

V x ~ y~ A -~ 3 Y ~ V x c A .[Yx]

fGr alle Typtupel

IP^

(VxA ~ 3yB) -~ 3y(VxA ~ B)

fGr A aus

~,~ ,

T^,

falls y nicht in A auftritt, und

M^

(VwA~B) ~ 3X,W(Ax<X Aw[WX] ~B)

falls

x,X,W

Die Theorie

nicht frei in HA w + A C +

A,B

I~+M^

fiir A,B

aus

Ta ,

auftreten.

wird wie in [2] mit

HA~

bezeichnet.

Definition 7. r sei eine Menge neuer Funktionale und T' eine einfache Erweiterung von

~ + r . Dann hei2t eine Formel A aus

b ezG~lich ^ interpretierbar in T', wenn es zu T^+ r

A^ ~ 3vVw~

HA w + r

Terme b aus

gibt, die nur in A frei auftretende Variablen enthalten,

~v[b]

so da2

in T' herleitbar ist. Sei wieder ~ ein Typtupel. Als

(8imultane) Bar-Induktion vom

T~p ~ verwenden wir folgendes Formelschema BI~

Hyp 1 ^ Fqvp 2 ^ Hyp 3 a Hyp 4 ~ Qc,x[O~

mit Hyp 1

Vc~

Hyp 2

VcVxAy<X(Pcx[Cly,y] ~ P c [ C t X ] )

Hyp 3

vcvx(Pc[ClX] ~ Qc[ClX])

Hyp 4

vcvx(VUQc,x[C.xU,Nx ] ~ Qc[ClX])

Pcx[Cly,y]

Als i n t e n s i o n a l e i n t u i t i o n i s t i s c h e das System

Analysis vom TFI~ ~ b e z e i c h n e n wir

HA+w+ BI~

Wir wollen in diesem Paragraphen die intensionale

intuitio-

nistische Analysis in Anlehnung an Howard [5], w 3, und [6], w 2, beziiglich ^ interpretieren. Satz 2A. Jede in in

Unser Ziel ist: HA++w BIo herleitbare Formel ist beziiglich

TA+ SimBR a interpretierbar. Da nach [2], Satz 3, Korollar I jede in

bare Formel bezGglich

^ in

HA++SimBR~

herleit-

TA+ SimBR o interpretierbar ist, folgt Satz

2A aus Satz 2B. Jedes Axiom

BI~ ist in

HA++w SimBR~ herleitbar.

Zum Beweis machen wir zwei Vorbemerkungen.

^

66

J. Diller, Lemma

ist in

2. E und F seien

H. Vogel

pr~nexe

reine

Allformeln

aus

HA w . Dann

HA w + AC + IP^ h e r l e i t b a r (3yE 9

3zF),~

Hiernach

3Zvy(E 9 Fz[Zy])

k~nnen

wir wie Howard

[5 ] die P r ~ m i s s e n

der B a r - I n -

dukt i on ~quival ent umf orm en. Lemma 3. Es sei Q ' = vz~

;

ist in

HA w

~,Q

aus

fGr

Hyp i (9.i)

Qcxy ['

, , ]

T ;

r,s,y,z

~

3Y,R,L,S,H

(9.i)

folgende

Formel

die Indizes

ist

Vc,x,rAx'<x(P'[ClX',x',r]

(9.3)

Vc,x,r(P'[clx,x,r]

(9.4)

Vc,x,w(Vu~Q'[C*xU,Nx,wu] Beweis.

ist

die ~ b r i g e n

Definition

Lemma (i=1,2,3,4).

4.

Hyp'

P' ,Q'

Es sei (9.1)

2 und

P~P^

,

Y(clx)

< x

Y(clx)<x

folgt

(3.2),

Damit

3

sei

der F o r m e l n

(9.i)

herleitbar

~ Q'[ojx,x,F'ex] (1.4) folgt

ClxlY(clx)

P'[clY(clx),Y(clx),R(clx)],

Behauptung

und s c h l i e S l i c h Dies

ergibt

aber nur u n t e r

=

daraus mit

wegen

(9.3)

(3.1)

die

der z u s ~ t z l i c h e n

.

Y(clx)_~x

. Aus

(3.3)

erh~lt

(9.4)

folgt

~ Q'[elx,x,H(F(olx)x)(clx)x]

man auch in diesem

,

Fall die zweite

Behaup-

ist das Lemma bewiesen.

Beweis 3 und setzen

F'

3).

. Wegen

daher

Vu~Q'[O*xU,Nx,F(elx)xu]

tung.

, und

~ Q'[clx,x,F'cx]

die zweite

Es sei nun

und mit

Q~Q^

mit Typen wie in Lemma F:= B ~ Y G H

die K o n j u n k t i o n

HA w + SimBR~

i=I

aus Lemma

Dann sei

bezeichne

ftir

Allformeln

Y,R,L,S,H

Definition

.

reine

Q'[clx,x,S(clx)x(L(clx)x(Y(clx))(R(clx)))]. Annahme

und in

pr~nexe

P'[clx,x,L(olx)x(Y(clx))(R(clx))]

erste Behauptung,

, , ]

VuQ'

Hyp' ~ VuQ'[ e . x U , N x , F ' ( O . x U ) ( N x ) ]

. Mit

P~xr[

, so da2 die ]~quivalenz

und

Aquivalenzen

Dann ist in

Beweis.

in

~ Q'[clx,x,Hwcx])

ist (Hyp ] )^ Da

(vgl.

Hyp' ~ Y ( c l x ) < x

(9.2)

Dann

-~ P'[c Ix,x,Lcxx'r])

8. Zu Termen

die Verkiirzung von F

mit

Variablentupel.

,

(wir lassen

G:= ~ c l x [ S c x ( L c x ( Y c ) ( R c ) ) ] .

ClY(clx)

mit

-- Q'[clx,x,Scxr])

3Y (9.1)

herleitbar

folgen

sei

~_=3yQ'

fort):

Vc P'[ clYc,Yc,Rc]

sind,

und

seien disjunkte

herleitbar

(9.2)

HA +

P' = V s~

i=1,2,3,4

(9.1)

in

mit

A

+

wobei

1~_= 3rP'

yon Satz 2B. Wir v e r w e n d e n D:= ~y[F'cx]

. Wir lassen

in

die B e z e i c h n u n g e n Dcxz[

, , ]

yon Lemma

wieder

die

67

J. Diller, H. Vogel Indizes fort. Die ^-Ubersetzung der Formel

vc,x(Y(clx)<x ~ Q'[elx,x,F'(clx)x] ) ist dann (10.1)

Vc,x,z(Y(clx)<x-

O[clx,x,z])

und die ^-Ubersetzung der Formel q'[ e Ix,x,F' (ClX)X]) ist (10.2) 3)~%-VVc~z(Ay~

,

vc,x(VuQ'[C*xU,Nx,F'(C*xU)(Nx)l

D[C*xU(Clx)xyz,Nx,V(clx)xyz] ~D[cLx,x,z]).

Nun ist HA w + SimBR~ eine Erweiterung von ~ + SimBR~, in der das Deduktionstheorem gilt. Also ist die in Satz I formulierte Fan-Regel in diesem System als Formelschema herleitbar, und das heiBt: (10.1)

^

(10.2)

~ Vz D [ O ~

H A + SimBR d herleitbar. Wegen Vz D[0,0,z] ~ Q'[0,0,F'00] ^ [0,0] folgt hieraus 3y Q ' [ O , 0 , y ] ~ qcx ist in

(10.1)

^

(10.2)

~ Q^[O,0]

Nach Lemma 4 folgt damn in

Hyp,~

Q^ [ o , o ]

und

.

H A ~ + SimBR o

,

also auch 3Y,R,L,S,H

Hyp' ~ Q^[0,O] .

Dann ist nach Lemma 3 in HA+~ + SimBR~ die Formel ist Satz 2B und auch Satz 2A bewiesen.

BI~ herleitbar. Damit

Wir ziehen einige Folgerungen aus diesem Ergebnis.

von

Korollar I. T + SimBRc.

HA~ + BI~ + SimBR~ ist eine konservative Erweiterung

Dies folgt aus Satz 2A, da alle Formeln aus ^-Ubersetzung Gbereinstimmen.

~ + SimBR~ mit ihrer

Korollar 2. Eine Gleichung a=b zwischen Funktionalen a,b ist genau dann in HA++w SimBR~ herleitbar, wenn a und b dieselbe Normalform im Sinne yon Tait [12] besitzen. Beweis. Nach Tait [12] besitzen alle Funktionale aus ~ + $imBR~ eine eindeutig bestimmte Normalform, die die in [ 2], w 3 unter 2) bis 4) aufgefGhrten Eigenschaften hat. Dann gelten die Ergebnisse von [2], w 3 auch fGr ~ + SimBR~. Insbesondere ist ([2], Satz 4, Korollar 4) eine Gleichung a=b zwischen Funktionalen a,b in ~ + SimBR a genau dann herleitbar, wenn a und b dieselbe Normalform besitzen. Korollar I ergibt dann die Behauptung.

in

Korollar 3. Ist eine Gleichung a=b zwischen Funktionalen HA+~+ BI~ herleitbar, so ist a=b bereits in ~ herleitbar.

a,b

68

J. Diller, Denn dann sind

dieselbe

Normalform

a,b

Funktionale

besitzen.

Also ist

w 3. Beweis von Spector's klassische

Typen,

. Dabei ist

also das System

das Fragment aufgebaut

yon

aus a=b

Resultat

~ , die nach Korollar in

fGr die. intensienale

Analysis,

angegeben

hat,

ven dem Specter

Ext

w 8. In Troelstra's

Zw die klassische

Zahlentheorie

HAw+{Av~A I A Formel yon HAm} ist die Extensionalit~tsregel

Terminologie

[13],

in [11 1

ist das System

1.6.12,

ist

endlicher

, und

Zw, dessen Formeln nur aus Gleichungen

sind.

2

T^ herleitbar.

Analysis

Das System der klassischen eine Funktionalinterpretation Z~ + Ext + A C o ~

H. Vogel

Z~w

ist

vom T y p o

G6 yon Specter Z~

Ext

[11]

das System

W

WE - H A % c

mit

Ext-R'

anstelle

Howard betrachtet

yon

Ext-R

in [5] zus~tzlich

. zu jedem Typtupel

9 das

Auswahlschema

(11)

Vx,X~SY~A ~ 3FVx A x y E F x , F ( N x ) ] ,

das als Spezialf~lle Auswahl seits in

DC

sowohl

enth~lt.

ACo~

als auch das Schema der abh~ngigen

Wie man sich leicht Gberlegt,

HA m aus folgendem

Schema

folgt

einer vom Wertverlauf

es seinerabh~Lugigen

Auswahl: CDC

Vx,xO~sy ~ Ax[XIx ] ~ 3F~

In Systemen,

in denen man endliche

ein 0bjekt vom Typ 9 kodieren CDC

Axy[FIx,Fx ] Folgen von Objekten vom Typ 9 in

kann,

z.B.

in

HA m + Ext

, ist

(11) zu

~quivalent. Ffir jede Formel A yon

die Formel,

HA m sei ferner die negative

die man aus A erh~lt,

wenn man vet jeder Gleichung von

einem Typ ~ o und vor jedem Existenzquantor eine doppelte Negation

.~

Version A-

einfGgt.

3 (also auch vor jedem v)

Das System

Z+

sei dann das System

W

Z w + { A ~ A -^}

.

Howard hat in [5], w 4, die Spector'sche tion auf das System Interpretation

Z + ~ Ext + (11) ausgedehnt. w bezfiglich ^ fGr das volle System

sienalen klassischen

Analysis

durch.

Satz 3A , Interpretationssatz. Ist A in

ZW+ + CDC

herleitbar,

Wir fGhren hier die Z+ + CDC W

einer inten-

Unser Ziel ist also Fiir jede Formel A v o n

so ist

interpretierbar. Wegen Satz 2A folgt

Funktionalinterpreta-

Satz 3A aus

A-

bezGglich

^ in

HA m gilt: ~ + SimBR

J. Diller, Satz

3B.

Ist A in

H. Vogel

Z + + CDC

69

herleitbar,

so ist

A-

in

HA + + BI

herleitbar. Diesen Lemma

Satz b e w e i s e n

wit zun~chst

5. Fi~r jede Formel

Avon

fi~r die Zusatzaxiome

HA

ist

( A ~ A-A) -

zu

in

Z . W

HA +

W

W

herleitbar. Beweis ~A-~A-

(vgl.

[5],

herleitbar,

Lemma

4A).

In

HA + w

sind

A - ~ A -^

und

also auch

--^

A Nun ist Formel

~ ~A A -^ eine Formel

3vVwB-

~3Vl...~3VnVWB

~3vVwB-

, also

zu

--

~A

mit B aus

, und diese

-^.

Insgesamt

T^. Dann ist

ist in

A -A-

HA W ~quivalent

erhalten

wir

A - ~ A -A-

die

zu , und alas

ist die Behauptung. Wir wenden uns Lemma

jetzt

6. Fiir jede Formel

~(VF~

[7], p. 352,

Howard

Theorem

(11.1)

V F3y~Ax
(11.2)

Vx , X . . 3 Y Ax[XIx ]

aus denen

fort und kiirzen

Dann folgt

HA W + BI~ h e r l e i t b a r ~ Ax[XIx])

[5], Appendix,

(ii),

Theorem

indem beide

. I, und Howard-

Annahmen

und

,

abgeleitet

des Lemmas.

wird,

Ax
Q =~B

Wir lassen

A[x,elx,cx]

gegen~ber

den V o r l a g e n

ab-

in

AxxY[

, , ]

die Indizes

mit B ab. Es sei

.

(Hyp I) u n m i t t e l b a r

(Hyp 3) sind fiir diese P und Zu

zu.

sind.

Beweis

P=

A ist in

Axy[F Ix,Fx ]

ein W i d e r s p r u c h

ges chw~cht

CDC

AxyIFIx,Fx] ^ V x , X ~

Dies versch~rft Kreisel

den Formeln

(Hyp 4):

aus der Annahme

(11.1).

(Hyp 2) und

Q trivial.

v u Q[C*yu,Ny]

ist wegen

(2.1),

(2.2)

~quivalent

ZU

Vu-(B ^ A [ y , c l y , u ] ) und dies

,

ist p r ~ d i k a t e n l o g i s c h

gleichwertig

mit

-(B ^ 3u A [ y , o l y , u ] ) Dis Annahme ( 1 ] . 2 ) ~3u

ergibt

A[y,ely,u]

Aus den b e i d e n

letzten

also

. Mit

Q[cly,y]

. Formeln

BI~

folgt

folgt nun

mit p o s i t i v e r Q[O~

Aussagenlogik

, im W i d e r s p r u c h

~B zu

,

70

J. Diller,

Ax
. Damit Lemma

ist Lemma

H. Vogel

6 bewiesen.

7. Ffir jede q u a n t o r e n f r e i e

Formel

B ist in

HA w + M ^

(vgl.

w 2) herleitbar: ~Vy~

B

Beweis. wandelte

~

3y~Ax
[x] Y auf die Pr~misse

Angewandt

Markov-Prinzip 3Y,W,X ~ A x < X

Hieraus

folgt

Vw B

~Vy~

ergibt

das abge-

~^

B[Yx,Wx]

.

pr~dikatenlogisch

3Y,X ~Ax<X

Ffir

Vw B [Yx] . Y y:= m a x I Y x l x < X l + I folgt hieraus 3y ~ A x < y

V w By[X]

,

und das ist die Behauptung. Mit

Lemma

6 und 7 b e w e i s e n

Satz 4. Ffir jede Formel Vx,X ~3Y Beweis.

(12) Dann

Ax[XIx ]

A aus

~3F

HA w ist in

HA++BI

herleitbar"

Vx Axy[FIx,Fx]

Es sei

~[x,X,Y,v,w]

A ^ = 3vvw ist

~

wit

Vx,X~.3Y

Ax[XIx ]

Vx,X..3Y,v

in

HA +

~quivalent

Vw ~[x,Xlx,Y,v,w]

Setzt man in Lemma 6

vwA

zu

.

ffir A und das Tupel

Y,v

ffir Y, so erh~lt

man hieraus

Hieraus

~VF,V

3y~(Ax
Vw ~ [ x , F 1 x , F x , V x , w ]

folgt

mit Lemma

7

~VF,V ~vy,w woraus

~[y,Fly,Fy,Vy,w]

sich p r ~ d i k a t e n l o g i s c h ~3F,V

und w e l t e r

ergibt

Vy,w ~[y,Fly,Fy,Vy,w]

wegen

~3FVy

,

(12)

A[y,Fly,Fy] ^ ,

so dab in

HA + ~3Fvy A[y,Fly,Fy] w ist Satz 4 ohne Rfickgriff auf

folgt,

Damit

Z +W

Der Satz 3B folgt und Regeln

von

Z

gehen

und das ist die Behauptung.

bewiesen.

nun durch H e r l e i t u n g s i n d u k t i o n . durch

die n e g a t i v e

Version

Die Axiome

in h e r l e i t b a r e

W

Formeln

und direkt

herleitbare

Regeln

von

HA w fiber. Wir zeigen

dies

J. Diller,

nur fur G l e i e h h e i t s a x i o m e ist

(a=b)-:= ~ a = b

Daraus

C ~a ~ =b ~ ~

, und aus

~ F-[a] - ~ F - [ b ]

ergibt

sich aber w e g e n

Die n e g a t i v e b a r e n Formeln, Damit

Version

der Z u s a t z a x i o m e CDC~

bei d e n e n A n e g a t i v e

D i e s e r Beweis

~0

folgt

die B e h a u p t u n g A - - A -^

C-

ist nach

sind die n a c h Satz

Version

. Dann zun~chst

einer F o r m e l

A--~

auch den Beweis

die F o r m e l n

Aussage

Lem-

4 herleit-

ist.

In Satz wird.

2A auch Satz 3A vollst~Lndig bewiesen.

vereinfacht

Lemma 7 e n t s p r e c h e n d e

mit

als zum Beweis von Satz 3B g e b r a u c h t

ist Satz 3B und w e g e n Satz

rem 4B. Wenn man statt

F[a] ~ F[b]

F-[b] -- ~ F - [ b ]

und die F o r m e l n

4 wird also m e h r b e w i e s e n

71

einem G l e i c h h e i t s a x i o m

~a=b

ma 5 h e r l e i t b a r ,

H. Vogel

trivial,

yon Howard

A--A D

und statt

[5],

verwendet,

Theo-

wird die

L e m m a 6 beweist

man

wie h i e r ~(VF3x ~Axy[Fx,F(Nx)] Erg~nzend rem

(v),

5. In

GegenstGck

In

Vx,X~

c(~Q[XIx,x]

A

yon Satz

Zm + CDCo ist

Beweis.

S e t z e n wir wegen

erw~hnen wir das H o w a r d - K r e i s e l

entsprechende Satz

^ Vx,X~3YA)

~

folgt

:= ~Q[X,x]

[7], P. 352, T h e o -

4.

BIr h e r l e i t b a r .

aus

(Hyp 4)

~ ~ Q [ X * x Y , N x ])

~ ~ Q [ X * x Y , N x ] , so folgt mit

CDC~ h i e r a u s

(2.5)

3FVx(~Q[FIx,x] Mit vollst~ndiger

Induktion

-Q[0~ und aus

~ ~Q[FINx,Nx]) erh~lt

man

~ 3FVx - Q [ F I x , x ]

(Hyp 3)

somit

-Q[0~

, ~ ~(Hyp 1)

. Dies

ergibt

die

Behaup-

tung. Dieser Objekten

Beweis ist

vom T y p c

in

S c h l u 2 y o n DCc a u f rung abh~ngt. Hiermit oder neutrales, System formuliert,

ein

unabh~ngig 0bjekt

BI~ i n

ist

[7]

auch fur

jedenfalls das sich

Das S c h e m a

wahlschema,

CDC

vom T y p c

ob man e n d l i c h e kodieren

anscheinend

die

klassische

kann,

yon einer

Analysis

fiber die negative Analysis

erweist

das der B a r - I n d u k t i o n

Version

w~hrend der Kodie-

ein intensionales unabh~ngiges besonders

mit B a r - l n d u k t i o n

sich dabei

Folgen yon

solchen

yon Extensionalit~ts-Gesetzen

fach in der i n t u i t i o n i s t i s c h e n ren l ~ t .

davon,

als das k l a s s i s c h e

am u n m i t t e l b a r s t e n

ein-

interpretie-

entspricht.

Aus-

72

J. Diller,

H. Vogel

Literatur L1] ]

Barendregt,H.: Pairing without f~ltigt, Utrecht 1973.

conventional

restraints.

Verviel-

Diller,J., Nahm,W.: Eine Variante zur Dialectica-lnterpretation der Heyting-Arithmetik endlicher Typen. Arch. math. Logik 16 (1974), 49-66.

[3]

Diller,J., SchGtte,K.: Simultane Rekursionen in der Theorie der Funktionale endlicher Typen. Arch. math. Logik 14 (1971), 69-74.

[4]

GSdel,K.: Uber eine bisher noch nicht benGtzte Erweiterung des finiten Standpunktes. Dialectica 12 (1958), 280-287.

[5]

Howard,W.A.: Functional interpretation of bar induction by bar recursion. Compositio Zath. 20 (1968), 107-124.

[6]

Howard,W.A.: Ordinal analysis of bar recursion vielf~ltigt, Chicago 1969.

[7]

Howard,W.A., Kreisel,G.: Transfinite induction and bar induction of types zero and one, and the role of continuity in intuitionistic analysis. J. Symb. Logic 31 (1966), 325-358.

[8]

Kreisel,G.: On the interpretation of non-finitist Symb. Logic 16 (1951), 241-267.

[9]

Kreisel,G.: Proof by transfinite induction and definition by transfinite recursion. J. Symb. Logic 24 (1959), 322-323.

of type zero. Ver-

proofs,

I.J.

[10] Luckhardt,H.: Extensional G~del functional interpretation. Springer Lecture Notes 306, Berlin-Heidelberg-New York 1973. [11] Spector,C.:Provably recursive functionals of analysis. Pure Math., vol.5, ANS, Providence 1962, 1-27

Proc.

Symp.

[12] Tait,W.W.: Normal form theorem for bar recursive functions of finite type. Prec. 2nd Scandinavian Logic Symposium, Amsterdam 1971, 353-367. [13] Troelstra,A.S.: Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes 344, BerlinHeidelberg-New York 1973.

NON-EXTENSIONAL TYPE-FREE THEORIES OF PARTIAL OPERATIONS AND CIASSIFICATIONS,

I.

Dedicated to Kurt Sch~tte on the occasion of his 65 th birthday

Solomon Feferman I-/

w

Introduction(t~oboth parts I, II).

w i a. General approach. The purpose here is to legitimize the use of some new~ simple, relatively unrestricted comprehension principles which among other things permit mathematically significant instances of 'self-application'.

Besides the familiar

formal-logical interest in the development of type-free principles off and on during this century~ there is the more recent and specific mathematical interest in theories of (global) structures of structures 2_/ as in categorical formulations of algebra and topology.

Thus one speaks of the category of all groups, or all topological spaces~

or even of all categories.

Such talk is currently accounted for in set theory only

by the use of mathematically unnatural and seemingly irrelevant hypotheses or distinctions.

Admittedly these serve all practical purposes, so any proposed substitute

should be at least as inclusive.

As an example of tests for adequacy~ one should

also be able to talk about the category of all functors between any two given categories (which is a structure of structure-preserving maps); the realization of this together with the previously mentioned global structures turns out to require the crucial step below.

Recursion theory and its recent generalizations provides a large body of experience with theories of operations permitting self-application. features of these are that they are

i) non-extensional and

Distinctive

2) partial.

A_~di)~

i_/ Research supported by NSF Grant GP~43901 X. A preliminary form of this paper was circulated in notes entitled "Investigative logic for theories of partial functions and relations~ I and Ii". 2_/ These have well-known isolated antecedents back to Burali-Forti and the wellordering of all ordinals.

74

S. F e f e r m a n

thismeans specific

we deal not with functions themselves but with presentations definitions

(or codes for such).

of them by

The word operation is used here for this

kind of representation 3 with the word function being used for the associated extensions.

A d 2), this is natural in recursion theory where we think of operations

as algorithms.

Any input

(argument)

can be processed according to a given algorithm,

but the procedure need not terminate with a value. F

may be extended to a total function

x ~ D m(F))

but

complicated.

F

by convention

One meets related situations

F*(x) = 0

for

mathematics.

in

may well be of interest).

set theory, functions

and classes are interchangeable,

ticular classes may be treated in terms of characteristic theory of partial operations

notion of partial classification, two prescribed values.

For example,

are of no interest and only complicate matters

(though extensions by other presentations

In extensional

throughout

is more

of functions by power series have natural partial domains of

artificial extensions

non-extensional)

(say

is no longer given by an algorithm and its description

analysis presentations definition;

F

Of course each partial function

functions.

Any

in par(possibly

induces in the same way a corresponding

i.e. a partial operation which takes on at most

We may think of this as a rule for membership and non-member-

ship but which supplied with any given argument does not necessarily terminate with an answer as to how that argument its extension

is to be classified.

is a class in the ordinary sense of the word.

cursion theories the induced total classification ditions(depending classifications classification.

on the theory),

Thus our present purposes

defined by essentially

The foregoing

In (generalized)

re-

from

b

to

if c

b, c

are total

is given by

dictate consideration

a total

of theories

of

which go beyond those induced by operations;

is the crucial step mentioned above. gous to that in constructive

is total~

usually satisfy good closure con-

but never the condition:

then the set of all operations

partial and total classifications

are reasonable

When the procedure

this

In this respect, the situation here is analo-

mathematics where classifications

richer means than operations

('species')

may be

('constructions').

only serves as a guide for finding the kinds of theories which

candidates

for our purposes.

We do not start with recursion-theoretic

S. F e f e r m a n

75

structures or impose any conditions of constructivity.

However~ the systems

T'

which we end up studying turn out to have models of recursion-theoretic significance and to include constructive and semi-constructive theories formulated previously (in [ F ]).

By permitting them to be non-constructive these also serve as extensions

of any already accepted theory are obtained both secure such relative to

w lb.

T'

The main conservative extension results which and show their use to be a matter of convenience

T.

Plan of the work.

universe

T 9

M .

Suppose given a theory

We wish to expand the language

T

s

and a model

of

T

~

of

T

with

by notions of (partial)

operation and classification and relations of application appropriate to each. Consider the former: (fjx,y) ~ A write from

to

in this case. M.

Each

is an

M

to

s

M

Assuming

~

and

A

and ha~ value

~

y ; we

represents a partial function F=[(x,y) Ifx~y }

A(or ~)

is

s

if every

is representable in this way. ~ = al,...,a n

s

In other words~ if

from

M,

the following

f~,_a

an assignment to

to distinct pairs ~

in

M,

(~,a)

and put

where

A : we ~(x,y,~)

(f~,~ , x,y)s A

just

(This is analogous to the step to the Bernays theory of

The question of interest is, to what extent is it possible to arrange

(I) for formulas build

x

is infinite it is trivial to find an adequate relation

~ ~(x,y,~).

classes.)

is defined at

~ ~ Vx,y[fx-~y ~ ~(x,y,a_)].

simply assign distinct elements is an s

f

M 3 , with

(~,A):

Vx ~ < l y r

in case

of application is to be a subset of

with constants

should be satisfied in

(1)

f

We would say that

partial function in ~(x,y,~)

A

just in case the operation

fx~y M

the relation

~ of the expanded language

s

One way to do this is to

in (possibly transfinite) stages, as follows.

a given stage just in case

f

is of the form

f

a

We put

for some

has been established at an earlier stage that

~(x,y, a_) holds.

works satisfactorily if the reason for taking

f~, a x _~y

(f,x,y) ~, a

in

A

at

for which it

This procedure

at a given stage is never

78

S.

Feferman

violated at a later stage; that is insured if

(2)

~

A 1 ~ A 2 and (9~,AI) ~ ( x , y , ~ )

We can then obtain

A

so that

(~, A)

more, by a slight modification, A is a function

g

implies ( ~ , A 2 ) ~ ( x , y , a ) .

can be chosen in such a way that each

Even

f = f~,_a

a ; in this case we shall write

result

.3_/ The combination (i) - (3) is immediately formulated

ga I... a n

as a comprehension principle (~) for ~ . With

U

expressing the unicity condition

~ ~ the first main theorem (3.2 below) is that for any s

some elementary conditions, the theory of

~.

f : ~ ~ "(~y)r

for the

for

-~ , i.e. if

satisfies (l) for all monotonic

applied to the parameters

(3)

is monotonic in

T.

T

: T + ~ + (~)

This may be strengthened by replacing

T

satisfying

is a conservative extension

(C) by certain selection principles

(S). The induced theory of partial classifications in membership

(4)

(i) (ii)

Then

c

p

and non-membership

~

x p c ~-~ c x-~ 0

x~c~cx--1.

is called a total classification if

Vx~[r162

for any monotonic of

~o ' ~i )"

has the relations of

defined by:

Vx [xpcVx~e].

immediately derives a comprehension principle for

(5)

T

From

(~)

one

p (and ~), namely

~ Zc[Vx[~c ~ ~o(X)] A Vx [x~c ~ el(x)]}

~ o' ~ I

"

(We have here omitted dependence on the parameters

Such schemes as (5) were first introduced by Gilmore [ G ], who

established their consistency by defining

R ~R

in stages.

3_/ This 'internal' representation of abstraction is stronger than just incorporating formal terms k x-(~y)~(x,y,~).

a

S.

77

Feferman

There is the following relationship with the generalizations theory via inductive U + (C)

definitions

defined from

~

partial classification

[B1

conditions.

]) for

induced by

A

and every hypere!ementary

The converses

(~, A)

This generalizes

The details of statements

for

is a ~

is a

also hold under

a situation found by Nepeivoda HI

~

set over

of these statements

~ = (w, +,., O) , where we obtain the

respectively.

[M]: in the model

in the proof of 3.2 every inductive set over

total induced classification. suitable

due to Moschovakis

of recursion

[N](cf. Bloom

, and hyperarithmetic

and proofs of these relationships

sets will be

included in Part II of this paper. There is also a close connection of of total operations formulated by Chauvin adjoins a single cation of

f

to

is taken to be

9 ic.

'undefined' x

element

is taken to be

. y

cannot obtain closure under

in

~

and

~

conclude that

like

(5) with

T~

~, ~

replacement

principles

of

T~

and

as closure under

the latter

fx~- y,

otherwise

it

in a language

in

~ .

~,p

to formulate

s

~, ~).

where

~o' ~i

The main theorem

extension

of

T

induced in

a stronger

Here we make use are monotonic

(4.1) allows one to

and hence of

T 9

by inductive generation principles

and infinite direct products.

There are

(94c)

and by

(94d) 9 on total classifications,

Many of these operations

as well enjoying reasonable properties

serves as a basis for a new theory of structures

be explained in Part II of this paper.

T

-c, b U e, b x c, etc. we

we can carry out all the desired operations

apply to partial classifications T~

with a theory

and the result of appli-

in the special case of set theory

including exponentiation

Thus

~

f,x,y e M

in place of

is a conservative

strengthenings

T~

outside of when

T

Roughly speaking,

This brings us in 9 4

but no__~_tnecessarily

interesting

In

cb

of partial classifications

of principles

[ C ]o

While the total classifications

can be shown to satisfy such conditions

T~

for the simplest

*

Plan of the work (contd.)

theory

T

of structures~

(cf. 94b) 9 which will

A principal point is that one distinguishes

naturally between ~artial and total structures

of a given kind~ depending

on whether

78

S.

Feferman

the basic domains and relations are partial or total classifications,

resp.

In this way we may speak of the (partial) category of all total groups, or of all total categories, considered

etc.

In the case of set theory

T,

the categories ordinarily

(both 'small' and 'large') turn out to be total categories.

The theories

T

and

T~

are provably non-extensional for total operations

(3.6) and universal classifications

(4.7) resp.

given in the Appendix of a subtheory of total operations

T_

In this connection,

an example is

which satisfies extensionality for

(and indeed where any operation with non-empty domain is total).

It is of interest to compare this with Scott's model [Sc] of the K-calculus satisfying a related statement of extensionality.

The mathematical role of extensionality

will be touched on both in ~4b below and in Part II ; it is much less important than ordinarily thought.

In addition to what has been mentioned above, Part II will contain a reformulation of the principles S4+BF

(cf.[H,C]).

(C), etc. in (formally) modal terms, based on the system

The point is simply that for the special case of Kripke semantics

applied to the system of 'worlds' ~(~) = {(~, A)IA c- M 3 and (3, A) ~ U } , ordered by (~,AI)

< (~,A2)

({-~ c]~) in arbitrary

(6)

iff

~(~).

A I _CA 2 , monotonicity of

{

is a consequence of validity of

Then, for example, we get the following in place of (5) for

go ' ~i :

w~[r

In particular,

ACz(X)] .~c {~x[xpc ~O{o(X)] ^Vx[x~s ~ we have for any

~l(=)]).

~:

This includes as a special case a system shown t o be consistent by Fitch [Fi 2 ] . ~-/

4_/ In the notes refereed to in ftn. i, I started with such a modal formulation and derived the schemes (C) and (C~) presented below.

S. F e f e r m a n Remarks.

i)

Y9

One of the most developed type-free systems in the literature is that

of Sch~tte IS ] Ch. VIII~ following a system of Ackermann. middle is not valid in it.

For each formula

considered to determine a class if

~(x)

~(t) V ~ ( t )

The law

there is a term

of excluded ~ ~(x) ; ~

is provable for each

t.

is

Beyond

this similarity, it is not clear how the present work is related to that of [ S ].

2)

The approach taken here, giving conservative extensions of accepted

theories, is opposite to the Frege-Russell program of reducing all of mathematics to "logic" (in the form of general comprehension principles).

There is currently

no reasonable prospect for the success of such a program.

3)

Theories of partial classifications jibe with Russell's idea [ R ] that

each property inherently has a range o__ffsi~nificance within which it is true or false and outside of which it is nonsense.

This suggests that there should be some

basic theory of meaning which determines which formulas have sense and which are nonsense (not by a__ddho___~crestrictions as in the theory of types).

The present work

makes no contribution in this direction, though one might expect such a theory to account for some such principles as those given below.

4)

Remarks on connections with various logics of partial predicates, 3-valued

logic and modal logic will be made in Part II.

5)

Many of the features of the systems in w

seem to be anticipated in Fitch

[Fi l],which provides a complete and consistent combinatory system logical operators as elements.

However, the formalism and logic of

CA

including CA

as well as

its very definition are so different from the systems and their models given below as to defy

direct comparison.

80

w 2.

S.

Feferman

Preliminaries. We assume given any semantics

Qo,...,Qn,...

S,

specified by a countable sequence

of finitary propositional operations and quantifiers in the genera-

lized sense due to Lindstr~m [ L ]; E

is to include operations which serve to

generate those of the ist order predicate calculus (PC). a,b,c,...,x,y,z

Lower case latin letters

are used as formal v~riables and, where there is no ambiguity, as

elements of the domain of any interpretation; finite sequences are indicated by underlining, e.g.

~ = (al,...,a n)

or

a = a l , . . . , a n.

Constant symbols or

closed terms will generally be denoted by capital latin letters.

Let

s

be any countable sequence of basic symbols (functions, relations,

constants); this determines the notion of being an s with

$

determines the notions of

of satisfaction, ~

I=~ (~).

s

~ .

formula (for which we use

It is assumed that

s

given

s

s

s

, On ) .

with each

by (9, RI,...,Rn)

A formula (or

or any of its extensions

Ri

to be considered).

the

~(~)

R i ~_ R:~ for

Given an

s

01,...~0 n , the result

~ = (M,...) and relations

s

structure

of

s

..., On )

PC

to some

is said to be monotonic in

OI,...,On

if

i < i < n and

which are monotonic in ~'

Pi ' we denote

(M,..., RI,...,Rn)-

v

(~, R1,...,R n) k~(a) :) (ZR{,...,R n) k~(~_).

As is well known (by Lyndon's interpolation theorem) the formulas order

O!,...,p n

~

of the ist

are just those which are equivalent

in which all occurrences of atomic formulas

Pi(tl,...,tn ) 1

positive.

and

All notions such as

of the same number of argument places as

Ol,... , On - monotonic)

(1)

s

is expanded by adjoining new relation symbols

is denoted by RI,...,R n

~,~,8,...)

consistency, etc, are taken in the semantical sense relative to the

~ (for

If

together

includes the equality symbol =,

which is always to be interpreted as the identity relation. eonsequenc%

s

are

S. F e f e r m a n

We shall be concerned first

- s

- by a ternary

further expansion by two we shall also write accordingly

defined

(2)

(i)

symbol

relation

and the second

symbols

~(x,y,z) or

~,

and

(~,A).

every if

of expansions

q

s

of

s

-s

the - a

and

~ 9

In the first

for

s

; models will

All the models to be considered

case

are to

axiom:

to use a notion

inductively

s

of pseudo-term

= Y2 ]"

relative

to the language

as follows:

is a pseudo-term;

t I, t 2

~ = tl,...,t n

ar__~epseudo-terms

is a sequence

to the left,

(3)

then

tI t2

of pseudo-terms

i_~s a pseudo-term.

then

t I ... t n

is to be read by

i.e. we take

t I ... t n = :(tI ... tn_l)t n 9

In the same vein, throughout

to_t

is written

to abbreviate

Given a pseudo-term

(i) (ii)

e~uals~

(tlt 2 ~ y)

further

t~ t I, t 2 :

for

totl..,

b_~y definition~

t, the formula

(t ~ y) = : (t =y)

The following terms

with two kinds

V f ,x , Y l ' Y2 [fx ~ Y l A fx ~ Y 2 ~ Y l

s

(4)

for

(~, ~)

It is convenient

associating

binary

be denoted

(Unicity).

If

relation

xy~ z

satisfy the following

(ii)

particularly

81

when

t

tn"

As in (3) we use

and similarly

(t ~ y)

for

is explained

' ~ :'

inductively

by:

is an s

= : Zu I, u 2 [t I ~ u I A t 2 ~ u 2 A UlU 2 ~ y]

definitions

'= : '

and abbreviations

.

will be used frequently

for pseudo-

82

(5)

S.

(i)

(t~) = :~y(t-~y)

(ii) ~(t) : : ~ (iii)

(iv) (v)

Note that

Feferman

[t-~yA~(y.)]

(t I -~ t2) = : Vy[t l ~ y

<--~t2 ~- Y]

(tl m t 2) = :Vx[tlx~-t2x]. Tot(t) -- : Vx Z y (tx-~y) 9

(tI t2) $

implies

tI $ .

as a part of a larger formula, but the intent is always

In the case of the language

is considered

(ii) could be ambiguous when

s

clear.

all models

(~, A , Y,~ )

to be con-

s i d e r e d w i l l also satisfy the following axiom:

Vc Vx-~ [x~c Ax~c].

D ~ (Dis~ointness).

We assume throughout the following that a unary operation

' , and a binary operation

s

contains a constant symbol

(,).

By

Z~

0,

we mean the following

set of axioms:

(6)

(i) (ii) (iii)

Vx[x' / o] Vx Vy [x' =y' ~ x = y ]

Vxl' Yl' x2'Y2 [ (x~, x 2 ) = (Yl'Y2 ) ~ xl = Y! A x 2 = Y2 ]"

It will also be assumed throughout the following that which contains

Z

o

among its consequences.

T

is any set of

As will be seen, various results below

actually need only weaker hypotheses~ e.g. that all models of

The numerals i~2,3,.., For

(7>

n > 2 , n-tuples

s

T

are infinite.

are introduced by definition, i = :0 ' , 2 = :i' , etc.

(Xl,...,Xn)

are explained by successive pairing to the right,

(Xl,''',x n) = (xI, (x2 "'', x n >>"

88

S. F e f e r m a n

With each formula

of

cludes all the free variables

s

q,~)

of

and list

z

of variables which in-

we assume associated in some effective way a

I

numeral which is denoted by re, z_

z) 9(~

or

This is to be done in such a way

that

(8)

if"

We simply write

F~I

(~, z) { (~/,w_)

when the list

We put for each such

then

z

r

We shall also use

~,

(io)

E

and all

Fr

=

~/ r~(w)-I

.

has been indicated in advance.

F~ = : (1, r~).

(9)

Thus

F{(z_?

(o,o).

are pairwise distinct.

84

S. F e f e r m a n

w

Comprehension and choice principles for. partial operations and induced

classifications. w

Formulation of the principles.

The purpose of the closed terms

(i)

F@

In w

we consider mainly formulas i n ~ ( = )

is indicated by writing

lUl...lUnlX.~y~(x,y,u ~ .... Un) = : Fr

for each 'g'

@

all of whose free variables are in the list

x,y,ul,...,u n .

Here

is supposed to act like the Hilbert epsilon symbol; in most applications of

the schemes to be considered the choice of use the iota symbol and

X_ulx.Ey@(x,y,u)

(i)

'#'

in place of

or

y

will be unique, and then we may

'g'

We also write

~w_~x-~y@(x,y,w)

for

Fr .

@(x,y,u)

for

Then we put

lUi+l.. ,lUnlX.Sy@(x,y,u) = : (lw_lx.syr

i (for i ~ i < n)

(2) (ii)

sy@(x,y,u) = : (lwlx.~yr

.

Note that the r.h.s, of (2)(i),(ii) are pseudo-terms.

(3)

Finally, put

XUi+l...XUnXX't(x,u) -- : lUi+l...~UnXX-sy[t(x,u)=Y]

for each pseudo-term

t

and

l
.

The main schemes and axioms to be considered are the following. (C) (-~ - Coml~rehension scheme) in

~

For each

r

o f ~(--)

which is monotonic

:

Vu_ 3 f{~x.~y~ (x,y,u)--f ^ Vx~!yr (x,y,u)§ (S) (-- - Selection scheme)

Just as in

'~y'. (E) (Extensional representation) Vx{ Tot (x)+Ex+ Ax-_--ExAVy[x "__y+Ex~-Ey] .

(C)

] A Vx,y[ fx--y§162 (x,y,u) ]} except that

'~!y'

is replaced by

.

S.

We most usually apply e-monotonic

(C)

Feferman

85

in the following Slightly weaker form, for each

@(x,y,~):

(C)' Vu3f{kx.cyr

.

It follows from the instance V~(kx-Sy@(x,y,~)+)

.

(C)@

for

r

of

(C)

that

In particular, for each pseudo-term

t , V~(kx.t(x,~)+)

9

This reflects the informal idea (e.g. in ordinary recursion theory) that for each ,

kx.t(x,~)

is (or names) a rule, hence exists (as a syntactic object) though

for any individual

x , t(x,~)

We denote by stance

(C)@ for

(C-) ~

may not be defined.

the scheme obtained for each

by deleting the initial conjunct

is called the parameter-weakened form of dependence of

f

on the parameters

~

(C) .

~-monotonic @ from the in'Xx-sy@(x,y,~)~f'

The parameter-weakened forms

are obtained in the same way from

(C)' , (S), respectively.

principles;

(S)

and

(S_)

F@)

which is

This makes a significant difference in

the consequences of the schemata.

The SChemata

This

One loses thereby the explicit

by an operation (namely

itself one of the objects of the theory.

9

(C,)' and

(S-)

would more usually be called uniformization

'selection' was used so as to reserve

'U'

for the unicity property.

Notions of classification are induced from operations as explained in the introduction, namely

(~)

(i)

(x~c) : : (ex~O)

(ii)

(x$c) : :(cx~l)

(iii)

Thus any object

C~(c) = :VX[XpCVX~C]

c

.

has associated with it a partial classification by (i),(ii)

which is a full or total classification if Heuristically, if we regard

fx= y

c

is total (with values 0 or i).

as analogous to the relation

from o.r.t. (ordinary recursion theory) then the sets

{f}(x)= y

X c = {xlxpc}

correspond

to the semi-recursive (i.e. recursively enumerable) sets and those for which ~(~c)

holds correspond to the recursive sets. The comprehension and selection schemes for operations lead directly to

86

S. F e f e r m a n

eQrresponding 4(x,y,~)

schemes for these induced classifications

to be

[(r

y=0) V(r

y=l)]

simply by taking

for any given

r162

This leads us to write

(5) ~[r

r

=: Xx.~Y[r

y=0 V 4z(x,~)A y=l ]

For simplicity, we state here the parameter-weakened (C!) (P,~

- Comprehension

of ~ ( ~ )

,

scheme)

(SP)_ ( P ~

latter

r is

related

x~c]A to

the

and descriptive set theory. (Sp)

'X'[r

If

p,p

~ :

40 , 41

(cP):_

principles

of

The stronger parameter-dependent

9

reeursion

forms

theory

(Cp )

and

(Cp)_ , (Sp)_ , respectively, by adding as initial

41(X,u) ] = c' are considered simply as two binary relation symbols outside o f ~ ,

~(p,~)

by taking

40 'r

p,p .

The scheme

(C p)

schemes

(Cp)

to be formulas o f ~ ( p , $ ) in ~ ( p , p )

, where ~

The following is obvious (assuming

Z0

and

(SP)

to

which are monotonic in both

is simply in the ist order

with equality, was first introduced by Gilmore

(i)

9

[x~c§162

reduction

then we may restrict the parameter-weakened

Lemma

as in

[xpc +r

so-called

are simply obtained from

clause

@0(x,~), r

+Vx[(xpc*-~o(X,~))A (x~o~+r

- Selection scheme). For each

Vu~dVx{[r The

For each pair of formulas

both of which are monotonic in

~u ~o{Vx~[%(~,~)A r

forms first:

PC

[ G] s.

arid using

p,p

as defined i n ~ ( ~ ) )

3. l-

Dp

(dis~ointness)

is a consequence of

(ii)

(C p)

is a consequence of

U + (C)'

(iii)

(SP)

is a consequence of

U + (S) 9

U

(unieit~).

The same relationships hold for the parameter-weakened Given a theory

T

in ~ ( = )

forms.

(always assumed to contain

Z O)

we shall be

eoflcerne~ principally with the theory (6)

~~ =

: 9

+ ~+

(~)

9

STo be more precise, Gilmore's scheme was intermediate in strength between (CL) and (sP), also using abstracts {Xlr as terms to give some dependence on parameters.

.

S. F e f e r m a n We shall also be interested in designation

w

87

T + U + E + (S) , but do not introduce a special

for it.

The main consistency

and conservation

theorems.

Theorem 3.2. (i)

Fpr any model

~

of

Z0

there exists an expansion

mo~elof

~+

(ii)

is a conservative extension of

T

(~, A)

which is a

(~) . T .

Proof. (ii)

i~ an immediate corollary of (i), which we proceed to prove.

~-monotonic

@(x~y,~)

and assignment

a

to

~

in

M

Given any

we shall define

A

(o r=)

in such a way that I)

[email protected] i = (F},al,...,a i)

for any

i , i
.

Thus we put 2)

fr

: (F@,a I ..... an)

A sequence of relations write

fx~Dy

for

.

A -CM 3

(f,x,y)E%

.

is defined by recursion on ordinals This is not to presume that

%

D 9

We also

has the unicity

property; That will be proved later. 3)

(i)

fx~0Y~: x=a i

(ii)

for some r ,~=al,...,a n

and

y=(Fr

fx= + l y ~ :

and

i
we have

f=(F@~al,...,ai_ j ,

I .... ,ai) ,

fx= y

or for some

~-monotonic

r

and

~ ,

f=f@,~

and (~,%)~ r247 (iii)

Al = : UB<X%

Obviously 4)

A

C A

A = : A~

We write

fx~-y

for limit

whenever

where for

~

~ < ~

I . Put

is the least ordinal for which

(f,x,y)EA .

%

It is to be shown that

= %+i (~,A)

" is a model of

+ (~). First~ if fx} y .

fx~-y

either

or there exists

~ <~

Which of these cases holds is uniquely determined by

the first ease, if

fx=y I

pose

fx~y I .

(~A~)

fx=0Y

fx=~§

but

satisfies

then

@(x,z~)

Y = Yl "

By definition

with

fx~ +ly

but

f . Obviously in

Consider now the second case and supy

and similarly for

is the unique Yl

at

m .

z

such that

Thus if

~ =

88

S.

then

y = Yl

But in any case if

tonicity of

~ , so again

Now to show that and suppose that Consider any

~

x

in

of

r

solution

z

but then

fx=cy 0

with

which Remark.

then exist f@,K

fx=y .

~

@(x,y,~)

Let

(~,A)

9

If

be monotonic

fx~y

Let

Y0

f , we have (~,A)

fx+ . fx#0Y

f .

be the unique

then by 3)(ii) Hence

F~=

in

fx= +ly 0 ;

Now to complete

, so there exists

must satisfy

r

~ from

by monotonicity. [fx*~!yr

with v >~

Then

so U is established.

f = f@,a , thus 9

by mono-

does not necessarily

fx=~+ly with

we do have

(~,A)

~!yr

(~/A,A v) ~ > 2 Y r x

with

hold in the model.

9

9

At

But there may

In other words, the domain of

~!y@(x,y,~)

and the set of

x

with

.

Addendum.

Connections with models in the literature.

By 3.1(ii),

(C O )

holds in

is needed to form a model M

, let

~ @(xpYl, ~)

is symmetric

(~, A) ~ ! y r

fx~Dy .

lies between the set of

~yr

~ < V

~ .

By the form of

but

~ r

Note that

the least

for which

to

(~,A)

by 4), which is a contradiction.

fx=D+ly

(~,A)

(~,A)

is an assignment M

then

The case

holds in

in

the proof, suppose < ~

v < D

Y = Yl "

(C)

Feferman

(~,A) (~, R , R)

need only be assumed infinite.

elements and

a

C~o,~l, ~ in

M .

of

.

M

If we extract from the proof just what of

in~(0

, p) , it is seen that

Under this hypothesis,

corresponding

Then define

(C2)

R CM 2

to each

R 2 C~ _ M

choose pairwise distinct

0 ,o-monotonic

(or

p~ , $ )

0 , ~-monotonic

@i(x~)

@i(x,~)

by induction on

as follows: (i) (ii)

R0 = R0 = : empty x0~+le~=~:

x0 e

or for some

c = c r162 x~D+l~=~:

xO e

or

a and c

(M,~

,~) > r162

is of this form

C@o'@l'--a and

(M, R~ ,~) p ~l(X,U)A~0(x,u) (iii)

Rl = : U
and

RI = : U <~R

for limit

~ 9

and

~ ,

(i=O,l)

S. Feferman Then take

R = U R

for 3.2) that an infinite

, R = U ~

(M ,R , R)

set

It is easily seen (by specializing

is a model of

and the semantics

is that of ist order V

showed that for

sets.

(C ~) .

one studied by Nepemvoda

i the -~'~-sets

D O + (C~)

9

the argument

In the case that $~

is just

M , this is essentially the same as the model given by Gilmore

for his proof of the consistency of

essentially

89

c = 0,1,2,...

In the case that

~

PC with = , the model

= (~,+,-,',0) (~R

~ ~)

[ N ] and further by Bloom [BI].

the sets

X c = ~Ixpc} = {xl(x,c)ER} ~" CZ(c)

and that those which satisfy

As stated in the introduction,

is

They range over

are exactly the hy~erarithmetie

a generalization

given in Part II of this paper, relating the model

of these results will be

(~, A)

defined in the proof

of 3.2 and the induced sets with the notions of inductive and hy~erelementary over ~

in the sense of Moschovakis

Theorem 3.3.

(i)

For each model

which is a model of (ii)

x~_--y

3)'(ii)

.

fx= +ly~=~fx= y

(~,~) ~

3)'(i),(iii)

.

extension of

A = : A~

~ .

Write

The definition

or~z(fx=

z)

of

M .

Tot (x) 3) of

A sequence

for ~

from all

(M , ~ )

~ Tot(x)

of the proof of 3.2

and either

~-monotonic

r

and

~(x,y,~) while V~y(~,~)~ and Tot (x)

and 3 ~

~

and

r

{Tet~(y)

and

x~Zy

such that

ty (w)&x~w]& ~ h w [ T o t v (w) A w 4 y A x ~ w ] }

are the same as 3)(i),(iii)

where

T .

, which is thus distinct

respectively.

creasing and we again take 4)

(~, A)

(ii) as follows:

for some

(~) f = E ~y
there exists an expansion

be any well-ordering

(x~y)

in the clause

(#)f = fr

or

Let ~

(~,A)I=

is m o d i ~ e d o n l y

Z0

E = (0,0)

is defined by induction on for

of

of the proof of Theorem 3.2. i) and 2) are taken

We shall also use

F~al...a i , l < i < n

and

~

is a conservative

Proof. This is by a modification

AD(or~ ~)

[ M ].

U + E + (S) .

T + ~ + E + (S)

over unchanged.

sets

O

is least with

A

= A +I .

The sequence

A

is in-

90

S. F e f e r m a n

It is now showu that fx=cy

and

fx#Dy

.

Tot(x)

If

for

fx#0Y

To prove

(~,A)

U

in

(S)

3.2, but now using

For

above.

= W

if

fx~y

for

(fx~ +ly)

and

and

od(f,x,y)

fx~y I .

= ~

and

W<WI

If

fx=0Y

~

or

= ~i

is excluded

so

fx~0Yl, "

Then

~ = WI

"

(#), (@). in

(~A)

is similar to the proof for

We thus turn to the proof of

E .

(C)

in

First, we have

(x) ~ Ex+

If not, then by

(@)

so

Ex= y

6)

Tot(x) ~ x~Ex

For let

Ex=y

W <~

with

Tot

we can determine

and again

Ex~

in

~y

(x) . a

y

If 3 z ( E x ~

such that

z)

then certainly

Ex= +ly

.

Then

Ex%

.

Ex= +ly

.

(~, A)

by 5); then

from that we see

7)

fx=y

z) ; similarly

holds

(t)

choose the least

x~y

take

y = YI' by

The proof that

od(f,x,y)

Write

exists.

, suppose

-~3z(fx=

U + E + (S) .

We put

~ <~

Otherwise

is excluded by

Tot

(x) .

(~,A)

Now it is clear that

5)

Tot

then such

we argue as before. ~i < ~

is a model of

, so

.

~ = od(E,x,y)

exists.

Vu~[xu~_DvAyu~_Wv]

.

Thus

(@)

applies

at

~

and

It follows by monotonicity

that

9 Tot(x)

and

x~x I ~ Ex = Ex I .

To prove this, let WI~B

.

similarly

Then

Yl

if

B ~i

Ex=y

, od(E,x,y)

fulfills "

the conditions

This completes

In contrast with the proof quires the axiom of choice. Ist order restrict that.

= W , EXl-Y I

PC (or any other satisfying ourselves

to countable

od(E,Xl~Y I) = ~i

describing

y

, so

"

Suppose

Y = Yl

;

the proof.

of Theorem

However~

uniquely

and

3.2, the proof

if the semantics downward

structures

of Theorem

re-

given is just that of the

L~wenheim-Skolem

in 3.3(ii),

3.3(i)

so

AC

to

~ o ) we can

is dispensable

for

S.

w

91

Feferman

Some elementary consequences.

Lemma }.4.

The following ar__~econsequences

of

(Zo) ~

for each pseudo-term

t(x,!). (i) ~.t(x,~)~ (ii) (kz.t(z,!))x ~ t(x,!). This leads us to the following abbreviations

(i)

(i) CConstants)

K = : ka, x.a

(ii) (Substitution) (iii) (Identity)

(v) (composition)

(vii) (Successor)

S = : kf, g,x.fx(gx) I = : kx.x

(iv) (Application)

(vi) (Pairing)

for certain closed terms.

A = : kf,x.fx

c = : ~,g,x.f(gx) P = : ku,x.(u,x) Sc = : kx.x' 9

By Lemma 3.4~ these terms provably behave according to their respective definitions.

As is well known, every

up by application from the definitions

K,S

k~x.t(x,u)

is ~ to a closed term built

and the function symbols of

s

In that sense~

(iii)-(vii) are superfluous.

Next we specify operations requiring the more general form k~kx.sy~(x,y,~) in their definition. (2)

(i) (Projection) For

i= 1,2,

Pi = :~x'~Y[~Wl'W2[x: (wl'w~)AY = wi] w ~ w l ' w 2 [ x = (wl'w2)^ y = 0]) (ii)

(Definition b y cases)

D = : kal, a2,bl,b2 9 E Y [ a l = a 2 A y = b l ) V

(iii) (iv)

(Inverse) J = : k g , x . Ey(gy-~x)

(Domain)

Dm: : ~ , x . ~ y

(fx ~ ^ y : 0 )

(al~ a 2 A y = b 2 ) ]

92

S.

(v)

(Range)

(vi)

Feferman

R n = :kf,x. Ey(~u(fu ~ x ) A y = 0 )

(Existential quantification)

Q ~ = : kf.sy [~x (fx-~0)Ay= 0 V Vx ~z(fx-~zA z ~0) A y = l ] .

(vii)

(Universal quantification)

QV = : kf. By [Vx(fx-~0)Ay=0 V ~x~z ( f x ~ z Az ~ 0 ) A y=l]. Again it is clear how these behave in consequence of the principles however it is worth noting a few additional points.

First, the

(Zo) ~ ;

are defined

P. i

in such a way that

(Pix)$

for all

x; of course

Pi (Wl' W2 ) = wi "

Nex%

for

D j we shall also indicate its action by f (3)

Dala2 blb2

= ~b I

if

al = a 2

L

if

a I # a2 .

b2

This is always defined. Write

g-i

for

Jg ; thus

g-i

itself is always defined.

Here

(~)

leads

us to

(4>

Thus if

(i)

~:y(gy ~- x) ~ (g-lx)~

(ii>

g-lx ~ -~ g(g-lx> -- x.

Vx ~ < i y (gy-~x)

then

Under the stronger hypothesis

(i)'

~(~

g-i (S)

is an inverse to

~ x) ~g-lx p, p

and

CI in s

(i) ~p(omf) <~fx (ii) xp(~f) ~zu(fu~x).

Pat

(6)

in the standard sense.

we can strengthen (i) to

Finally using the definitions of

(~)

g

QI = : ~c,u. Q~x.o(x,u).

we derive:

S. F e f e r m a n

93

then the following can be derived from the given principles .

(7)

Cl(e) ~ Cl(QlC )AVu[up(Qlc ) < ~

In other words~ Q1

(x,u)pc].

gives ~ro~ection of (~ -induced)

relations.

The following is a form of the Reeursion Theorem.

Theorem 3.5.

in

We can find a closed term R

for which the followin~ ar___~e~rovable

(Zo) ~ . (i)

Vf(Rf ~)

(ii)

Proof.

Vf, y[(Rf)y ~ f(Rf) y].

Let

G=kf~x,y.f(xx)y.

Gfxy = f(xx)y.

By Lemma 3.4, for all

In particular,

(i),(ii) hold for

Gf(Gf)$

and

f,x, Gfx $

and

Gf(Gf)y = f(Gf(Gf))y.

Thus

R = kf.Gf(Gf).

The consequences

of

(Zo) ~

that we have obtained via the definitions

(i)

and (2) include the axioms for an enumerative system as defined by Friedman [ Fr ] (modifying axioms of Wagner and Strong). that these imply non-extensionality.

It was stated in [ Fr ] p.l17

A proof of this may be found in [ F ]

w 3.4 , but is repeated here for convenience and to note that it can be formulated more strongly as failure of extensionality for total objects. 6_/

Theorem 3.6.-~Vf,g[Tot(f) A f ~ Proof.

g ~f=g]

Assume to the contrary that

this to decide whether any given 0 6_/

=KO

(or

kx.O)

and

f

follows from

(Zo) ~ 9

Vf,g[Tot(f) A f ~ g ~ f = g ] .

We shall use

is total and then diagonalize.

H=kfXx.D00(fx)O.

7_/ Then for any

Let

f

Gilmore showed non-extensionality for his theory [ G ], but.in his counterexample the classifications were nowhere defined. Under

(~)

the simpler

H=kf,x.~y(fx

~ A y = 0)

would serve the same purpose.

94

S.

l)

(i)

(iv)

Tot(H),

(ii)

Feferman

Vx[Hfx i ~-~ fx $ ],

(iii)

Vx[fx i ~ H f x ~ O ]

and

Tot(f) <~ mot(m).

By hypothesis and (ii), (iii) we have

2)

Hf -~ 0

whenever

Tot(Hf), so

Tot(f) < ~ H f ~ o

Using definition by cases, take f

Nx ~ J l

3)

x = 0

L0

so that

Nx ~ x

for all

x. Using

D again with

F=kf.D(N(ff))O(Hf)O*

we get

4) otherwise. Then by i) we have 5)

(i)

and

Tot(F)

(ii>

Tot(f) ~ F f

~ N(ff).

Hence 6)

FF $

and

(FF) ~ N(FF)

gives us a contradiction. This proof makes essential use of operations defined by abstraction,

i.e.

with reference to our schemata by explicit dependence on parameters.

Question

.

over

?

Z

Remark.

o

Is extensionality

(for total objects) consistent with

Fitch [Fil] seems to provide an extensional enumerative system

To the extent that comparison is possible is no conflict with 3.6 because in

U + (~_)

(cf. Remark 5) of w

(CA).

it seems there

(it appears) definition by cases is not available

CA.

Addendum i.

Connections with a system for total o~erations.

The consequences

of

Z

obtained via the definitions o

(i)~ (2) above are here

95

S. Feferman

compared with a system of axioms introduced by Chauvin [ C ] pp.50-51. 8_/ The latter are formulated in the language of ist order PC with equality using as non-logical symbols a unary predicate symbol

O, two binary operation symbols

( , ) and [ , ] , and finally two constant symbols Chauvin's axioms by

0

and

TO; these fall into three groups indicated below.

interpretation of this language is given which fulfills TO III except III 5,6.

Namely,

*

w

is an object not in

in case

x ew,

arguments in while

(iii) ~,

(x,y)

A familiar

TO I, II and all of

(i) the range of the variables is

that acts as 'the undefined' , (ii)

and (iv)

[x,y] = *

i. ~/ I shall denote

~ U [*}

O(x)

where

holds just

acts as a pairing function when restricted to [x,y] = [x}(y)

in all other eases.

when

Here

x ~w

[x}(y)

and

y ~w

and

[x}(y)$,

is the application operation

of ordinary recursion theory. The groupings of axioms of and

lll(logical).

in particular

~

TO

are called I (P~eliminary), II (Combinatory)

Group I contains expected axioms cancerning x(-70(x)).

0

and pairing,

The nature of Group II should be clear.

In the

last group III, i-4 provide for propositional operations and the relation of equality.

III is (equivalent to) the following:

Vp{Vx,yE0(x) A0(y) ~ 0([p,(x,y)])] ~ ~qVx [(0(x) ~ [q, xl = 0v[q, xl = l) A ([q,x]

= o <~ ~ ( O ( y )

A [p,(x,y)]

Axiom III 6 is formed dually (under the same hypothesis). are not realized in the above interpretation. open in [ C ].

= o))]}.

Naturally, III 5,6

The consistency of

TO

is left

One method of establishing that is to use Kleene's generali-

zation of recursion theory to functionals of finite type in [ K ]~ and in particular recursion in the type 2 functional

2E

which gives existential

8_/ I learned of Chauvin's work after circulating the preliminary form of this paper referred to in ftn.l. ~/

These are used here instead of the constants truth-values.

'v' and 'f' in [ C ] for the

96

S.

quantification over [x}(2E,y)

when

~.

Feferman

Then instead of (iv) above one takes (iv)'

x,y c w

and

{x}(~,y)$

, and

[x,y] = *

otherwise, i_~

The work here permits one to prove the consistency of

TO

in another way~

namely by interpreting it in a certain conservative extension (Zo) ~ ~*

In general, for any

of ~

as follows.

~

and theory

~

The language of ~ *

symbol [ , ] and a constant

*

(1")

all axioms of

(2*)

(x = *) v (y~= * )

(3*)

w--z

(2*)

-~z(xy

Write

~

in

O(x)

for

((x,y)

of

s

, a binary function

(x ~ *).

The axioms o f ~ *

are

0,

= *),

<~(x/*)A(Y/*)A(z/*)A[x,Y]

= z

-- z) ~ [x,y] = *

Obviously every model of

~

is consistent if ~

It may be checked that

(Zo) ~_ .

(Zo) ~ _

form the extension

includes

relativized to

<~

s

[x,y] =

is.

can be transformed into a model of ~ * TO

, so ~ *

is interpretable in

For this in particular, we may use the consequence (7) above, giving

projection

QI ' to verify axiom

TO III ~

These two consistency proofs of of the model given for

TO

and thence its dual

III 6.

are closely related, by the connection

(Zo)~_ in 3.2 with Moschovakis' theory of inductive and

hyperelementary relations over

~ = (w,P,

',O) (w

Addendum above) and thence

with Kleene's recursion in 2E . It is likely that same strength as

TO.

analogous to that of

(Zo)_~ (considered in ist order Moreover, TO.

(Zo) ~

PC with =) is of the

should have a simple finite axiomatization

Thus in this case it is a matter of taste whether one

deals with partial or total operations. It is less elegant to deal with total , operations in theories T_~ , for in that case one must also relativize all axioms of

T

to

O.

When the semantics is stronger than that of ist order

PC (as will

From correspondence with Chauvin, I understand that he knew the consistency of TO could be established in such a way, but did not include it in [ C ] for philosophical reasons.

S.

Feferman

97

be considered in w 4c below), there does not seem to be any direct simple axio, matization for T The crucial difference between and axioms of ~

~*

and

~

only shows up when the logic

are restricted to be constructive.

In general, there will be

no constructive interpretation of ~ *

Addendum 2.

Some consequences here.

and obstacles for induced classifications.

We assume

(Zo) =

Recall the definition

(5) in w

of

~l(X,~)).

The following are some obvious examples to consider.

~(~o(X,~), (These abbrevi-

ations are temporary and will be superseded in w

(i)

(i) (ii) (iii) (iv)

(Universal classification) (Unordered ~air) (Complement) (Union)

V = : ~(x =x, x ~ x)

{bl, b 2] = : x [ X = h l V X = b 2, x ~ b l A x ~ b 2]

-c = : ~ [x p c , xpc]

Uc = : x[~z(xpz A zpc),

Vz(xpzV zpe)].

The r.h.s, of each of (i)-(iv) is a pseudo-term which is provably defined. Further consequences

are:

(2)

cl(v) A Vx(xpv)

(3)

c-i({bl,b2]) AVx[xp[bl, b2] <-~X=blVX=b2].

(4)

C-l(e) ~ 61(-c) ^ Vx[~p(-c) <~ -,(x~)]

(5) Note existence of a total classification the definition of schemes

(C_)

(bl, b 2]

or (CP)._

We may note also:

s

which behaves according to

is also derivable using just the parameter-weakened However, without extensionality we cannot obtain

ordered pair in the usual way from this. is taken as basic in

c

and

Z

O

That is one reason why ordered pair

98

S.

Feferman

, The followin~ are consequences

Lemma 3.7-

Z ~ + ~ + ~ + (~):

of

(i)

(Universal choice)

~f Vc [~y(y pc)-~ (fc)pc]

(ii)

(Extensional representation ) Vc[C~l(c) ~ C~I(Ec)Ac ~- Ec AVd(c -~d-~ Ec-~Ed)) . Now for the obstacle to working only with ~-- induced classifications 3 con-

sider the following

(6)

(temporary definition).

(f

:b ~ c )

= : Vx[~pb~Zy(fx ~ yAy pc)].

A fundamental operation of abstract mathematics (or

cb).

is that of exponentiat!on

b

c

In the present framework we would want to obtain such as the result of

an operation on

(7)

b~c

with at least the properties:

Cl(b) ^Cl(c) -~ cl(bc) ^Vf[fp(bc) ~ (f :b ~ c)].

But if this held then for

(8)

b=c=V

we would have

Cl(d) AVf[fpd

d=bc

satisfying

(f)].

~Tot

That (8) is impossible to obtain is seen from the following.

Lemma 3 . 8 . ~ d [ T o t ( d ) Proof.

AVf[df=O

Assume to the

diction

contrary

by diagonalization

namelyj take

d

for

H

*~Tot (f)])

that

there

holds

exists

such

in

(Zo) ~ . d.

We t h e n r e a c h

9 just as in the proof of Theorem 3.6 from 2) on ;

and

0

for

0

Another way of seeing the obstacle is that if we try to define form

f[~o(f,b,c),

choice for when

~i

~l(f,b,c)]

in which

Cl(b) A CI(c).

G

where

~o

is

(f :b ~ c),

b

e

in the

there is no obvious

is positive and which is equivalent to ~ ~o(f,b,c)

This obstacle would not be met if (partial) classifications

were not treated as defined in terms of new axioms in an extension of section.

a contra-

s

~

but rather introduced by appropriate

This is the procedure taken in the next

S.

~4. ~4a.

Feferman

99

An amplified theory of classifications. The new scheme and conservation theorem.

The language symbols

s (-~)

~ , ~

is expanded by the adjunction of two new binary relation

to form

are ~K-monotonic;

~-~,~

o Here we shall be interested in formulas which

these need satisfy no monotonicity conditions on

the latter are maintained on previous axioms. are analogous to the parameter-dependent

The basic new axioms to be taken

scheme

is convenient to make use of closed terms

-~ , though

(Cp)

G~0,~ 1

indicated in ~3a.

def~d

Here it

as follOwS for each

pair ~i(~,Ul,.O.,Un) of ~(--,~-for~i~ (i)

a~O,~l :: ~Ul'''"~un (2' r~2")l%Ul'""Un)

"

Their purpose is indicated by

(2)

~[r162

(~) s

= : Gr162~ .

(Amplified B,~-comprehension B,~

scheme).

formulas which are monotonic in

V

^ uac{X[r

For each pair B

and

~o(X,U),~l(x,u)

of

~ :

~- c A (~-[~0(x,u) A ~l(X,UD ]

~[z~c <--> ~o(Z,~] ^ ~ [ ~ c <--> r Remark.

The notion of partial operation and application is not expanded.

the mesming of the pseudo-terms A

X[~0,r ] --~ C in ~2.

or

for

-~+

terms of

G~0,~l u-- and thence of formulas such as

^

(x[~0,@l]) $ is explained in

~(-~) Just as initially specified

There would of course be an alternative broader possibility,

Join a new symbol Then -~ and

~+ ~,~ (~)

or

Thus

-~+ and introduce a parameter-dependent

could be defined in terms of above would be derived from

-~+ as (~+) .

p,~

namely to ad-

schema

(~+)

were defined in

In following such a

formulation one would want to be careful that the dependence on parameters is still given by operations in the sense of ever, though more powerful, in this way.

-~ and not in the broader sense of

-~+

How-

it seems that no mathematical purpose would be served

Our main interest here is in a theory of partial operationS and

100

S.

Feferman

classifications where the means of definition of the latter are essentially richer than the former, while at the same time the former can still be used to operate on the latter.

As we shall show this will be accomplished by the given scheme.

Suppose

~

is a theory in

~=)

~:

(3)

where

Dq

.

We write

:~+D~+(~),

is the disjointness axiom for

~,~

stated in ~2.

In particular, we

shall work with theories of the form

Theorem 4.1. (~i,A,Y,Y~ (ii)

If

Proof.

(i)

For any model

(~,A)

which is a model of ~

contains

of

(Zo) =

there exists an expansion

D ~ + (~q) .

(Zo) ~ , then

T~

is a conservative extension of

T .

This uses the same ideas as the proof of 3.2, specialized to the member-

ship relations as in the Addendum of ~3b. for each

~,~-monotonic

~i(X, Ul,...,Un)

l)

Cr162

With the

G~O,~ 1

defined above, take

and each assignment

: : ar162

~

to

~

in

M

,

hence

2)

c

a

= (2, r ~ j , C ~ l ~ ,al,... ,an )

and

3)

(~o'~r a_) ~ (~O,~l,b_)---->C~o,~}

2) and

3) are really all we need to know from

~ C~O,~ b-

(Zo) =

for this proof.

We now proceed to define a transfinite sequence of binary relations by recursion on

xnc

~)

orxn~

~

just as

R

for (x,o) ~Y

were defined in Remark 2 of ~3b.

Write

or x~ lc for (x,o) c Y

and x~c

(i) (ii)

, E

Y,

Y0 = Y0 = : empty . i

i

xT~+l e <==> : xq~ c c -- cr162

and

or for some

(~A,Y,7)

~

r

~0,~ 1

monotonic in ~ ~ ~ , an~ A "Vr

(~,a_)

.

a_ ,

S.

, 2x=:u<>j

Y~=U<~Y

(iii)

Fef erman

101

.

Then it is easily checked as before that

~,A,Y,Y~

is a model of

D G + (C ~)

when we put

5)

y=UY

Corollary 4.2.

If

T = T + ~ + E +

(Sp)

of w

3.7 (ii). and/or

Y=UY

is an ~-theory containing

, then

T~

Z0

and

T = T

is a conservative extension of

We could introduce an amplified

Remark 2. to

T

and

T .

~,~-selection scheme

and an extensional representation statement

from a model

(~A)

of

(Zo)~ .

(S)

Eq

However, there is no obvious way to build a model of

(S)

or

analogous

analogous to

Dq

with

E~

For, an attempt to carry out such

a proof in the manner of 3.3 requires the introduction of new operations (beyond those given by the application relation cesses defined in terms of

q

and

~

A ) in order to handle the selection proat any given stage

~ .

This point will be

of significance below. [] The following definitions will be used throughout the following: (5)

(i)

(b _~ c) = : V~[x~b ~ x~c] .

(ii)

(b-o)

(iii)

(f : b ~ c) = : Vx[xNb - ~ y ( f x

(iv)

C~(x) : : ~ ( ~ c

(v)

CL : : x[C~(x),x ~ x] .

Thus it is proved in

= : bCcAcCb

(Zo) ~

v ~c)

In particular,

CL

c

~- y A yNc)]

.

.

that for each

called a total classification; any sification.

.

x , x~CL <

> C~(x)

. Any

c~CL

is

whatever may be regarded as a partial clas-

is one such (which is not total).

102

S.

~4b.

Feferman

Some elementary consequences of the amplified schemata.

In this section we consider only consequences 1st order

PC

with equality.

two formulas monotonic.

~+ ~ ~-

~

g(-~,~ ~

These will be approximations

drawn in the classical

formula we shall associate

to

~

and

~

which are both ~,~-

~ ~ , respectively,

(A form of this procedure was introduced in

in the

[ G ] .) First

in prenex disjunctive normal form,

(i) where each

Qi

(2) where

(Z0) ~

with the same free variables as

sense explained below. put

Given any

of

is

>

~

or V

~ ~ < ~' = V

(3)

QlWl "'"

~ <

and each

Ql'Wl

>

and V' = ~ .

~ : :

Finally we put

~0 _ SlqS 2

I

~-Sl~S2

O'm~m iw(~ @ij

"'" Qm'Wm ~i(

~ij

and

~ik

k is atomic.

~j ~ @ij

v : ~ik)

For each atomic formula

@ , let

if

@

is an ~(--~) atomic formula

if

@

is

Sl~S 2

(where

sI , s2

if @

is

Sl~S 2

(

sI , s2

: : aaWl "'" %~mi

(ii)

j

"

Then

are s "

. "

) .

k

~- = : Ql'Wl ... Qm'Wm ~ ( i

W @iJ V W ~ i k ) . j k

The following is then trivial.

Lemma 4.3. (i) iables as

hold.

(/~) and

~-

and

~-

are both

9,~-monotonic and have the same free var-

(~-~~~)

are consequences of

are useful when also

(~ -* ~+)

D~ .

and (if possible)

( ~ -~ ~-)

Roughly speaking, this occurs under certain conditions on the variables

which insure that Sl~--s2 <

and

~ .

(li) ~+

~+

s2~CL

> ~ (Sl~S2)

Sl~--s2

In anticipation of such we write A

(~) for any formula

.

holds whenever we meet

^

§

_

~(x,u) = : x(~ (x,u),~ (x,u)) ~ .

in

~+

or

~- ; for then

S. F e f e r m a n

I0S

Note that no Russell paradox arises in the use of (5). For the case that is

-7(Xqx)

and

ns c o n s e q u e n c e s o f

c

is

~(x)

(Zo) ~

;

it

we have

only fono~

The s i m p l e s t c a s e where we can u s e

~(x,~)

that

c~c

~+ , ~-

Sl~S 2

appearing in

of occurrences of the

ui

Suppose

vu{

has

s2

not being excluded).

case to assume that each Lemma 4.4.

~

~nd

c~c

Vx(x~c < - > x ~

.

i n t h e way s u g g e s t e d i s t h a t

. By this is meant that every

is elementary relative to the ~arameter s

atomic formula

(i)

VX(X~C <----> X~X) and

equal to some

ui

(other kinds

Thus it is sufficient in this

ui~CL . In other words:

~(x,u)

is elementary.

uiNCL~Vx[~(x,E ) <

Then

-> ~+(x,~)] A Vx[~ ~(x,E) <

> ~-(x,~)]]

l
D ~ , and

A Vz[z~x^~(x,u_) <. > r

f~ u i ~ c = ~ ( x , ~ ) ~ c ~ i
is a consequence of

(Z0)~

The following definitions give some obvious examples of abstraction with respect to elementary formnlas (now superseding the definitions in ~3c, Remark 2):

(6)

(i)

(Universe)

V -- : ~(~ :, x)

(ii)

(E~pty)

(iii)

(Unordered pair)

(iv)

(~-cla~slflcations)

(v)

(Union)

(vl)

(Intersection)

(vii)

(Complement)

(viii)

(Product)

(ix)

(Projection)

A = : q(x ~ x) {al,a2] = : x(x = aI V x = a2)

av : : ~(~x -- 0)

b U e = : ~(~b

V x~c)

h n c : : ~(x~b A x~c) -c = : ~(x~c)

b X c = : x~Yl, Y2[X = (yl,Y2) A Yl~b A y2~c] ~c

= : ~y[(x,y)ue ]

104

S.

(x)

(Converse)

(xi)

(Exponentiation)

c = : x ~ Y l , Y2[X = (yl,Y2) A (y2,Yl)~C]

(We also occasionally write By Lemma 4.4, Further,

if

V,2~CL

CL .

bc = : x(x : b - ~ c) . cb

for

-c

and and

a , aI , a2 ,

{al, a2)~CL

b U c , b 0 c ~ -c , b X c , ~ i c

Moreover,

and

~ c~ , and

However,

(bc) , because in all those cases

p onentiation,

bc

all

even when it is not assumed that

c~CL , all of these follow their defining conditions for

Vx[xB(b N e) <

av~CL o

all of these behave according to their defining con-

ditions under the given hypotheses. b~CL

bc .)

and for any

b,c~CL , then

fall under

Feferman

> xqb A xqc]

~ = ~+ .

is a consequence

of

q , except

In other words, (no) j

e.g.,

In the case of ex-

we can also obtain

(7)

~t,c{b~CL ~ Vf[f~(bc) We now form extensions

<

> (f : b ~ c)]]

of the operations

U ; n ,•

.

~ and exp ; we also form A

a version

~

elementary

of the power class operation.

~ .

Nevertheless,

in each case, except

(8)

for

we shall obtain

(Extended tu}ion)

(ii)

(Extended intersection)

(iii)

(Disjoint s ~ ) = : Zcf

(Direct product)

Len~na 4.~.

(i)

(Power)

> ~+)

[_)x~c x =: ~.)c= : x ~ y ( x ~ y

~c(fX)

~

and

A yqc)

x = : ~-] c =: ~ V y ( y ~ c

-~ x~y)

: (x,y) A x~c A (fx -- z) A y~z]

~x~c(fX)

= : gVx[x~c

x~c

= :

-~ ~ y , z ( g x

~- y A fx -~ z A y~z)]

e c = : ~(x~CL A x c c) .

The following are consequences

of

cNCL A c __. CL -~ ((Jc)NCL A ((-]c)NCL

(ii) x~(Uc) <--> ~ y(x~y A y~c)

(Zo) ~

with non-

(~ ~ <

= :

= : w~x,y,z{w

= : ~cf

(v)

(~ <

x~(x,u)

~ , under simple suitable assumptions.

(i)

(iv)

All of these use

> ~-)

S.

Feferman

(iii)

c~CL~

(iv)

c~CL A (f : c ~ CL) ~ (Zcf)~CL A (-~-cf)hCL

(v)

w~(Zx~efX) <

(~i)

cNCL ~ V g [ g N ( ~ x N c f X ) < - - >

Vx[xNc ~ (gx)G(fx)]}

(vii)

x~(ec)

.

Vx[x~/-]c <

<

>~(y~c

c

we have

> x~CLAx~c

(~ c)hCL .

this "weakness"

of

~ c

~x~y)]

> ~ x,y(~ : (x,y) A x~c A y~(f~))

Remark. There is no obvious definition of on

105

@e so that under very general conditions

Since operations are treated independently of classes~

is not significant;

the mathematical role t h a t

classically is here taken over by exponentiation. We have now established counterparts

~

plays

[]

of all the basic set-theoretical

erations by means of partial operations under which

CL

op-

is closed, except for P

as Just noted and for passage to equivalence classes, as we shall now discuss. For partial classifications o_~n c , and write

(9)

Equiv(e,c)

c

and

e , we call

e

an equivalence relation

, if it satisfies

(i)

e c (c X c)

(ii)

Vx[xNc -) (x,x)~e]

(iii)

Vx, y,z[(x,y)~o A (y,z)nc~ (x,z)~e a (y,x)nc] .

We then define

(io)

(i)

[x] e : : ?[(x,y)~e]

(ii)

c/e = : ~ x ( x ~ e

Note that for a certain

Lemma 4.6.

G~O,~ I

A z = [X]e ) . we have

[X]e = z

The following are consequences of

(i)

e~CL~ [X]e~CL

(ii)

c~CL-~ (c/e)~CL

(iii)

yq[x] e <----> (x,y)~e

just in case

(Zo)j 9

G~O, ~l x ~ z .

106

S.

Feferman

(iv)

zq(c/e) <

> ~ x ( x ~ c A z = [X]e)

(v)

Equiv(c,e) -~ Vx, y{(x,y)~e <

> xqc A [X]e -- [y]e ] .

Thus every equivalence relation is reduced to the single relation co-extensiveness. however,

q

In fact, we can prove failure of extensionality

(even) for universal classifications,

Proof.

- 7 V c [~x (xqc) ~ c = V]

in analogy to Theorem 3.6.11)

is a consequence of

(Z0)~

Suppose, to the contrary, that V c{~x(xqe) -~c = V] .

H = kf 9 ~ y ( f x [Tot(f) < - - >

of

In classical (extensional) systemS this is just the identity;

is not extensional.

Theorem 4.7.

~

~ y) .

Hf = V] .

Then for all Replace

0

Hf~

f , by

V

and

Let

[Vx(xqHf) <

>Tot(f)]

so

in the proof of 3.6 above and then

proceed to a contradiction as before.

Addendum.

The Frege-Russell program revisited.

There is no question here of fol-

lowing this program in the sense of reducing all of mathematics to principles of logic.

Rather we wish to re-examine the idea for initiating this progr~un by

defining such notions as cardinal number and ordinal number in terms of the equivalence classes for the relations of 1-1 correspondence between classes, reslm~~ isomorphism ~

well-ordered structures.

the present context have just been discussed.

The general obstacles to this in We shall now sketch to what extent

the program survives. i-i (1) (b-e)=: ~ f ( f : b

(Recall that ~ x ( f x -~ y) .)

f-i

> c) = : ~ f [ f q ( b c ) A f-lq(Cb)AVx(xqb ~ f-l(f~:) _~ x)] . onto

was defined in ~Sb

in such a way that

In order to obtain an equivalence relation

evidently restrict

b

to lie in

f(f-ly) = Y E

whenever

from this we must

CL ; in that case it follows from

(b ~c)

c~CL .

(2)

ECD = : ~ b , e [ z

= (b,c) A h ~ C T

A ~

Ab

~ e] .

ll) This also adapts and strengthens a corresponding result in [F ], ~3.4.

that

S.

N o w we m a y easily prove in on

CL

.

(Zo) ~

107

thit

ECD

is an equivalence

CD is a partial classification

cardinal numbers.

9 (CL/EcD)

=

Two cardinal numbers

~ one u s i n g the operation

.

which is not total.

We can consider two versions ke

z , w

are thus the "same"

of Cantor's

9 c[0,1)

We may call its members the

t h e o r e m that

and the other

~

proved b y forms of the standard argument that there is no f : c -----> [0,i} c , nor is there onto i-i f(f : c > [0, i} c) and ~ f ( f of the classical statement : if

f

with

A ~B

where the classical argument breaks terms of m e m b e r s h i p

in

A , B

f : c

:c l - l > ~ c )

A , does not h o l d here ; iO, l} V c V

.

>~c onto

.

K < 2K

.

for cardinals

Thus it is easily

f

with

Furthermore,

is a counterexample.

f

of

B

onto

It is easy to see

since the required map is defined in

. cannot b e pushed much b e y o n d

this point since we lack the a x i o m of choice and the p o s s i b i l i t y in terms of membership.

ever familiar approach)

z -= w

.

, then there exists a map

down,

if

Note also that the counterpart

It seems that the classical theory of cardinals

erations

relation

This leads us to take

(3) CD

Feferman

of definirg

Even the theory of finite cardinals

runs into troubles.

do not f o r m a total classification

In particular,

op-

(from which-

the finite cardinals

and the induction scheme cannot be established

in its full generality. W h e n we turn to w e l l - f o u n d e d further difficulties.

(4) the

Wf(c,r) formula

removable included

to take

_tea

Vx[x~eAVy((y,x)~r

given here has

even if

structures

(c~r)

we meet

If we define

= : ~b~CtAb ~(c,r)

and w e l l - o r d e r e d

cNCL

and

'bNCL'

r~CL .

care of the other

-~ y~b) -~ x~b] - + c - b }

in a negative

On t h e o t h e r

positions

of

position

which is not

hand, the hypothesis

b

in

~ .

Thus

bNCL

~+(c,r)

is

will

,%

not be equivalent

to

~

and

~=

.~ w ~ c,r[w = (c,r) A e ~ C L A r ~ C L A W f ( c , r ) }

not b e h a v e provably according to its definition. holds we cannot in general carry out induction

on

Moreover, r

even if

Wf(e,r)

with respect to all

does

108

S.

properties formnlable in ciples.

~(=,~,~

Feferman

, because of the limited comprehension prin-

The following section gives a means of incorporating such principles into

a (consistent)

extension of

(Z0) ~

(and

W~

more generally).

This will include

the natural numbers as a special case.

~e.

Inductively generated classifications.

We should now like to see under what

conditions we may obtain the existence of a classification numbers. (i)

N

for the natural

A fully satisfactory definition should have as consequences:

(i)

N~CL

(ii)

O~N A Vx[x~N ~ x' ~N]

(iii)

~(0) A y x [ # ( x ) ~ ( x ' ) ]

-~ Vx~(x)

(possibly with parameters)

of

, for each formula

~(x) ,

~-~, ~

The consistency of (i) with the principles considered so far is no problem. simplest way to obtain this is to take 0,'

~

having their standard interpretation9

to be an then

~-structure with

V

The

M = ~

serves the purpose of

and

N .

It is also not difficult if we want to show even more that the adJunction of (i) maintains the eonservati~ extension property. a simple modification of the proof of Theorem 4.1. which is an

s

(2)

x~0N<=> : Yz_cM[o and

Then

~,~

theory containing

~o 1, < = >

:

-,

(x%~)

D R + (~)

Given any model

(Z0)_~ , designate

~ z~

N

as

(~,A)

(3,0)

of

T

and put

Vy(y ~ z---->y, c z) ---->x ~ z] ,

.

are defined as before for

is a model of

This may be accomplished by

~ > 0 , and the proof that

proceeds in the same way.

By

(~, A ~ Y ~ Y~

(2) , it is automatically

a model of (i). It is another matter, if we want to try to derive the existence of isfying (i) from the principles already accepted.

xVz[z~CL

A 0~z A

sat-

The obviouS choice would be to

try (3)

N

Vy(y~z ~ y'~z) ~ x~z]

S.

as the definition of rences of

z

the formula N

as

x~(x)

N 9 The clause

manageable. Wf

Feferman

'z~CL'

10g

is put in to make the other occur-

But now we have Just the same sort of difficulty as with

in (4) of ~4b Add. just preceding,

we do not see how to insure that

N

namely that for this choice of behaves according to its def-

inition.

Thus the possibility of deriving (1) by suitable definition of

hopeless,

at least in the 1st order

a consequence of

(Z0) ~

N

seemS

PC . However, we shall see below that (1) i s

in a semantics with stronger quantifiers.

We now turn to the corresponding questions for a more general class of inductive definitions.

Let

(3)

L : : kbkr 9 (3,b,r)

This is brought in for the following scheme; informally c

of

(IG)

b

inductively generated under the relation

Lbr

represents the part

r .

(Inductive generation) (i)

Vo, r (Lbr~)

(ii) V b , r,c{C~(b) A C~(r) A Lbr = c -* C~(c) A

W[x~b A Vy((y,x)~r ~ y~e) ~ xnc]] (iii) V b , r,c,u[C~(b) A C2(r) A Lbr -~ c

[Vx[x~b A V y ( ( y , x ) ~ r ~ ~(y)) -~ ~(x) ] -~ Vx(x~c -~ ~ ( x ) ] ]

of ~ ( % ~,~

.

Suppose

is an

theory which contains

If

(~l,A)

is any model of

of

~

for each formula

Theorem 4.8. (i)

(ii) Proof.

s

for

(~I,4Y, ~Y

+ (IG) .

T ~ + (IG)

or

(Z0)~_ .

~ , then we can find a model

is a conservative extension of

By modification of the proof of 4.1.

b#r . Y ~ ( ~

changed.

~(X,U)

~f)__ and

Vx[x~ e V x~ic]

%(~ 9

or

~' )

In

~ 9

(M~A) , Lbr --~ (3,b,r)

are defined by induction.

for any Write

4) (i), (iii) in the proof of 4.1 are left un-

4) (ii) is modified in such a way that we also arrange:

110

S. F e f e r m a n

4) (iii)'

xq~+l(Lbr ) ~ >

CI (Lbr) & xq~(Lbr)

or

x e ~ ] [ Z I Z ~M_ & Vz[zq b &Vy((y,x)q r = > y

CI (b)

and

e Z) = >

C~ (r)

and

z e Z]]

and x~#+l(Lbr ) ~

C~(Lbr)

& x~(Lbr)

or

C~(b)

and

C~(r)

and

x~+l(~br) The proof then proceeds as before to show that for (~,A,Y,Y~

satisfies

D q + (~q) .

In order to derive

(IG)

duce a generalized quantifier

where (5) If ulas

B CM

, R C M 2 9 and

includes

J

of structures.

For this (M;B,R~m)

m e M . _CM & Vz(z e B & Vy((y,z) e R ----~yeZ) ----~z e Z ~ .

@0(x,u) , @l(y,z,u) 9 we have a formula

finally any

These are of type

Q~ , and the symbolism is given by

that for any structure

,

is automatically satisfied by 4) (iii)'

in the sense of LindstrSm [ L ].

(M,B,R,m) e ~ <--> : m e ~ [ Z I Z ~

, Y = U Y

in an appropriate extended semantics, we introQ~

we need only specify a class

(IG)

Y = 0 Y

s

q~

, then for any form-

Q~x,y,z[@0(x,u),@l(y,z,u),w]

~ = (~,A,Y,Y-~ , any assignment

a

to

u

in

M

such

and

m e M 9

(6) ~ ~ QJx, y,z[@0(x,~),O1(y,z,~),m] < ~ > (M,[xl~ Theorem 4.9.

@0(x,~)],[(Y,Z)l~

If the semantics

$

@l(Y,Z,~)}, m) e ~ .

contains

Q$ , then

(IG)

is a consequence of

(z0)J The proof of this is straightforward. Remark.

Theorem 4.8 is a corollary of 4.9 and 4.1.

does not already contain

Q~ , simply adjoin

To see that the principles for

N

Q~

to

If

T

is given in

S

which

~ .

may be taken as a special case of

(IG),

put

(7)

~=

: LB~

where

~=

~[x=o

vSy(x=y')1

and

R-- : u H y ( u =

(y,y')) .

S.

Addendum.

Feferman

111

Connection with a previous system of operations and classifications.

In [ F ] I introduced a constructive theory

TO

of partial operations and total

classifications.

The language used was that of ist order

relation symbols

~ , ~ , and

easily seen that

of

TO

xhc

by

A second theory

using an additional constant

(1)

vf[(f:

~ ~)

Using the preceding latter

theory

TI

(Zo)~,

can

by defining

C~(c)

It is

as is done

The operation existence axioms

namely

kuZ~x- gy~(x,y,~)

with

with axiom

~ 1) A (Q(N)f ~ o

interpret

m1

in

< - - > ~ x(xGNA f x = O ) ) ]

(Zo)~ + (IG)+ (V ~ )

r e m a r k and w h a t we saw a t t h e b e g i n n i n g

can b e m o d e l l e d on

(x,0) .

of "Borelian" character was introduced in [F ]~

Q (N)

we

It was shown in [ F ]

of this

section~

.

the

~ .

how to use (IG) to develop notions of constructive

ordinal classes (or tree ordinals) N .

(ZOn) ~

~ (Q(N)f_ ~ o v ~(N)f

By taking Q~ for Q~(~)

x' was defined as

[C~(c) A x~c] .

require only a weak part of

existential.

~ j

is interpretable in

here and interpreting

( , ) , and constant

0 ~ K ~ S ~ D , P ~ P1 ' P2 ' Z , L , and

for each elementary TO

with equality~ basic

C~( ) , a binary pairing symbol

symbols which are here denoted C~ = k ~ 9 ~ ( x , ~ )

PC

01,02~...

Thus the principles studied here with

and of transfinite types

(IG)

N

over

added are prima facie rather

rich mathematically.

The portions of constructive and classical mathematics which

are accounted for by

TO

~4d.

and

TI

are mapped out in general terms in

The situation over set theory.

If we take a suitable system

T

[ F ], w167

of set theory

for grantedj the deduction of the natural numbers~ cardinalsj and ordinals is already taken care of.

Our only interest here would be to see to what extent good

properties of sets carry over to classifications in a theory of operations and classifications over 0 ~ ' , and pairing

T . ( ~ )

Let

s

have

=

and

c

as basic symbols.

are defined in the usual way.

We assume

The extension of useful

properties is facilitated by the following strengthened separation principle.

@

112

S.

(Sep)

(Strong separation).

Feferman

For each formula

VuYa~bVx[x~br

@(x,u)

>x

of

~(~,~

c aAr

:

o

This leads us to consider

+ (Sep) for

T

in the language

Theorem 4oi0. Proof.

If

T

s of

contains

First note that if

some limit ordinal

ZF .

ZF , then

S ~

ZF .

with

S

K , then the model

satisfy Sep.

Now consider any sentence

finite and

S~

sentences ZF S .

in

, and let

so that Thus

from

~

(RK, e )

~

these assertions

such that

T#~@

for

are

.

be the set of s

Let in

the existence of a limit

K

S .

for which

of the universe with respect to all the

Then by the preceding we can define

A , Y , and

~

in

can be proved to satisfy all the remaining sentences in

and hence

@

holds

(in the universe),

We can also formulate a strengthened ~(~, ~ , ~

K

Y , and

and

~,

is a model of

(~,A,Y,Y~

+ (~ep)

[]

in the obvious way.

is inaccessible

the conservative below.

Ss

of

~=

T .

provided by Theorems

Moreover, ~

extension of

i.e., we have deduced

T .

language

(zFc)~

U [@] .

(~,A,Y,Y~ ~

Remark 1.

when

ZF

is an elementary substructure S

T , Joe.,

~ = (~,A~Y,Y~

By the reflection principle we prove in = (RK, g )

is a conservative

~I is a standard model of

3.2 (i) and 4.1 (i) automatically provable in

~

~=

is consistent.

replacement principle

It follows from 3.2 (i) and 4.1 (i) that

(RK~e)

is a model of Tq

+ (Rep)

.

T , then for some

A

As a corollary,

~owsver, the addition of

extension property]

(Rep) in the

(Rep)

does not maintain

the reason for this will be seen in Remark 2

S.

Feferman

Lemna 4.1_I. The following are consequences of

u

(i)

Vx[xuc ~

x ~ b] .

(ii) ~ _ { V X [ ~ SxO(y,uD

113

Z~

.

B a Vx[x ~ a <--> xue]] ~(x,u)

-~ @(X, UD] -@ ~X@(X,U)] for each formtila

Both (i) and (ii) are direct consequences of (Sap).

o_~ ~ ( - - , ~ )

(i) expresses that any partial

classification which is bounded by a set is represented b y a set.

The converse is

A

immediate in the sense that if

~ = x(x 6 a) , then

(ii) is the principle of transfinite induction on if we take

aqCL e

and

~x[x6 a <~>

xq[] .

for the full language.

Thus

N = : ~ , then the principle (i) of S40, i.e., (IG) specialized to

is a consequence of

~

N ,

. More generally, insofar as we are only concerned with

inductively generated sets, the principle (IG) is dispensable. Just as well strengthen Theorem 4.10 to

However, one can

T # + (IG) , providing for inductively

generated classifications in general. Note that the range Thus

U

of

ka- ~ , i.e.,

ka. x(x e a) , is itself in

CL .

is a total classification which represents the universe of sets qu_~asets.

Remark 2. over

U

If we introduce the variables

'X' 'Y', 'Z' , ~

to range by convention

C~ , then all of Bernays' axioms for sets and classes [ B ] are derivable from , except for replacement.

That is, of course, derivable if we add (Rep).

we can prove the consistency of

ZF

in this system by familiar arguments.

But

First,

using class variables we can formulate the notions of satisfaction and truth in the universe of sets. true.

By (Rep) one can establish that all the axioms of

are

~gen with induction available in the extended language by 4.11 (ii), one

can show that every consequence of than

ZF

ZF

ZF

is true.

and hence than Bernays' system. 12)-

Thus

ZF~

+ (Rep)

is stronger

On the other hand, though

Z~

does

not give (Rep), it goes beyond Bernays' system in expressive powers by providing a reasonable theory of the relation

X~u

where

X

need not be a set.

12)Note, incidentally, that all of this holds equally well if we interpret the class variables to range over

C~

and work in

ZF

plus (Rep) restricted to

.

114

S.

Appendix.

Fef erman

A theory of extensional total operations.

Given

s

we denote by

(ES)

the following set of sentences in

(ES)

(i)

Vf'x'Yl'Y2[fx ~ Yl A fx ~ Y2 ~ Yl = Y2 ]

(ii)

Vf[ ~ x(fx$)

~ Tot(f) ]

(iii)

Vf, g[Tot(f)

A f

(iv)

For each

~g~ f

~(x,y,u)

~{V~y~(x,y,~)

s (=) .

= g]

which is ~-monotonic,

~ ~f Vx, y[fx = y <--> ~(x,y,~)]} .

This system is introduced by way of comparison with Theorem 3.6 and the question directly following it. Let

T

be any s

Theorem. (i)

For any infinite

s

~

we can find an expansion

(~,A)

which is a model of (ES). (ii)

If

T

has only infinite models, then

extension of Proof. in s

Let

~

We define

A~_C M 3

(f,x,y) ~ A 2)

(i)

A0

and

~

is a conservative

T .

be a well-ordering of

and assignment

T + (ES)

to

~

in

M . Associate with each s-monotonic ~(x,y,~) M

an element

by transfinite reeursion. Tot ~ (f)

for V x ~ y ( f x

Again

m~,~

in such a way taat

fx -~ y

is written for

: y) .

is empty.

(ii) fx -~+l y <--_.> : f~ - ~ and G)

(~,A)

or ~ ~z,w(fz ~S)

kV~:~(z,w,a)

and (+) ~ ~ v < ~ , h { ~ A v )

~,a,

f = f~,a

A ~(x,y,~)

bv~;~(z,w,k)

Vz,w[ (~Av) I:@(z,w,b) : > and (@) ~ ~g ~ f ~ , _ ~ g

and for so~

~d

(~UlAk) ~: ~(z,w,a) ]]

= f~,h and (~,A) k Vz ~ '~(z,w,h) A Vz,w[~(z,w,b_)

~(~,w,~)]}.

S.

(iii)

~

= ~J
for limit

Fef erman

118

k .

Let 3)

A = : A

We write

where

fx ~ y

Vx, y[fx ~ y < ~ >

A

= Aq+ 1 9

for

fx ~

and

Tot

for

Tot(f)

(f) , and

f

g

for

gx = y] .

(ES) (i),

(ii) are easily proved for

It may be noted even more

(~,A)

that 4) ~ x , y ( f x ~-y) ----> Tot (f) . We now consider only (iii), f = f~,a

and there is a unique

--7~z,w(fz --~w) and

and

od(g) = V

fx = y

and

(~Av)

f .

for some

gx -~ Yl

so

contradicted

Tot(f)

If

f ~ g

Then for some

fx ~- +l y

and

SuppoSe

.

for some

If

V < ~ .

Then

, then

~ = v and

is not possible.

~;,b

gx -~V+ly , then

This contradicts

gives a contradiction.

r

such that

g : f~,b

(~,Av)~-~(x,y,b)

(~,A)~(x,y,a)

f -< g

od(f)

we have

by hypothesis.

, and whenever

f ; similarly

.

which we denote

Note that if

Y = Yl

so

~ < V

for

V .

so

fx = +ly

Similarly,

~

Tot +l(f ) .

I= Vz~'.w~(z,w,b_)

gx=y

Suppose

(iv).

gx -~v+ly

2)(ii)(~)

so for

g-< f , then (~)

Hence

f = g

is

and

(ES) (iii) is established. For (ES) c

to

(iv), suppose

u , (~,A)~

@(x,y,u_)

Vx~:y@(x,y,c_)

.

is s-monotonic Let

~

and that for an assignment

be least such that for some

~,a

we have

5)

(=,A)FVz~:w~(z,w,a)

Among all

~,a

and V z , w { ( ~ L , A ) ~ ( z , w , a )

"

By 4) i f

V < ~ .

But then we would have 5) satisfied at

are met so that

~ .

~z,w(fz

~-w) , t h e n

Hence - 7 ~ z,w(fz ~-- w)

fx = +i y ~ >

fx -~ y <-=> fx - ~ + #

the proof.

(D~,A)l=@(z,w,c_)~

satisfying 5), choose that one with the <-least

f = f~,a

the choice of

:>

and

Tot~(f)

.

(~,A)~(x,y,a)

fx -~ y < = >

and v

f~,a "

od(f) = V

in place of

Let

f o r some

~ , contradicting

Now all the conditions .

.

(~), (~),

(@)

But then

(~,A)~ ~(~,y,c_> by 5).

This c o . f e t e s

S. Feferman

116

Some consequences

Addendum i. Assume

T

contains

and limitations

I , all constant functions,

closed under composition, themselves

Tot

represented

inverse,

in

holds of every

(x ~ c) = : (cx = O) . - , X , ~ l , and ~ and

~-definable function,

P ' P1 ' P2 ' Sc , and and

Q~

Q V (though

D .

Tot

Q~

QV

C , J,

C~

holds of Whenever

whenever

Tot(f)

semantics

includes the generalized

C~(c)

and

V

C~(c) = : Tot(c)

and every

C~(c)

and

[al, a 2} .

C~

quantifier

Q~

then

, then

It follows that there is no class

Addendum 2.

~(V)

subsets of Tot(x)

~ .

that ~ x ~ f ( f

C~

~ Op

x .

sented, Vf~x(fx

when

-~ x)

x -~ f)

xy -~ z

and

defined for

D and

x,y,z ~ ~ (~)

Hence there is no

n

weak instances of comprehension

=>

Op

.

of

such that

~ (~)

f = g] 9

of all operations.

and some arithmetical kx(x-l)

M = the set of all

such

Thus

Op

Besides the

operations

are repre-

The first recursion theorem

such that is avoided.

nO ~ 0 , nx -~ 0

for

It appears that only

are satisfied in Scott's model.

Thus the interest in it lies in a different direction from the kinds of theories studied here.

.

is a class with more than one

Vf, g[f,g c Op & f ~ g

x ~ 0 , and thus paradoxical diagonalization comparatively

~ c <--> C~(x)]

In [So ] Scott has defined a simple

system of representatives

0 , kx(x+l)

holds.

If the

does not exist as a class.

certain analogues of

including

b

There is a naturally defined subclass

and

C~(Uc)

is also closed

natural model of the k-calculus with domain

gives an extensional k-calculus,

c = bV

There is a relation

for each

, then

(IG).

Comparison with a system of Scott.

and mathematically

is closed under

C~(Zcf)

It can be proved with the Russell argument that ~ 3 c V x [ x

and that

are not

and

~x[x c c - ~ C ~ ( x ) ]

Vx[x ~ c - ~ C ~ ( f x ) ] ,

under inductive generation satisfying

element,

is

(ES)).

Next, in comparison with ~4 b), define

u,

These are only indicated.

Z0 .

First, in comparison with ~3 c), including

of (ES).

S. F e f e r m a n

117

Bibliography

[B]

P. Bernays, A system of axiomatic set theory, I., J. Symbolic Logic 2

(1937) 65-77. [B~]

S . L . Bloom, A note on the predicatively definable sets of N. N. Nepe~voda?

IBM Research Report RC 4829, #21499, May i, 1974. [C]

A. Chauvin, Theorie des objets et the~orie des ensembles, Th~se, Universite

de Clermont-Ferrand (1974). IF] Lo~ [Fil]

S. Feferman, A language and axioms for explicit mathematics, in (Proc. 1974 Summer Res. Inst., Monash) ed. J. N. Crossley~ to appear. F . B . Fitch, The system

CA

of combinatory logi c , J. Symbolic Logic 28

(1%3) 87~97. [Fi2]

F. B. Fitch, A consistent modal set theory, (abstract), J. Symbolic Logic

31 (1966) 701. [Fr]

H. Friedman, Axiomatic recursive function theory, in Lo ic Collo uium '6

eds. Gandy and Yates, North-Holland, Amsterdam (1971) 113-137. [G]

P.C.

Gilmore, The consistency of partial set theory without extensionality,

in Axiomatic Set T h e o ~

(1967 U.C.L.A. Symposium), Proc. Symposia in Pure Math.

XIII, Part II, ed. T. Jech, A.M.S., Providence, 1974, 147-i~3. [H,C]

G. E. Hughes and M. J. Cresswell, An Introduction to M

~

Methuen,

London (1968). [K]

S . C . Kleene, Recursive functionals and quantifiers of finite types, I.,

Trans. Amer. Math. Soc. 91 (1959) 1-32. ILl

P. LindstrS"m, First order logic and generalize d quantifiers, Theoria 32

(1966) i~-195. [M]

Y . N . MoschDvakis, Elementa

I dn uction on Abstract Structures, North-

Holland, Amsterdam (1974). IN]

N. N. Nepe~voda, A new notion of predicative truth and definability, (in

Russian), Mat. Zametki 13 (1973) 735-745. 13 (1973) 493-495.)

(English translation Mathematical Notes

118

[R]

S.

Feferman

B. Russell, Mathematical logic as based on the theory of types (1908),

reprinted in From F r e e

to GSdel

ed. J. van Heijenoort, Harvard University Press,

Cambridge (1967) 150-182. IS]

K. Sch~tte, ~eweistheorie

[Sc]

D. Scott, Data types as lattices (lecture notes, Kiel Summer School in

Logic~ 1972), to appear.

Springer, Berlin (1960) .

KOMBINATORISCHE KONSTRUKTIONEN MIT BEWEISEN UNO SCHNITTELIMINATION

K~rt SchQtte zur Gelegenheit seines 65ten Geburtstages

gewidmet

Walter Felscher

In dieser Arbeit wird untersucht, man an Beweisen vornimmt, Prinzip,

yon welcher Art die Manipulationen sind, die

um Schnitte zu eliminieren.

Oabei handelt es sich um, im

sehr einfache kombinatorische KonstruKtionen mit Beweisen, die zu diesem

ZweeKe als FunKtionen betrachtet werden, welehe auf 8~umen definisrt und deren Werte endliohe Mengen sind; die Elemente dieser Mengen sind abstraKte Terme oder Formeln.

Im ersten Tell der Arbeit wird daher mit solchen AusdrucKsmitteln allgemeine Begri%f des Probeweises

erKl~rt;

zun~chst der

die wegen der vorausgesetzten

abstrakten

Situation etwas langwlerigen technischen Oefinitionen sollten unmittelbar einsichtig sein,sobal~ man sie in den bekannten Sequenzenkalhulen deutet. S von Probeweisen,

Alsdann werden Mengen

die unter gewissen einfaohen KonstruKtionen abgesohlossen sind,

axiomatisch als Mengen yon Beweisen definiert. Eliminationstheorem bewiesen,

besagend,

FOr Mengen ~ yon Beweisen wird das

dass S unter Eliminationsverfahren

abgeschlos-

sen sei. Unmlttelbare Spezialf~lle davon sind die Eliminationss~tze der minimalen, intuitionistlschen

und KlassQschen Pr~diKatenlogiK einschliesslich allf~lliger infini-

t~rer aussagenlogischer Operationen. Eliminationslemmas

yon TAIT

88

Zum Beweis des Theorems wird ein Analogon des

bewiesen;

jedoch wird im Onterschied zu TAIT die

120

W.

Behandlung

yon @ u a n t o r e n

hen zurOcKgefOhrt, benBtigt

werden.

nicht

handelt

Von z e n t r a l e r

von denen

einmal

den A n w e n d u n g e n

als

yon E i n f O h r u n g s r e g e l n , formeln

Kleineren

dann

vonder

der Art A werden Pr~missen

fOhrung endet,

P

--

dutch des wurde.

Oabei

Gestalt

v und

Oie Pormeln v

positiven positiven

die reehte

Kopie v

Beweise

Grades

an die Stelle zerfallen

einer solehen

Hauptformeln

der andere

anderen

in zwei Arten,

Formel,

istj

von Neben-

so ist v

Heuptformeln

der Art B h~nnen mit m e h r e r e n

S~ , S I ,von

[vs) besagt:

denen

mit der E i n f O h r u n @

enth~lt

der eine mit der Ein-

von v

als Hauptformel

w von v einen

Beweis

von S 1 so, dass Sw

v* -EinfOhrung

der a b s c h l i e s s e n d e n

neben

ver allem die A n w e n d u n g e n

Grades

vonder

enthalten

w

denselben

Beweisbaum

Teilbeweis •

ist dabei w

sogar

ist w v o n d e r P

w

sie etwa im Fall

erlauben;

dass

geh~rende

gelegene

sehen gleish Gestalt

ist v o n d e r

Oer Beweis

entstehen,

Selte

formalen

des R e s u l t a t

man die deft zu v anderen

Pr~misse

mit denselben

von S I , w ~ h r e n d

der jeweils

F~llen

, suggeriert

der Art A , so gibt es zu j e d e r N e b e n f o r m e l

S w in S und eine P r ~ m i s s e

steht,

Pormulierung

Oie A b g e s c h l o s s e n h e i t s e i g e n s c h a f t

von v als Hauptformel,

Beweisb~ume

ist die Abge-

in z w e i f a c h e r

Formeln.

und der S c h n i t t r e g e l n

aus der Menge S zwei T e i l b e w e i s e

und ist v v o n

, pp.113-115

als rechte

mit n u t einer Pr@misse, werden.

deren

sind jeweils

Hauptformeln

treten. linKe,

88

einen Art A o d e r B , wenn

ersehlossen

ein Beweis

Regeln

bei denen

A und B ; ist v etwa die

ven Beweisen,

und OisjunKtio-

auch nut finite

for die E l i m i n a t i o n s v e r @ a h r e n

handeln,

linKe und einmal

Grades

KonjunKtionen

um des Folgende.

Beweise

struktureller

unendlicher

@uantorenlogiK

von S M U L L Y A N

es sioh im W e s e n t l i c h e m

gegeben,

genau

Bedeutung

(vsJ yon Mengen

Eliminationstheorem

Oie Formeln, v

auf d i e j e n i g e

sodass f o r die f i n i t ~ r e

schlossenheitseigenschaft abstraKte

Felscher

Kepie

w(y]

des M i n i m a l K a l k u l s die E i n f O h r u n g s r e g e l n ist v i e l m e h r

Nebenformel yon w .

hat wie der mit P

Sequenz

ersetzt

ist,

dureh w

besondere

deutlich,

der bei einer

solchen

F~llen

in der

y .

dass S e h w i e r i g K e i t e n ,

AufmerKsamKeit

die H i n z u f O g u n g

ent-

; in den q u a n t o r e n l e g i s e h e n

mit einer E i g e n v a r i a b l e n

macht

endende

, also die auf

In den a u s s a g e n l o g i s c h e n von v~

w

die aus Pw

mit einem Term t , und die N e b e n f o r m e l

des E l i m i n a t i o n s t h e o r e m s

Kritisch

yon Sw d i e j e o i g e

jener N e b e n f o r m e l

w[t)

Gestalt

Regelanwendungen

erfordern,

neuer Formeln Gelegenheit

wie

Kaum dedurch

els H a u p t f o r m e l n

eintretende

Wegfall

W. F e l s c h e r

von Nebenformeln,

aiso die Verklelnerung,

Im zweiten Tell der Arbeit wird gezeigt, gelten, wenn neben den aufbauenden

Pr

nicht die Vergresserung des Beweisergebnisses. dass Ellminationstheoreme

auch dann noeh

logisehen Regeln auch noch abbeuende Regeln vom

Peirce-Typ auftreten, wie sie yon CURRY in dem man v o n d e r

121

63

M, a~b ~-~ a

Umst~nden ist des Eliminationsverfahren

for den Spezialfall betrachtet wurden, auf

M-~

a

schliesst.

Auch unter diesen

noeh fundiert, wenn auch die LQngen der durch

Elimination gelieferten Beweise erheblich starker wachsen als des ohne abbauende Regeln der Fall ist.

Teil

I

Sei @ eine uhendliche regul~re Kardinalzahl, zahl; als finit~rer Fall wird derjenige bezeiehnet, Menge, sei

aufgefasst als kardlnale Anfangsin dem @ gleich m i s t .

eine involutorische BijeKtion von F auf sich.

mit Werten in @ definiert,

die jedem v aus F seinen Grad

Menge aller Elemente yon Grad o , die dann auch Atome

c~ (ff)

~O~allev:

veL

genau dann, wenn

die Menge P-F B , die d u r o h

(fvJ

auf

F-F

o

o

{vF) Fist

wenn

yeA

heissen. Weiter gelte:

R , die durch

vertauscht wet-

werden

kIv)

rechten und linKen H~Iften R und L v o n

@-{o}

definiert,

die jedem v

zuordnet

, so ist KCvJ endlich, und es gilt

dann eine Menge abstraKter Terme

Konjugation

sei F ~ die

zerf~llt in zwei disjunKte Teilmengen A und

ist eine Funktion k mit Werten in

seinen Verzweig~r~gsgrad

Ivl zuordnet;

v*eR

der Nicht-Atome vertauscht

Auf F sei eine GradfunKtion

,

F zerf~llt in zwe• disjunkte Teilmengen L u n d den:

~fff]

Ivl ~ I~*I

Sei F eime

[oder Formeln);

K{vJ ~ K{v*) die Abbildung

F miteinander vertauscht,

, welche die

heisst auch die

.

Ein B a ~

T •

eine geordnete Menge <E, ~ >

mit Kleinstem Element e T derart,

dass jedes von e T verschiedene e aus E genau einen unteren Nachbarn besitzt, weiter jedes e aus E unterhalb eines maximalen Elementes liegt und, schliesslich,

for jedes

122

W. F e l s c h e r

maximale'e' bier

nut endlieh

betraehteten

der oberen

8~ume

Nachbarn

eine

GradfunKtion 9

alle

e' aus

die K n o t e n


s

sei

o

>

als die

dann

besteht

derart,

o

dass

unterhalb Kleinste

noeh

ITI

sind.

for jedes

yon

= leTl

seien

e die M e n g e

@ hat.

Ordinalzahl

Weiter

AuF

m mit

le'l

. Oie E l e m e n t e

aus

einem

Baum

T

= <E,

> und

<

einer

dass

oder

s

(e)

= <2,j>

oder

s

(e)

= <3,0>

9 und d a n n

enth~it

N[e]

ein E l e m e n t 9

oder

s

[e)

=

9 und dann

enth~lt

N[e]

ein

le]

= I

emit

diejenigen

s

o o o

o

mit

dass

alle

man

zu < T , S o >

mit

den i n d u z i e r t e n

%

eines

T',

T" von

bestimmt

Probeweis

ist und

s1 9

q

for die

gilt:

sI

for

alle

eeE

S ist

ein

~

e

mit

einer man

= <2,j'>

Folgt

folgt:

enth~lt

es gilt

alle

den W e f t

j'

T'

Knoten, <3,o>


ist

T partiell

Teilmenge

>

< j

die noch

von

Schnitten = <3,1>

reehts. Ordinalzahl

. Welter ~

nicht

>

den W e r t , n~mlieh

definierte

heisst

, T"

den W e f t

, in d e m < T 9

van F u n d

logischer

Sole)

kleinste

= <E',

0 3 = IT"I

E von

auFbauender

Abschw~chung

die

Elemente,

Element.

oder

> verstehe

Ce)

zwei

= <3 9

= IT'I

sq(e] ;

oder

N[e]

ist,

die V o r n a h m e

a o, 01 , o 2 , o 3

Sextupel

deFiniert;

o

Sole)

leer

for

Rangzahlen

auf d e r G r u n d m e n g e

von S an d e r S t e l l e

o

die w e d e r

vier

,

s

T wie

und E'

Knoten,

ITI

links

T aus

N(e]

enth~It

diejenigen

emit


wenn

f o r die A n w e n d u n g e n

sind

Knoten

SKelettes

e yon

alle

~

= <2 9

dann,

, und d a n n

diejenigen

j ; die

Ordnungen,

sowie

Ein

(e)

o

genau

j < e

Abschw~ehung

Knoten

E" e n t h ~ I t

rang

s

einer

Teilb~ume

mit

sind

vom Grad

so,

Shelett

und z w a r

<3,1>

emit

Schnittrang

Jedes

auch

auf E deFi-

= I

ergeben,

man

for

e heissen

s (e] o

~Or

NIe)

< m

oder

dem

alle

E definiert

= o

o

die Knoten

ist

@-verzweigt,

N@ehtigKeit

lel

vorhanden

T .

Sehnitttermen

Unter

e T < e < e'

[b]

Knoten

sind

man

erKl~rt;

Funktion

Regelnj mit

e eine

indem

s

Die

emit

in dem S i n n e

yon

Skelett

Ein

nierten

NCe] von

viele

<3,J>

definiere = <E", <3,o> ergeben.

den S c h n i t t -

ein S K e l e t t

Punktionen

das

j

sind,

Ergebnis

W.

s2

•

{Or al3e eeE Oe{iniert; aller

Teilmenzen

daher

e hSchstens

kurz s3

ist

s2(e)

for

alle

s3(e]

ist

ist

ein

alle

esE d e { i n i e r t ,

ist

eine

au{

N[e] je

so schreibt gilt:

Nachbarn,

unter

so die

yon. F , im F a l l e

sie

unter

de{inierte

Elemente

= o

, so

so(e]

= 1

, s3(e)sA

heissen

Parc~neter

die

1 , <3,o> = 1

Werte

deren

mehrere die

> . Hat

s2(e](e]

oder

sozar

1 oder

Werte

aueh

yon e b e i <3,1>

ein

e'

oder

nut

ein

auch kurz s4[e]

lie{ern; oder

Element

yon F - F o ,

einen

s4[e]

Teilmengen

enthaltj

von e , •

Sohnitt#erme von e . Hat e h 6 c h s t e n s

s 1 (e]

= s2[eD(e] so

N(e]

sowie

, s3[e]eB_ _

so

Falle

oberen

die s (e] o

Nachbarn

;

= s2[e)

= {e'}

und = s4(e}

~

= s2[e]

sl(e]

= s2(e]v

Sq(e)(ei]~F

= <2,j>

, so

N(e]

= {e',e"]

,

Is4Ce)[e'] sowie

< Is3Cell

= K ,

I~41eI%~I

= s2(e][e)w

NEe)

,

,

= {ell

i
< Is3Cell,

{s4(e][e~]

}

{s3[e]}

{s4(e)[e'],s4[e](e"]) I = j

= k(s3(e)]

{SZ(e] }

Sl[e i] = s 2 ( e ] [ e i ] v sl(e]

, cardCs4[e]]

ws4(e]

= k(s3(e]]

9

F

Ivl

:

sl[e']

card[N[e]]

sowie

;

lie{ernj

Element

<2,j>

Elemente

Nebenterrne

{o~llev~hlel

s (el o

man { O r

Werte

s o die

yon F h e i s s e n

men {Or s4[e](e']

t e'eN[e]

so s c h r e i b t

sole]

Funktion,

s4[e)[e']

= 1

U<s2[e](e']

in die Menge

{alls

So(el

So[e]

von {e} wN(e]

j

n e c h dem, ob N [ e ]

auch die

=

von s 2 [ e ) ( e ' ]

die

{Or

<2,j>

s2(e](e]

Haz,@##erm yon e

der

au{tretenden

eine Funktion

dass

Elemente

Element

•

oberen

esE d e s

yon F s i n d ,

weiter

einen

; die

und h e i s s t s4

s2(e]

von F s o ,

123

Felscher

, s4[e)[e')*

g

F

= s4(e][e"

sl(e']

= s2(e]Ce']

v {s4[e)[e,)}

sl(e"]

= s2(e](e")

v {sq(e][e"])

sl[e)

= s2[e](e)

)

,

und

= e',

124

W. F e l s c h e r

So(e)

= <3,o>

, se

NCe)

= {e')

sowie

So(e)

= <3,1>

, so

NCe)

eT des

Kleinste

Probeweise

heissen

Element

SKelettes.

= s2Ie)

sl(e)

= s2(e)

,

v(s3(e)),

s3Ie)~R

slIe')

= s2(e)

sl(e)

~ s2[e)

wenn s i e

~o 9 01' ~2" ~

-

,

v{s3(e)),

v o n T , so h e i s s t

~uiva~ent,

R~gzahlen

Als die

s3(e)~L

sl[e')

= (e')

sowie

Ist

,

s l ( e T)

dasselbe

auch

SKelett

Ergebnis yon

das

und d a s s e l b e

S , Zwei

Ergebnis

haben,

eines P r o b e w e i s e s erklSrt man diejenigen seines

Die Rolle der Mengen A und B erklSrt sich nun aus dem verschiede-

nen Verhalten der beiden FSlle s [e] = 1 . Die U n t e r s c h e i d u n g zwischen den Mengen A o und B geht zurOcK auf SMULLYAN a- und y-Formeln,

68

; die Menge A entspricht den S M U L L Y A N ' s c h e n

die Menge B den S M U L L Y A N ' s c h e n

B-

und ~-Formeln.

Die folgenden einfachen K o n s t r u K t i o n e n erlauben es, aus gegebenen P r o b e w e i s e n neue zu bilden. Sle

Ist S ein Probeweis und e einer seiner Knoten,

dadurch erKl~rt,

dass man for Tle

nimmt und die FunKtionen yon S a u f weis, der auch als F2CS,e)

Rest~ktion

so sei die

den T e i l b a u m aller Knoten e' yon T m i t e

Tie restringlert;

_< e'

dann ist Sle w i e d e r ein Probe-

notiert werde. Sei w e l t e r S ein Probeweis mit dem Baum T

und sei v in F ; man verl~ngere T zu einem Baum T'

, i n d e m man unter des Kleinste

Element e T yon T noeh einen neuen Knoten eT, legt; damit erK15rt man einen Probeweis S' mit dem Baum T', indem man die F u n k t i o n e n yon S fortsetzt:

s~IeT.)

veL

, s 89

,

S~CeT.)

s~(eT,)

= v

= <3,1>

fells

veR

, s~IeT,)

im z w e i t e n

9

BSumen T ~ V ~ S ol ( e o)

T1 ,

Falle

als

und kleinsten

v ~ S s l1( e 1]

: sei

Summe T y o n T ~ u n d T 1 , i n d e m und noch

slCe T) v { v }

ein

neues

kleinstes

Seien

e o,

eI

Element

Absc~chung

nun S ~ , S 1 P r o b e w e i s e

, u n d es g e b e

Do = s ~ [ e o ) - { v ) men d i e s e

9

; im ersten Falle notiert men S' als

F4(S,v)

Knoten

falls

= s1[e T)

. Der Probeweis S' heisst dann die V e r l ~ n g e r u n g yon S dutch

mit v gemSss <3,o> respeKtive gemSss <3,1> F3(S,v)

=

= <3,o>

,

v in

F so,

D 1 = s l1[ e l ) - { v * }

BSume a l s

eT unter

ein

geordnete

e~ , eI

setztj

mit

dass

Men b i l d e

die

Mengen n e b e n e i n a n d e r damit

erklSrt

leg~

man e i n e n

W. F e l s c h e r

Probeweis

S , indem man auf den Teilb~umen

Obernimmt

und dann welter

s 2 [ e T)

= s 3 [ e T]

erkl~rt Art

, s4[eT)[e

man e i n e

definiert

Konstruierte

setzt

o]

Funktion

ist,

Probeweis

T 1 von T die Funktionen

Sole T) = <2,1vl>

= v , s4[eT){e

G, , d i e J

so~ern

T~

125

noch

for

Iv]

1)

alle

< j

, sl[e T) = 0 o ~ 0 1

= v

FOr jedes

Tripel

giltj

S~ S 1

von

v,

S~ , S 1

j

mit

j

< 8

der soeben beschr~ebenen

der Wert G..[v,S~ J

1]

sei dann der

S 9

Von nun an werde vorausgesetzt, Knoten von Probeweisen

endliehe

dass alle Ergebnisse

Teilmengen

von F sind.

und Parametermengen

Dies bedeutet

an

im Besonderen,

dasa im nicht-finit~ren

Fall Iogische Regeln nicht in Ketonen-Form

Eine endliche

M von F heisst vom Typ I , wenn sie genau ein Element von R

enth~It

Teilmenge

(Fall des MinimalKalKuis]

Element von R enth~lt Typ I respeKtive

vom Typ II , wenn die Ergebnisse

tive vom Typ II sind; Probeweise,

d~ss niemals

So(e]

<3,1>

der Obergang von s1[e']

auch nicht durch Abschw~chung dutch Abschw~chung enthalten

gilt,

zu st(e]

Ein Probeweis

aller seiner Knoten vom Typ I respeK-

da sonst bereits

nicht unterliegen,

s3[e]es2[e]

nichts Neues lieferte. verl~ngernj

dann verl~ngert

ein

heisst vom

vam Typ I , so kann vorausgesetzt

gem~ss <3,1>

gem~ss <3,1>

KaIKuls].

die diesen Einschr@nKungen

Ist S ein Probeweis =

werden.

~ sie heisst vom Typ II , wenn sie h~chstens

(Fall des intuitionistischen

vom Typ III heissen.

zugelassen

sollen

werden,

gelten mOsste und

Oaher Kann man ein S vom Typ I Probeweis

werden,

S vom Typ II

wenn ihr Ergebnis

KBnnen

ganz in L

ist.

Ein wichtiges

Hilfsmittel

for alle Eliminationsverfahren

ist das

Ausfege-Verfahren. Se• S ein Probeweis mlt dem Ergebnis M , se• dass

lwl<j

weise,

for alle weW

. Zu jedem weW sei

S w mit dem Ergebnis

M w v [Cw - {rw} ] v {w*}

M w u C w v {r w]

. Sei

C =U
Oann gibt es einen Probeweis aus S , den S

w

und den S w

so ist auch S a v o n

struKtion

von S a h~chstens

weW >

durch K Anwendungen

ond sei

w'

so,

S w Probe-

k = card(W-C]

[M-W]v C , und S a entsteht

der FunKtionen

Sind S und die S

diesem Typ;

und seien S

j<8

und S w mit dem Ergebnis

S a mit dem Ergebnis

k darauf folgende Abschw@chungen. Typ,

M r M-W w-

W ~ M , sei

w

G. und h~ehstens j

, S w von einem fixierten

liegt der Typ I vor, so werden zur Kon-

2 Abschw&chungen

ben~tigt.

126

W.

Sei n~mlich W

o

W' = W-C

Felscher

; sei

die Menge aller wgW'

mit nicht r cC w w

W i die Nenge miler weW'

mit r sC , nicht w w

W 2 die Menge miler wsW'

mit rwEC w

W Z der Rest yon W' Sei

r

w*gC

w

: w*

w

.

<w[i)[ i eine Abz~hlun g van W'

eine Folge

<Sil

S.z = Gj. [ w ( i ]

'Si-I"Sw[i]

. Nan setze S_I = S und definiere rekursiv

i yon P r o b e w e i s e n dutch

]

for

S.l = G.(w[i] j 'Si-i" S w[i) ]

w(i]~W 3

for nicht w [ i ] g W S

Ourch Induktisn zeigt man, dass for jedes i
Hi ist, wobei

= (n - { w ( j ]

uj = rw[j],

alle w wegen

I j ~ i }] ~ U
w [ j ) e W I , uj = w(j)

M w ~ M-W . Sei des for i-1 bewiesen.

(HI_ I - {w(i]}]

v [(Cw[i]~

Hi_ i

= [M - { w [ j )

- {will}

{uj}

{rw[i]}) I

j~i

- {w(i]*}]

.

}] v U
also much in keinem C w liegt. Oa rw[i]gCw(i] [Cw[i] v {rw[i]}]

Ergebnis

- {w[i]*}

[Hi_ I - {w[i)})

bier l~sst sieh Hi_ i

Cw[i] Ni

- {w(i]*}

= Cw(s

: Cw[i]

vN i {w[i)}

, for

{ui}

I i
sonst.

Hier

Als >

; dies

ist

I j
>

, da w [ i ]

Falls nicht

= Cw[i]

Ergebnis

MwC_H.m ~Or

nicht

in

C ,

wegen w[i]eW 3 , gilt w e i t e r

N.z = ( { C w ( i )

Ni

Oebei gilt noch

gilt

{uj}

{w(i]*}

gilt

j ~ i >

Falls w [ i ) g W 3 , so ergibt S i

{w[i]*}

FOr

w(i]cW ~

{rw[i]}

for

v o n Sk_ i e/he

w[i]gW 3 , so hat S i das

{rw[i]})v{w(i]*})

wie zuvor sehreiben.

w(i]aW 1

- {w[i]*}

[M-W'] v U
mit

I

erh~lt

Teilmenge

gilt N i =

w(i]eW 2

man d a h e r von

j

gilt

HK_ i

[M-W'] uC

= (M-W) v C

sodass man durch hBchstens k A b s c h w ~ c h u n g e n mit den Elementen u i den Beweis SK_ 1 zu einem Beweis S a v o n

(M-W]v C

auch die S.i P r o b e w e i s e d•

verl~ngern

Kann. Liegt ein f i x •

Typsj folglich ist Hk_ I eine Menge des fixierten Typs,

und die eventuellen A b s c h w ~ c h u n g e n ver@ndern den Typ nicht. KSnnen diese A b s c h w ~ e h u n g e n

2 Abschw~chungen

h~chstens ein Element w

o

Liegt der Typ I vor, so

nur mit Elementen aus L geschehen~

so Kann hBchstens eines der u. in R liegen. l hSohstens

Typ vor, so sind

auftreten.

Es i s t

aus R enthaltsn

liegt der Typ II vor,

Im Falle des Typs I kBnnen eber auch nur dann

ngmlieh

N vom Typ I

,

sodass

W'

Kann~ for alle Obrigen w aus W' muss dann w

,

W.

in R liegen

sodass neben w ~

9

l~elscher

127

hinzuzufOgen ist. Weitere u i k~nnen abet nicht euftreten, folgt

uigR :

(Cw

es gilt dann weW I , u i = r w

( r w } ) v {w*}

R~(C

W {r )) w

als ein

hbchstens for eines dieser w das Element w

denn aus wsR , w* ~ u i ,

, und da C w u { r w)

vom Typ I 3st,

ist.

w

i mit

und N gegeben,

o
eine Folge von Probeweisen;

im Falle

K > I

Folgen

<Mil

seien lm Falle

i und

Kleinsten Knoten der zu den S i geh~renden B~ume T i s~(e o) = M v W

gegeben

so,

, im Falle

dess

und aus vgB alle

i
verm~ge die

Ti

Kann

nut dadurch auch vom Typ I sein, dass rw das eine Element aus

Eine h~ufig gebrauchte KonstruKtion von P r o b e w e i s e n ist die folgende. <Sil

u .l

aos v s A

{olgt

K > I

folgt

k = k(v)

. Man K o n s t r u i e r t

M,W,v

respeKtive

=

,

, im Fall einen

verm~ge

oardm)

k = q

=

k{v

auch

neuen P r o b e w e i s <Mil

i
>

zu e i n e m neuen Baum T , i n d e m man s i e

i

,

<

S , die i
; mit e• als den

,

Ivl

k = q

> , v

for

sowie

formale

geordnete

Mengen M

auch

Sei weiter ein v in F-F ~

W = {w O}

<wil als

K = I

sei weiter im Falle

i s1(e i) = Miv{w i} ,

aueh k

<wil

Sei

alle

Iwil

< Ivi

S~z~rne d e r wie folgt.

for

<sil

i

Man s u m m i e r e

Mengen n e b e n e i n a n d e r

legt

und nooh Bin neues kleinstes Element e T unter die e i legt. FOr alle schon in den T i auftretenden Knoten von T definiere man die F u n k t i o n e n yon S als diejenigen der e n t s p r e c h e n d e n S i . Alsdann setze man

MU{v} U Mi

, s 2 ( e T]

<Mil

i
, s 3 ( e T]

= M , s 3 [ e T]

> , wobei

noch

= v , s4(eT)(e i)

So(e T) = I u n d

= v , s 4 ( e T]

= W

M~ = M

k = q

= wi

for

,

im Falle veA

im P a l l e

veB

s q ( e T]

dann

auch

= s2[e T)V{v}

s1(e T) =

s2(eI](e

T]

=

, s2(eT](ei)

.

Sei nun einer der Typen I, I I o d e r sehreibt, wie man yon einem Prebeweis

III fixlert. S

Oie folgende KonstruKtion be-

der mit einem Sehnitt oder mft der An-

w e n d u n g e i n e r logischen Regel endet, zu e i n e m neuen Probeweis S ~ Obergeht, einen P a r a m e t e r v aus dem Ergebnis von S entfernt,

indem man

sofern bereits geslohert ist, dass

es weitere P r o b e w e i s e gibt, welche die um den Parameter v verkleinerten Pr~missen des letzten Schlusses yon S liefern~ mit einem A b s c h w ~ c h u n g s p r i n z i p

Oiese E n t f e r n u n g des Parameters v wird ausserdem

kombiniert, o a s e s

in d i e s e m Palle erlaubt,

aus Ab-

schw~chungen der P r o b e w e i s e for die Pr~missen auch eine A b s c h w & c h u n g von S ~ zu gewinnen. Sei also S mlt dam Kleinsten Knoten s gegeben,

se• NCf) = (fil i
mit

=

128

fi ~ fj So(e)

for

i ~ j

= I , such

H = U
So{f)

v ~ s3(f)

= 1

.

M = (s2{f]w

Ho]

-

sei

; ist der fixierte

i,v)

des Parameters

v

. Sei ves2{fJ{f)

gegeben. sei

wi

Im Fells =

i

k = 1

s4{f)(f

i)

von F , von Probeweisen

sei

, Mi

W = s4{f)

,

=

Typ I oder II , so sei such die Menge

i , <wil ,v)

und, for

eine Folge von Teilmengen

Im Fslle

So(f)

dann die formale Summe der <sil

vermBge
Zusa~cnensetzung

= <2,j>

von diesem Typ.

Fs(S,<Si I i<2 >,

damit die

i

K > 1

[(s2(f)(f)v HJ - { v } ) u {s3(f)}

respektive

So{f)

Sei nun weiter eine Folge <sil

; im F s l l e

{s2{f)(fi)~ H i ) - {v)

s3(f)

odor

des letzten Absstzes

{v}

F5(S,<Si I i,
Felscher

Sei
i , nicht v~H

mit den Bezeichnungen

W.

i , s3{f)

= 1

i vermBge M,W,

. Im Falle

gleich Gj+1{Wo,S~

sei

Solf)

Oie PunKtion

= <2,j> F5 liefert

der S i gom~ss dem letzten Schluss yon S , unter Auslassung

und HinzufOgung

neuer Parameter

in den H.

9

1

Im Anschluss endlichen

an TAIT

Teilmengen

erkl~rt man ein Axiomensystem ~ sis eine Menge von

68

M von F

M, N, v

so, dass for alle

gilt:

o

wenn

Nw{v}

in A

Sei nun A lest gegeben. III

fixiert,

und

Nv{v

Sei S e i n e

} in A

, so umfasst

Menge von Probeweisen,

und seien die Probeweise

MvN

eine Menge aus _A .

sei einer der Typen I, II,

sus S slle von diesem Typ. S heisst eine

Menge yon Beweisen aus A , wenn for jedes S sus S und for jeden maxima/on Knoten e von S der Wert slCe) ein Axiom aus ~ ist tionen sbgeschlossen (s)

(sss) (sv)

KonstruK-

ist:

Seien S und S' Beweise aus S , deren B~ume beide je nur einen Knoten e und e' haben,

und s e i

Beweis

S"

slCe)

,dessen

in A vorNandenen (ss)

, und wenn ~ unter den folgenden

= My{v}

Baum n u r

, s~(e')

einen

= Nv{v*}

Knoten

Mengen ist, welche in

hat

M~N

. Osnn l i e g t

und d e s s e n W e f t

enthalten

Sei S e ~ und sei e ein Knoten yon S . Dann liegt F2(S,e) Sei S E ~ und sei veL .

Dann liegt F3(S,v)

Sei der fixierte Typ yon I verschieden. fixierte

in

dort

eine

sind. in

in

Sei ScS

Typ gleich II , so sei das Ergebnis

liegt F4(S,v)

in ~ ein

und sei veR

j ist der

yon S in L enthslten.

Dann

der

W. Felscher

CvS)

Sei SeS

, sei <Sil


>

i eine Folge yon Beweisen aus S

so, dass

Fs(S,<Si I i,
liegt Fs(S,<Si I i,
~vs]

Sei S e S

i,v)

= 1

Dann e x i s t i e r t derart,

s3{eo)*

'

i,v)

definiert ist. Dann

S~ =Sle ~ , S 1 =Sle I .

= s 3 { e 1)

s3{eo)eA

'

Se•

~

zu jedem w aus s4{e o) e i n e w i n N(e 1) und e i n Beweis Sw i n S

dass $w

das S K e l e t t

yon S l e w

{ s l ( e w) - { s 4 ( e l ) ~ e w ] } ) v { w * }

und das E r g e b n i s

hat,

FOr d i e Beweise S , S" aus (s) w ~ h l t

man a l s F I ( S , S ' )

Beweise S" ;

w~hlt

f~r

, seien v und

in S

, s e i e n e o, e I Knoten von S ,

s 0 {e 0 ) = So{e l j

129

S~ und w aus ( v s )

e i n e n d e r gemgss (s) v o r h a n d e n e n

man a l s F [ S ~

e i n e n d e r gem~ss ( v s )

6

v o r h a n d e n e n Beweise Sw

Ist S e i n e

schaft

als

Menge von B e w e i s e n vom Typ I , so bezeichnet man die f o l g e n d e Eigen-

das

Linksprinzip.

Ist S e ~ , e ein Knoten von S mit Sole)

wenn

s3(e)~A

,

so

s4(e) C L

wenn

s3(e)s

,

so

s4(e](e')~L

= 1

, s3(e)cL

, so gilt:

for mindestens ein e' in N(e)

.

W e l t e r nennt man for endliches K die folgende E i g e n s c h a f t von S die

K-Beschr~nktheit yon A-Nebentermen.

Ist SeS

, e ein Knoten von S mit s {e) = I ,

--

s3(e)cA

, so gilt

0

card{s4(e)) ~ K

Oamit formuliert man das

Eliminationslemma.

(I)

Sei S e i n e

Menge von B e w e i s e n eines fixierten Typs; im Falle

des Typs I gelte das LinKsprinzip.

Sei

j < @ und sei ~ a b g e ~ c h l o s s e n

gegen die F u n K t i o n Gj . Sei S e ~ , s e i e n e o, e I Knoten yon S , S ~ = Sle ~ ,

sl

= s1~1

J s~i

v au~ F , I v l

= J

. s~i

sl~

o)

: o or{v}

, s l o e 1) :

D l~{v*}

, nicht veD ~ , nicht v * ED 1 . F~r die S c h n i t t r ~ n g e o~

S~ , S1

geite

o~ < j 0

oI < j '

des E r g e b n i s s e s yon Gj§176 SS

'

olo

von

. Dann gibt es in S einen Beweis S ~

0

mit einem S c h n i t t r a n g unterhalb von J s

liegt in der A b s c h l i e s s u n g yon S ~ , S 1

unter den F u n K t i o n e n F I .....

W. Felseher

130

und G. J

F8

(2]

die

Sei

Rangzahlen

o 51 ,

o o2 ,

deren

S~

von

I

S~

I

o2,

d3

sind,

f1(o,o,z]

= z , fl[O,O,Z)

fl[x,y,z)

= fl[Y,X,Z]

und

gilt

= z ,

mit

%2(0,0)

, f~(x,y,z)

= I

,

= f3[Y,X]

9

, x ~ %3[x,y)

fl(x,y,z']

wenn

x'

< x

so

fl[x',y,z+k)

§ 2K

~

fl[x,y,z)

f4[x',y,z+k)

+ k+2 ~

f4(x,y,z)

so g e l t e n

_

oI _

s

+

k

~

f3(x,y)

II und

III

for

oi f 2 [ o 2 , o 2) ol

0 3 ~ f3(o3,o3) dabei Kann

ist man

die

Typ

I

f o r den

Typ

II

s

d = do + dI

Typen

f o r den

jeden

Typ

,

o~ ~ o~

for

den

Typ

Ill

,

;

, d o ~ c a r d [ D o ) 9 d I = c a r d ( O I)

9 Im B e s o n d e r e n

w~hlen

fl[x,y,z)

=

[3k)'[x

f~(x,y,z)

=

(2K+2)'(x

f2[x,y)

=

(k§

f3[x,y)

=

k'[x

Hier

< f4(x,y,z)

Absch@tzungen

[Ol,Ol,d)

o2 ~

, f4(x,y,z')

f2(x,y)

f 1 ( o 1 ,o~ l , dI]

w <

~ 1

die

= o

f3[x,y)

so

~

Funktionen,

,

< z

k+q

fl[x,y,z),

= f2[Y,X)

z'

+

Rangzahlen

, f2(x,y)

wenn

f2[x',y)

sich

z

, f3(o,o)

x ~ f2(x,y]

< fl[X,y,z)

den

de~inierte

endlichen

= s

, x+z ~ ~[x,y,z)

Sind

x,y,z

berechnen

aus

SI

von

for Ordinalzahlen

Ordinalzahlen

Oann

primitiv-reKursiv

I

oI ,

und

f3[x,y)

x+z ~ fl(x,y,z)

f o r die A - N e b e n t e r m e .

von

uS

f2[x,y),

Werte

Schranke

o 1,

o o3

f4[x,y,z),

k eine

bezeichnen

net

die

der

2-ad•

oder

+ z

fl y) # y]

+ z + 1

# y]

+ und

nat0rliche

# y)

9 Summa und P r o d u k t

Summe [ w o b e i

yon O r d i n a l z a h l e n ,

es g l e i c h g 0 1 t i g

der ~-adischen

Normalformen

ist,

ob d i e s e

erkl~rt

wird].

und # b e z e • vermittelst

W.

Zum 8eweis der Behauptung KonstruKtion

ist symmetrisch

hung vom Typ

[a)

~1 # ~I

01 # ulI = o

tens

o oI > o 9

do+d I

in S ~ , S I , sodass es bequem ist,

Abschw~chungen

e o, so lies

, sos

So[e e] = I So[eo)

= 1

S[ BOO 9 Sle I bbaa)

veL

S unteF

Iv S)

OoV01

Kraft.

So[e o] = <2,j'>

o

. Ist eoo der obere Naehbar van aus S]eoo

,Sle I

einen Beweis S' van mit s3[e o) zu S ~ verzu L .

mit j'<j

, so lies

einen Beweis S' van

lies

die Induktionsannahme

((s2(eo]VSq[eo]]

(s2(e o) - { v } ) w s 4 [ e o ) W 01 . dann den Beweis S ~ van

aus

- {v}] v 01 erh~It man aus S'

Ole Abgeschlossenheit

[s2(e o) - { v ) ) w {si(e

o

van

} v O1

" . Gilt nieht

ves41e o)

Dies let im Besonderen siIeo)eL

Gilt aber ves4Ie o)

die Abgeschlossenheit

Abschw@ehungen,

, so bleibt das vorangehende

, also auch

, so zerlegt man

in

[v S) lies

s4(eo)~ L s 1leaD)

denn

wegen des Llnksprin-

= [s2Ie o) - { v } ) v s4(e o)

dane einen 8eweis S" vun

= 0 a . Aus ihm erh~lt man S t durch hSchstens

van denen im Falle

aus R geschieht.

Argument

dann der Fall, wenn der Typ I vorliegt,

yen _S unter

[s2[e o] - { v } ) V { s 3 [ e o ] )

Pormel

ist.

, sB[eo)eA

wegen veR gilt dann zips.

des

. Indem man netigen Falles noch mit v absehw~cht,

--

veR

eine dieser Ab-

9 den man durch eine Abschw&chung

einen Beweis S" van

bbab)

SoCe o] = <3,1>

der obere Nachbar van e

DO

h~ehstens

Liegt der Typ I vor, so gehert s3Ce o) notwendig oder

Ourch h~ehs-

man ihn zu einem Beweis S t van

Oberhaupt

die Induktionsannahme

Kann.

Axioms.

van S unter

van R .

s3(e o]

oder

[Oo - { s 3 [ e o ) } ] v 01

Ist e

verl~ngert

mit einem Element

= <3,0>

l~ngern

Oiese

sie Ober eine Wohlord-

enthaltenen

Liegt der Typ II vet, so geschieht

v ~ si[e o)

So[eo)

bba)

Konstruiert.

zo ~Ohren.

sehw~chungen

[bb]

[1) wird der Beweis S t reKursiv

In diesem Falle gibt es wegen der Abgeschlossenheit

O o V 01 .

[ba]

131

in S_ einen Beweis S' eines in O o V 01

(s]

[hi

Felscher

des Typs II hSchstens

dI

eine mit einer

W. Felscher

132

[bbb]

So[e o] = I ,

s3[eo]eB

N(e o] = {eoi I

Sei

i
vcs1[eol ') , die Menge wie den Rest 12 nicht

leer,

Beweise

ieI 2

I wegen

SI von Z

unter

[v S)

trachten,

(s2[eo](eoi]

dass 11 l e e r ,

legt

so l i e F e r t

Folge einen

vcL

also

][e

O~ w 01

Beweis S'

leer

O•

for 1~I 1 ,

Sleoi

noch als

sl[eo•

for =

Abgesehlossenheit

von

Sw der Fall zu be-

ist.

Betrachtet

man d i e F o l g e

- (v})

die

Ergebnisse

wie zuvor zer-

yon ~ u n t e r

[ v S] aus d i e s e r

u {s3[eo]]

= O~

, so{ern

Typist.

Liegt der Typ I vor, so ist mit

tens d I Abschw@chungen. auch

, S l e I for icI I

Es bleibt

die Abgeschlossenheit (s2[eo][eo]

Ist 11

.

, wobei {Or ieI 2

yon

leer.

, so-

)} v 01 . Man betrachte

oi

sehreibt.

12 n i e h t

, v ~ s4[eo)[eoi)

Sleei

die Ergebnisse

selbe gilt, wenn der Typ II vorliegt.

v*eR

aus

for is ~ , S:z

letzten

i
vom fixierten

o

I ~ aller i mit nicht

1 I v 12 nicht

aus dieser Folge einen Beweis

Sleoi

[bbba]

Sleoi

- { v } ] U { s 3 [ e o ] } ~ 01

warden,

ist

- (v}] U { s 4 [ e

der Beweise

liefert

1 in die Menge

die Induktionsannahme

der Beweise

o

ves2[eo][eo]

- {v})V(s4[eo][eoi]}

[ s 2 [ e o ] [ e o]

0

Man zerlege

, wobei man for diese

[s2[eo][eoi)

So[e e) = <2,j'>

11 aller i mit ves2[eo][eol]

so liefert

dann die Folge

oder

OI~L

O o V {v} eueh D o vom Typ I ; das-

Aus S' erh@lt man nun S w dutch

Liegt der Typ I oder II vor,

, sodass

diese Abschw~chungen

h~chs-

so folgt aus vcL

s~mtlich

mit Elementen

aus L geschehen. [bbbb]

vER

wegen Beweis

FOr den Typ I kann dieser Fall nicht des Linksprinzips Sw

hBchstens

hOchstens

1 ~1 > o

v*

[d]

o oI > o

v = s 3 [ e o]

S o [ e o]

Typ I v o r ,

folgt.

d I Abschw~ehungen,

Nieder

da eus s3[eo]gL

erh~It

von denen

man aus S' den

im Falle des Typs II

aus R geschieht.

Oieser Fall

~ s3[e I ]

= <3,0>

so e r h ~ l t

11 ~ o

eine mit einem Element

(c)

[da)

dann

eintreten,

ist symmetrisch

zu [b] 9

.

oder

S o [ e o)

man S w aus S eoo

= <3,1> dutch

so muss veL , a l s o

.

Ist

h~chstens

v eR

gelten,

eoo d e r o b e r e N a c h b a r von e ~ , d I Abschw~chungen. s o d a s s wegen

D1 s

Liegt

der

diese

W.

Felscher

133

Abschw~chungen

nur mit Elementen

aus L gesehehen.

Kann h~chstens

eine dieser Absohw&chungen

Lie~t der Typ II vor, so

mit einem Element

aus R gesohe-

hen.

[db]

sole o) = 1 dass

o~ > o

und wegen

{da)

A geh~rt,

ist

Sei nun e

Wegen

Iv I = Iv*l > o

gelten

muss; wegen (c) Kann man auch

waiter

[vs)

so bestimmt,

S w = Sle w [dbaa)

nicht

yon e

dass S W

, nioht

{dbab)

St

von

nicht ves2{e o)

,

. Man definiere

das Skelett,

annehmen,

Elemente

veA

v, v

zu

vorauszusetzen.

eine Menge

W =

also auoh den Rang,

[s2{el)(ew ) - {sq(el][ew)})v

von

{w*)

liefert.

v ~s2{el](e 1) .

Man wende das Ausfege-Ver~ahren einen Beweis

o

= v*

so-

~ for jades wEW sei e w in N{e I) und S w in

hat und das Ergebnis ves2[e o)

kein Axiom,

s3(e l)

es keine Einschr~nKung der Allgemeinheit,

sq(e o) - (s2(eo)V s2(el]{el)) naeh

0 1 U {v*}

So(e ~ ) = I . Oa eines der beiden

der obere Nachbar

oo

ist dann

9 W

auf Sleoo

und die S W , S w an; es liefert

s2{eo]V s2{el)[e I) = O o W 01 v es2{el][e I) .

Sel W e die Menge aller w aus W mit v ~ s 2 ( e l ) [ e wj , sei W b der Rest yon W. FOr wow a liefert

die Induktionsannahme

se Sw% und S w~ mit Ergebnissen und

s2{eo)V

,

Wendet

S w mit woW b und die Sw% von

s2[e o] u {s2{el)(e w) - { v ~ ) ) ~ { s q [ e l ) { e w ) }

((s2{elJ(e W) - {v~})

v* ~ s4(el){e w)

S2(e o) v [s2(el)(el)

aus Sle ~ , S w und aus Sle ~ , S W 8ewei-

- {sq(el){ew)})v{~}

man das Ausfege-Verfahren 9 S W~

mit woW a

- {v*}]

= 0~

, da

v~ ~ ~

auf SIeoo

an, so erh&it

und

, W , die S W ,

man einen Bowels

St

01 .

#

{dbba)

ves2(e o]

,

nicht v es2[el){e I]

Oie InduKtionsannahme

liefert

aus

Sleoo

, Sle I

einen Beweis

S' mit dem

Ergebnis

(s2[e o) - {v}) U s4[e o) v s2[el)[e 1) . Man wends das Ausfege-

Verfahren

auf S', W

und die S

[s2(e O) - {v}) v s2{el][el) [dbbbJ

v~s2(e o]

w

, S W anj es liefert

Sw

= O o V 01 .

v es2~elJ[e I)

Man Konstruiere

S' wie im vorangehenden

Fall;

(s2[e o) - { v } ) u sq{e o ) v {s2[el){e I) - {v*}) wie im Palle

einen Bowels

{dbab).

Man wende

das Ergebnis . Welter

das Ausfege-Verfahren

lautet jetzt

definiere

man die Sw~,S W~

auf S', W und die S

oS W w

134

W.

mit

wEWb

(s2(eo]

und d i e Sw%, Sw% m i t - {v}] ~ L~,2(el](e I)

Oamit ist die Konstruktion tung

leicht,

jedenfalls

T i < o~

f2(T2,0~)+1

, f 3 ( T 3 , 0 3 )1+ 1

o F +d 1, [bbb)

o~ +1 ,

o~

mit 11 ~ o

fl(Ti,~,d+1)

mit

Funktion

der S w%

dutch f1[o~,~W,d+1]

9 f1(~

,max[T

f2[T2,o~),

f1[Tl,O~,d+k)+2,

von S j a b g e s c h ~ t z t Im

der S~ , iEI I , abgesch~tzt van S w wie gewOnecht.

dutch fq(T,ol,d+k] I

zw < ~I

;nach

dutch Im Falle

und die ersten

den K Anwendungen

der

w

I weWa),d+1

) + k um 2 , in jedem Falle abet hachstens

sodass man auch dann noah unterhalb

van

bleibt.

besitzen,

folgt

aus

h'(x'

# yJ + h

x'

auch

=

x'

h'[[x'

# y

# y)

angegebenen

<

§ 1)

x # y ~

des Eliminationslemmas

theorems,

wie er van TAIT

duzieren.

Man definiert

= x

explizit

lehren einfache Anwendungen

< x

Vermittelst

XK[X]

68

];

for

,

FunKtionen

die verlangten

der OrdinalzahlarithmetiK.

[x' # y)

# y]

+ 1

<

x # y

Eigen-

Zum Beispiel

, also

9

l~sst sich nun der Beweis des Eliminations-

dazu Normalfunktionen

: [2K] x

z > I

h.[x

und FEFERMAN

68 , Theorem 3.4 , gegeben wurde, ~ und, ,fOr endliche

. Oann haben die Funktionen

vierte X 1 , und for die erste Oerivierte 55 , p.88

van S ~ d u t c h

Warden

man daher die obera Schranke

Abschw~chungen,

Oass die for f1' fl 'f2' f3 sehaften

o

zeigt dann

d i e gewOnschten F u n k t i o n s w e r t e .

sich for den Typ I noch h~chstens

um K dutch die folgenden a I {I [~ "~ ,d)

mit

erh~It

I max( f l { T , O l , d + K ] sie vergr~ssert

auch d u r c h

van S' abgeseh@tzt

G. im Auss J

InduKtian

werden d i e R a n g z a h l e n

, ~Iso aoch die Rangzahl

(dbbb) wird die erste Rangzahl Rangzahlen

{a] liefern;

ffl' fl " f2" f3

van S' dutch f1[T1,ol,d+k],

wird die erste Rangzahl

T i < o~

Zum Beweis der Behaup-

dass die Funktionen

so d i e j e n i g e n

(bbab)

, also

Beweis S ~ vop

auoh in allen anderen F@llen ergeben.

abgesch~tzt, In

durchgefOhrt.

im Falle

in [bbaa] die drei Rangzahlen

mit

einen

= OoU 01 .

van S w in allen FAllen

Abseh~tzungen

f 3 [ T 3 , o ~]

~(x]

- {v*}]

dass sie diese Abseh~tzungen

zum Beispiel

Felle

wsW a an~ es l i e f e r t

[2] sieht man aus den Anfangsbedingungen,

die gewOnschten

durch

Felscher

~I van ~ g i l t

ist daher die z-te Oerivierte

k , auch XK

k k alle dieselbe X 1 [l+x) Xz

= ~1 [ x ]

van XK

reprodutch

erste Oeri-

[cf.

BACHMANN

gleich der

W.

z-ten Oerivierten ~z von ~

Menge yon Beweisen eines fixierten Typs~ im Falle

Typs I gelte das LinKsprinzip.

des

135

. Oamit formuliert man das

Sei S e i n e

Eliminationstheorem.

Felscher

Sei --j SS. die Teilmenge aller 8eweise

aus S , deren Schnittran Z unterhalb von j liegt, sodass SS --

aus den

----o

schnittfreien Beweisen besteht. Sei Beweis S ~

j = a # z

,

sei S in --j SS.

mit demselben Ergebnis wie S , der in der A b s c h l i e s s u n g Yon S

unter den F u n K t i o n e n

F 1 . . . . . F 6 und Ga.

03

als den Rangzahlen von S

S~

die A b s c h ~ t z u n g e n

o~

~

9z(o i]

zur B e h a n d l u n g des Falles

~ Xk(U]

zur VerfOgun Z hat.

liegt. Nit

j : z

, pp. 214-215

o~

oI,

o2,

o3

voo

oI, ; ist k eine

z XK(o S)

~

, so ist S } schnittfre•

, Obernommen werden, x < u , y < u

, f~(~(x],~(y),z) ~ ~(u]

,

sobald man

die Ab-

f2(~[x),~[y)) ~ ~ [ u )

mit den im E l i m i n a t i o n s l e m m a explizJt angegebenen H u n k t i o n e n

Oiese A b s c h ~ t z u n g e n gelten abet, well zum Beispiel

# ~(y)] + z

K'((2K) x # (2k] y] Spezialfall

i = 1,2,3

for alle x,y,u mit

fl(~(x),9(y),z] ~ 9(u)

s

(3K).(9(x]

z = o

a' < a

so gilt auch

W@hlt man im Besonderen

Oer Beweis Kann w S r t l i c h von TAIT 68

mit

@elten for die Rangzahlen

S c h r a n k e for die A-Nebenterme,

schStzungen

. Oann gibt es in -SS - a einen

~

= (3k].(~ x # m y ) + z K'2"(2K] max(x'y]

z = o

~

< 2

j :

a+l

u

und

(2k) max(x'y]+l

K'[XK(X) # Xk(y)) ~

(2K) u

ist hier die Absch~tzung von o I

sch~rfer als dieJenige van F E F E R N A N beim Obergang von

~

~

88

, Lemma 3.8

dutch

=

~(o 1]

Im etwas

Im finit~ren Fall Kann man

zu a an Stelle der FunKtion ~ genauere FunKtionen g mit

endlichen Werten verwenden,

in die dann allerdings w i e d e r die Anzahl d der Elemente

im Ergebnis von S eingeht.

Eine solche FunKtion g muss den Ungleichungen

genOgen.

g(x,z]

~

fl(g(x-1],z+l],g(x-l,z+l],z)

g(x,z)

>

sup
Nit dem A n f a n g s w e r t

zum Beispiel

g(x,z]

I x' < x

g(1,z)

= E <[6K)Z(z+i)

= z I o~•

>

leistet das for die angegebene Funktion fl >

136

W. F e l s c h e r

Teil

II

Sei ~ die Mange aller Terme einer fast gegebenen Term-Algebra. ~ besteht dann aus allen formalen AusdrOcken,

die man erh~it,

indem man aus einer Menge X van Variab-

fen mit O p e r a t i o n s s y m b o l e n fl ' ~sI , Terme f1[X]

bildetj

dabei ist f IX] der Term,

den man aus airier Folge X bereits K o n s t r u i e r t e r Terme dutch A n w e n d u n g des Operationssymbols fl gewinnt.

Im Besonderen bestimmt jades O p e r a t i o n s s y m b o l

Ordinalzahl n I als seine Stellenzahl, mOssen yon der L~nge n warden,

i

fl eindeutig eine

und Folgen X , welche zu Termen fl[X]

fOhren,

sein. Ein Term t Kann nun auch als eine F u n k t i o n betrachtet

die auf einem geeigneten Baum definiert ist und die D p e r a t i o n s s y m b o l e der

Stellenzahl n

den Knoten mit n I

oberen Nachbarn zuordnet,

und O p e r a t i o n s s y m b o l e der

1

Stellenzahl o oder Varlablen den maximalen Knoten.

Ist e ein Knoten des Baumes van t ,

so Kann man die t besehreibende FunKtion au~ den T e i l b a u m aller Knoten e' mit

e < e'

einschr@nKen~ diese r e s t r i n g i e r t e FunKtlon besehreibt dann den S u b t e r m van t , der an der Stelle e gelegen ist. Daher Kann man den Baum van t auch dazu verwenden, die Subterme van t mit den Stellen zu indizieren,

an welchen sie auftreten;

Sinne sell damn auch yam Baum der Subterme van t g e s p r o c h e n werden.

in diesem

Die 8 e s c h r e i b u n g

van Termen durch B~ume macht Klar, dass man jedem Term einen K o m p l e x i t ~ t s g r a d zuordnen kann, wobei Variable den Grad o erhalten und der Grad van f IX] zahl oberhalb der Grade aller Glieder van X ist. s&mtliehe S t e l l e n z a h l e n n

die Kleinste Ordinal-

Van nun an werde vorausgesetzt,

dass

pasitiv sind; damit ist eln Term f IX] yam Grad 1 genau 1

1

dann, wenn alle Glieder der Folge X V a r i a b l e sind.

Man setze w a i t e r voraus, dass die Menge der O p e r a t i o n s s y m b o l e f

in zwei

Teile zerf~llt, also auch die Indexmenge I in zwei disjunKte Teilmengen I o und 11 . O p e r a t i o n s s y m b o l e fl mit IEI ~ heissen van erster Art, und solche mit ~ I 1 heissen van z w e i t e r Artj

ein Term heisst van erster oder z w e i t e r Art, wenn er mit e i n e m

Operationssymbol

dieser Art beginnt.

Seien nun L u n d zwisehen L u n d

R zwei disjunKte Kopien van T ; sei

die natOrliehe Bijektion

R 9 die j e d e m Term der elnen Kopie den gleichen T e r m der anderen Kople

W.

zuordnet;

sei

F = LvR

.

Felscher

Als Gradfunktion

in Bezug auf T ; dann wird 8 die kleinste aller n

, ieI

Auch die Elemente

137

auf F definiere

unendliche

man den Komplexit~tsgrad

regul@re

Kardinalzahl

oberhalb

yon F sollen noch Terme genannt warden,

jedoch

1

sollen Endomorphismen Kopien definiert

[oder Substitutionen]

sein,

sodass auch die Ergebnisse

selben Kopie angehBren.

Sei nun

{o,1}xI

Teilmengen ~ und B so, dass for jedes das andere in B liegt.

nur zwischen

IEI

=

Termen je einer der beiden

solcher Endomorphismen

A v B

eine Zerlegung

eines der Paare

Oamit erkl~re man die Zerlegung

F-F

--

{

IX) i n A l e g t ,

falls

( f

(I)EL

eA ] oder

[ f

<1,1> in ~ und

= A~B

IX)mR

--

beiden P~llen gehSre f

, indem man

I

[X]

und

eA

1

) gilt; --

Oamit gelten die Bedingungen

zu B .

[f],

[fff]. Man setze weiter veraus,

grad K[v)

so definiert

Art seien eine endliche den folgenden

; ml

positive

Eigensehaften

und

m"

dass for ;eden Term v v o n

sei, dass

[fv) und

Iv F] gelten.

zweiter Art ein Verzweigung~FOr ;eden Index I v o n

Zahl K[I] sowie vier Folgen ~'I , ~

gegeben:

tone Folgen von Ordinalzahlen K[I]

und

l

(ff],

in disjunKte

0

I

in den anderen

,

wieder der-

kill 2 n I

mit L~ngen

haben disjunKte

;

K'[I],

Wertbereiche

i'(I],

und Werte in

[o,n

als n

falls l

; K'(I]§ ]

;

=

B~ und

B"

1

Wertbereiehe

FOr Terme f IX] 1

mit

sind striKt mono-

1"[i]

1

haben disjunKte

, B~ , B~

m[, a[, B~, B[

K"(i],

erster

und ihre Werte seh8pfen

von erster Art definiere { [X]~B

und als K(I]

l

das ganze Intervall

man nun den Verzweigungsgrad

falls

%

f [I]~A

;

[o,n I]

aus.

K(f (X]] l

dann gelten auch bier

[fv]

1

und (v F ] . Alle fortan betrachteten genOgen:

Gilt So[el

deuti Z bestimmter

Probeweise

= q , sz(e]

= flip]

sollen der folgenden yon erster Art

Term f (X) vom Grad I , ein eindeutig

Termschluss-Bedingung

, so sind definiert bestimmter

ein ein-

Endomorphismus

g

I

von ~ mit

g[fl[l]]

stimmte Wohlordnung

= f1(gl] <eil

= f [~]

sowie,

i yon N[e]

wenn

$1[gX]EA

so

wenn

fl(gl]sB

so

s4(e)

=

im Falle

f1[~]sB

, eine eindeutig

be-

, und es gilt

{g[X[m~[s])],

g[X(a'~(t]]]*

I s
g[l[i]]

for

i : B'[s] I

, s
g[l(i]]*

{Or

i = B[[t]

, t
s4[e)[e i] = {

,

188

W.

Liegt flip] und

B[

Folgen

in L

als die linken ~"I

flip)

[respektive in R]

und

B~

9

Felscher

so bezeiehnet man die G l i e d e r der Folgen

[respektive rechten] als die rechten

B[

dann erfOllt, wenn

ss[e o]

[respeKtive linken] direkten Subterme von

sei, ist die Bedingung

eine Teil{olge von B~

und a'~

[vs] for Mengen von Beweisen stets

von erster Art ist.

Oie folgende Oefinition sell zu einer neuen Art von P r e b e w e i s e n fOhren, eine neue Art von Regeln erlauben. seien beide etwa aus L . t , wenn

T(o]

T[i-1] ist. erkl&rt,

: t ,

L-Subterm

Sei t ein Term und sei x eine Variablej

Eine endliehe Folge = x

und, for

T =


o
T(i]

die

t und x

i heisst x-Folge von d i r e K t e r S u b t e r m yon

In einer solchen Folge T heisst t auch L-Subterm von t ~ sei bereits

dass

T[i-I)

so heisst T(i] reohter]

~[n)

1

direkten Subterme und die Glieder der

Unter der zus~tzlichen Annahme 9 dass a~

eine Teilfolge von

~'

L - S u b t e r m oder R - S u b t e r m von t ist.

L-Subt~rm

[respektive R-Subterm]

d i r e k t e r S u b t e r m von

T[i-l)

[respeKtive R-Subterm]

d i r e k t e r S u b t e r m von

T[i-l]

ist~ ist

von t wenn

Ist dann

von t , wenn

T[i) linker

T[i-1] R-Subterm,

T(i]

rechter

T(i-1)

so heisst

L-Subterm, (respektive T[i]

[respektive linker)

ist. Es heisst x uniform in t , wenn x in j e d e r x-Folge

von t L - S u b t e r m von t ist oder wenn x in jeder x-Folge R - S u b t e r m von t ist; im ersten Falle heisst x auch L-uniform,

Ein P e a r

im zweiten R-uniform.

heisst eine

Peirce-Form

,

wenn

[p] eine x-Folge von t existiert,

[pp] die Terme einer jeden x-Folge von t , mit A u s n a h m e von x selbsts alle von erster Art slnd, deraus,

(ppp) x u n i f o r m in t ist, T[i)

~ x , gilt:

Term w der Folgen ~ T'[i'+1]

= w

Ist

ist

, pa'~

[pv]

for jede x-Folge T und for jeden Term

T(i) in A , eine x-Folge T

T(i]

: f1[~]

, so gibt es for jeden

von t und ein i' mit T [i ] = T(i)

P = < t s x > eine P e i r c e - P o r m ,

Peiroe-Subterme

,

so schreibt man tp for t und Xp for

x ; die Terme in den x-Folgen yon t heissen die P e i r e e - S u b t e r m e F o r m P beisst einfach, wenn Xp R - u n i f o r m

TIi]

von t 9 Fine Peirce-

in tp ist und wenn alle von x v e r s e h i e d e n e n

auch L-Subterme von tp sind.

Man erweitere nun die Oefinition eines Skelettes, den Wert 4 zulgsst;

gilt

So[e]

indem man for s

= 4 , so enthalte NCe] ein Element.

o

auch noeh

Alsdann erweitere

W. F e l s c h e r

men die Definition eines Probewelses, Gilt

So[e) = 4

indem man die neue Sorte yen Skeletten zul~sst.

, so schreibe man for s2(e](e]

s 3 und s 4 definiert,

139

wieder s2(e)

und mit dem oberen Neehbern e' von e enthalte s4(e)[e')

ein Element, for des dann selbst Sq{e) geschrieben werde. so[e) = 4

,

so

; welter seien denn

N(e) = {e'}

Endlioh gelte:

geneu

Fells

, und es gibt eine eindeutig bestimmte Pelroe-Form P

und einen eindeuti~ bestimmten Endomerphismus g yon T mit s3(e) = g[Xp)* ,

s3(e)eR

s4[e) = g(tp)

s4(e)EL

oder

sowie

,

s4[e) = g(tp)*

sl(e')

=

sl(e) FQr P r o b e w e i s e

falls Xp R-uniform in tp

,

Sq{e)eR

s2(e) ~ {s4(e),s3(e]

,

falls Xp L-uniform in tp

}

,

s2(e) ~ {s3(e]}

vom Typ I o d e r vom Typ I I

werden a u s s c h l i e s s l i c h

einfeche

Peirce-Formen

zugelassen.

Im Folgenden sollen Probeweise dieser neuen Art betrechtet werden.

Zur Oefini-

tion einer Menge ~ von Beweisen wird eine lest gegebene Menge P von Peiree-Formen voreusgesetzt,

aus der s~mtliehe Peirce-Fermen P stemmen sollen, die en Stellen e

mlt So(el = 4

yon Beweisen eus ~ euftreten.

nicht-einfache Peirce-Formen in P euf,

Treten im Falle des Typs III euoh

so muss voreusgesetzt werden, dass for je

zwei Elemente P und Q von ~ die Menge der in tp euftretenden Varieblen disjunkt zur Menge der in tQ auftretenden Varieblen ist, und welter muss ~ in folgendem Sinne ebgeschlossen

sein: sind P und Q eus ~ , ist t Peiree-Subterm yon tp , •

zu~leich

Xp Z-uniform in tp und t Z-Subterm von tp

mit Z gleich R oder Z gleich L , so

llegt euch

aus t~ entsteht,

in ~

, wobe• tQ[t

indem men RQ Oberall

dutch t ersetzt.

Welter mOssen for die neuen Probewe• FunKtion F 5 euch noch FunKtionen FSo und P51 beschriebene KenstruKtion bedeutet,

neben der im ersten Tell erKl~rten betrachtet werden.

Oie dutch F5o

dess man yon einem Probeweis S , der mit der

EinfOhrung eines Heuptterms erster Art endet, zu einem neuen Probeweis Obergehen Kenn, sofern nut an Stelle der Pr~missen des letzten Schlusses von S •

Probe-

140

W.

weise S i gegeben entsprechenden {1(g'l] mit

sind,

{i ~ fj

die start der Nebenterme

Nebenterme

enthalten.

Felscher

des dutch ein'es Substitution

Sei also S mit Kleinstem

for

i ~ j

des Hauptterms

, So(f]

und g aus der Termschluss-Bedingung.

= {t (p)

Sei <Sll

von S die

g' trans{armierten

Knoten f gegeben,

= 1 ' s3[{)

{t[gl)

sei N(f)

Hauptterms

= {{il

yon erster Art und mit ft [~]

i eine Falge von Prabeweisen

den bei der Bildung {ormaler

Summen einge{Ohrten

marphismus

K = 1

i
Bezeiehnungen;

sei g' ein

mit

Endo-

p

mit

von ~

s
s
. Im Felle

t
und

I im Falle

w. = g'[X[i]]* 1

II , so sei auch die Menge Fso[S,<Sil speKtive F51

i,g')

{Or

i = B"(t] t.

i , <wil

KonstruKtion

i

s [{] = 4 o

tution trans{ormierten

Peirce-Schluss

Knoten { und e ; sei der zugehBrige verschiedenen o s1[e)

Probeweis, l~ngert

Endomorphismus.

endet,

Oie andere,

dass man van einem Probeweis zu einem neuen Probeweis

redurch S ,

Obergehen

die durch eine Substiyon S enth~lt,

Seien also S und S ~ Probeweise

von ~

mit Kleinsten

Peiree-Form

und g

, der for die van Xp

Nit einer Menge H sei das Ergebnis

{ells Xp R-uni{orm

in tp , und gleich

. Oann sei F51[S,S~163

derjenige

indem man den Baum von S ~ um einen Knoten e T ver-

van S ~ dutch 9

Typ I oder

M,W,z'({t(I]]

des letzten Sehlusses

} {alls Xp L-uni{orm in tp

s~(e T] : g ' [ X p ]

vermSge

S ~ , dessen Ergebnis

mit @ Obereinstimmt.

und die Funktionen

i = B 1'(s) ,

Oann sei

' g'[{1(l)]

Sei g' ein Endomorphismus

H v {g'[tp],g'(Xp)*}

,g'(Xp]

{Or

und sei P die dadurch bestimmte

der aus S ~ entsteht,

S~(e T] = H fortsetzt

{ortsetzt.

i

*

[s])], g'[l(~'~(t])]

. Ist der fixierte

von diesem Typ.

und Nebenterme

s ({) : 4 o

Variablen

gleich

H v {@'(tp]

Haupt-

wi = g'(l[i]]

, t
hat zum Inhalt,

indem man einen zweiten Probeweis

mit demselben

sei

M ~ {g'[{t(l]]}

dar mit einem Peirce-Schluss hann,

K > 1

die {ormale Summe der <sil

vermSge
beschriebene

sei W die Menge aller g'[l[~

,

o T] = s [e T] = 4 , sl[e

s~(e T] = g ' [ t p ]

respektive

H u { Z' [ X p ] * } o

s4(e T] = g ' ( t p ] *

und diesam Schluss d i e P e i r c e - F o r m P und den Endomorphismus g' z u o r d n e t .

Die durch F5o und F51

b e w i r k t e n K o n s t r u K t i o n e n machen d e u t l i c h ,

E i n { 0 h r u n g von Haupttermen e r s t e r

A r t sowie d i e Vornahme yon Peirce-SehlOssen in dem

Sinne r e i n {ormalen C h a r a k t e r haben, e l s s i e nur yon den Terman { t [ l ) abh~ngen, i n denen b e l i e b i g e

dass d i e

Substitutionen

vorgenommen werden kBnnen,

respektive tp

W.

Felscher

Sei v ein Term aus R . Unter einem
yon drei endlichen

n

paarweise

mit

q _> o

Folgen

Peirce-Datum fur v versteht man ein Tripel

~ =
und den folgenden

verschieden

141

n,

~ = <l

Eigensehaften:

und alle Terme erster Art;

die

jedes

n,

Glieder

P

~ =

a n v o n _a s i n d

ist eine Peiree-Form, n

und t n ist

ein

v o n Xp

verschiedener

Peirce-Sobterm

yon tPn

; jedes

gn ist

ein

Endo-

n morphismus

von ~ mit

]* = v ,

gn[Xp

g n [ t n]

= an

falls

[Xp

term von tpn] oder

[Xp

n

L-uniform

und t n R-Subterm

von tp ) , gn[tn ]* = a n n

n Z-uniform

und t n Z-Subterm

u n d t n L-Sub-

R-uniform

n

van

tp

, Z = L

oder

Z = R . Ist

c ein

falls Xp n

weiterer

Term aus

R,

n

so versteht

man unter der

, wobei

~'

Transformation ven

=
n

dureh e

und g~ sich van gn

das Peirce-Oatum

nur for die Variable

Xp O

unterscheidet

und g~[Xp

]* = c

gilt,

und welter

~ =
n und

g~[t n) = b n

n

respeKtive

g ~ [ t n]

Nunmehr

erkl~rt

und vorgegebener

[vS] [v~]

und

[vs)

gemOss den

angegebenen

man Mengen 2 yon Beweisen

Menge P van Peirce-Formen

EOtlen

gilt.

bei vorgegebenem

w~rtlich

wie zuvor,

Axiomensystem verschOrft

Beweise

aus S

der FunKtionen , so liegen

F 5 , FSo

, F51

die S , <sil

auch die Funktionswerte

von S , S ~ = S l e ~ , S I = S l e I . Sei

So[e o] = Sole I) = I , s3[eo]*

= s3[e I] , s3[e o) = v , yeA .

Oann existiert

zu jedem w aus s4[e o) ein e W in N[e I] und ein Beweis

dass S W das SKelett

[s1[e W] - { s 4 [ e l ] [ e w ] ) ]

v

Sei

, sei

nun

[i,e, und

ausserdem

vEL

q > o ] Peirce-Oatum sei



die

von Sle W

{w*}

S W in

und das Ergebnis

hat. D e = s l [ e o)

for

v

Transformation

gebe es for das zuvor bestimmte

oder

i

in S

Sei Se2 , s e i e n e o, e I Knoten

derart,

aber

wie folgt:

Sind von den Argumenten und S ~

[vsP)

= bn

Ist

-

{v}

, Sei



D nR o

nicht

leer,

von

durch

c

e w in N[e 1] einen Beweis

[O o

(c}] W [sl[e w]

im[~]]

[D O

{c)] ~ [sl[e w)

[{v*) V

V

sofern

ein

nicht

so s e i

, Zu j e d e m

leeres

c daraus, w~s4[e o]

S~ in ~ mit dem Ergebnis

im[~] im[~]]]

V

[{c) ~ im[~]]

v ~ S l [ e w]

,

,

142

W.

Felscher

oder, falls es kein c gibt,

oder

O~

[ s l [ e w)

O

(sl(e

0

W

im[~))

]

-

({v*}Vim[a)]J

sofern

--

Oann gibt es in S auch zu jedem

wes4[e

--

von S~ hat

seheidet,

)

v e s l [ e w)

einen Bewels S %W , der das Skelett

0

und dessen Ergebnis sich yon demjenigen van S~

d a s s d a s E l e m e n t s q ( e l ] ( e w)

entfernt

dadureh unter-

und d u t c h w

ersetzt

worden i s t .

Oamit kann man auch noch eine Funktion F6o erKli~ren, indem man for S ~ und S%w aus [vs P) als F6o(S~

~)

einen der Beweise S %w

w~hlt.

Ein Beweis S sell sohlank heissen, wenn for keinen seiner Knoten e met

s [e) = 1 o

ein Nebenterm zugleich Parameter ist, es sei denn, dass der Typ I vorliegt, for den P ein Nebenterm aus R auch Parameter sein daz-f. Es folgt aus [Vs), dass jeder Beweis S aus einer Menge ~ von Beweisen mit Hilfe von F 5, F 4, F 3 transformiert werden kann. o Ivon

S

: ist e m i t

in einen sehlanken Beweis in

Oas sieht man sogleich dutch Induktion Ober die Rangzahl

So(e) = 1

der Kleinste Knoten von S , so kann man (v~) derart

auf die oberen Nachbarn von e anwenden,

dass Keine Nebenterme mehr Parameter bleiben,

und danach kann man eventuell verloren gegangene Parameter dutch AbschwBchungen wieder hinzufOgen.

Oamit formuliert man das

Eliminationslemma mit Peirce-Daten. Typs;

fixierten

und s e i

~ abgeschlossen

Knoten yon s l [ e e)

S

,

S~

v o n S~ '

Datum f o r

v

positiver

Lange

Olm R Gibt

sofern

j

veO

,

S1

gilt

0

Menge v o n s c h l a n k e n

des

Typs I

gelte

die

= Dl ~ { v * }

nicht alle

so se•

, auch S ~

o

Glieder

die

c in mit

, so s e i a

~ aus

v eD 1

dieses

von a seien

n

--

Transformation

Oln R

,

Schnittr~ngen

so g i b t

eines

Sei

, Sei

d ~1 < j

,

Beweisen

L•

Gj

sei

, nicht



Element

,

do ~ < j veO

das

Funktion

, s 1 = sle~

gelte

q j

es e i n

Seine

gegen

sl%

=

= DO , s l [ e 1)

o , d~ 1 o~

aus

im Falle

Sei

Se~ , s e i e n e

R ,

Ivl

= J

. FOr die , Sei

D

0

yon

, Ist

o, .

e1

Peirce-

leer,

i,e,

c ein



yon j

@

Sei

ein

, die

yon

Element

dutch

es i n ~ B e w e i s e

unterhalb

<

SchnittrSnge



Datum n i c h t in

j

c

,

S ~ und, in

der

W.

A b s c h l i e s s u n @ von S ~ , S 1

143

Felscher

unter den F u n k t i o n e n F 1 ..... F6o, Gj

liegen

und die die Ergebnisse

(~]

[o

im[~)) v im[b] v (01

o o

haben.

Ist

v

[{c}

-

{c})

u im[~]]

respeKtive

C{v} v

im[~]]

u D1

01mR

leer, so gibt es in S Beweise S ~ und,

sofern

veO

--

auch S ~} S~ , S 1 (~]

[0

C~]

o

mit S c h n i t t r ~ n g e n unterhalb von j , die in der A b s c h l i e s s u n g von unter den FunKtionen

F 1 ..... F6o, Gj

llegen und die die Ergebnisse

im[~}} v 01

o

Co e

[{v}

respektive

v im(~]]

u 01

haben.

Die Beweise

S~ , S1

und deren Rangzahlen noch allgemein-reKursiv.

W~hlt man hier im Falle

veO

Ergebnis wie Gj+I[v,sO,s1]

Sw , S~

und ihre Rangzahlen berechnen sich aus den

das leere P e i r c e - O a t u m <~,~,~>

o

, so hat S ~

dasselbe

.

Die Beweise S ~ und S ~

werden reKursiv Konstruiert.

Man nimmt zun~chst an

dass die Konstruktion for alle Beweise S 11 mit einer Rangzahl Beweise S ~ und for alle P e i r c e - O a t e n

o~ I < ~

for alle

ausgefOhrt sei; alsdann nimmt man in einer

zweiten ReKursion an, dass die Konstruktion mit S 1 for alle Beweise S ~176 mit einer Rangzahl

o~ ~ < o F

und for alle Peirce-Oaten ausgefOhrt sel.

e Sei zun~chst o I = o .

O~nn ist O ~ ein Axiom,

und da die Terme a n elnes Peirce-

Oatums positiven Grad haben mOssen,

Kann nut das leere O a t u m vorliegen.

I oI = o

schliesst man wie im Falle

[a) des @ e w S h n l i e h e n Eliminationslemmas.

o1 I > o

wendet man die Konstruktion

fern

s 3 [ e o)

gew6hnlichen werden s o l l e n .

oF > o Oberhaupt

. Man b e t r a c h t e definiert

Eliminationslemmas, Im F a l l e

(ba)

ist.

daraus reKonstruiert man S ~

zun~chst

den F a l l ,

Diese Situation

dessen Unterf~lle

schliesst

Im Falle

au~ S ~ und die P a r t i a l b e w e i s e von S I an,

(w

die in den oberen Nachbarn von e I enden;

S e i nun

Im Falle

dass

entspricht

vermBge

(v~).

v ~ s 3 I e o) g i l t , dem F a l l

(b)

sodes

und B e z e i c h n u n g e n

hier

Obernommen

man genau w i e im g e w B h n l i c h e n

Fall,

w o b e i man

in der InduKtion das v o r g e g e b e n e P e i r c e - D a t u m verwendet.

Ebenso verf~hrt man im

144

Falle

W.

[bba]

, sofern

a n = s3[e o] gibt.

[bbaa]

Felscher

, n~mlich nieht vss4{e o]

, gilt und es

auch kein a n mit

Tritt aber ein an mit dieser E i g e n s c h a f t auf, so betrachte man

die v e r s c h i e d e n e n anm , die in s4[e o]

liegen und v o n d e r

Gestalt

anm = gn(tnm]

for

direKte P e i r c e - S u b t e r m e von t n sind; da S ~ schlank ist9 sind sie von allen a n , versohieden.

N u n m e h r vergr~ssert man das gegebene Peirce-Oatum,

anm der Folge _a hinzufOgt, noch g u m

lauter

gleiche

ebenso _P um alle

gnm'

gnm=

diesem neuen P e i r c e - O a t u m w •

gn "

Ist

indem man alle diese

verl~ngert

c vorhanden,

wie im frOheren Fall

und schliesslich

so s o h l i e s s t

[bba]. Ist e nicht vorhanden,

so wendet man die I n d u k t i o n s e n n a h m e ebenfalls wie frOher auf Sleoo warden im Ergebnis von S' alle a t EA n

und alle a

n

nm

entfernt.

9 folgt aus der Oefinition der Peirce-Formen,

Wegen

nm

Man betrachte nun den Fall

Beweis mit dem H a u p t t e r m s3[e o) erh~lt man S w respektive S ~

[bbab), n~mlioh ves4[eo], a n = s3[e o)

9 so wendet

[v~)

liefert dann einen

9 wobei v als N e b e n t e r m verschwunden ist. Aus •

, indem man v oder c gegebenen Falles dutch A b s c h w ~ c h u n g

Tritt aber ein a n mit a n = s 3 [ e ~ ] auf, so v e r g r ~ s s e r t man das Peirce-

Oatum auf dieselbe Art wie zuvor. die Konstruktion [v~]

Oaher hat man somit

[~) w i e d e r auf Sieoo und Sle I an; im Ergebnis van S' liegt dann

v nebst aller weiteren Elemente van sq[e o) . Anwendung von

hinzufOgt.

a eA , also auch n

auftreten.

sodass der Typ I nicht vorliegen Kann. Gibt es Kein a n mit man die KonstruKtion

, Sle I an; dabe•

zusammen mit der Termsehluss-

Bedingung 9 dass ella E l e m e n t e aus s4[e o) unter den a bereits S w oder S ~w erhalten.

man m i t

Ist c vorhanden,

so wendet man bei der Induktion

[w167 mit dem neusn P e i r c e - O a t u m an; alsdann rekapituliert man vermSge

den ursprOngliohen Schluss, wobei der N e b e n t e r m c verschwindet;

ung mit v oder c erh~lt man daraus S w und S ~ bei A n w e n d u n g der Konstruktion

[~] s~mtliche a

sodass man damit bereits S w e r h g l t j

dutch Abschw~ch-

. Ist c nicht vorhanden 9 so entfellen o

und a

nm

, die von v v e r s c h i e d e n sind,

b e i Anwendung d e r K o n s t r u k t i o n

[~w

entf~llt

Oberdies v , und man erh&It S ~w

Im Falle v = s4[eoJ[eo•

(bbb] betrachte man als Unterfall

(bbbaJ

, dass es Kein i mit

gibt; dann ist die frOher in [bbb] erkl~rte Menge 12 leer 9 also 11

nicht leer. Gibt es kein e n mit

an

s3Ce o] , so wendet man die I n d u k t i o n s a n n a h m e

mit dem gegebenen P e i r e e - O a t u m for alle i , i
W. Felscher

145

verwendet man zur Bildung von S ~ Oberall die KonstruKtion S~

0berall die KonstruKtion

verm6ge

[v~]

.

(~]

, und zur Bildung von

[~j] ; aus den entstehenden S~l erh@it man S w und S w

Tritt ein a n mit

a n = s3[e o)

auf, und 1st a u s s e r d e m e vorhanden,

so vergr~ssert man das P e i r c e - O e t u m und schliesst wie seeben. so w~hle man eines der anm , und dazu findet man ein • mit

Ist c nicht vorhanden,

anm = Sq(eo)(eoi)

Bildet man unter V e r w e n d u n g der I n d u K t i o n s a n n a h m e nun des zugehBrlge Si , so ist dort a

nm

entfallen.

Oaher erh~it man S w und S w

von e ~ , die nicht in s2(eo](eoi) ist zu bedenKen,

auftreten,

, i n d e m man allf~llige P a r a m e t e r

dutch A b s c h w ~ c h u n g hinzuf0gt.

Hier

dass der Typ I wegen des nicht v o r h a n d e n e n c a u s g e s c h l o s s e n ist;

liegt der Typ II vet, so muss v der einzige in R gelegene P a r a m e t e r gewesen sein, sodass auch dann alle jene A b s e h w ~ e h u n g e n mit Elementen aus L geschehen. fall

(bbbb)

bleibt die MBglichkeit,

nehme z u n ~ c h s t

an,

dass

es e n t w e d e r

dass Kein

v = s4(eo)~e an mit

.) el

for ein i gilt. Man

an = sl(e ~ ) gibt,

an = s3[e o] gilt, jedoch XPn nicht d i r e k t e r S u b t e r m yon tn ist. so verf~hrt man wie im Falle vor,

[bbab].

Als Unter-

oder

dass

Ist c n i c h t

zwar

vorhaqden,

Ist e vorhanden und liegen die Typen I I o d e r

so wendet man bei der Bildung der S~ l

die Konstruktion

und zwar for

a n ~ s3[e o] mit dem gegebenen,

Peirce-Oatum.

Oamit bleibt v e i n

Nebenterm,

for

[5] for alle i , •

a n = s3[e e)

und verm~ge

III , an,

mit dem v e r g r B s s e r t e n

[v~) f0hrt man dann s3[e o]

wieder als H a u p t t e r m ein, wobei v verschwindet;

man erh~lt S ~ und S w

v oder c w i e d e r durch A b s c h w ~ c h u n g e n einfOhrt.

Liegt der Typ I vor, so wendet man

bei der B•

der Ss

for i mit

v = s4(eo)(eoi)

die Konstruktion

, i n d e m man nun

(5} anj for

die restlichen i , bei denen v noch als P a r a m e t e r zu e . auftreten meg, verwendet man 01 die Konstrukt•

(5) falls S w gesucht ist, und die KonstruKtion

sucht ist; aus den entstehenden S i die MBgliehKeit,

dass

c nicht vorhanden,

a n = sl[e o)

[w167 falls S w

erhalt man S w und S 55 vermSge gilt

(v~} 9

Es bleibt

und dass v als eines der a nm auftritt.

so verf~hrt man wie im Falle

III vet, so wendet man bei der Bildung der S~ for alle i die i

KonstruKtion

an; dann l~sst man bei der folgenden A n w e n d u n g von

term c verschwinden und erh~lt S w und S 55 dutch a n s c h l • der Typ I vet, KonstruKtion

so wendet man be• der Bildung der Ss

(~]

Ist

(bbab). Ist c vorhanden und liegen

die Typen I I o d e r (w

ge-

for i mit

anj for die restliehen i verf~hrt man wie •

(v~} den Neben-

Abschw~chungen. v = s4Ieo)(eoi)

Liegt die

ersten Tell dieses

14@

W. Felscher

Unterfalles

{bbbb].

Es bleibt ein weiterer im gew~hnlichen obere Nachbar

Eliminationslemma

Peirce-Schluss

Peirce-Oatum

vermOge

so sei

Endomorphismus

hlxo]*

oder

Z = R . Oaher ist

und Fs1

Z-Subterm

von tPn

einem neuen Endomorphismus = h(t o)

So(e oj bestimmte

und Xp

Man vergrOssere

weshalb

falls

von s4(e el entfernen

S 5 ebenfalls

= <3,1>

Peirce-Form

und beh~It

Z , Z = L

f{Xp

) = v , f{t n = n n Peirea-Oatum dureh s4{eo), auf $1e' und Sle I an. Bei

for

f'(t n)

ausgefghrter

InduKtion

Peirce-Datum

sowie

der x O in

h'(t O] = f'(to(tn]).

also b n

evident.

Gilt

um die durch

auf SIe' (v~)

wendet

,Sle I

Sei nun

So(eo]

Zur Konstruktion

g j alsdann

verm6ge

ist s3[e o)

h(t el = s4(e o) ,

so folgt

v ~ ss(e o} ersch6pft.

Induktion.

um s4(eo)

Peirce-Schluss

f', der sich aus h

for 0 und h' das Transformationsresultat

nut noch

P und den Endomorphismus

mit diesem vergr6sserten

Tell von gn zu

= sq[e o) . Ist nun h' der Endomorphismus,

, so ist die InduKtion

Peirce-Datum

nochmaligen

Wegen

mit demselben

von f ein Endomorphismus

und sonst mit h Obereinstimmt,

dutch eine evidente

das vorgelegte

for den dann

f'It n) = g~{t n) ; n a c h

sind alle M6glichkeiten

So(eo)

= s4(e el j seien

n

nun das gegebene

Kann man dutch einen Peiree-Schluss

Oamit

in tp

und s4[e o) durch f'[tO{tn ]] falls

h(t O)

f'(t n] , also b n * , sendet Mithin

Z-uniform

se• h der

Peirce-Oatums.

und wende dann die InduKtion

an die Stelle

durch f'[tn )*

dutch f'[tO(tn))

h{to]*

Teil yon h mit dem auf t n wirKenden

und f

tritt

zusammensetzt,

daher ersetzt

oder

{sodass der Typ III

Peirce-Fermund

Teile des gegebenen

% zusammenfassen,

.

<,tO[tn)>

und gnJ

s3(e el = a n

eine Peirce-Form, deren Auftreten in Beweisen n Oa die Variablen von t o und t n voneinander verschieden sind,

Kann man den auf t o wirKenden

der Transformation

man die InduKtionsannahme

an und fOgt dann den gegebenen

an. Gilt aber

h(t O) = s4(e o)

der

So[e o) = 4 ; se• e' der

n

ist.

f(tO{tn]]

n&mlioh

und SIe I

die durch

= an .

v ~ s3(e ej zu batrachten,

der a n , so wendet

n

aus ~ zul~ssig

8ioeo

auftrat,

Sle'

die zu a n geh6renden

ist dann t

Gilt

nicht

auf

O =

mit

und gn

(v~]

a cR n

und

des Falles

van e ~ . Ist s3(e el haines

mit dam gegebenen

vorliegt),

Spezlalfall

v = ss{e o}

= 4 , so erhalt

von S ~w

s4(e o) = 4

vergr6ssert

. Es ist klar,

man

bestimmte

man die KonstruKtion an und erh~lt

man

daraus

dass es genau

(~) S w167durch dieser Fall

W.

Felscher

147

ist, der die EinfOhrung yon Peirce-Oaten Oberhaupt notwendig macht. Es bleibt nun der Fall

sole o) = I

einzugehen.

zu behandeln.

Oabei wird es nGtig, auf das Verhalten yon sole 1)

Bezeiehnet man die oberen Nachbarn von e I dutch elj , so wird man ver-

suchen, die InduKtionsannahme auf Sle ~ erhaltenen Beweisen dann S w und S ~

und die S]elj

vermSge

(v~)

anzuwenden und aus den so

zu rehonstruieren.

Ergebnissen yon S I heine Peirce-Oaten zu transformieren solange unproblematisch,

Da in den

sind, bleiben diese I n d u K t i o n ~

wie das Element c aus

D I~ R

auch noch in allen Sl[elj)

auStritt, respehtive solange, wie zusammen mit

01 n R

auch alle

sind.

Man nehme daher zun~chst an, dass diese Verh~ltnisse vorliegen.

struktion von S w und S ~

leer

Oie Rehon-

aus den durch die Induktionsannahme gelieferten Beweisen ist

dann evident, sofern nicht ein,

s1[elj)n R

So[e I) = I , s3[e I) = v*

gilt. Tritt dieser Fall aber

so entspricht die Situation dem Fall [db) des gewBhnlichen Eliminationslemmas,

Im Falle v~A

Kann man dle dort verwendeten Bezeiehnungen Obernehmen.

Kann man das Ausfege-Verfahren

nicht sogleich auf Sleoo anwenden,

erh&it.

Im Falle

(dbab] ist S[eoo ebenso zu ersetzen;

induKtive KonstruKtion der Sw%, S w% sogleieh so einrlehten, dabei transformiert werden. zu transformieren, F~lle zurOck.

Im Falle

und den Fall

[dbba)

dass v mit v*

[dbaa]

ersetzt man die

dass aueh die Pe•

sind in S' ebenfalls die Peirce-Oaten

(dbbb] fOhrt man wie frOher auf die vorangehenden

und der Index o m i t S W = S]ew

frOheren

S %w

(dbba)

durch Beweise S w %

wieder die S % w

und fegt dann aus. Im Falle , indem man S' aus

und S %W

betrachtet.

entspreohend vor; zur Bildung der S B w je nach dem, ob bestimmten S %w

ves2(eoB(e w]

(db) so zu

dem Index 1 vertauscht wird. Im Falle , welche die InduKtion aus

Sw und Sle I liefert; alsdann betraehtet man start der alten S w gelieferten

{5) aus Sleoo und

ausserdem muss man die

Gilt vsB , so sind die frOheren Bezeichnungen im Fall

~ndern,

[dbaa)

sondern muss an

dessen Stelle einen Beweis setzen, den man mittels der Konstruktion Sle I

Im Falle

(dbab)

Sle ~ , Sleqo

die dutch (vs P]

verf~hrt man gem~ss dem

honstruiert und statt der S w und S w

Im Falle [dbba]

geht man dem frOheren

verwendet man die Konstruktionen

gilt oder nieht. Auf diese S~

wendet man dann das AusTege-Verfahren

kombiniert man jetzt S' aus (dbab] mit den S %w

oder (w

und die vermSge

mit Slelo an.

aus (dbba).

(~)

[dbab]

Im Falle

(vs P] [dbbb]

,

148

W.

Es bleiben das Auftreten formation und

nun die Ausnahmef~lle

anderer

der Peiree-Oaten

Sole I] = I

liefert

Elemente

oder

Vorhanden

fOhrt.

sffnd. Vermittelst

Iv P]

vermSge

sind v und v ~ entfernt (v~)

vorliegenden

sl[eljJ~R

der InduKtion

wieder

weiter

betrachte

ist, w~hrend

dass nicht

CSSl[elj)

an, dass

Peiree-Oatum

man den Fall,

es aber wenigstens respeKtive

s3[e I] ~ v*

gilt.

Sleoi

, Sle I

FOr die eljo. mit

nicht

v*ss~[e~., ~Jo)

Sle o

9

S Jl &

das Resultat

SleljI

Induktionsannahme [~) Oaten

jedoch noch Schluss,

Nachbarn

ejl mit

geblieben.

man nun den bei e 1

liefert.

Oie geschach-

sich dann vermeiden, Peiree-Oatum

Ergebnis

sowohl

wenn

liefert

v wie die Ele-

man S!j , indem man im[b] --

ein zweites

nicht

dutch Abschw~ch-

setzt man

S& Jl

elj von e I gibt so,

leer ist; man nehme zun~chst [w

liefert

oberen

SlelJo

"

liegt respektive [w167 mit gegebenem

Nachbarn

S w , Slelj 2

S~ J2

Naehbarn

S~3o

Mal an und definiert aus

gilt oder c gar nicht

mit dem gegebenen

S~ , aus denen man wie zuvor S ~ erh~lt.

v und v ~ sind verschwunden,

Aus den S~ Jo

c # sB[e 1]

einen oberen

in s1(eqJl)

FOr die restlichen

transformiert 9

dass

der KonstruKtion

mit leerem Peirce-Oatum

vorhanden.

Beweise

. In ihren Ergebnissen

repetiert

Oie Konstruktion

8eweise

v ss1[elJl ) , for die aber e n o c h sei

Subterme

von c abet erhalten

in dessen

sl[eljJ m R

aus

ist,

S!j

der S! l~sst J

aus ihm erh~lt

, Sle I

vorliegenden

o

mit dem gegebenen

einen Beweis,

sind;

Beweise

und damit S ~

(w

Sleoi

Mal an und konstruiert

durch Abschw@chung, einfOhrt

s3(e 1) = c

eoi von e ~

Mit den oberen

ein zweites

Nebenterme

dass

einfOhrt.

Als n~chsten vorhanden

allf@llige

Oie ~onstruktion

entfernt

so S w .

zur KonstruKtion

aus Sle ~ , Slelj

mente von im(aJ -ungen

worden,

der c wieder

leer ist.

dann n&mlich

und erh<

aus S ~ , Sleqj

, oder auch direkt

Schluss,

telte Anwendung

[w

aus

von c o d e r

bei der Trans-

Nachbarn

Peirce-Oatum

man den bei e

man nun die InduKtionsannahme

leerem Peirce-Oatum

den Fall,

die v aufbauenden

repetiert

wird,

zun~chst

Mit den oberen

transformiert,

durch den v wieder eingefOhrt

Vermittelst

gilt.

in denen das Fehlen zu SchwierigKeiten

Man betrachte

(~) mit dem gegebenen

die Peirce-Oaten

von e I wendet

zu behandeln, sl[elj)~R

sole I) = <3,1>

die KonstruKtion

S', i 9 in denen

in den

Felscher

Konstruiert

elj 2

S~ 32

s1(elJl)n R Peirce-Oatum

von e I wendet

als Resultat

. Dann bleiben jedoch

FOr die eli1 mit

ist

aus

man die

der RonstruKtion

in S~32

die Peirce-

st[eli 2) - (v*)

man S w167verm~ge

leer

[v~)

wieder

W.

Felscher

149

Nun blelbt noch die MBgliehKeit zu diskutleren, setzungen des letzten Falles gilt

ss{e I) = v*

dass unter den sonstigen Voraus-

. Oies b e e i n t r O e h t i g t die Bildung von

S ~ nieht, die unverOndert Obernommen werden Kann. Man betrachte als ersten Unterfall den, dass c in s1(e I) vorhanden ist. Oann ist e P a r a m e t e r und kann nut dann in einem sl{elj) fehlen, wenn

v*eB

OisKussion der FOlle

gilt und mehrere Pr~missen auftreten.

{dbaa} - {dbbb)

for

vEA,

Betraehtet man die

so findet man, dass eine InduKtion

mit diesen Pr~missen nur be• der Bildung der Sw%, S w% in {dbab), und damit in (dbbb} auftritt. Is~ nun S

w

oder S w v o n d e r

Art, dass c n i c h t

in seinem Ergebnis auftritt,

,

so

findet man die neuen Sw%, S w% , indem man die I n d u K t i o n s a n n a h m e ein zweites Mal, und zwar mit der Konstruktion < ~ )

und leerem Peirce-Oatum,

Als letzter Unterfall bleibt, dass s 1 { e l ) n R ganz wie soeben sehliessen. [dbbb) for veB

Im Falle

v*EA

leer ist.

auf S w

S

w

oder S ~, S w anwendet.

Im F a l l e v*eB

Kann man dann

lehrt die OisKussion der F~lle

[dbaa) -

, dass dort eine InduKtion mit der PrOmisse von S I nur be• der Bildung

von S' aus Sle ~ , Slelo die I n d u K t i o n s a n n a h m e Oatum anwenden,

in (dbab]

, und damit in [dbbb)

, vorKommt.

ein zweites Mal, n&mlich auf S ~ , Slelo

Hier muss man

und mit leerem Peiroe-

um dos neue S' zu erhalten.

Dos beendet den Beweis des E l i m i n a t i o n s l e m m a s mit Peirce-Oaten.

Oie Rangzahlen der hier Konstruierten Beweise S ~ und S ~ ihre AbhOngigKeit

von den Rangzahlen der A u s g a n g s b e w e i s e S ~ und S I wird durch Funk-

tionen beschrieben,

die w e n i g s t e n s

so stark wie die A c K e r m a n n ' s c h e FunKtion wachsen

und die deshalb nicht p r i m i t i v - r e k u r s i v sein kBnnen. sich auf die Rangzahlen o 3 zu beschr~nKen,

kSnnen. Eine F u n k t i o n f3 liefert,

muss

wachsen sehr schnell;

da die

' welche Absch~tzungen

Um dies zu sehen, genOgt es,

01

, 02

nur noch g r 6 s s e r werden

o 1 o~ ~ f 3 [ o 3 , o 3 ]

%~ o 1 0 3 ~ fS{OS,a3]

nun neben den im @ew6hnlichen E l i m i n a t i o n s l e m m a angegebenen Un-

g l e i c h u n g e n ouch noch d e r U n g l e i c h u n g fS(f3(x',y)§ genOgen.

dann

~

fs(X,y)

x' < x , y' < y

Betrachtet man nOmlich den so genannten A u s n a h m e f a l l mit zweifach anzuwenden-

der InduKtionsannahme, s•

§ 1

etwa in seinem ersten Fall

x, x i, y, yj

die o 3 der S~

durch

die

f(xi,Y)

o3

von

Sle ~ , Sleoi

s3{e I} = e , So(e I} = I , und , Sle I , Slelj

, dos o 3 von S ~ also durch

f{x.,y}§ l

, so werden abgesch~tztl

150

W.

for die a 3

Felscher

der S!j erhOlt man Absch~tzungen dutch f[f(xi,Y)+1,y j) , for das a 3 von S ~

also eine Abschatzung dutch

f[f(xi,Y)+1,yj)

+ I

Falle liegt hier also eine Ungleichung der Art

Bereits im einfachsten, f[f[x-l,y),y-1)

sie for die Ackermann'sche FunKtion charakteristisch ist. Anfangsbedingungen

y < f3[o,y)

Anfangsbedingungen tionsbeweis,

und

x _< f3(x,o)

63

Es ist zu vermuten,

Inspiziert man dan Elimina-

(s) , p. 211

oben , ebens

eine

die dasselbe Wachstum dar BeweislOngen nach sich zieht.

dass die UnmBglichKeit

einer primitiv-rekursiven

hier nicht in der zufOllig gewahlten KonstruKtion

Des Eliminationslemma R1imT'nationstheorem.

Oa f3 ausserdem den

for Sequenzenkalkule mit Klassischer Peirce'scher

Regel gegeben hat, so bemerkt man dort im Fall geschachtelte InduKtion,

vor, wie

$enOgen muss, sind auch die

der AcKermann'schen Funktion $egeben.

welchen CURRY

< fix,y)

finit~ren

AbschBtzung

sondern prinzipiell begrOndet ist.

zieht nun auch nach sich das

Sei S e i n e

Menge von schlanKen Beweisen eines fixierten Typs;

im Falle des Typs I s

das Linksprinzip.

Oann 8ibt es zu jedem S aus S

ein S j in S mit demselben Ergebnis wie S , des schnittfrei ist und in der Abschliessung yon S unter den Funktionen

F I ..... F6o

liegt.

Der Beweis ist klar.

Als Beispiele von Regeln,

deren HinzufOgung z u den Oblichen logischen Sequenzen-

kalKulen $mmer noch Schnitteliminationss&tze

in GOltigkeit belBsst, findet man daher

neben der bekannten Peirce'schen Regal etwa mit einfachen Peirce-Formen den Obergang yon

mit

nicht

M,

(x + y)vz

M,

[ x + y) A (x + z)

einfachen M,

~

x

Peirce-Formen 1 X

~

den X

~

x

zu

M

zu

M

0bergang

oder

X

VOIq

yon

ZU

M

M, ~ ~ ~ x

~

x

zu

M

M, ~ [ x ^

y]

-~

x

zu

M

M

~ ~ x , x

zu

M

~

X

~'

-~

X

9

VOIq

X

9

VOFI

x

yon

W. F e l s c h e r

151

Referenzen BACHMANN

55

: Baohmann,

H.: T r a n s f i n i t e Zahlen

(Ergebnisse der Math. N.F.

Berlin-G6ttingen-Heidelberg CURRY

63

: Curry, H.B.:

I)

1955 .

F o u n d a t i o n s of Mathematical Logic. New York 1963 .

FEFERMAN

68

: Feferman,

S.: Lectures on Proof Theory. S u m m e r School in Logic, 7o , I-Io7

SMULLYAN

66

: Smullyan,

R.M.

Lecture Notes in Math.

(1968)

: First O r d e r Logic

(Ergebnisse der Math. N.F.4@~

B e r l i n - H e i d e l b e r g - N e w York TAIT

68

: Tait, W.W.:

P r o c e e d i n g s of the

1968 .

Normal Derivability in Classical Logic. and Semantics of Infinitary Languages, Notes in Math.

72

, 2o4-236

[1968]

The Syntax Lecture

C A L C U L A B I L I T Y OF THE P R I M I T I V E RECURSIVE FUNCTIONALS OF FINITE TYPE OVER THE NATURAL NUMBERS ( a

revised

version

Yoshito

We

consider

arithmetic T-Z

in

has,

notion

of Z(t)

the

:

'

T-z

(ii)

For

(iii)

By proof

meral

n

a

notion

equal

is

to

a

w ,

neutral 73

of

)

intuitionistic

.

types,

natural

restricted then

a

This

Theory

more

restricted

by

number.

'

Z.

:

Vx~

~ each

closed

term

inspection without

of

cut

a

nor

t

of

proof

type

of

0,

Z(t)

induction,

T-Z

in

of

t

T

= ~

F- Z ( t ) Z,

,

.

we

for

can

ob-

some

nu-

.

As than

usual

is

remark

(i)

a

t

N-HA

( Troelstra

:

induction We

tain

0

of

type

the

type

Hanatani

variant

finite

besides Z

And

a

all

)

such

a

proof

calculation

calculability

of

without

the

cut

term

t

nor

,

induction

we

have

is

thus

the

none

other

effective

.

Acknowledgements This The

original

University

'

and

published

67,

71

cal

results

and

tude

to

The

author

a

version

result

was

presented

Toky8

Ky8iku

in

1966

Hinata

author

67

The by

ShSji

sincere

positive form

to

thanks

the '

in

version,

for also advice

and

of

here

) his

his

those

result.

Dr.

to

at of

the 1965,

of

Tait

metamathemati. sincere

inestimable

which

equally

beginning from

etc.

to

former

master-thesis the

differs

express

go and

author's

applications

Maehara

criticism this

as

method the

the

theorems

wishes

Professor

to

.

of

Daigaku

( cut-elimination

author's his

revised

of

The

for

is

Jean-Yves encouraged Professor

gratiguidance. Girard the Justus

Y.

Diller this

for

Benedikt

I.I.

of

the

type

for

from

theory

T-Z

type

induction

I.Z.

sion

of

zen

the

) with

the type

the

bound

of

the

following

(

a 1 ~

{

Tp

type

~.

g,

T c Tp

Tin0) note

p, (

[

~

them

advice

by

we

that

on

our

Z

V-elimination

the

is

T-Z

has

notion

~~ is. befor

Z

and

that

the

be

by

the

ver-

ZJ

terms

the

ZJ

defined

many-sorted

calculus

and

should

N-HA

T-

on

also

as

of

.

take

structure

theory

constant

sequential

Tp

type

sub-system

term

~

g ~

~,

T

of

Tp

}

classes

by

x~

Tml)

0

~

Tm

Tin2)

N

~

Tm

0 0 ~0

)

of

Tm~y

(Gent-

.

That

is,

3-introduction same

this

as

that

of

version.

inductively

by

the

. any

as

are

.

We

(~J 1, ~ 2 , - - - ,

terms the

type

is

classes

the

abbreviate a n ~

~ ),

and

union Tma

defined

of

the

terms

inductively

by

of

the

: of

,y~r

c Tp

denote

the

clauses

variables

r)

)---)

Tm e

seven The

its

denote

~rn--

cIass

The

following

and

.

,

c~ 2 . . . .

Tma

Mr

clauses

0c

The U

the

structure

following, (

type

type

) the

of

to

restricted

predicate

framework

. We

TpI)

In

and

finite

finally

.

finite

~~ is

more

unary

concerning

two

a

restricted

variable The

Tp2

the

find

first-order

of

N-HA

intuitionistic

application

the

an

T-Z

logical

Gentzen's

35

from

is

of

version

many-sorted

structure by

and

to

revision

this

functionals

schema

Description As

a

difference

usual

valuable

~

is

recursive

were

careful of

N-HA

0 , represented

the

his

T-Z

essential

sides

version,

revised

theories

primitive

The

the

formulation

The the

which

first

Difference

153

suggestions

Peppinghaus the

Formal

on

kind

formulation

writing

i.

his

Hanat ani

each

,z~

type , w~Z

u ).

belong

to

Tm~

,

( we

de-

154

Y.

Tin3)

K {~, T

Tin4)

S p , ~ , ~

Tin5)

Rcr

Tin6)

t

In

the

( ---

t n )

languages

(E)

as

,

our Z

follows

or

for

Vx~

Vy ~

Vx~ axioms Vx ~

Vy'r

V x (' V y l r , , 0 - - , , )

(DRZ),~

Vx ~ V r l ~

Axioms

axioms

are

have

an

L

[=,

of

.

~

T)

r

We

abbreviate

equality

Peano

' = '

and

( Nx=O

for

Z

(zP1)

z(o)

(zpz)

vxO(Z(x)

(ZE)

Vx0Vy ~

the

sub-

any

formula employ .

(E)

,

(D)

respectively

an

,

(P)

,

(Z)

. and

So

~ y=x)

V z c~ V w rr Ka,

T

,

(x=y

i z=w

Sp,C,,T

'

n xz=yw)

R~

:

r,,Txy=x

)

-(r

( S p.,,,rXyZ=xz(yz

VzP

~ x:y

~ Z(~x)

)

)

)

e~ xrz)

z )

)

)

^ Z(y) of

in the

A y--z

)

A D A [Nx]

sequence

L[=],L[Z]

and

:

Sequent-schema Vx0(

,

:

Vx 0 ~

0 ],

by

Z]

0 ( R~, x y ( N z ) = r (

(P4) Axioms

,

( R a xy0:x

0 ( Nx:Ny

manner

term

p

,

:

Vx 0 Vy

finite

Tm

by =

(P3)

We

we

without

(K

(DR1)o_

A

0 .... ) (tu) e

t n

for

VxIP'"--T)VyP

A[

,

t 1 t 2 ---

V z ~' ( x = z

(DS)p,~,r

(IndZ)

~, )

any

VY ~ ....

Defining

(Z)

~ ) ~

( p

,

denote

aquality

(EZ)~

(P)

.-

)

:

(x=x)

(DK)a,r

"

r

language

Vx~

r

)

~-

u

as

(E~)~

(D)__

a

t

non-logical

Axioms

(SAP)or,

p,

a

'

without

)

( (

--

constants

denote

The ( IndZ

,

predicate I Z

T

( ~, ( ~, 0T ' u e Tm(5

T m -

( t 1 t 2 ) - --

one

a ,

Tm

s

We

usual

c

following

unary

with

Tin(

Tm

e

e

As

is

e

Hgnat ani

the

notation

T - Z Ff0

of

V--A

)

induction

),

L [=,

~ Z(x)

Z(t) Z]

--

3x0

and 'F-' for

on

t of

:

(t=x

any the

example

axiom-formulae

Z

^A term

) e Tm

0

deducibility means of

T-Z,

. in

the

that

there

such

that

Y. Hanat a n i

the

sequent

(IndZ)

F 0 , F--A

supplementally

.

We

denote

sequence

.

sistent

,

T-

is

of

N -HA

denote

i.e.

by

without

T-

the

the

axioms

Consistency

of

case

the

schema

of

beginning

where

means

that

If-T

F

is

empty

T-Z

is

incon-

to

Proof.

Evident

.

If-T

+NJ

Gentzen

35,

Corollary

and

the

the

(Z)

nor

predicate

induction-free

logic-free

If-T-F- t=u

example,

apply

Prawitz

the

71 T-

1. 3. 3.

For

In

the

' t and

u

fact,

Z

subsystem

equational

calculus

, for

consider

any the

normalization

t , u c Tin. intermediate

theorem

for

sysNJ

(see

) . is

by

also

consistent

Remark.

measure

~=~

If-T-

~

also

result

Closed A CTm

by

and

closed

of

is

term

and

class

T

Nt=0

for

any

CTm

CTm

=

terms

with

t=u

is

. ' The

property (see

show

axiom

73)

Lemma

theorem

for

derivations

.

reduction

interpreted

of t h e

Troelstra

this

the

(E2)

by

proof

.

is

system.

then There

.

directly,

lf-T

as

in

proving

induction

on

some

consistent sketch

of

the

.

term

those

~- INt=0

term

models

if-T-

1. 3. 4.

Z. 1.

a common

can

If-T-

, when

Church-Rosser

the

Main

0

of t h e

If-T

normalization on

Lemma

for

other

We

t c Tm

system

to

the

many

of

any

a model reduce

satisfied

then

the

~

For

i. 3. 2.

give

2.

containing

without

0

Lemma

sort

T-Z

T-

T-F- t = u

rules

of

.

denote

I. 3. i.

t ~ Tm

subsystem

with

T-

Lemma

is

the

' T-ZF--'

identical

corresponding

Let

with

schema

w

Let

the

T-Z ~-A

that

essentially

i. 3.

a

LJ

.

Z)

are

in

accepted

by

Remark

We

tem

deducible

the

sequent

(Ind

is

as

155

is

a

a

denote

of

type

U{

CTmal

term the a

containing class The

a ~ Tp}

no

of

the

inductive and

by

variable closed

terms

definition the

six

.

of

clauses

and CTm

156

Y.

CTml)GTrn

CTmd)

,

each

place

of

Tm

in

2. 2.

Main

Theorem that

T

For

F-

being

exactly

as

Tml)

- Tin6)

hut

whith

o

result

.

Hanat ani

.

any

t e

CTm

0

,

there

exists

a

numeral

fi s u c h

t=5

The

following

follows

from

this

by

Lemmata

i. 3. I.

,

1.3.3. Corollary.

For

numeral

fi

such

Remark. tive,

proof

actual

2. 3.

t e CTm

that

Every our

each

any

If-T-F-

actual

will

Our

of

proof

(a)

t

(b)

T - Z F- Z ( t )

(c)

T-+

(d)

T

an

of

the

exists

cut-eliminition

give

example

Steps

, there 0 t=6 .

t c

and

process

effective CTm

one

only

being

calculation

one

effec-

process

for

0

proof

follows

the

steps

: (a)~

(b)~

(c)~

(d)

;

E CTm 0

In

the

(ZPl)

+ (ZP2)

~-t=fi

above

for

Remarks

(i)

(c)

is

(if)

(d)

implies

(iii)

(b)

does

(iv)

Our

proof

The

(i)

step In

say

that

'

type

0

each

type

a

0 ),

(a)

give

:

.

any

term

T- + (Z)~-

points

E Tm

0

Z(t)

of

of

fact,

terms

for of

any

the

u

form

such

that

K0,

UXT

. with

the

the

(a)~(c)

In

the

(if),

following

[ T-ZF-Z(t)}=

{ t

(iii)

above

relation

and

the

for

some

consis-

:

E Tm 0 [T-~

t=n,

f i } c T m 0.

proof_.

. the

T-ZF- Z(t)

'

fi for

(c')

(a)

together

will

view

to

imply

not

0 c {t e Tm0

2. 5.

stands

)

(b)

example

but

CTm

t

not

(b)

T-

A Z(x)

numeral

equivalent

( for

of

(t=x

.

satisfy

tency

some

schemata,

2. 4.

T-Z~- Z(u)

~-- 3 x 0

The

point

a

predicate

'

equivalence

between

represents

' t

of

the Fa

proof in

of L [Z]

is this such

(b)

and

a calculable step that

is

to

'T-ZF-

(d),

we

term assign F a (t)

can of to '

Y.

represents The

' t

is

assignment

form (ii)

to

Step As

the

by

an

apply

we

see

of

proof

by

is

a

to

restate

induction

on

the

interpretation

(iii)

Step

2. 4

,

term our

the

strong

tool,

this

a proof

of t y p e

step

in

cr .'

a general

definition

step

in

of

of

CTm.

we

of

not

Z

only

a

that

a very

method

, followed

of

mean

the

. Our

theorem

does

need

essentially Z(t)

interpretation

. This

for

is

T-Z

cut-elimination

calculus

much

second-

we

weak

use

a

system

for

.

(c)~(d) have

with

of

from

of t h e

too

We

(i)

second-order

predicate

proof,

calculable

us

the

(IndZ)

application

order

enable

157

(b)~(c)

elimination of

a hereditarily

will

suitable

Hanat ani

.

only

to

respect

to

consider each

the

purifications

predicate

of

symbol,

the

given

remarking

that

m

the

axioms

of

the

axioms

(ZPI),

the

cut-elimination

should and

3.

remark

the

3. 1.

T

awe

any

For

every

a e Tp

in

L [Z] (t)

Lemma the

of

e Tma_

3. I. 2.

- CTm5)

,

whith

evidently

properties

of

we

T- +(ZPI)+(ZP2)

.

main

reduction

F

(t)

.

with

t e Tm

inductive

a denotes

rules

a

:

,

(F~(x)

CTm,

we

T-Z ~ F o-

of

common

it is

main

definability

let

following

any

(F)

in

for

. The

following

Vxa

T and

LJ

tool

~(b)

and

the

Z(t)

:

the

1. 1.

F a

by

=

(t)

collection

CTml)

,

for

(a)

of

3. t

the

main

symbol

T

Definition

definition

Lemma

. Our

explicit

step

of

through

tive

the

_Proof

F

predicate

theorem

of

F 0

no

(ZP2)

consistency

formuIa

Then,

contain

steps

corresponding

establish

our

T (t) u e

are

all

formulae

Tm

as

:

F 0 (0)

(F2)

:

F0~

and a

T-Z ~

0 (N)

to

step

F a (u)

the

(a)~(b) ~T-ZF-

induc.

FT(tu) ,

for

.

provable

.

(FI)

D F T (tx))

follows

in

T-Z

, where

corresponding

(F)denotes to

158

Y.

(FB)o

.

(r4)p,,.

~

(FS)~ Lemma

3. 1. 3. As

by

3 . 1. 1

:

F(~,~_o)(Ko,~

:

F ( (p

:

F(o

and

is

~

,

it

3. 2.

So_ m e

Lemma

3. 2. 1.

T- +

formula

A[x"]

Lemma

3. 2. 2.

Let

(IndZ)'

:

A[0]

,

(IndZ)"

:

A[0I

, Vx0(Z(x)

any

for

where

A

stands Then

Remark.

The

These pect 3. 3,

axioms

T-+

in

T-Z,

(FI)

T-Z

and

to

formula

for

Z(t)

in

and

induction

3 . 1. 3

prove

any

(IndZ)"

] ),

t e CTm is

. evident

3. I. 2.

--A[t]

and

L[--,Z]

(IndZ)"

be

t,

u ~ Tm

as

follows

t7

"

:

--A[t] )),

Z(t)--A[t]

L [=, Z]

,

and

t,

for

any

term

T-Z~(IndZ)" is

evidently

are

therefore

the

strongest

form.

equivalent

with

res-

of

T - Z F- ( F 2 )

T - Z F- ( F 3 )

Fa

and

axioms we

3 . 1. 2 .

and

.

by

(DK)

apply

are

and

3. 2. 1 and

(IndZ)" .

evident,

T-Z~- (F4) applied

(DS)

rest

are

to

is

by

A

by using

To

to

are

= Fa(x),

3. 2. 2

similar

they

immediate

respectively

admitted

The

as

,

to

the

prove the

case

the the

(F5)

ease of

(F3)

and

The

only

pri-

.

4.

Proof

4. 1.

Description The

mitive

of

the

quantifier

the

VZ

formulae

are

1)

t , Tm 0 ~

2)

A

is

(b)~(c)

of f o r m a l

is

added

on

The

step

language

symbols

predicate

of

any

only

A[u]

any

and

o (A oA[Nxl

Lemma

A [ z ] = F a (RaxYz) (F4)

t=u,

o AINx

for

)

(Z)

T-Zt-

defining

F-

(IndZ)'

of

of

definition

T-Z

~ )

0 -- ~ ) ( R o

definition

of

in

schema

Proof

of

by

(IndZ)'

for T-Z~

schemata to

(Z)

Vx0(A

-- ~ ) ,

remains

properties

e Tm 0

, (p_.),p_T)(Sp,~

T - Z F- F a ( t ) ,

evident

3 . 1. 2 .

)

~,_~) (~,0

t ~ GTma

3 . 1. 1

H a n a t ani

are

terms The

theory extension

the

e Tm 0

T

,

2

of

variables

additional

and clauses

L[=,

Z]

(denoted the

by

@ ) of

corresponding of

unary

universal

inductive

definition

:

r

is

a formula logical

an

.

not rules

a formula, containing are

all

the

V2~ rules

V2r

A

of

LJ

is

a

formula.

extended

to

159

Y. Hanat ani

this

language

V2 @ A

sequence and

and

of

additional

occurrence

B

'

rules

V2.E

stand

for

:

but

@

denotes

the

result

~(t)

in A

for

F

containing

We

V2

:

A[~tx0B],r--@ V2~A ,F--O

a formula,

formulae

A[Ax0B]

L J2

the

FAA(~F) F --V2@

V2-1 : where

and

of

and

at

|

most

substituting

denote

this

, for

one

a finite

formula

B[t ] for

logical

, each

framework

by

. The

4. 2.

non-logical

Remarks

on

axioms

the

of

weakness

T

2 are

and

the

exactly

those

of T

cut-eliminability

of

L J2 The LJ

. The

from

the

that

a

logical

weakness

as

definition

of t h e

successive We

for we of

firstly

is

difference not

Our cial Lemma

of

4. 2. LJ2

4. 3.

And

: let

Let

the

our

LJ~-

Proof. fact

provable

of

(~(0)

^ Vx0(~(x)

4. 3. 1.

that

the

in

T 2 .

comes

example

excluded

.

in

use

the

and

theorem

of t h i s

classical

55),

and

tool,

version

limit

of t h e

secondly

intuitionistic

that

versions

.

is

to

obtain

some

spe-

@

on

transformed

Then,

@ ~ in

T-2

~ ~(Nx) of

the

o 3x0(t=x

substituting

as

follows:

^ ~(x)

) ).

Z "(t)

for

each

.

T-2~-F"-

@*

complexity axioms

formula

)

respectively ~

.

second-order

results

T - Z F - F -- @

]By i n d u c t i o n

F--

the

the F

the

L [=, Z]

T-Z

V2~

in

is

strength

provable

theorem

in

denote

Z(t)

for

of

:

Z "(t)

0 ~ denote

V2-I

(Takeuti

the

@ be

~

be

calculus

cut-elimination

for

restricted

of

F ,

the

classical

Let

occurrence Lemma

e0

following

F- F - - @

F',

to

extension

remark

rule

theorem

to

Interpretability

Z "(t)

the

the

of t h e

following

up

for

. We

proof-theoretic

application

cases

formulae

the

weak

predicate

the

between

essential

a very

second-order

ensured

induction

is

the that

explicitly

transfinite

is

in

Concerning

remark

the

a

L J2

application

suppose

L J2

L J2

framework

of t h e

(__Z)" a n d

deduction,using (IndZ)"

are

all

160

Y.

Corollary Proof

4 . 3. Z .

.

By

4. 4.

T-Z

4. 3. 1

(b) ~(c)

axioms

are

in

all

of

T-2

T- + (ZP1) Proof

5 . 1.

Explicit

Lemma

of

5. 1.

:

where

all

As

its

Remark were

step

by

the

.

The

~

of

based

is

But

we

5 . 3~

a term

:

(Z)

implies

(2')

:

u

satisfying

proof

5. 4.

in

Proof We

cy

of

T-

shall

.

Given

(Z')

(i) by

such any

that type

expli-

U ~- A [ t ] , .

calculus form, for

even a

(ZP2)

LJ

when

more

L J,

this

is

and im-

. the

logical

detailed

(ii)

F- 3 x 0 ( t = x

satisfying

t=u

base

reasoning

.

Z(u)

both

(1)

) and

by

Then

(2)

:

. when

T-F-

t=u

(ZP1) can

^ Z(x)

,

we

prove

the

following

:

,

+ (ZPZ)~

be

none

considering

Z(u),

other

than

a

numeral

.

s=s

the for

transformation each

of

occurrence

the

Z(s)

in

it.

.

show

Evidently a

the

.

substituting of

a ,

true

accomplished

Immediate given

.

satisfies

V-prenex

then

+

~

(l')

of

system

intuitionistic of

u e Tm 0

implies

Proof

sequent

.

+ (ZP2)

(ii)

(Z)

t c Tm a and

also

T- + (ZP1)

because

T-+

theorem

T- + (ZPI)

be

above

non-

have

concerning

of

a

an

need

(2)

(1)

is

are

+ (ZP2)F-

(i)

U

on

lemma

will

the

we

the

.

formula

T- + (ZPl)

proof

But

,

^ Z(x))

the

cut-elimination

.

T - Z F- Z * ( t ) .

in

4. 2,

(c) ~(d)

(1)

The

by

have

A Z(x))

formulae

F- 3 x 0 ( t : x

axioms

Suppose is

the

there any

is

we

.

reduction

there

4. 3. I, 3x0(t=x

subsystem

for

U

Main

5. 1,

the

A

classical

5. 2.

By

and

(ZP2)

non-logical

mediate

,

(ED)

stands

Proof.

.

I. 3. 4.

Therefore

+

Any

U ~-3xa A

.

definability

definability

(ED)

and

(ZPZ)-

L [ = , Z]

5.

4. 2

T - Z F- Z ( t )

T-ZF-(ZP1),

logical

cit

,

consistent

.

Suppose therefore

is

Hanat ani

cut-free

that the

(2) latter proof

implies is of

(2')

impossible Z(u)

in

or

the by

inconsisten1. 3 . 4 .

T- + (ZP1)

+ (ZP2)

.

Y.

By

the

in

it is

lae

subformula

in

either

the

obtain through This

in

least

the

L[=]

or

6.

Remarks

(i)

The

follows

:

sion

T-Z

on

original Let

denote

concerning

Z

,

I)

are

all

provable

with

the

type-free

3

but TC-Z

~- Z ( t )

(ii)

The

proof

the

sical

formal

via

the

If-TC-

above

, the

(iv)

Still

through

the In

any

formu-

evidently in

of

was

type-free

L[Z]

the

,

proof

.

essentially

as

(combinatory)

without

is

an

the

ver-

axioms

e CS by

to

a the

numeral the

same by

say

stratifiable of

t e CS 0 ,

sort

the of

is

of

be

of

such

all

to

one

the

clas-

reduce up

to

CS~

type ,

the E0

, and any

If-TC

in

~-t={~

passing

CSa

and

only

.

theorem

theorem,

denote

~

for

that

the

normalization

the

of

: Let

calculable

union

.

.

Then n

Church-Rosser

preserves

to

.

present

order

55)

TC-

numeral

our

induction

following

ours

TC-F-t=s

choice

in

(Takeuti the

as

the

was

as

0,N,K,S,R ~,

transfinite

result

(__F) a r e

constants

choice

the

unique

terms

where

caused

version

: Any

fact,

TC-Z,

unique

This

terms

reduction

following

can

purely

result

atomic

an

to

more

corresponding that

.

logic-free

, there

we

the

.

sub-system

in

results

closed

t e CS 0

this

the

difference

Takeuti's

of

are

parts

version

essentially

explicitly

The class

was

theories

Professor

(iii)

, its

for

detailed

calculability

so

formula

Then,

2)

except

or

redundant

of

and

in

proof

L[=]

every

.

version

TC-

a

in

original

TC-Z

proof,

, because

such

the

,

(F)

L[Z]

wanted

the

161

cut-free

purely of

we

a

From

proof

what

in in

.

one

elimination

implies

of

principle

end-sequent

at

Hanat ani

for by

stratifiability

, we

the

of

If-TC-,

the

fact

have

the

:

Lemma.

Let

restricted

to

If-TC-

If-TCCS

.

~ t=u

/CS

Then

~

denote for

any

If-TC-/CS

t,

subsystem

u e CS

5- t = u

~ ,

If-TC-

,

.

162

Y. Hanatani

References

G.

Gentzen

1935

Untersuchungen

fiber d a s

Mathematische K.

logische

Zeitschrift

39,

Schliessen.

pp.

176-210,

I. II. 405-431.

G~del

1958

I~ber

eine

bisher

noch

nicht

benfltzte

Erweiterung

des

finiten S t a n d p u n k t e s . Dialectica Y,

pp.

280-287.

I-lanatani

1966

Calculabilit4 de

type

des

fini s u r

Annals

of the

Science, S.

12,

fonctionnelles les

nombres

Japan

Vol.

3

r~cursives

naturels.

Association

N o . 1,

primitives

pp.

for

Philosophy

of

19-30.

Hinata

1967

Calculability

of p r i m i t i v e

recursive

functionals

of

finite type.

D.

Science

reports

Vol.

No. 226,

9,

of the pp,

Tokyo

Kyoiku

Daigaku,

S e c t i o n A,

42-59.

Prawitz

1971

Ideas

and

results

Proceedings sium.

Ed.

of the by

J.E.

(North-Holland

Wo W ,

ait

1967

Intensional

1971

in p r o o f

theory.

Second

Scandinavian

Fenstad

interpretations

of s y m b o l i c

Normal

form

for

235-307.

of f u n c t i o n a l s

Logic,

theorem

Sympo-

Amsterdam-London

P u b l . C o . ), pp.

Journal

Logic

32, bar

pp.

of finite t y p e I.

198-212.

recursive

functions

of

finite t y p e Proceedings sium.

Ed.

of the by

J.E.

(North-Holland Go

Second

Scandinavian

Fenstad.

Publ.

C o . ),

Logic

Sympo-

Amsterdam-London pp.

353-367.

Takeuti

1955

On

the

Journal pp.

fundamental of

the

249-275.

conjecture

Mathematical

of Society

GLC

I. of

Japan,

7,

Y. Hanat ani

A. S.

1973

163

Troelstra Metamathematical Arithmetic Springer New

York

and Lecture

Investigation

of

intuitionistic

Analysis. Notes

Vol.

344.

Berlin-Heidelberg-

.

5,1, r u e 78240 France

de

Montaigu

Chambourcy

OBSERVATIONS ON A RECENT GENERALIZATION OF COMPLETENESS THEOREMS DUE TO SCHUTTE

For K. Sch~tte on the occasion of his 6~

th

birthday

G. Kreisel

Summary.

The generalization in question is contained in [i~] which will be

referred to as [KMS]. function symbols or

It concerns the language of ordinary predicate logic without = 9 (i) two kinds of valuations

p~ namely

'total' ones (defined

for all formulas of the language considered)~ and 'semi-valuations' in the sense of Sch~tte's article [22]~ (ii) the complexity

~

validity (of a formula) for valuations

is considered provided

C~

of the valuations used~ that is~ ~

satisfies

some simple closure conditions depending on the exact choice of data determining the valuations of the kind in question -- and not only 'logical' validity~ that is~ validity for the class of arbitrary valuations 9 and (iii) instead of finite (derivation) trees 3 socalled ~ - f o u n d e d

trees are used~ built up according to suitable rules

Rp~ in particular~ rules with cut correspond to the total valuations mentioned in (i) and rules without cut correspond to semi-valuations.

As in [KMS] the familiar

completeness theorems~ for finite (well-founded)

trees built up by use of the finitary

rules

and logical validity (for valuations

p)~ are generalized to ~-founded

p-valuations of complexity

The generalization also applies to

Rp

trees and e-logic.

~

resp.

These generalizations suggest a reappraisal of some work in the fifties

on 'constructive' models by Mostowski [18] and Vaught [24]~ in which the complexity of a model is measured by its valuation on the atomic formulas. -- The significance of the difference between rules with and without cut is analyzed in 'extensional' terms by showing that9 for

~

of suitably low complexity~ for example~ for recursive

9 there are formulas which are true for all total valuations

C~

~ but not for

all such semi-valuations. The observations fall into 3 groups.

The most controversial ones concern

defects of traditional views~ for example9 (i) of the restriction to finite~ and~ in particular~ formal derivation trees (which would make the use of infinite founded trees teratologieal~

~-

if not 'illegitimate') and (ii) of so-called operational

semantics which provided -- in effect~ if not in intention -- the only permanent~ not merely heuristic~ significance for cut free rules:

this semantics can now be

compared to the usual model theoretical interpretation refined by 'definability' requirements

~.

observations~

'related' because earlier work was presumably influenced by~ and

Only slightly less controversial are the related historical

certainly formulated in terms proper t~ traditional views.

The least controversial

observations are about open problems stated by essential use of the concepts involved in the generalization.

Some of these problems are new~ some are more precise versions

G.

Kreisel

168

of questions scattered in [KMS]. It is certainly possible to read this article with very little previous knowledge of the subject provided one simply ignores the issues arising from the literature specifically cited.

But the principal aim is to serve the needs of those readers

who have a detailed knowledge of

our (venerable)

subject which goes back

> iOO

years; readers who wish to test to what extent this detailed work agrees with the expectations they -- or~ for that matter~ its founders -- have had of our subject. In short~ the article is intended to have pedagogic use for the so to speak logically over privileged (with genuine problems of their own); a class which is created by progress~ and therefore liable to be neglected by those who follow uncritically (once)

i.

'reasonable' pedagogic traditions.

Backsround.

Completeness proofs for (the usual rules of) predicate calculus

were given in the first half of this century by G~del and Henkin.

As far as clarity

of the arguments is concerned~ neither proof leaves anything to be desired. only room for improvement was in the discovery of significant refinements.

The A first

step was made in Hilbert-Bernays where (a version of) GSdel's proof of the completeness theorem was formalized in first order arithmeti~ but without special stress on the complexity of the predicates involved -- except that~ trivially~ they are definable in the language of arithmetic.

Around the middle of the century the

complexity was stressed explicitly in publications by Kleene [8] and myself [iO]~ in two different forms (which later turned out to be equivalent): both Kleene and O I ([iO] pp. 275-276) used the arithmetic hierarchy~ ~2; I also used a peculiar circumlocution of what we should now call 'recursive in the jump' ([iO]~ p. 7 8 and footnote i on p. 39).

Both were concerned with the complexity of the satisfac-

tion relation on the atomic formulas~ my main stress in [I0] being on the 'negative' results~ establishing a conjecture in Hilbert-Bernays II~ p. 191 ~ that there are consistent formulas of predicate logic which have no recursive model.

Evidently~

such a negative result is optimal for the measure of complexity on atomic formulas. Soon afterwards Hasenjager [6] improved the 'positive' result by formalizing Henkin's proof; he showed that any consistent formula has a model for which the satisfaction relation on all formulas (that is~ in Henkin's terminology~ a complete and consistent O extension) is A 2. Both the positive and negative results were steadily refined; v 0 perhaps most satisfactorily in terms of Ersov's hierarchy for ~2 sets; cf. [7]. Having been skeptical from the very start about the value of the piddling business of 'formalizing' convincing proofs~ I was very much taken by the possibility of formulating the interest of then-current formalizations in terms of definability theory, and coined the term 'basis theorem' [ii] for the type of result I was after. (It is too late now to change this mediocre terminology.)

It remains open whether

significant further improvements are possible by closer attention to the (metamathematical) principles of proof needed for establishing the 'definability' results:

166

G.

Kreisel

if so~ it seems certain that an imaginative classification needed~ and that current favourites 0 to ~2-predicates) are not adequate.

In the middle fifties a bunch of new completeness Hintikka~

Sch~tt%

and probably others.

as 'simplifications'

[i])

papers.

proofs appeared;

by Beth~

in view of the pellucid character

But already then (cf.~ for e x a m p l %

my review of

it seemed that -- in effect if not in intention -- a detailed model theoretic

foundation of the choice of lo$ical rules was being attempted; one starts with the (model theoretic) precisely~

of counter model (to the formula

A

considered)~

finite if and only if there is no such model. obstructions a complete

Precise details~

counter models~

It turns out that the possible

Of c o u r s %

'natural'

'sophisticated'

conditions).

'sophisticated'

data

or only on the atomic

of formulas were noticed by almost everyone

But the distinction

insignificant

for a given domain of individuals~

determines

'sophisticated'

value in the context of infinitary

between the 'extreme' and the

unless (something

after all~ the satisfaction

The use of the

are of interest

only the terminology differed (model sets~ semi valua-

data remains

the data is involved;

procedure provides

predicate on all f o r m u ~ s

the relevant classes

who looked at the procedure; tions~ consistency

and this then yields

which are given below~

somewhere between the two extremes discussed a b o v %

that is~ between the satisfaction

formulas.

and builds up a tree

involved and which is

to getting an infinite tree are easily enumerated~

set of rules.

in as much as they show that this determining

in terms of [KMS]:

notion of infinite countable model or~ more

of formulas which codes up to isomorphism all the models

ones.

is

or even its restriction

It seems fair to say that they were presented

This was quite unconvincing

of Godel's and Henkin's

of such principles

(first order arithmetic

like) complexity

of

predicate on the atomic formulas~

the satisfaction

predicate on all

data turned out to have some expository

languages

tions which are so to speak non-archimedean

[9]~ where simplifications

or distinc-

in ordinary predicate calculus have a

chance of becoming noticeable. Early in the sixties Schutte [22] choice of data discussed

proposed a much more imaginative

in the last paragraph;

ordinary predicate c a l c u l u %

but analogues

to be p r e c i s %

of those data for two particular

tions of 'applied' many sorted predicate calculus~ theories of types with and without cut. jectured

Takeuti~

is -- still -- not well known~

Sch~tte then observed~

of semivaluation

to those

'applied' calculi~

If a formula~

sense)~

(Since the significance in Appendix

of the notions of total and

that they are complete for validity

So Takeuti's conjecture

of the language considered~

(in the modified

formal

the latter~ con-

it is briefly discussed

for his natural extensions

all total~ resp. all semi valuations.

formula-

known as the impredicative who formulated

that the two theories have the same set of theorems.

of the conjecture I.)

use of the

not for the case of

is equivalent

is true in some semivaluation

it is also true in some total valuation.

to:

in

G.

Kreisel

167

Evidently~ this does not require that every semivaluation can actually be extended to a total valuation (though~ of c o u r s %

in general not of the same complexity).

But it raised genuine model theoretic questions concerning such extensions;

'model

theoretic' because structural properties of models are involved~ not merely the existence of some model satisfying conditions formulated in the language considered (and thus 3 by completeness~ reducible to a 'proof theoretic' question).

It also

raised the possibility that 3 once we have learnt to make use of our model theoretic knowledg%

Takeuti's conjecture would be almost as obvious as -- the truth of --

Gentzen's

Hauptsatz~ and thereby refute the particular consequences which Takeuti

(and others) obviously expected from a proof of the conjecture. reasonj Sch~tte's formulation~

But~ for this very

in terms of total and semivaluations~

provided a

(possible) new and perfectly legitimate use of these notions~ as a tool for correcting misconceptions about the nature of the conjecture. However 3 if both these notions are to be principal objects o f study, one will look for contexts in which they are no_._~tequivalent (not even w.r.t, their 'logic'); not~ of c o u r s %

contexts manufactured for this purpose~ but as venerable as possible.

Such a context is the theory of models of low (recursion theoretic) complexity~ and the generalization discussed in this article belongs to this subject.

2.

Minimal semi-valuations and cut-free rules.

The exposition below~ of the

'simplified' completeness proofs that appeared in the fifties~ should be regarded as an instance of the expository principle which dominates (the exposition of) mathematics in this century.

One starts with the proofs that the pioneers stumbled

on~ either by experimentation in the subject concerned or in attempts to solve problems outside it (for e x a m p l %

algebraic manipulation or algebraic theorems

inspired by geometric problems); and then one tries to find concepts to reformulate those proofs appropriately; known as: utiles.

Trouver le bog cadre~ d~gager les hypotheses

To the inexperienced the principle seems delicate because it suggest an

infinite regress:

one sees no reason why this choice of appropriate concepts should

not be subject to an analogous analysis (why for any explanation one should not want a further explanation).

As Bourbaki stress~ at least implicitly~ the discovery that~

occasionallyj we have no need for this may provide genuine information about the notions considered and about the structure of our thinking; the parallel in the natural sciences is obvious. It is in the nature of the expository principle above that changes i__nnterminology may be essential. We consider two sets

F+

and

P-

of formulas of predicate calculus without

and without function symbols (except so-called and put

P = F + U P-.

0 -- ary ones~ that i %

The logical operations used are:

~ ~ v~ 3

(and

=

constants) A3 ~

V may

be considered as 'defined'). -- The 'basic' question in the fifties was to give simple criteria for the existence of some realization (of the language of

F) which

108

G.

Kreisel

is a model of

{F : F C P +) [J {~ F : F C P-] ,

in terms of the formal underivability of -- what is usually written as -- F + ~ A more

'structural'

F-.

question is to code up all models of (*) of some suitable

kind, where, as with all structural questions, the proper choice of kind of model and of the data determinin$ those models is an essential part of the problem. We consider term models (also called F

'canonical' in [14]) in the language of

enriched by familiar Henkin constants~ eF~ associated with existential formulas

3xF

in

the enriched language.

Furthermore~

the models are required to satisfy

(according to the meaning of 'Henkin constant'): known, if (~) has any

that of the language generated by card

F.

F[x/eF] v ~

3xF.

As is well-

model at all, it has such a 'Henkin' model; its cardinal is F

or, equivalently~

the first infinite cardinal

For our principal aim (explained in para. i), namely definability refine-

ments, a more important fact is this:

The logical complexity of (Tarski's) adequacy conditions for truth (or satisfaction) and

3xF

in Henkin models is reduced:

if

D

is the domain

is, say, closed, we have

T(e3xF ") -> T(rF[x/eF] l)

instead of:

T(r3xF ") -~ (3a C D)

NB.

Sat(a : rF1) .

Readers who are interested in such matters will easily find the relation between

the term models above, introduced by Henkin, and older term models in languages obtained from

F

by adding suitable function symbols~ either by themselves or by

looking up the literature, for example, [14], Ex. 2 on p. ii0. The data used to determine those term models of (~), that is, the classes of formulas of the language generated from made to depend on

(F +, F').

F

which are assigned truth values, are

They are semi-valuations in the sense of Sehutte [22]

in which (*) holds, but -- in addition -- required to be minimal.

Roughly speaking,

the formulas in question arise directly in the lo$ical analysis of the formulas in f

or, more precisely, of subformula occurrences.

The difference from Schutte's

notion of positive and nesative part is dictated by the minimality requirement: a given subformula occurrence FI, F 2

F I v F2

(with the value:

is assigned a truth value (true), and for an occurrence of

with value:

true) only

F[x/eF]

is assigned a value.

for

true), only on____eeof the 3xF

(again~

Since formula occurrences

G.

Kreisel

are involved 3 the obvious I representation

Exercises.

169

is in tree form.

Before turning to the representations

of (minimal)

semi-valuations,

some easy facts should be verified -- or looked up in the literature,

for example,

in [KMS].

(a) Any semi-valuation 3 say SV (tacitly,

for term models of the kind

considered

here), can be extended to a total one; in fact, the class of total ones

compatible with SV are just those which agree with SV on the atomic formulas which are given a value in SV, and are

otherwise

tion SV can be of much lower (recursion

arbitrary.

theoretic)

complexity

compatible with SV; and almost equally trivially some recursive

semi-valuation~

contrast: increase

but in no recursive

the restriction this complexity,

(of arbitrary

much less precise: of a minimal

semi-valuations)

in

SV

a semi-valua-

than any total valuation

(F +, F-)

are true in some

total valuation.

at least for finite

minimal one which is contained

(b) Trivially,

(c) Perhaps,

in

to minimal ones does not

F, since for any

and primitive recursive

SV in

there is a SV.

(d) This is

there seems to be a quite sharp sense in which no proper subset

semi-valuation

has the (obviously)

essential

properties

of semi-valua-

tions.

(i)

Brutal representations

branchings

of minimal

of the trees will occur even if

formula gets the value:

false);

semi-valuations; F

'brutal' because

infinilte

is finite (unless no existential

equivalently,

infinite sets of formulas are

'put'

at the nodes. NB.

If

F

is finite,

is also finite; For each ~(F)

and

F C F, stage

On(F)~

F

~

of the representation

their elements are pairs,

true and false resp., leads from

the height of the trees used in brutal representations

it is determined by the lo$ical complexity

to

and sequences

F'.

Specifically,

+

involved

~-), the second is

o+

or

sequence of

O~ i.

whether

or

F0

3x ~ (F 0 v FI)

FI

or

Here

and

mines whether

~ , 3

vI

or

indicate,

gets the value false~ hence each

Fo[X/eGi]

or

of

by two sets F

with values

the sequences consist again of pairs:

s

Fl[X/eGi]

the

the mth stage of the analysis VO' Vl~ Vs if

is given a value (but not both),

value true for all Henkin constants

F'

E ?.

n, coding the logical analysis which

(to indicate whether

v0

is determined

of subformulas

of length

first element is

of the formulas

where

F0 v F2 and

vs

s

is an infinit~

is given the value true, is used when

Fo[X/eG i] v FI[X/6G i]

assumed to be in order

~, and

gets the s

deter-

is given a value.

i Opinions differ; but it would seem that the criterion of obviousness or elegance is useless here since the business is so simple that, realistically speaking, we understand any representation that comes to mind. Presumably a good choice can be made if one thinks about concrete implementation by a computer program.

170

G.

dO(+F)

= ~F}

if

F C l~+

The intended definition readerj

and

-

for

Oo(F ) = IF}

the well-foundedness LK, %

genesis

do(F ) = ~ .

Representation

vs

to the familiar

% > ~ -- and like the less familiar

languages

infinitary

LK~%~

in

are well ordered outwards).

of minimal

semi-valuations

of cut-free rules (of proof).

from the use of

are well founded is obviously related to

of the subformula relation (in contrast

for card

which strings of quantifiers (ii)

F C F-; otherwise

were envisaged).

The fact that the trees considered

languages

if

d+ and J- (n > O) is clear (enough for the intended n n -- subject to modification if~ for example, some computer

necessarily

implementation

and

Kreisel

in (i)~

We consider

in general

by means of finitary trees: finite sets

F.

As is evident

the set of distinct minimal

semi-valuations

which make

(~)

each

F : F C F+

true and each

has the power of the continuum.

F: F C F-

In any c a s %

false ,

at some nodes of the

sentation there are infinite sets of formulas~

'brutal' repre-

provided some existential

formula

is given the value false.

Exercise.

Given

semi-valuations

F+

satisfying

Nevertheless

and

F-~ show how to determine whether any or all minimal

(~) are infinite.

all minimal

semi-valuations

mined by

(F +, F-); specifically

formulas,

say

FN+

Warninss. fifties,

and

FN, at each node

familiar~

all minimal semi-valuations the property:

those trees, cultivated

the only point that needs verification

are coded (since completeness

if (~) has a model at a l l

sets seems pointless

or unordered

the sets

ability)

refinement

footnote

1 concerning much more delicate refinements

recursive

operations

when operations 2For example, F + U (B,B}

is that

only a tree with

iN, FN

theorem)

satisfying

are to be regarded

(but permitting repetitions)

in the present context,

(of the completeness

requires

since the

the tree codes some semi-valuation

(b) The familiar question whether

as ordered (as finite sequences) 'ordinary'

'coded' by binary trees, deter-

N.

(a) The rules for constructing

are perfectly

(~) above).

can be

by their infinite paths, with only finite sets of

or as

that is, for the on_._~e(defin-

discussed

in this paper; cf.

than such things as primitive

or -- as in [KMS], Part II -- the effect of those distinctions 2

on trees are involved.

the use of 'sets' with repetition ~- F" 0 (A~A}

derive

F + U (B}

and a contraction

~- ?- U CA}

as the rule Rep, discussed at length loc. cit.

rule:

from

has much the same effect

G.

Kreisel

171

Perhaps the most familiar construction is this (which differs from the literature of the fifties only in the use of constants of constants

c13 c2~ ...

N

O~ there is one node

is at level i

2n~ N

'successors ', if

if

FN

where 3xA

is

FN+

is

~ G~ A v B

cG i

often.

N'

or

3xA

FN+

is

(NB:

A[x/eG i]

~ G

A v B, namely

FN

and

F+ ~

F-.

If the node

or

~xA

namely


FN

--and < ~ ,

B} ~

FN .

is placed at the end to ensure that all formulas

2n+l~ N

has only i 'successor' at level

2n+2;

we have reSpo

is the first Henkin constant s.t.

A[x/cG i]

to ensure that

3xA

does not occur in

~N;

will be analyzed infinitely

the word 'successor' or 'descendant' conflicts with the familiar tree

orderin$ of nodes to

is at level

is placed before

FN+

(FN~ ~N>.

<~, A[x/eA]} ~-- FN ;

(Note that -- the analysis of -- F + N are eventually analyzed.) N

resp.

< }~ 'equipped' with

2n+l) if

~+F N [-
If the node

(F~ ~},

has

'successor' (at level

2

for our term models in place

indexed by numbers instead of formulas).

are regarded as sequences of the form At level

eG i

N

where the path from < }

to

N

'precedes' the path from < }

if and only if it is an extension of the latter.)

For any

N~ at odd or even levels~ if

+ resp. FN~

FN

is atomic~ then

N

has

just i 'successor'~ namely ~+ + <%, Fp

resp

Finally 3 the construction terminates at

+

<% N

if

i~+ N

and

PN

have an atomic

formula in cormnon.

Exercise.

Verify that there is a level

nO

beyond which the tree has only

atomic formulas at its nodes if no existential formula appears in any Adapting the terminology of tree of

F+ ~

F-

[KMS]~ we call the tree:

w.r.t, minimal semi-valuations~

canonical

FN. 'refutation'

in symbols

TCF

(r+r -) The seneralized completeness theorem says that~ for a class

C

(specifically:

of

172

G.

infinite paths of suitable formulas)~

Kreisel

closed under~

say~ primitive recursive

opera-

tions

T CF

is C-founded

(r+,F)

makes

if and only if no minimal semi-valuation

(F : F C F +} U ('~ F : F E F-}

In view of Ex. (c) on p. 6~ is a special case if predicates C

theorem i is large enough to be a 'basis' for socalled strict ~i

C

~

classes

C-founded binary recursive

are well-founded

and therefore

finite.

we have axioms (in the usual cut-free derivations

F N+ ~

of

FN

of inferenc%

except when

one requires

A[x/t]

For in this c a s % formulations

N~

logic)~ and the

of

N

is

3xA

has an even level and

for

trees)

at the terminal nodes

of predicate

are the usual rules (since ordinarily

_t that does not occur at all in

~A~ dictated by our interest

Readers

device.

First~

say

which are

all variables 3xA

~; in other words~

~+F N

in the particular

nor in

FN

in

term models

by Henkin).

Exercise.

is

of subsets of

trees (like our canonical refutation

F N+

N

should verify (by themselves

ture) that usual derivations

aN

The usual completeness

from the immediate descendants

for some

place of the specific considered

true.

'minimal' can be dropped.

or 3 equivalently 3 for

such that

~C

can be brought

one standardizes

or by inspection of the litera-

to the present form by the following

the given derivation by use of distinct variables 3

'quantified'

(3~)

at the node

that are never quantified.

N

where the

Then one replaces

aN

aN by

differ from cA

if

F~

and the other variables

arbitrarily by cG,. -- NB: This standardization l operation (for the usual topology on binary trees) and~ in particula~j

is a continuous

does not assume the trees to be finite. Readers will have noticed the following seneralization (for classes

C

of the kind above):

tree built up locally according F+ ~

F-

If

true~ that is~ F + ~

F-

This is so because an infinite path in recursively

from an (hypothetical)

Warnin$

C

'pure' 0~ i

Sur%

theorem

C-founded

which makes

IF : F C F +} U

is valid for all semi-valuations

~C.

can obviously be defined primitive

It would be quite simple minded to assume operations

theoretic

our proof~ resp. refutation

trees (just as formulas

letter alphabet).

T

SV C C

under primitive recursive

operations defined in familiar recursion sent context.

recursive)

SV.

(cf. also para. 3 below).

that closure of

of the soundness

is a (primitive

to the usual cut free rules with the end formula

then there is no semi-valuation

U [~ F : F ~ F-}

even

T

or~ for that matter~

terms is appropriate

in the pre-

trees can be recoded by suitable

and sequences

of formulas can be coded in a

But the trees and paths so obtained are

'sparse' (for familiar all purpose measures

under

'special' and~ p e r h a p %

of density).

Nothing is easier

2

G.

173

Kreisel

here than to concoct precise 'sensible' problems on refinements in familiar terms~ and thereby create a so to speak vested interest in the subject of those refinements. When it comes to choice of problems~ delay is usually preferable to error~ as (apparently:

3.

only) experience shows.

Open problems.

Granted the interest of models of relatively low complexity

mentioned in para. 13~ there are at least 3 areas of (potential) application of the observations in para. 2. First of all~ for infinite sets the familiar finiteness theorem plexity).

F+

we should expect some generalization of

(for total and semi-valuations of restricted com-

The paradigm for the generalization is provided by

founded but infinitary 'proof' trees.

Curiously~

~-logic~ with well-

this type of problem was not

mentioned in [KMS]. Secondly~ there is the grey area of those classes (i)

C

for which:

C-founded binary trees (built up according to the usual rules with and

without cut) are in general not well-founded or~ equivalently~ (total or semi) valuations

eC

formulas valid for

are not logically valid~ but

(ii) exactly the same formulas have

C-founded proof trees built up by means

of rules with and without cut. This grey area should be of interest~ in analogy with the heuristic value of cutfree formulations of ordinary predicate logic which were used in the discovery of Herbrand's theorem~ Beth's definability theorem or Craig's interpolation lermna. -NB:

This problemj mentioned and discussed in some detail in [KMS]~ may be deceptive

inasmuch as an improper choice of the classes

C

is liable to trivialize the probl~m

for reasons given at the end of para. 2~ for example -- but only as an example7 -if

C

is required to be closed under all recursive operations. Thirdly~ there is the extension (of 'logic') to mathematical theories which is

mentioned in [KMS] in the course of some 'reappraisals'~ but without an adequate bibliography.

9

V

.V

For example~ the apparently relevant papers by Pljuskevlcus [19] and

Rogava [21] were overlooked. 3Certainly~ there is a massive literature on recursive(ly presentable) models~ with satisfactory positive results when a model is given by its satisfaction relation on all formulas~ for example~ generally for the case of recursively decidable theories~ and 3 less easily~ for specific theories with prime models~ perhaps most recently by Harrington [5]~ there are satisfactory negative results of the kind mentioned in para. i~ for formulas which have no model with a recursive satisfactian relation on its atomic formulas. As pointed out in [KMS]~ the (negative) results of Mostowski [18] and Vaught [24] are less convincing~ by [KMS] and para. 2 above the validity predicates for recursive total and for recursive semi-valuations are certainly arithmetical since the property of~being a C-founded tree is arithmetical if C is the class of recursive (or~ say~ ~ sets.

174

G.

Kreisel

It is perhaps worth mentioning that the problems above~ involving infinite 'proof' figures~ are not intended to bear on the idea that our reasoning is best represented by infinite data, an idea with which I have flirted occasionally and which some of my betters~ for example, Zermelo [ ~ ] and fury.

have embraced with much sound

The simple idea behind the problems is the observation~ coming from

experience with ordinary predicate logic and its finite proof figures~ that the latter have generally been of marginal and at best heuristic use -- certainly

not

often for any (realistic) analysis of reasoning except for its most external features. A subsidiary idea is this:

traditional proof theory is full of 'transfinite inductions'

tacitly restricted to predicates of low logical complexity;

there is just a chance

that this restriction may have a sensible use in some context~

like the C-founded

trees above, where one has to do with figures which are not well-founded~ but Cfounded for

~.

C

of low complexity.

Adequacy of 'semantics' for predicate lo$ic:

an attempt at speaking for

the 'silent majority', SM for short~ of logicians (and mathematicians);

especially

in connection with the wide-spread view that this SM is apathetic or even hostile when it comes to foundations and~ more generally, towards logic. unconvincing;

The view is

such logical problems as the independence of the parallel axiom excite

the SM~ and there is little doubt that a proof (of the asymptotic version) of Artin's conjecture on zeros of polynomials in p-adic fields~ by Ax-Kochen and Er~ov, would have caused less exci~ment if it had not used logic.

Familiar

'explanations' of

this view which are current in the logical literature, range from the great depth and difficulty of (logical and~ in particular)

foundational problems to their lack

of meaning or precision; in short 3 however much the explanations may differ~ they certainly have a dramatic character.

The depth of foundational problems is commonly

supposed to be such that it is even difficult to find any coherent

'account' (of

the area of experience for which foundations are to be given), let alone a correct analysis or theory. Generally there will be some restrictions on the type of 'account', at least tacitly; in the case of predicate logic two types are familiar, namely model theoretic and operational semantics; the latter was touched on in passing by Gentzen (on the meaning of logical operations being determined by rules for their 'use')~ developed in an imaginative way by Lorenzen in

his operative Lo$ik and his dialosues (where

the 'use' consists in playing certain two-person games with the odds in favor of the defense)~ and somewhat more polished by Prawitz. 4

The area of experience invol~ed

gFor details, see~ e.g.~ App. i of [13] ~ where in particular various different ways are pointed out how this kind of thing can be done almost mechanically, starting with various versions of familiar realizability.

G.

Kreisel

is here supposed to consist of the linsuistic logical reasoning~ too often~

in familiar

from a genuinely empirical

restrictive arguments

and codified

178

evidence

point of view this

because we use constantly mathematical~

to prove logical

actual acquisition

theorems.

of knowledse

provided by our ordinary

formal systems.

Consequently~

is concerned

As cannot be repeated

'evidence'

-- there is something

proofs is prima facie implausible

'need' of further evidence.

(NB:

Of course~

proofs although~

in fact~ they are often

tion of set theoretic

axioms and auxiliary

proofs is the selec-

-- explicit -- definitions

to which those

are applied.)

Both model theoretic and operational to the 'precision'

blems orj in effect~ considered w.r.t,

are unimpressed:

-- minded

accounts have the -- unquestionable

logician in starting with

adequacy conditions:

the relevant

adequacy conditions.

completeness

semantics).

Each

a reason which would certainly

is sound and c o m p l e t e

(semantics)

'account' verifies

but~ in the case of operational

pro-

(of the rules

its -- own --

logicians

as Brouwer

appeal to both is that there are

for which the $iven

that is~ this evidence

--

'fundamental'

and soundness

Not only the SM but also such vociferous

not only many different meanings

principle~

in

these purely logical inferences may

but the obvious heart of the mathematical

attraction

'special' about

and certainly

omitted in practice:

inferences

set theoretic

the view that -- as far as

purely logical (first order)

be said to occur also in the mathematical

is itself highly

in particular~

'linguistic

evidence'

leaves the semantics undetermined

semantics~

in

it is wildly undetermined~

and simply not even remotely connected with the meaning we have in mind in ordinary reasoning when we use logically complex expressions However~

at all.

there is a subtler adequacy condition which forces itself on those of

us who have enlarsed our experience beyond ordinary or mathematical studying Gentzen's work:

There simply is no doubt about the

practice~

free rules (which of course still leaves doubts about an adequate analysis interest).

Certainly~

there is no evidence

that I know for supposing

found those rules by use of any kind of operational to it as a passing afterthought

semantics;

of this

that Gentzen

he may have referred

at a time -- the thirties -- when the slogan

meaning of a word is its use' was at least not as hackneyed as later. and I believe I am speaking here for a SM (albeit hypothetical) with cut free rules -- operational

by

'raw' interest of cut-

'the

Nevertheless

--

which is familiar

semantics was best in at least one respect~

for

Tin such circumstances~ at least the SM would suppose that what these differenL meanings (compatible with the $iven evidence) have in common would be 'superficial'~ and significant progress would require selection among these meanings; naturally in terms not included in the evidence considered. (Of course~ it is not disputed that -- the analogue of -- an operational semantics is perfectly natural for such things as lo$ic free numerical computation. It is the 'grand' scheme of regarding all mathematics as essentially computational that is in doubt -- and even the more 'modest' idea that computational features are senerally significant for mathematica~ reasoning.)

176

G.

Kreisel

the simple reason that it was the only semantics which gave a suitably distinguished place to cut-free rules.

In contrast, for model theoretic semantics these rules

had initially heuristic value, but no obviously permanent place. theorists), using clumsy jargon familiar to mathematicians~

The SM (of model

spoke of not understanding

cut-free rules though, of course, they knew as well as anybody what these rules were (was sie sind); they did not know any convincing use for those rules within the model-theoretic scheme (was sie sollen). The generalized completeness theorem discussed in the present article provides a candidate for such a use.

Of course it is coherent and capable of development;

that is not the issue at all.

What is needed to establish the proposal, is indicated

in the open problems of para. 3.

Or, put slightly differently, what has to be

established is that

the expressive power of first order language and its formal laws are not only (occasionally) useful in areas of mathematics which consider arbitrary structures, but also in those that fall, for example, into the'grey area' of para. 3.

It is probably superfluous to add that there is a quite separate area of study, of structural properties of proofs, which at least the SM does not expect to be analyzed adequately in model theoretic terms.

Remark.

The reader will have noticed that intuitionistic rules and their

semantics are not considered here at all.

Some reasons for this omission are given

in Appendix 2.

APPENDIX i:

Takeuti's Conjecture

The account below is autobiographical~ easy to write but b e c a u s %

not only because such expositions are

it would seem, the average (informed) reader has probably

had much the same experiences with this subject. Originallyj

the interest of the conjecture (for type theory without the axiom

of infinity) was said to consist in its connection with the consistency of type theory with the axiom of infini~

; this was later refined to an equivalence, by

primitive recursive methods~ with socalled i -- consistency [12].

Like most people

I felt very skeptical about the value of this connection for a consistency proof~ I although it may fairly be said that it was of value to Takeuti in the case of ~lanalysis. Sch~tte's discovery [22] of a relation between the conjecture and 'model theoretic' problems concerning semi and total valuations certainly

did not remove that skep-

ticism; once one considers this sort of problem in model theoretic terms at allj one

G.

Kreisel

177

is almost bound to speak of~ say~ e-models of the theory of types, and once one does this one has a trivial consistency proof anyway; cf. the obvious model theoretic proof of Gentzen's Hauptsatz (by use of the completeness of cut-free rules and soundness of rules with eut).

And certainly~ if one is interested in model theoretic

problems at all~ the particular one corresponding to Takeuti's conjecture would be low on anybody's list.

If anythingj the relation with model theory reduced my

interest (wrongly, inasmuch as I am interested in the present paper). When various proofs of Takeuti's conjecture were found~ consisting essentially of a closer inspection of Henkin's exposition of 'general' models for type theory (by Tait~ Takahashi~ also Prawitz)~ I was interested -- necessarily more so than by the originally proposed u s %

for a consistency proof:

surely there was something

that could be done in this direction~ and I mentioned possible directions to various people (obviously~ the circumstances behind the particular choices are of no general interest~ and they were ephemeral anyway). A first attempt was to find something of interest in the socalled theory of species~ the formally intuitionistic analogue of impredicative analysis.

Here one

had the old stand-by of 'explicit definability results' established by Prawitz [20]. One of my own favori~s was closure under Markov's r u l %

which I sketched very briefly

[12]~ and Girard [3], then a student at Paris, and others later worked out properly. A -- to me -- much more interesting direction,

suggested by Takeuti's conjecture,

was a so to speak radically nonmodel theoretic problem~ nowadays called normalization procedure (in contrast to the original normal form theorem) namely whether certain specific (cut elimination) procedures terminate.

In

1969/70 I

learnt a bit, from

Prawitz, about natural deduction systems which I had previously ignored on being assured that they were 'equivalent' to the ones I knew; cf. [26] to see how far this is from the truth.

Whatever doubts remained about any special significance of

natural deduction formulation%

it was at least clear which normalization procedures

were to be considered.

1969/70 Girard,

Also in

who had learnt GSdel's functional

interpretation from Shonfield's book Mathematical Lo$ic, worked out such an interpretation for the theory of species by use of functionals with a Erima facie somewhat peculiar type structure.

Again~ whatever doubts remained it was clear that (i) the

work should be brought to the attention of proof theorists~ in particular at the Oslo Symposium 1970 ~ and

(ii) if the work was basically correct it should be easy

to adapt it to a proof of normalization of the theory of species formulated style of natural deduction. normalization,

6

in the

I myself was particularly taken by one application of

to finding the realizations of existential theorems provided by a

6For people with practice in such systems; (ii) had a striking proof since I mentioned (ii) to Martin-L~f (and Prawitz) in the train to Oslo and gave Girard's text to Martin-Lof~ who already during the Oslo conference convinced himself that (ii) could be carried out.

178

G.

Kreisel

given proof [13] J later extended by Mints normalization

problem had displaced

This impression questions

[17].

turned out to be wrong in the following sense:

about self-referential

sentences

for cut-free analysis

know for which classes (of generally undecided be proved in analysis

At this stage it seemed that the

the normal form problem.

itself -- whereas

sentences)

the termination

dures considered cannot be so proved (for the classes answered

for certain

it was necessary

Takeuti's conjecture

of the normalization

in question).

accordin$

in contrast

to lo$ical form).

to procedures

by Girard in [4] where~ Finally~

however~

to classical

[23] for analyzing

kind of structural

properties

analysis)

referred

is left implicit.

APPENDIX 2:

procedure according

to

was made in Chapter III of Statman's

properties

of proofs of

~l-theorems;

the

to at the end of the main text above.

Intuitionistic

To what extent are the considerations intuitionistic

The work was greatly extended

use of the normalization

structural

proce-

to the content (of the end formula~

the basic procedure

a really convincing

logical form (adapted dissertation

according

proce-

This was

in [16] for certain useful classes by means of primitive recursive

dures which we called normalization

to

can

Rules

of para. 4 inadequate

because they neglect

rules altogether?

Put differently: Should we do better if we looked for a view that applies equally to classical as to intuitionistic

rules? (though obviously such a view would not be close to

actual reasoning at the present time where

the relative

importance

of the two kinds

of rules differs enormously). On (i)

present knowledge

I am doubtful

For Kripke models

in finding analogues

results

that there is great difficulty

to para. 2 even if some slightly subtler notions are needed

than minimal semi-valuations. in the models

for two reasons:

it seems hard to believe

themselves~

(There may be a difficulty

if one is not interested

but merely wants to use them as a tool for certain formal

such as closure properties

since such results

tend to require the formaliza-

tion of model constructions in the theory considered.) (ii) Heyting~

For the originally which involves

intended

logic seems simply uninteresting. look at proofs~

intuitionistic

interpretation 3 by Brouwer and

proofs -- and not expanding universes

in particular~

If one wants to know about proofs~

at their structural

properties

only at such statements 3 as those of first order logic~ quite i__nndirectly~ namely only via the interpretation NB:

as in (i) -- predicate

This view is of course absolutely

one should

and relations;

not

into which proofs enter

of the logical operations.

orthodox (for Brouwer's

--

original intentions).

G. K r e i s e l

179

REFERENCES [I]

E. W. Beth, La crise de la raison et de la logique, Collection de logique math. A, No. 12, 1957; rev. JSL 23 (1958) 35-37.

[2]

J. Y. Girard, Une extension de l'interpr~tation de Godel ~ l'analyse et son application ~ 1 .i . . / ellminatlon des coupures dans l'analyse et la theorie des types, pp. 63-92 in: Proc. Sec. Scand. Logic Symposium, ed. Fenstad, Amsterdam 1971.

[3]

, Quelques resultats sur les interpretations fonctionnelles, pp. 232-252 in: Springer Lecture Notes 337 (1973).

[4]

Three valued logic and cut elimination: the actual meaning of Takeuti's conjecture, Dissertationes math. (to appear).

[5]

L. Harrington, Reeursively presentable prime models, JSL 59 (1974) 305-509.

[6]

G. Hasenjager, Eine Bemerkung zu Henkins Beweis fur die Vollstandigkeit des Pr~dikatenkalkuls der ersten Stufe, JSL 18 (1953) 42-48.

/

.

1

[7]

0 G. C. Jockusch, Hi-classes and boolean combinations of recursively enumerable sets, JSL 39 (197~) 95-96. See also recent numbers of: Algebra and Logic.

[8]

S. C. Kleene, Introduction to metamathematics, Amsterdam, 1952.

[9]

H. J. Keisler, Model theory for infinitary logic; logic with countable conjunctions and finite quantifiers, Amsterdam, 1971.

[I0]

G. Kreisel, Note on arithmetic models for consistent formulae of predicate calculus. FM 37 (1950) 265-285 and Part II, Proc. XI Int. Congress of Philosophy, 14 (19531 39-39.

[ii]

, A variant to Hilbert's theory of the foundations of arithmetic, Brit. J. Phil. Sc. 4 (1953) 107-127.

[12]

Church's thesis: mathematics, pp. 121-150 in: Amsterdam, 1970.

a kind of reducibility axiom for constructive Intuitionism and Proof Theory, ed. Myhill et al.,

[15]

, A survey of proof theory II, pp. 109-170 in: Logic Symposium, ed. Fenstad, Amsterdam 1971.

[14]

G. Kreisel and J. L. Krivine, Modelltheorie, Spinger Hochschultext 1972.

[15 ]

G. Kreisel, G. E. Mints and S. G. Simpson The use of abstract language in elementary metamathematics; some pedagogic examples, pp. 38-131 in: Proc. 1973 Boston Logic Colloquium ed. Parikh, Springer Lecture Notes 453; vgl. PPS.

[16]

G. Kreisel and G. Takeuti, Formally self-referential propositions for cut-free classical analysis and related systems, Dissertationes math. i18 (1974) 4-50 .

[17 ]

G. E. Mints, On E-theorems, Zapiski 40 (1974) 101-118; siehe auch PPS.

[18]

A. Mostowski, On recursive models of formalized arithmetic, Bull Ac. Pol. Sc., cl. III, 5 (1957), 705-710.

[19]

A. J. Pljuskevlcene, A sequential variant of R. M. Robinson's arithmetic system not containing cut rules, Proc. Steklov Inst. Math. 121 (1972) 121-150.

9

v

.V

Proc. Sec. Scand.

180

G. K r e i s e l

[20]

D. Prawitz~ Some results for intuitionistic logic with second order quantification rules~ pp. 259-269 in: Intuitionism and Proof Theory, ed. Myhill et al.~ Amsterdam~ 1970.

[21]

M. G. Rogava~ Sequential variants of applied predicate calculus without structural deductive rules 3 Proc. Steklov. Inst. Math. 121 (1972).

[22]

K. Schutte Syntactical and semantical properties of simple type theory~ JSL 25 (1960) 305-326.

[23]

R. Statman, Structural complexity of proofs~ Dissertation~ Stanford~ 1974.

[24]

R. L. Vaught~ Sentences true in all constructive models~ JSL 25 (1960) 39-53.

[25]

E. Zermelo~ Grundlagen einer allgemeinen Theorie der mathematischen Satzsysteme~ FM 25 (1935) 136-146.

[26]

J. Zucker, Cut-elimination and normalization, Annals of Math. Logic 7 (1974) 1-112.

[27]

E. G. K. L o p e z - E s c o b a r and W. Veldman, I n t u i t i o n i s t i c completeness of a r e s t r i c t e d s e c o n d - o r d e r logic, this volume.

[28]

D. Prawitz, Comments on G e n t z e n - t y p e procedures notion of truth, this volume.

and the classical

P.S. At least two papers in this volume - [27] and [28] which were added to the list of references above at the last minute - are d i r e c t l y r e l e v a n t to the present paper, for the f o l l o w i n g reasons: [27]: The p a r e n t h e t i c a l w a r n i n g in (i) of A p p e n d i x 2 (of the present paper) does not apply if, instead of Kripke models, some such variant as that of [27] is used which can be proved to be complete by formally i n t u i t i o n i s t i e methods. - Remarks on [27] (to avoid any misunderstanding): The use of [27] just m e - ~ o n e d is not affected by the w e a k n e s s e s of its general i n t r o d u c t i o n concerning c o m p l e t e n e s s of formal rules, say, R w.r.t. (intuitionistic) validity or provability. Contrary to p. 2 of [27] this, obviously, does not require each ~ (method of) proof, #, to be actually formalized by some R - d e r i v a t i o n , or - in Heyting's language cited lee. cit. - to be 'embraced' by R; it is enough that some R - d e r i v a t i o n , p o s s i b l y quite d i f f e r e n t from ~, has as endformula the p r o p o s i t i o n proved by ~. More generally, it can not be assumed that possible uses o f [27] will depend on any (close) r e l a t i o n between its semantic and the o r i g i n a l l y intended m e a n i n g of i n t u i t i o n i s t i e validity - quite apart from the fact thatthe latter suggested itself before anybody had much experience of the subject (and so its r e l e v a n c e has to be tested in the light of later experience). [28]: Many readers may find the present paper more useful after first reading [28], whe#e the completeness of out-free rules w.r.t, validity for s e m i v a l u a t i o n s is proved in more detail than here (and also reading Part Z of [KMS] for its - c o r r e s p o n d i n g - details on the completeness of rules with out w.r.t total valuations, and on the incompleteness w.r.t, the latter of cut-free rules, for example in the case of r e c u r s i v e valuations). - W a r n i n g s on some implicit assumptions in [28] which are capable of being tested, but have not been checked: (i) The particular, usual choice of cut-free rules may be uniquely determined in a much less s u p e r f i c i a l sense than that of [28], p. 2, i. 3; namely - as e x p l a i n e d in [KMS] - up to c o n v e r t i b i l i t y by means of suitable rules p r o v i d e d completeness for a s u f f i c i e n t l y wide class of c is required (in analogy to 'glamor' theorems of modern m a t h e m a t i c s where global analytic conditions, here taken to c o r r e s p o n d to a suitable

G. Kreisel

181

choice of r ensure an algebraic structure). (ii) Para. 4 of [281 (pp. 14-15) assumes that the role of socalled side-formulas should be analyzed in terms of constructivity in its modern sense, which requires restrictions on the (metamathematical) methods of proof; in particular, the strategy of the present paper, which makes dist~ctions only in terms of definability or 'complexity', is inadequate. But this assumption has not been tested. (Indeed, before accepting the assumption, one would first test the strategy of [KMS], Part If, where not the complexity of the derivation trees themselves is used as principal tool of analysis, but definability and other properties of the transformations of proof figures into proof figures). - These warnings evidently do not affect the following more limited aim of [28 ] and of the present paper w.r.t. cut-free rules (and of [KNS], Part I, w.r.t, the usual rules with cut): to show how 6iven encodings, by means of finitely branching trees, of all minimal semivaluations (resp., of all countable total valuations) indeed d e ~ r m i n e the rules in question. As a by-product, the relevant completeness properties of those rules are evident, once one has convinced oneself that all valuations of the type considered are included in the given encodings. P.P.S. (added at the very last minute). The paper by G.E. Mints Finite investigation of infinite derivations, Zapiski 49 (1975) 67-122, contains the ~etails ~of tr~usformuations sketched in Part II of [15] and extensions of the E-theorems of [17], for example, to the functional interpretation introduced by GSdel in Dialectica 1958 (and much besides).

STRONG NORMALIZATION FOR ARITHMETIC (VARIATONS ON A THEME OF PRAWITZ) Daniel Leivant Mathematisch Centrum, Amsterdam The Netherlands

We refer to GENTZEN's natural deduction system for intuitiouistic arithmetic (GENTZEN [36] w

PRAWITZ [71] III.I), for which we give a variant of PRAWlTZ's ([71]

app.A) proof of strong normalization. The main point of departure of this note from PRAWITZ's treatment is this: we define (an analogue to) "strong validity" (called below "stability") explicitly in arithmetic, instead of giving an inductive definition. The raisons d'etre of this variant seem to be: I. It is an alternative, which some people might like (e.g., I do). 2. The formalization of restricted versions of the results within arithmetic is direct (compared to TROELSTRA [73] IV.5,I.4). 3. A more economical measure of complexity on formulae, for the definition of "strong validity'~ is obtained (2.1). As a consequence the role of implicational complexity in normalization proofs is stressed (7.3.). 4. This kind of treatment is applicable to infinitary derivations, where inductive definitions seem to fail altogether.

(This matter will be treated in detail else-

where). In section 6 we indicate an even simpler variant of the proof, but a variant which works only for the disjunction free fragment (and for which 4. fails). A number of metamathematical corollaries (such as consistency of Peano's

Arith-

metic) require normalization only for the negative fragment (i.e.- where B and v are not used). The reader wishing to look only at this fragment may obtain a simple proof of strong normalization for it by reading only the following: l.l(i)-(iii);

1.2; 1.5; 1.6.

2.1; 2.2(i)-(iii); 2.3-2.9. 3.

except 3.12(ii).

In 4.2 consider case (i) only, and take i(H) := w(H0) + w(Hl). Cases [a], [b] in the proof become one trivial case, and case [c] concludes the proof. 4.3. is also trivial. 5.1; 5.2. I am indebted to J. Diller and to J. Zucker for a number of helpful remarks.

D.

183

Leivant

I. REDUCTION STEPS; STRONG NORMALIZABILITY

The reductions assumed

of 1 . 1 - 1 . 3

are defined by PRAWITZ

to satisfy the convention

on parameters

to be replaced by others in course of reductions 1.1.

Detour reductions

z0

z 1

(i)

A0

AI A0&A 1

Z. l A.l

>

([65],[71]).

of PR[71]

All derivations

1.2.4, and parameters

are

are

so as to conform to this convention.

(i=0,1)

A. 1

[A]

A

Z

[A]

(ii) A§

A

B

Here [A] is a set of some open assumptions charged"

of Z of the form A, which are "dis-

(or "closed") by the indicated § Also, no open assumption of A A in [A] (compare LE[73] p.14 and ZU[74] p.36). Z

is discharged

Z(a) Aa

Z(t)

(iii)

> VxAx

At

At

(iv)

Z

[A 0 ]

[A 1]

Z

A.i

A0

A1

C

C

[A.] i A. i c

AoVA

1

>

c

(v)

Z

[Aa]

Z

At

A(a)

[At]

3xAx

B

A(t)

B

B

>

(i=0,1)

184

D.

Leivant

1.2. induction reductions

if the normal form ~ of t is ~, and where

Am l

a(a)

Z

Ao

A(Sa)

Ao

rt,o

(i)

F t's is a logic free derivation for t=s. (Cf. e.g. TR[74] 2.2.33, 2.3.]I for the

t=o

existence and uniqueness of the normal form

At

At

of t and for an eventual formalization of Am Z

A(a)

AS

A(Sa)

(ii)

this reduction).

Am A(a) A(Sa)

Ao

if ~ is Ss.

As A(s)

At

F t'Ss

A(Ss)

t=Ss At

].3. Permutative reductions A

A

A

(i)

- -

A

(Fi) i

BE

A

w

-

-

p ~

[Fi)i

>

p

E

B BE

B

B

Al

A2

Al

A2

A

A

A (Fi) i - p B

A

(ii)

vE A (Fi) i - - p

> ~

(Fi) i p B vE

B

B

where P is an elimination rule. Note that the premise A of p in (i), (ii) above is always the left, i.e.- the major one. 1.4. Semi-proper reductions [A 0]

[A I]

A0

g1

[A.] (1)

AoVA |

B

B

>

(2) B

(i=0,1)

[Aa] >

3xAx

z

B

B

[Aa]

1 A.

A B

(These reductions preserve the derived formula, but may alter the set of open assump-

D.

Leivant

tions, except if the indicated set of discharged they are identical with PRAWITZ's

185

assumptions

([71] 3.3.2) immediate

is empty,

in which case

simplifications).

1.5. Inner reductions If A is a proper subderivation

of E , A > A' by one of the above,

and E' comes

from E by replacing A with A', then E 9 E'. We say then that ~ ~ E' by an inner re-

duction; A > A' by I.I-1.4 we call a main reduction. 1.6. A is strongly normalizable (s.n.), > A2 >

... > An is impossible.

minimal n satisfying 1.7. Remark.

if there is natural number n such that A ~ A I

If A is strongly normalizable

The treatment below may be modified

of permutative

reductions,

(for the case

of §

generalization

to apply to a more general definition

where 0 is allowed to be any inference rule except induction

such a reduction may, however,

of the derivation).

we write ~(A) for the

the above condition.

For applications

is superfluous.

alter the set of open assumptions

of the strong normalization

theorem, however,

this

We therefore prefer to treat the restricted definition,

allowing a greater clarity of the proofs.

2. IMPROPER REDUCTIONS,

STABILITY

2.1. A measure of complexity The measure ~ on formulae is defined by recursion on their length: ~(A)

:= 0 for A atomic

~(A&B)

:~ ~(AvB)

~(VxAx)

:~ ~(3xAx)

~(A§

:= max[~(A)+],

:= max[~(A),~(B)] := ~(A~) ~(B)]

For a derivation A with a derived 2.2. Improper reductions

formula A we also write U(A)

:= ~(A).

*)

Assume that the notion "stability"

and the reduction-step

>o are defined

for deri-

vations A such that ~(A) < n. For A s.t. ~(A) = n we then define

(i)

EO

El

AO

AI

>~

E. A.l l

(i=0, I)

A0&A I

*) Similar notions have been used byH.R. Jervell,

by P. Martin-Lof

and by R. de Vrijer.

186

D.

EA] (ii)

B

A EA] E

-to

A§

whenever

J

,

is stable

A

B

Z(a)

E(t) At

Aa

(iii)

Leivant

for every term t

VxAx E

(iv)

A.

E

l

(i=O, ! )

A, 1

AoVA1 E (v)

At

>o

At

3xAx

Note that these reductions have a combinatorial

do not preserve

the meaning of derivations.

They only

role in the proof of strong normalization.

2.39 We write A >'r A' if for some n _> 0

A -- A0>r A l'zr ...>~ An - A', where>~ is either

or>-. 2.4. Stability It is seen outright A >>

that if A >~ A' then U(A') <- ~(A). Hence the definition

A' uses the notion of stability only for derivations

may define: A is stable if A > ~ A '

from A by substituting

stable derivations

2 96. A is stable under substitution 2.7. Lemma. A is stable i f f A ~

F s.t. ~(F) < ~(A). So we

~ A' is s.n..

2.5. We write A ~-+ A* if A* is obtained free in A and then substituting

of

terms for parameters

for some open assumptions.

(s.s.) if A ~-+ A* ~ A* is stable9

A' implies that A' is stable.

2.8. Lemma. EVery s.s. derivation is stable, and every stable derivation is s.n. . 2.9. Lemma. I f A' is a subderivation of A, and A is s.n. then A' is s.n. and

~(~') _< ~(A). 2.7-2. 9 2.10. Remark.

are immediate

from the definitions.

If A is stable,

then every >~-chain

starting with A is finite (this is

easily proven by induction on the usual logical complexity of the derived formula of A).

D.

Leivant

To prove the converse one needs, prima facie, the fan theorem, uniform bound required

in the definition

servative over Heyting's Arithmetic alternative

characterization

so as to obtain the

of s.n.; the fan theorem is, however,

(TROELSTRA [74]). Note that, in any case,

of stability

3. TREATMENT OF INTRODUCTION

187

conthis

is H 1 I"

INFERENCES AND INDUCTION

ao(a I ) p where p is an introduction-rule an atomic (Post)

If A =- ~

3.1. Proposition.

rule or the replacement rule, and A 0 (and A I) are s.s., then A is s.s.. Proof.

By 3.3, 3.7, 3.11 and 3.12 below.

3.2. Lemma. If A0,AI are stable then so is A B

A0 A

A1

-- A

B A&B

Proof. By induction on ~(A0) + W(Al).

If A > A' then this reduction

is necessarily

an

inner one,

i

v

A0 A' z A

A1 B A&B

where v(A~)+~(Ai)

A O A'

say, then

< ~(A0)+~(AI) , hence A' is stable by induction hypothesis.

A'

is stable by assumption.

By 2.7 A is stable.

If A ~o A'

D

3.3. Lemma. I f A0,A l of 3.2 are s.s., then so is A. Proof9 Immediate from 3.2. 3.4

Definition.

D

Let EA] be a derivation,

"

where [A] is a set of open assumptions F

A

of A

F

of the form A. We say that A is s.s. at [A] if for every stable derivation A' [A] is A stable. 3.5 9 Lemma. Let EA] A be s.s. at EA], EA] A ~ [A]' A' where [A ] ' is the set of copies of ele-

ments of [A]. Then A' is s 8. at EA]' Proof.

Immediate by induction on ~(A). (Note that the same P is substituted for every F occurrence A ~ [A] in 3.2, and that no assumption of F maybe discharged inA in[A].) D --

3.6. Lemma. I f

A

E~] i s s . s . B

a t EA], then

188

D.

Leivant

[A] A

Z -

B

A§

is stable. a v

Proof.

By induction

on v(A).

If Z > E' ~ ~

then v(A')

< v(A), A' satisfies

dition of the lemma by 3.5, and we are done by ind. hyp..

the con-

If

F Z ~o 2' = EA] A

(F

is stable)

B

then Z' is stable,

since A is s.s. at [A] by assumption.

3.7. Lemma. If E A]A is s.s.,

Hence by 2.7 Z is stable.

then

B

[A] A = l B

A§ is

S.8.

Proof.

.

Let

EA**] Z ~-+ 2" -

A, B

A* § B* A is s.s.,

hence A* is s.s. at [A*],

3.8. Lemma. If a is free in Z(a),

so by 3.6 Z* is stable.

So 2 is s.s..

Z > Z', then a is free in Z' (if it occurs there) and

Z(t) 9 Z'(t) for every term t. 3.9. Lemma. If A ~+ A* and a does not occur in any open assumption of A then A ~-+ A*[t/a]

for every term t.

The proofs

of 3.8 and 3.9 are immediate.

3. I0. Lemma. If a is free in A(a) and A(t) is stable for every t then Aa A(a) -

Aa VxAx

D.

Leivant

189

(if at all a correct derivation) is stable. Proof.

By induction

on v(A)

(as in 3.2).

If

A' (a) E >E'

-

Aa VxAx

then ~(A')

< v(A) and by 3.8 a is free in A' and A'(t)

the induction

hypothesis

by assumption. 3.11.

A' is stable.

By 2.7 2 is stable.

is stable

for every t. Hence by

If E ~o l' - A(t) then 2' is stable outright

D

If A(a) is s.s. then so is Aa

Lemma.

A(a) E

Aa

-

VxAx

Proof.

Let A*(a) I ~--+ E* -

A a

VxA*x By 3.9 A~-+ A*(t) required. 3.]2.

for every t, so A*(t)

is stable.

By 3.]0 then E* is also stable,

D

Lemma.

A (i) I f A t i s

A A F t's A A At ond At t=s . (ii) I f AA is s.s., then so are A , A AvB BvA BxAx As A (iii) If A is s.s. then so is ~ p where 0 is an atomic (Post) rule. Proof.

s.s., t h e n s o a r e

Similar

3.]3. Lemma.

to 3.2-3.3. (Note thatF s't is logic free, hence stable outright).

E is stable, and for every term t If A~

[At] is s.s. at [At], A(t)

then

[Aa] E

A(a)

H ~ A5

A(Sa)

IND

At

is stable for every term t. Proof.

By induction

on ~(I) + ~(A) + T(H), where T(H) is defined

T(t)

:= O,

if t is a term and for no term s

T(t)

:= T(S) + l, if ~ is SS,

~ = Ss,

as follows:

as

D.

190

T(H)

Leivant

:= T(t), if the main inference-rule

of the derivation H is IND,

with t as a proper term.

NOW if H > H' by an inner reduction in the proof of 3.10 H' satisfies hypothesis

then T(E') = T(H), 9(Z')+9(A')
the assumptions

and as

of the lemma. Hence by the induction

H' is stable.

If t = 0, H > H'

as in 1.2(i),

then H' is stable by 3.12(i)

and the assumed

stability of Z. If [Aa] A(a) H > H' = Ao

A(Sa) [As] Ft,Ss

A(s)

t=Ss

A(Ss) At

then T(H0) < T(H), so by the induction hypothesis s.s. at [At], so K' is stable by 3.]2(i). 3.14. Proposition. Proof.

By assumption A(t) is

The lemma follows by 2.7.

If %, A(a) of 3.13 are s.s. then so is ~.

3.14 follows 3.13 like 3.11 follows

4. TREATMENT

H 0 is stable.

3.10.

OF ELIMINATION-INFERENCES

4.0. Notations

and definitions.

For the sake of brevity we shall skip cases for disjunction-rules, be treated in complete analogy to the 3-rules. subderivation

of H, and h(H)

denote

which are to

Let H0(H I ) denote the left (right) main

the height of H (as a tree).

If H0,H 1 are s.n., we define for H the measure i(H) byi(H) := <~(H0),%(H0),w(HI)> (where <,,> is the lexicographic 2 .~(~0) + ~.X(n0) + ~(~I)).

ordering.

One may also think of the ordinal-measure

Let Z z 3xAx" Z [At] . stable under z at [At] if A is stable, A IS 0 At

Z ~ ... ~

then

BxAx

[ 2t ]

and whenever

is stable.

A

4. I . Lemma. If [A]is stable under Z at [A] then (i)

if [A]), [A~' then A' is stable under Z at [A]'/ A

A'

(ii) if Z ~ Z' then A is stable under Z' at [A]. Proof.

(i) is analogous

to 3.5. (ii) is immediate

from the definition.

D

D.

4.2. Main lemma. Let H ~ H0~HI s A

)

191

Leivant

be given s.t. either

p

(i)

p

is an elimination-rule other than 3E, and HO,HI are stable; or

(ii)

p

is 3E, [A(a)]

Hl(a )

H0

~

3xAx

B

say,

B

Hl(t) is stable under HO at EAt].

~0 is s.n., and for every t Then H is stable.

Proof. By induction on i(H). I.e., we assume that every @ satisfying the lemma and i(@) < i(H) is stable,

the conditions

and we prove that H > A ~ A is stable

of

(which

implies that H is stable by 2.7). Case [a]: ~ > A by an inner reduction in ~0. Then w(A 0) < w(H 0) so i(A) < i(H). If (i) applies to H (and to A) then A satisfies (ii) applies - by 4.](ii).

Case [b]: H > A by an inner reduction < i(H). A satisfies if (ii) applies. Case [c]: argument

the conditions

of the lemma by 2.7, and if

So by the induction hypothesis

A is stable.

in H I. Then A 0 = H O, w(A I) < v(~ ]) so i(A) <

the lemma's conditions by 2.7 - if (i) applies,

By the induction hypothesis

(i) applies,

A is stable.

and H > & by a main detour reduction.

is similar for &E and VE).

H!

I EA]

H ~

H0 ] [

L

F

B

HI

A§

>

A

H 0 and HI are assumed stable,

[A] F

~

A

B

so

ii I [A] H 0 >o

hence A is stable Case [d]:

A

-

F

(2.7).

(ii) applies,

and H > A by a main detour reduction.

H0

[Aa]

At

H|(a)

H ~ BxAx

B B

H0 [At] ~

HI(t) B

and by 4.|(i) -

~

A .

Take the case p = §

(the

192

D. Leivant

By condition Case [el:

(ii) A is stable outright.

(i) applies and H ~ & by a permutative

II O

H ---

!xO

reduction.

r 1 (a)

3

B B

3E (II 1 )

[Aa] Fl(a) ro

B

(E 1 )

3xAx

C

H 0 is stable by assumption,

p

_-- &

3E

H 0 > F! (by a semi-proper

reduction, 1

cf.

1.4), so F| is 0

stable, and v(Fl) < ~(H0).-- H ] is stable by assumption, hence A is stable. A = F 0 is 0 a subderivation of H , hence it is s.n. by 2.9, and v(A 0) s v(H 0) while %(A0) <%(~0). So i(A) < i(H). To show that A satisfies condition

(ii) of the lemma, it remains to see that

whenever F0 3xAx > "'" >

(*)

@ At 3xAx

then @ [At]

r1(t)

-: E

B

(~1)

P

is stable. But if (*), then @ [At] H0 >

... Y

0

Fl(t)

=

;

B

so ~(E 0) < ~(H 0) and i(E) < i(H). H0 is assumed stable, hence H 0 is stable (2.7), while E I ~ H I is assumed outright.

Hence E satisfies case (i) of the conditions

duction hypothesis

E is stable. Hence A satisfies

and by the induction hypothesis

A is stable.

stable

of the lemma, and by the in-

case (ii) of the lemma's condition,

D.

193

Leivant

Case Eli: (ii) applies, and H ~ A by a permutative reduction.

(i) [Aa]

r0

rl(a)

3xAx

3yBy

(2) EBb]

(1) 3E

HI(b)

3yBy

C

[Aa]

(1)

(2) [Bb]

rl(a )

~[l(b )

3yBy

C

F0

C

3xAx

H 1 is stable at [Bb] under H0,

l

(2) 3E

(2) ~E

! A (a)

-- A .

I

(1) 3E

H 0 y FI, Hence (by 4.1(i)) H 1 is stable at [Bb] under

F I. We conclude that A| is stable, and that i(A) < i(H) like in case [el. It remains to show that for every t

Al(t) is stable at [At] under F0; i.e.,

that if

(*)

FO> ... >

0 At 3xAx

then @ [At]

[Bb]

rl_t.( )

(b) C

3~By

z: Z 3E

C is stable. But, like in [e], (*) implies that H 0 >

... > _0,a so z-0 is s.n., v(E 0) <

< ~(H 0) and i(Z) < i(H). (~(E |) is well defined, because Z | ~ HI which is stable by assumption, E satisfies case (ii) of the lemma's conditions by 4.1(ii),

so s.n.).

so by the induction

hypothesis E is stable. Hence A satisfies case (ii) of the lemma's conditions, the induction hypothesis A is stable. Case [$]: (ii) applies and H ~ A by a semi-proper reduction:

Hl

H0

~ 3 E

A

! ~ H

~ A

then A is stable by assumption.

This concludes the proof.

0

and by

194

D.

Leivant

~0 (~i) p where p is an elimination-inference,

4.3. Corollary.

If H = - A then H is s.s. .

s.s., Proof.

If p is &E, §

or VE this follows case (i) of 4.2 outright.

and HO,H j are

If O is 3E (or anal-

ogously - rE) then, if

H ~.+ H* -

N0*

[Ba] Hl*(a)

3xBx

A

3E

then a does not occur in any open assumption

HI~

of H, so we have that

[Bt] H1*(t )

for any t and any stable ~t" In particular,

H O* >

... >

0 Bt

>o

if

@ Bt

3xBx then Bt@ is stable, conditions

since H 0* is stable

(by assumption).

of case (ii) of 4.2 are satisfied,

5. THE STRONG NORMALIZATION

@ [Bt] So ~l,(t)

hence H* is stable,

is stable,

THEOREM

5.1. Theorem. Every derivatfon

H is s.s..

Proof. By induction on %(H). For h(H) = I, i.e., H is a singleton-derivation, theorem is immediate

from the definition

rule,

subderivation(s) elimination

rule, an atomic

(Post) rule or the

then H is s.s. by 3.1 and by the ind. hyp. applied to the immediate of H; if p is IND then H is s.s. likewise by 3.|3, and if p is an

rule - by 4.3.

5.2. Corollary.

the

of s.s..

If the main rule p of H is an introduction replacement

and the

as required.

(strong-normalization

Proof. By 5.1 and 2.8.

theorem). Every derivation

is s.n..

D.

195

Leivant

6. ANOTHER VARIANT OF THE NORMALIZATION-PROOF

6.0. For the disjunction-free

fragment we may give an even simpler alternative

as below. Note that disjunction

is eliminable

in intuitionistic arithmetic

proof,

(TR[73]

1.3.7, LE[73] IV.l). 6.1. Define the reduction relation > as follows. (I) Detour reductions

- like 1.1.

(2) Detour reductions

through BE:

Z0 A 80

ZI B

Z0

A&B

&0

A

BE 81

BE

A&B

~

gl

A

BE

BE

A&B Ak

A

Ak

9

BE

BE

A&B

A

A and similar clauses corresponding correspond

to the q-reduction

to the other detour-reductions.

(These reductions

of LEL73] IV.2.3 for a system with existential

instant-

ation). (3) Induction reductions

- like 1.2.

(4) Inner reductions - like 1.4. The definition of strong-normaiizability

and the measure v(A) follows as in I.

6.2. To the clauses of 2.2 add to the definition

80

z0

zl

A

B

improper reductions

through BE:

~0

A&B

A0 - -

BE A&B &k

of ~

A BE A

>o &k

BE

BE

A&B

A etc.

The definitions

of stability,

and s.s. follow.

6.3. The treatment of the introduction-rules that of the elimination-rules and permutative obtained.

reductions

is considerably

simplified

the same as in 3, while

(cases are treated separately

do not occur). Thus the strong normalization

Here we get all the corollaries

mutative reductions,

is now essentially

of normalization

because of the presence of reductions

theorem is

without referring through BE.

to per-

196

D.

Leivant

6.4. Permutative reductions of the most general kind (|.7) may be reinserted into the treatment without destroying its simplicity. Let p|,...,O k enumerate the instances of BE in a given derivation 4, and ~|'''"~k their respective major premises. Say that A. is s~7~ordinated to A~ iff A. occurs in -i -j -l the minor premise of pj. Define

oA(~i)

:= __max[o4(A j)_ I A._l is subordinated

h4(Ai)

:= the height of ~i in 4

~3(4)

:=

nr

:= Z{hA(~i ) I ~

Then, if 4 ~ 4' by a permutative

where i) = r}

reduction,

(usual summation)

relative reduction sequences

by induction on <~(A),~3(4)>,

A > A' by a non-permutative

.

then ~3(4') < D3(4) and ~(4') = ~(4).

Now, every derivation 4 is strongly normalizable allowing permutative reductions,

to A.]_j + 1

since

reduction ~ ~(&') < ~(A);

4 > 4' by a permutative reduction ~ ~(4') = ~(A) and ~3(4') < ~3(4).

7. FORMALIZATION OF THE PROOF IN ARITHMETIC

7.1. The formalization within arithmetic of our proof of strong normalization tine, except for one point: the arithmetization Let Stk(F) be a tentative abreviation is stable" (where derivations

is rou-

of the stability predicate.

for the formalization of "~(F) ~ k and F

are identified with their ~odel-numbers).

Strong norma-

lizability is seen outright to be formalizable as a E20 predicate (Sn say). Since

Sto(4) +-+ Sn(A) if ~(4) = 0, St 0 is also a E 02 predicate. If ~(4) = n+1 then 4 >o 4' is in general formalizable as a predicate of the form (3r < A') [Stn(F ) & F(&,4',F)] where F is a p.r. relation. 4 > > A '

is 2|0 in >~

is E 0I in St n.

hence A ~ > A '

Stn+l(r) ~ vA'EA>>A' § Sn(A')J, so Stl(A) is of the form V[E

§ E J which is classically a H3-predlcate;

we can see by induction that St

n

is classically equivalent to a H~+2-predicate.

7.2. Consequently, we may formalize within H~+k-arithmetic n) the normalization-proof

(where k is fixed for every

for all derivations A, satisfying:

~(A) ~ n". Some consequences

and for n e 2

"if A occurs in A then

of this are given by TROELSTRA ([73] IV.4).

D. Leivant

197

7.3. Our proof of normalization illustrates the essential role of implication in formulae complexity, since implication is the only logical symhol counted for the measure ~. By 7.2 normalization of derivations with a bound on the (negative) nesting of implications in the formulae (but with no bound on the alternations of quantifiers) is formalized within arithmetic. Thus, for example, Heyting's Arithmetic (HA) is not conservative over 0 Griss' positive arithmetic (NA. Cf. LOPEZ-ESCOBAR [74]) even for Nl~sentences, because the consistency of NA is provable in HA (in fact even in a simple fragment of HA).

REFERENCES G. GENTZEN [36], Die Widerspruchsfreiheit der einen Zahlentheorie, Math. Ann. I12 (1936) 493-565. D. LEIVANT [73], Existential instantiation in a system of natural deduction for intuitiouistic arithmetic, Report ZW 23/73, Mathematisch Centrum, Amsterdam, 1973. E.G.K. LOPEZ-ESCOBAR [74], Elementary interpretations of negationless arithmetic,

Fund. Math. 82 (1974) 25-38. D. PRAWITZ [65], Natural Deduction, Stockholm,

1965.

D. PRAWITZ [71], Ideas and results of proof-theory,

in: FENSTAD (ed.), Proceeding8

of the 2nd Scandinavian logic symposium, Amsterdam, 197|, pp. 235-307. A.S. TROELSTRA [73], Metamathematical

investigation of intuitiouistic arithmetic and analysis, Berlin etc., 1973.

A.S. TROELSTRA [74], Note on the fan theorem, Report 74-14, University of Amsterdam, Sept. 1974. J. ZUCKER [74], Cut-elimination and normalization, Annals of Math. Logic ~ (1974) |-||2.

INTUITIONISTIC COMPLETENESS OF A RESTRICTED SECOND-ORDER LOGIC Dedicated

to Kurt

of his

E.G.K.

w

INTRODUCTION,

icate calculus

Sch~tte

65 th

on o c c a s i o n

birthday

L O P E Z - E S C O B A R and W. V E L D M A N

The completeness of the c l a s s i c a l f i r s t - o r d e r pred-

is over

40

years old, n e v e r t h e l e s s most of the proofs

given for it are,

if not wrong,

at least misleading.

From the prelim-

inary discussions

one is often led to believe that what will be proven

is that every i n t u i t i v e l y valid formula of the ~ l a s s i c a l 2 r e d i c a t e ~alculus~

CPC, is derivable using the axioms and rules of CPC.

ever, what is shown,

is often no more than:

ValcPc(A) ~ > where

"ValcPc(A)"

valid

DercPc(A),

stands for "the formula

(i.e., true in all s e t - t h e o r e t i c

is an a b b r e v i a t i o n for "the formula Of course,

How-

A

A

of the CPC is f o r m a l l y

structures)", is derivable

and

"DercPc(A)"

in the CPC".

it doesn't take much to remedy the situation.

All that

remains to be shown is that every i n t u i t i v e l y valid formula of CPC is f o r m a l l y valid. ity of

A

tations,

The latter is justified on the grounds that the valid-

entails that

A

is true in all p o s s i b l e kinds of interpre-

including the s e t - t h e o r e t i c

structures.

The i_ntuitionistic predicate ~alculus, better.

IPC, has not fared much

To start with there is the t r a d i t i o n a l view that i n t u i t i o n i s m

is solely concerned with questions p e r t a i n i n g to specific m a t h e m a t i c a l constructions. terest

Thus,

from a t r a d i t i o n a l viewpoint,

there is little in-

in a t t e m p t i n g to c l a s s i f y those sentences which are intuition-

istically true i n d e p e n d e n t l y of the interpretation.

In addition,

Heyting has e x p r e s s e d the opinion that his system for the predicate calculus

(and the systems that have since been introduced)

are not

E.G.K. demonstrably

complete;

Lopez-Escobar,

for example

W. Veldman

in H e y t i n g

1966,

199 page

i02,

he

states:

It must be remembered that no formal system can be proved to represent adequately an intuitionistic theory. There always remains a residue of ambiguity in the interpretation of signs, and it can never be proved with mathematical rigour that the system of axioms really embraces every valid method of proof. In view of the fact that m o d e l - t h e o r y classical kind

of

mathematics,

there have

(intuitionistic)

such a m o d e l - t h e o r y (w.r.t.

if it is to be part

model-theory

is one of the

(w.r.t.

methods.

heuristic

use then any valid

istically

true

with respect E.W.

must

then be by

if the m o d e l l i n g (in the

sense

interpretations)

for any

probably

the

first

Predicate

sentence

A

person

Calculus.

In Beth

(0.2)

B - ValIPC(A)

~>

DerlPC(A) ,

(0.3)

ValidlPc(A)

~>

B - ValidlPc(A),

"B - ValIpc(A)"

that

and that

is an a b b r e v i a t i o n

"ValidiPc(A)"

combining

(0.4)

(0.2)

stands and

(0.4) no m e n t i o n

for

(0.3)

ValidlPc(A) in

Furthermore of the

intuitionisti-

of being

intuition-

also be valid

a modelling

1956 he

set out to

of IPC

B - ValIpc(A),

Note

theorem

the proof

who p r o p o s e d

~>

and

if

is to be of any

should

DerlPC(A)

models"

some

Obviously

requirements.

(0.i)

where

to obtain

a completeness

mathematics

use to

to the modelling.

Beth was

that

first

sentence

in all p o s s i b l e

for the I n t u i t i o n i s t i c prove

attempts

the m o d e l l i n g )

Finally

of great

for intuitionism.

of i n t u i t i o n i s t i c

theorem

cally a c c e p t a b l e

been many

is to be of any use then

that m o d e l l i n g )

completeness

has been

"A

for

"A

is true

is i n t u i t i o n i s t i c a l l y

we obtain

that:

--> DerlPC(A),

is made

in all

of the

Beth-models.

Beth

valid".

2~

E.G.K. Just

(0.3)

Lopez-Escobar,

as in the c l a s s i c a l

are of a quite

explicitly

defined

ValidiPc(A),

The

same

case,

as-constructions

counterparts, nature.

mathematical

although

definition. itionistie

different

and

obvious,

is true of

the proofs

(0.2)

constructs;

probably

ValidlPc,

W. Veldman

so it is not

closer

in

solely with

however

mathematical in the intu-

to the concept

immediately

and

(0.3),

no explicit

ValidcPc(A);

is much

(0.2)

is c o n c e r n e d

while

has

of

obvious

of proofs-

that

(0.3)

should

hold. As a m a t t e r cause

there

yields

(see Kreisel

principle

for p r i m i t i v e

Kreisel's

proof was not with

respect

to species

"S - ValIPc(A)".

is that the

to be more

easily

be-

of

of Kreisel

recursive

is yet another

of IPC.

It was

that

predicates

by K r i p k e - m o d e l s ;

(classical)

~>

interpretation, of

which

we shall

S - Vallp C

assumptions

in Beth ab-

over

can be shown to

S - ValIPC(A)

than of

introduced

is played

to v a l i d i t y

implication:

notion

of semantics

respect

An a d v a n t a g e

accepted

treatment

correct

point

(see the report

was a result

(which under a few r e a s o n a b l e

There

was a moot

1982).

ValidlPc(A)

formulae

(0.2)

Markov's

be equivalent)

tends

of

(0.3)

damaging

by

B - Valip C

in the proof

that

Even more

but with

breviate

out

1961).

Actually models,

it turned

was an error

Dyson/Kreisel (0.2)

of fact

(0.3).

(mathematical) in Kripke

for i n t u i t i o n i s t i c probably

completeness

1965.

formal

because

proof was

validity

for the

In the c l a s s i c a l

systems,

of the

a main part

fact that

given with respect

the first to this

type of models. Kripke

models

intuitionistic vative

theories.

extension

itionistically

are often

In addition,

properties,

acceptable

used to i n v e s t i g a t e through

some of those

(see T r o e l s t r a

the

of some

the use of some conser-

results

1973).

strength

can be made

intu-

E.G.K.

In the "valid

following

in e v e r y

The

abbreviate by

the

201

intuitionistic

notion

"K - V a l i P c ( ' ) " .

ValidiPc(A)

~>

in K r i p k e

1965,

justified

acceptance,

However,

even

intuitionistic

and

W. Veldman

implication:

is p a r t l y

an

we w i l l

Kripke-model"

(0.5)

versal

Lopez-Escobar,

S - Valip C

it has

its

if one

accepts

shown

and

although

it does

not h a v e

uni-

charms.

completeness can be

K - VallPc(A).

of

(0.5), IPC

Kripke

(i.e.

models

of

to be e q u i v a l e n t

are no use

(0.4))

since

(under

suitable

for

K - Valip C assump-

tions). In v i e w

of the

tuitionistie istie

proof

proof

with

to c o n c l u d e

notion

of

COMPLETENESS ity for

that

intuitionistic

state

the

FOR

of the

of

that

is not does

(logical)

unless

yield one

obviously

not

validity.

quite As

Is t h e r e

that an

an in-

intuition-

identifies

the

case.

correspond

a matter

intuitionistic

a mathematical predicate

such

that

(A)

the

notion

(B)

the

implication

[ValidiPc(A)

~>

(C)

the

implication

[ViPc(A)

DeriPC(A)]

intuitionistic

state

of

We to the

fact we

problem:

IPC:

VIPC(A)

to

IPC w o u l d

However,

S - Valip C

following

PROBLEM

formulae

it is c u s t o m a r y

principle.

S - ValiPc,

prefer

to

remarks

of the c o m p l e t e n e s s

of M a r k o v ' s

ValidiPc(A)

like

above

has

a semantical

~>

mathematics

without

notion

calculus,

of v a l i d -

say

VIPC(A)

character,

VIPC(A)]

is p l a u s i b l e , is p r o v a b l e

making

use

in c u r r e n t

of M a r k o v ' s

principle?

In this can be

solved

restricted

p a p e r we

will

try to

for a r e s t r i c t e d

second-order

show

that

second-order

language

in the

the

completeness

minimal

sense

that

logic the

~.

problem It is a

second-order

202

E.G.K.

variables

are

intended

definable

species.

are p r i m i t i v e towards order

to range

concepts.

is a

The format

over a subclass

It is minimal

problem

(conservative) of the paper

language

A calculus

w

~

w

A formal

w

Soundness

w

Explicit

theories.

w

A spread

which

w

Construction

of a u n i v e r s a l

w

Construction

of the

of a r e s t r i c t e d

as an extension

of the

semantics theorem

for for

generates

second-order

of the

spread

~.

w

"V"

spread

~.

w

"DerR"

w

The c o m p l e t e n e s s

w

Realizations

and the

explicit

IPC.

logic.

of

predicate

theories.

of explicit

theories.

Z.

~.

~, and

Kripke

models.

THE LANGUAGE OF A RESTRICTED SECOND ORDER LOGIC, can be b r i e f l y

of

described

The lanzuage

as follows:

~.

A denumerable

set

Var

A denumerable

set

Par I

For each

a denumerable

n,

calculus.

realization.

spread

spread

of

intuitionistic

spread

,,n,, and the

Symbols

second-

~.

w

I.i

it is a contribution

the r e s t r i c t e d

of the

falsity

~.

Some p r o p e r t i e s

~

nor

~.

w

of

negation

is as follows:

w

i.

neither

we believe,

extension

The

for

of the f i r s t - o r d e r

for IPC because

w

and the

because

Nevertheless,

the c o m p l e t e n e s s

logic

Lopez-Escobar, W. V e l d m a n

of i n d i v i d u a l of i n d i v i d u a l

variables:

v0, Vl,...

parameters:

set

p(n)

of

set

~ (n) of

a0,al, . . . .

n-ary predicate

variables:

n-ary

parameters:

p(n) _(n) (n) 0 ' ~I ' P2 ' .... For each

n,

a denumerable

predicate

E.G.K. Q(n) 0

^(n) ' UI

First-order

connectives:

universal

symbols:

Symbols

1.3

Pseudo-formulae,

1.4

not of

1965.

V

I~

of v a r i a b l e s

,

but u s e d

formulae

terms.

and s e n t e n c e s

occurrences.

Given

Ix0...Xm_iF

term.

only used

in the o p e r a t i o n

will

The a b s t r a c t i o n

Some n o t a t i o n a l

: U nin(n),

are d e f i n e d

in w h i c h

A sentence

as done

all o c c u r -

is a f o r m u l a

be omitted.

will abbreviate w i l l be u s e d

instead

occurring

ters o c c u r r i n g IXl...XnF,

variables

be c a l l e d terms

are

an

F

occurring m-ary

such that

in

F,

then

elementary

in the m e t a l a n g u a g e

ab-

and are

in

in

F.

F

duction having

FOR

of

variables

(parameters)

'(Aoi)'

then and

Parl(F) Par2(F)

Furthermore

then we d e f i n e

A CALCULUS

and p r e d i c a t e

'VPoP 0'.

is a p s e u d o - f o r m u l a ,

parameters

pseudo-formula

Par 2 : UnQ(n).

will usually

w

IR

conventions:

in the q u a n t i f i e r s

F

of

~, I, ~.

of s u b s t i t u t i o n .

Superscripts

If

a atomic

individual

straction

'~A'

in a b b r e v i a t i o n s :

is a p s e u d o - f o r m u l a

are b o u n d

are all the

the e x p r e s s i o n

'I'

V (2)

any p a r a m e t e r s .

x0,...,Xm_ I

P

n. (i)

quantifier

A formula

Abstraction

1.5

(i)

( , ).

1.2

in P r a w i t z

A, v,

quantifiers:

Second-order

without

20S

' ....

Propositional

rences

W. Veldman

~(n) ' U2

Auxiliary

Lopez-Escobar,

R,

the u s u a l

if

Pari(T)

With

~

T

is the

the set of p r e d i c a t e is the a b s t r a c t i o n

= Pari(F),

we a s s o c i a t e

introduction

set of i n d i v i d u a l parame-

term

i : i, 2.

a s y s t e m of n a t u r a l

and e l i m i n a t i o n

rules

for

de-

A, v,

204

E.G.K.

n, V (I)

and

3 (1).

Lopez-Escobar,

W. Veldman

For the s e c o n d - o r d e r q u a n t i f i e r

t r o d u c t i o n rule is standard

V (2)

the in-

(e.g. as in Prawitz 1965), h o w e v e r the

e l i m i n a t i o n rule is w e a k e n e d to: v p ( n ) A ( p (n))

(V(2)E)

A(T) where

T If

is any F

is a set of formulae of

d e r i v a t i o n in 'mere(A) '

w

R

n - a r y e l e m e n t a r y a b s t r a c t i o n term.

~

of

A

instead of

from ~_~A

F.

R

then

F~A

iff there is a

O c c a s i o n a l l y we shall write

@

AS AN EXTENSION OF THE INTUITIONISTIC PREDICATE CALCULUS,

Let us assume that the i n t u i t i o n i s t i c predicate caloulus,

IPC, has

been f o r m a l i z e d as a system of natural d e d u c t i o n w i t h the falsum symbol

's

as a p r i m i t i v e symbol and negation as a defined concept.

Then given a formula

A

of IPC let

A*

be the formula of

tained by r e p l a c i n g all occurrences of the atomic formula by the sentence

I ;

if

&

~ ~

obin

A

is a set of formulae of IPC then we let

&* = {A* : AEA}. is an e x t e n s i o n of IPC in the following sense:

3.I

THEOREM.

then

AU{A}

is a set of formulae of IPC and

A ~ iPC A

A* ~ A * .

PROOF. of

If

~

The only rule of inference of IPC not included in the rules is the rule for

plexity of

A

Moreover that if

&*~A*

I.

shows that,

~

However a simple i n d u c t i o n on the comfor any

A,

I~A

.

is a c o n s e r v a t i v e e x t e n s i o n of IPC in the sense then

n o r m a l i z a t i o n theorems make the following:

& ~ i P C A. for

~.

The latter is a c o n s e q u e n c e of To be a little more

specific let us

E.G.K. 3.2

DEFINITION.

iff

A

Lopez-Escobar,

A formula

is built up from

V, n, V (I)

and

A

of

~

is e s s e n t i a l l y f i r s t - o r d e r

and the atomic formulae by means of

A,

3 (1)

It should be clear that formula

I

A

205

W. Veldman

of IPC,

A

is e s s e n t i a l l y f i r s t - o r d e r iff for some

A = A*.

The d e f i n i t i o n of a normal d e r i v a t i o n in long to write down, however,

the d e f i n i t i o n

s e c o n d - o r d e r logic have been given

~

w o u l d take too

for the case of full-

(explicitly or implicitly)

in

Girard 1971, Prawitz 1971 and T r o e l s t r a 1973 so that it is a relatively simple m a t t e r for the reader to make the a p p r o p r i a t e changes r e q u i r e d for

~.

E i t h e r using the

(strong) n o r m a l i z a t i o n

for full s e c o n d - o r d e r

m i n i m a l logic or m o d i f y i n g the proof for 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

(V (2)

causes no p r o b l e m because

formula than

VPA(P)),

ization for

A(T)

is always a simpler

it is possible to obtain a (strong) normal-

~.

Because of the r e s t r i c t i o n we have placed on derivation

3.2

~

in

PROPOSITION.

tially first-order order formulae

~

If

V(2)E

a normal

has the following kind of s u b f o r m u l a property.

~

formula

is a normal A

derivation

from a set

then every formula

occurring

F

in

~

of an essen-

of essentially in

~

first-

is essentially

first-order. An immediate c o n s e q u e n c e of 3.2 is the f o l l o w i n g c o n s e r v a t i v e e x t e n s i o n result.

3.3

then 3.4

THEOREM.

If

AU{A}

is a set of formulae

of IPC and

A*~A

A~IPcA. REMARK.

From (the proof of)

3.1 we obtain that n e g a t i o n in

~

206

E.G.K.

Lopez-Escobar,

W. Veldmsn

behaves in the same way as does i n t u i t i o n i s t i c n e g a t i o n in IPC. example the following are theorems of

An

For

~:

(~AnB)

(A n B) n ~(A

(~B ~ ~A) ^ ~A)

~ 3 x A n Vx~A

An~A w

A FORMAL SEMANTICS

FOR

R,

Our formal m o d e l l i n g for

~

will

be in the style of Kripke 1965.

4.1

DEFINITION.

such that

K

r e l a t i o n on for all

A model-structure

is an inhabited set, K

and

DI, D 2

is a quadruple ~

a reflexive and t r a n s i t i v e

are unary functions on

Dl(e)

is an inhabited

subset of

Parl,

(.2)

D2(a)

is an inhabited subset of

Par 2,

(.31

if

~ s ~

then

DEFINITION.

unary function

M

M(e)

(.2)

if

A (M(e)

(.3)

if

e S 8

4.4 Dz,M>

4.5

such that

on

K

and

D2(~) ! D2(8)"

on a m o d e l - s t r u c t u r e such that for all



is a

~,8 E K:

is a set of atomic formulae,

REMARK.

that "A

DI(~) ! DI(8)

A model

(.i)

4.3

K

e,B ~ K:

(.i)

4.2



then then

Parl(A) ~ DI(~)

A (M(a)

has been v e r i f i e d by stage

such that

DEFINITION.

A realization M

Par2(A) i D2(~),

M(e) c M(8).

If the atomic formula

DEFINITION.

and

of

then we shall say

e".

~

is a structure

is a model on the m o d e l - s t r u c t u r e

Given that

~ =

~

= .

is a r e a l i z a t i o n of

E.G.K. ~,

aEK,

A

Lopez-Escobar,

is a formula

Par2(A)

~ D2(a) ,

symbols

~I=eA

either

A

then

"A

is an atomic and

or

A = (BvC)

and either

or

A = 3xB(x)

or

A : (BnC)

or

A : VxB(x)

or

A = vP(n)B(P n)

4.6

(or evident)

and

~ Dl(a)

at

~

and

in

~",

in

A (M(a)

~ ]= B or

C,

a ( D l ( a ) , ~ l=aB(a),

8 ~ ~,

and for all

n-ary elementary

~[=

B ~ a, ~

and for all

I=~C

~ I=6B(a)

B ~ e,

abstraction

whenever

~ I=BB,

whenever

~I=BB(T)

a (DI(B) ,

whenever

term such that

T

Parl(T)

is

~ DI(B),

Par2(T) ~ D2(B). A sentence in symbols:

DEFINITION. Va~(A)

formula

Parl(A)

B, ~ l=aC,

and ~or all

DEFINITION.

bols:

~I=

and for some

= ,

4.7

such that

is true

formula

A = (B^C)

and

~

207

iff:

or

an

of

W. V e l d m a n

A ~I=A,

A sentence iff

~ I= A

is formally

is true in a realization

A

iff for all

~(K,

~ l=aA"

of

~

is formally

whenever

~

is a realization

valid iff its universal

closure

valid,

in sym-

of

~.

A

is formally

valid. w

SOUNDNESS

THEOREM

FOR

length of the derivation 5.1

THEOREM.

and

Der~(A)

5.2

COROLLARY.

PROOF.

Let

If

A

R,

The usual proof by induction

on the

gives us the following:

is a sentence

of

~,

~

a realization

of

then ~ I=A.

~0

I

is not a theorem of

be the realization


such that: K (01 D~0)(~)

~.

=

{~}

=

{a 0 }

(0)>

208

E.G.K.

Then

•

5.3

is not true

REMARK.

ample

let

Then

I

There

~I

w

=

{Q~O),Q(O)}

M(O)(~)

=

{Q(O)}.

in

except Thus

THEORIES,

is a set of f o r m u l a e

Par.(F), 1

i : i, 2.

If in a d d i t i o n (i)

(AvB)

(ii)

3xB(x)

then

F

6.1

formulae

~

AEF

Also,

or

B(d)

A set

A(F

I

then

Parl(F)

has the

F

way;

for

for e x a m p l e

= UA( F Parl(F).

is c l o s e d u n d e r deri-

FI--~ A

and

following

Pari(A)

properties:

B(F

( F

F

whenever

although

nevertheless

mention

the r Sle of

i.

6.2

If

LEMMA.

to

conventions

in the o b v i o u s

for some

of f o r m u l a e Par.(A) l

individual

constant

F

is overcomplete

c Par.(F), -1

Note that we a l l o w the p o s s i b i l i t y plete.

For ex-

= {Q~0),Q~0)}.

the n o t a t i o n a l

whenever

F

I~

d,

an explicit theory.

is c a l l e d

A,

that

A(F

E F ~>

DEFINITION.

of

a theory

( F ~>

is true.

M(1)(a)

is a theory iff

of f o r m u l a e

in the sense

•

of

"the U n p r o v a b l e " .

to sets of f o r m u l a e

vations

be a t h e o r e m

s h o u l d not be c o n s i d e r e d

We e x t e n d

if

F

cannot

in w h i c h

that

as r e p r e s e n t i n g

formulae

A set

•

•

for h o w c o u l d the False e v e r be true?

single F

and h e n c e

@0

~i"

"the False";

EXPLICIT

~0

be like

c o u l d be c o n s i d e r e d

W. Veldman

D~O)(ct)

are r e a l i z a t i o n s

is true in

represent

Lopez-Escobar,

iff for all

i = i, 2.

that a t h e o r y

be o v e r c o m -

we do not m a k e use of the f o l l o w i n g

it b e c a u s e

it gives

some m o r e

is a set of formulae of

conditions are equivalent:

~

lemma,

information

on

then the following

we

E.G.K.

Lopez-Es6obar,

209

W. V e l d m a n

(.I)

F

is overcomplete,

(.2)

F

is a theory and

(.3)

F

is a theory and for some formula

(.4)

r

is a theory and for every sentence

IEF,

(AA~A)

A,

( F,

S(F.

S,

A SPREAD WHICH GENERATES EXPLICII THEORIES, Spreads are basic

w

constructions

in i n t u i t i o n i s t i c

consists of a spread

law

Z,

set theory.

Z =

w h i c h is a function from finite se-

quences of natural numbers to

{0,1},

which is defined on the nodes admitted

and a c o m p l e m e n t a r y

by

{<no , .... nk_l > : Z ( < n 0 , . . . , n k _ l >) = 0), (previously)

A spread

~,

i.e., on

c o n s t r u c t e d m a t h e m a t i c a l entities. (i -i)

function from the

set of finite sequences of natural numbers onto the set ural numbers such that

0

~

of nat-

is the code for the empty sequence.

will be used for the c o n c a t e n a t i o n function,
e.g.

> = < n 0 , . . . , n i _ l , m 0 , . . . , m r _ l >.

Greek letters: functions and [(i)

If

Ec,

and whose range c o n s i s t s of

To simplify matters we use a standard

i.e.,

law

e

~

e,8,..,

is the c o u r s e - o f - v a l u e s

function d e t e r m i n e d by

e,

= <~(0),...,~(i-l)>. is such that

a member of

will be used for number t h e o r e t i c

Z,

Vi(Z(~(i))

and write:

It will be later shown, a spread

~ =

mappings

F(n), Dl(n) , D2(n)

: 0)

then we say that

~

is

a(Z. in Sections

9 through 13 that there is

whose c o m p l e m e n t a r y law

Zc

such that if we set

F

=

Um(~ F(am)

Die

=

Um(l~ DI (~m)

consists of three

210

E.G.K.

Lopez-Escobar,

D2e then the following Condition

7.1.

UmEi~ D 2 ([m)

conditions

If

set of formulae,

:

n

W. V e l d m a n

are satisfied:

is admitted

Dl(n) ! Par I

by

and

Z,

then

7.2.

If

eEZ,

then

Die 9 = Pari(F e) ,

Condition

7.3.

If

aEZ,

then

Fe

Condition

7.4 9

If

eEZ

and

following

are equivalent:

(ii)

(A~B)

7.5.

the following ({) (s163

VxA(x)

and

If

(s

i = i , 2,

then the

and

BEF8]. Pari(VxA(x))

! Die,

i = i, 2,

then

E Fe ~

7.6.

If

vp(n)A(p (n))

and eEZ

If

and

A(a)

6 FS].

Pari(vp(n)A(p(n)))

! Die,

i : i, 2,

( F e, and

term such that 7.7.

aEDIs ~ >

are equivalent:

VBSEzVT[Fe!F 8

Condition

theory.

are equivalent:

then the following (s

is an explicit

Par I.(AnB) ~ Die,

AEF B ~ >

eEL

V88E~Va[F e i F B

Condition

i = I, 2 .

E F ,

VBBEz[F ~F B

Condition

is a (finite)

D2(n) i Par 2.

Condition

({)

F(n)

T

Pari(T) A

is an

n-ary elementary

! Pari(F$),

is a sentence

of

abstraction

i : i, 2 = > ~

A(T)

e F8].

then the following

are

equivalent: ({) (ii)

w

Der~(A), VaaE~(AEFe).

CONSTRUCTION OF A UNIVERSAL REALIZATION, The spread

previous

section

can be used to define

a realization

~

~ of

of the ~

such

E.G.K.

Lopez-Eseobar,

that for all sentences

S

(8.1)

~ I=S = >

The d e f i n i t i o n of

~ B

~

iff

e,6 ( ~

=

DIs,

D2(e)

=

D2e,

]M(a)

=

{B : B

211

of Der~(S).

is as follows:

DI(~)

W. V e l d m a n

and

Let

F ~ ! F B,

is an atomic formula and

B s F }

and then set

= We prove

< ~ , S , D I , D 2 , ~ >.

(8.1) a s s u m i n g that the spread

~

satisfies conditions

7.1 -7.7. 8.2

LEMMA.

If

Par.(A) c D. 1

PROOF,

--

~

and

A

(i = 1,2),

is a formula of

~

such that

then

i~

By induction on the logical c o m p l e x i t y of

Basis ste~.

A

is an atomic formula.

quence of the d e f i n i t i o n of Induction step.

that

and

Then it is an immediate conseI= .

Let us c o n s i d e r the case when

Assume thus that ~I=~ A . V6BE~VT[8 ~ ~

~

and

T

A.

A = vP(n)B(p(n)).

Then

is an

n-ary e l e m e n t a r y a b s t r a c t i o n term such

Pari(T) ! D i ( 8 ) ,

Then u s i n g the d e f i n i t i o n of

i = 1,2 = > ~

~I=8B(T)].

we obtain that:

212

E.G.K.

V 8~EzVT[F B ~ F

and

such that

T

Lopez-Escobar,

is an

Pari(T)

From the induction

n-ary elementary

~ Pari(Fs),

hypothesis ~I=$B[T]

Using condition

that

<=>

( F s.

B(T)

7.6 we then conclude

is similarly

The proofs

abstraction

i = 1,2 ~ >

we obtain

term

~6~(T)].

that ~ F

vp(n)B(P (n))

The converse

W. V e l d m a n

proven.

for the other

compound

formulae

are analogous

(and

well-known). 8.3

If

COROLLARY.

A

i8 a sentence of

(i)

~I=A

iff

V~aEZ(AEF ) ,

(ii)

~ I=A

iff

VaaE%3m(A E F(~m)),

(iii)

~ I=A

iff

Der~(A).

PROOFS.

Of (i),

(ii), immediate.

For

w

CONSTRUCTION OF THE SPREAD ~

next

5 sections

we shall adhere

~,

then

(iii) use condition

OF EXPLICIT THEORIES, For the

to the following

FO,FI,...

is an enumeration

of the formulae

~0,~i~...

is an enumeration

of the derivations

~0 U 9 1 U

... is a partition

into a denumerable

of the set

sequence

7.7.

Par 2

conventions:

of in

of predicate

of pairwise

disjoint

parameters

deDumerable

sets. ~i = {Qio~Qil ''''} ~ o u ~ l U ...

is an enumeration

is a partition

of

of the set

~i Par I

of individual

param-

E.G.K. Lopez-Escobar, eters into a denumerable

sequence

Veldman

W,,

of pairwise

213

disjoint

denumerable

sets

Mi = {ciU'Cil''''} ~k,s

is an enumeration

is a (i-i) mapping ~k,Z,3~

= ek,i,2~

The functions recursion

from

+ 2,

F(m), Dl(m) ,

on the length of (the finite

Basis step.

Xi

~2 x {1,2,3}

+ 1 = ek,s

Z(m),

of

onto

and

and

~

such that

~0,0,1~

D2(m)

sequence

= 0.

are defined by

coded by)

m.

Z(< >) = 0

r(< >)

r

=

DI(< >) = K0 D2(< >) = Q0 Reeursion

step.

F(m), Dl(m)

Suppose that

and

ural numbers

k, s

D2(m) r

m = <m0,...,mp_l >

have been defined.

=

We then determine

nat-

~k,Z,r~

and proceed by cases depending

on the value of

that if for some

its value is to be

E(m),

such that p

the convention

and that

i,

s,

E(m*~)

r.

We shall follow

is not specified then

and thus the finite

sequence

(coded by)

.,o^

m"s = <mo,...,mp_l,S> Case i

~.

r = i.

SHb~__!a. (i)

is not admitted by

If

if for some

Pari(F k) ! Di(m), q ~ p,

q

i E i, 2

is a derivation

then we set: Z(m*l)

=

0

r(m*l)

=

F(m)

Dl(m*l)

=

Dl(m)

U {F k}

then of

Fk

from

F(m)

214

E.G.K~

D2(m*~)

=

hand

for all

(2) and if on the other tion

of

Fk

from

Lopez-Escobar,

F(m)

W. V e l d m a n

D2(m) , q ~ p,

~q

is not a deriva-

then we set:

Z(m*~)

=

0

Z(m*2)

=

0

r(m*~)

=

r(m) U {F k}

r(m*9)

=

F(m)

Dl(m*l)

=

Dl(m)

Dl(m*2)

=

Dl(m)

D2(m*~)

= D2(m)

D2(m*2)

= D2(m).

S~b~e_ib.

If e i t h e r

Parl(F k) ~ Dl(m)

or

Par2(F k) ~ D2(m)

then

we define Z(m*O)

: 0

r(m*~)

: r(m)

Dl(m*0 ) = Dl(m) D2(m*0) Case

= D2(m).

2

r = 2.

In this s > 0

Case

3

B(x),

functions.

That

is,

for each

~(m*s)

=

0

r(m*s)

=

r(m)

Dl(m*s)

=

Dl(m)

U {Cpj : j < s}

D2(m*s)

=

D2(m)

U {Qpj : j < s}.

r = 3. case we c o n s i d e r

in the previous

Sub~ase_~a. some

the domain

we define:

In this duced

ease we enlarge

formulae

If

the f o r m u l a

might

have

been

intro-

cases.

mp_ 2 = i AI, A2,

F k = 3xB(x)

which

(and hence

F k = (ALVA 2)

then we define

Fk

~ r(m))

and if either

or for some p s e u d o - f o r m u l a

for

E.G.K.

Z(m*l)

=

0

r(m*~)

=

Y(m)

Dl(m*l)

=

D2(m*l)

=

in the case that for all

where

Lopez-Escobar,

W. V e l d m a n

21B

E(m*2)

=

0

r(m*~)

=

F(m)

Dl(m)

Dl(m*2)

=

Dl(m)

D2(m)

D2(m*2)

=

D2(m) ,

U

{A I}

F k = (ALVA2) ;

and in case

U

{A 2}

F k = 3xB(x)

we define

s > 0

tl, t2,..,

Z(m*~)

:

0

r(m*Z)

=

r(m) U {B(ts)}

Dl(m*s)

:

Dl(m)

D2(m*~)

=

D2(m) ,

is some

(previously

agreed

upon)

enumeration

of

Dl(m). Subcase

3b.

Failure

of subcase

3a.

Then we set:

Z(m*~) = 0 F(m*8) ='F(m) Dl(m*O)

= Dl(m)

D2(m*~) = n2(m). r(m), Dl(m) ,

Combining we obtain 9.1

the spread

REMARKS.

(A)

m

and

Dl(m) , mitted (B)

m*O

~

two conditions m*0

by

has been defined

= D2(m)

Z.

is admitted

law

~c

by

Z.

so that for any

m

are equivalent:

are admitted

D2(m*0)

into the complementary

~ = <x,xo>.

The spread

the following

D2(m)

by

Z,

F(m*O)

and for all

= F(m),

s > 0,

m*s

Dl(m*0)

=

is not ad-

E.G.K.

216

A node

m

such that

Thus

m

is a p r o c r a s t i n a t i o n

node. ~.

Or in terms

~p

w tion

for

8EZ

and not w o r t h

If

For a p r o o f parameter

theory

aEZ

then

Q

and i n d i v i d u a l 7.3,

namely

is an i m m e d i a t e

tion of

D.

of i0.i it s u f f i c e s

Condition

node

iff

~.

That

),

to o b s e r v e

parameter

that e a c h

consequence

~

F

satisfies Condition

for any

7.2 is

n-ary

Der~(Qcc...c (for

of c a ses

eondi-

i = i, 2.

that

e,

is a d m i t t e d by

8p = 0.

repeating.

= Par.(F

a procrastination

~(Z)

1 and

n Qcc...c).

is an e x p l i c i t

3 of the d e f i n i -

~.

Conditions consider

contains

7.4 - 7 . 7

are a l i t t l e m o r e

complicated

so we shall

them separately.

"v"

AND THE S P R E A D far too m u c h

LEMMA.

i : i, 2

following

properties:

SEA ~ >

~.

For any g iven

information

To every

P a r i ( F a)

(III)

we h a v e that

7.1 is i m m e d i a t e

LEMMA.

(II)

m*0

OF THE S P R E A D

i0.i

(I)

iff

SOME P R O P E R T I E S

as a lemma.

ii.i

node

is a p r o c r a s t i n a t i o n

stating

W. V e l d m a n

(A) h o l d s w i l l be c a l l e d

of f u n c t i o n s

worth

w

Lopez-Escobar,

a(E,

so we f irst

formulae

there corresponds

A, B

sEZ

we find t h a t

cut it down to size.

such that

a subfan

~

of

Par.(A) l

~

with the

AEr~,

SEA

> F a _c r 8 ,

8E~,

p = ~k,Z,l~,

8p # 0 ~ >

p = ~k,Z,3~,

8p ~ 0,

A = Fk

or

F(Sp)J--F k

or

FkEFa, (IV)

8(4,

F~ = 3xCtx) ~ >

<

there is an

E.G.K. individual

PROOF.

Lopez-Escobar,

constant

a

W. Veldman

217

C(a) ( F ( ~ p )

such that

and

(if

3xC(x)

(F(~p)

then

C(a) ~ F )

(if

3xC(x)

~ F(~p)

then

a ~ Parl(F ~) U Parl(A)

Let

q

or

U Parl(B).

be a natural number such that

Parl(A)

and

U Parl(B) ! %0 U ... U %q

Par2(A) U Par2(B) ! ~0 U ... U ~q. Let ~0

i

be such that

such that

A = F.1

and then determine a natural number

q.

The subfan

mined by the condition that (i)

B(ei,~o,l~)

(ii)

VkV~(e(~k,s

(iii) (iv)

of

~

is then deter-

iff

= i, i ~>

VkVs163

8(ek,~,l~)

= a(~k,s

VkV~([6(~k,~,3~) 8(~k,~,3~))]

= 0] v [6(~k,&,3m)

and

= fu(~(ek,~,3m),

= fl(~(~k,s

g' f0' fl

D2e ~ D28

= i),

+ g(8(~k,s

v [8((k,s

where the functions DIe ! DI8~

B(A

A

B({k,Z,&~))]),

are chosen so that

DI8

is sufficiently

(iii) ensures that

larger than

and so that (iv) ensures that the required instantiations junctions on existential

formulae are placed into

F B.

Dla,

of dis-

That such

functions can be found follows from the fact that the only time that constants from

Xp

are placed in

11.2

If

e(~

THEOREM.

Pari(A~B ) c Die ~

and

i = i, 2

Diy

(AnB)

(ii)

(AnB)

then the following

~ F ,

VBB~sEF e ~ r B

and

A(F B ~ >

is at the node

is a formula

equivalent. (i)

(y(Z)

BEF~].

of

~

~p.

such that

two conditions

are

218

E.G.K. Lopez-Escobar,

PROOF.

That

that

the

Let

A

Then

from

({) ~ >

FB's

are

be the

({i)

theories.

subfan

(ii)

and

is an

of

(II)

immediate

Thus

E

W. Veldman

assume

determined

of L e m m a

consequence

ll.1

(ii).

of the

We w i l l

according

fact

prove

to L e m m a

(i).

ll.1.

we o b t a i n

V B B ( A ( B ~ F B) and h e n c e

that

Using

the

monotonici•

there

is

a natural

of

r

number

and

PO

VB~EA(B We n e x t

prove,

If

A(m)

Basis

:

induction

0

can be

stated

Vs[~(m*~)

Furthermore F(m*s)

if

m

= F(m)

F , AI-~B.

Hence

k,

p = {k,~,r~. Case

1

Subcase

we c o n c l u d e

that

that

PO - l t h ( m )

~ PO

that

r(m), r , AI-~B.

then

B (r(m),

so

r(m),

p : lth(m).

Let

Then

F,

AI--RB.

the

induction

follows:

= 0 ~>

r(m*~),

induction

from now

theorem

(r(~P0)).

Then

as

fan

r ,

is a p r o c r a s t i n a t i o n

so the

procrastination Let

PO"

l t h ( m ) < PO"

step.

hypothesis

:

such

on

lth(m)

and

lth(m)

step.

Induction

by

the

node

hypothesis

on we

shall

AI-~B]. then

then

assume

A(m~O)

give

= 0

and

us that

that

m

F(m),

is not

a

node.

Z, r

be the u n i q u e

We p r o c e e d

natural

by cases

numbers

depending

such

that

on the v a l u e

of

r.

r = i. la.

A(m*~)

= O.

Then

the

construction

of

Z

tells

us t h a t

219

E.G.K. Lopez-Escobar, W~ Veldman F(m*l)

= F(m)

U {F k]

and

so the

induction

r(m),

Fk,

of

tells

hypothesis

gives

us t h e n

that:

But the

construction

Fk = A

or

F k E F e.

,A

Thus

as

lb.

A(m~2)

= 0.

Case

2

In this F(m).

case

3

Subcase C(a), ~uction

F(m)I--~F k

that

F(m*2)

= F(m)

and the

argument

is

node.

for

some

argument

s > 0

then

we h a v e

proceeds

as

= 0

A(m*s)

that

r(m*s) =

and

for a p r o c r a s t i n a t i o n

node.

r = 3. 3a.

For

some

F k = 3xC(x) hypothesis

s > 0,

(F(m),

Now u s i n g

condition

we o b t a i n

(IV)

A(m*s)

F(m~s)

= 0

= F(m)

and

for

some

U {C(b)}.

of L e m m a

r(m),

F r o m the

re,

AI-~B.

II.i

we c o n c l u d e

that

AI-~B

re,

or

r(m),

since

3xC(x)

E F(m)

3xC(x),

we h a v e

r(m),

Sub~ase_3b.

F k = (ClVC2).

formula

that

C(b),

F(m),.

But

or

r = 2.

The

Case

either

AI-~B.

Fe,

Then

for a p r o c r a s t i n a t i o n

us that

we o b t a i n

F(m),

Subease

AI--~B.

r,

re,

rei-~B.

that

in e i t h e r

case

A]--pB.

Analogous

to S u b c a s e

3a.

either

in-

E.G.K.

220

Lopez-Escobar,

We now consider the situation when

W. V e l d m a n m

is the empty sequence 9

Then

F ,A I-~B . From the latter it follows that theory we may conclude

w

that

"V" AND THE SPREAD

F I-R(AnB)

(AnB)

~,

any essential use of case 2 of the definition

second-order

quantifier

V(2);

F

is a

E F

In the case of

will be used for the quantifier

and since

"V".

,,n,, we did not make of the spread

~

We will only consider

the first-order

case being almost

As in Section ii we must first obtain an appropriate

fan of

(by essentially

12.1

LEMMA.

corresponds

To every a subfan

(I)

BEA ~ >

Q E D2B

(II)

B(A 2 >

F ~F 6

of

the same as ii.i

(III)

(IV)

the same as ii.i

(IV).

THEOREM.

such that

If

a E ~

and predicate p a r a m e t e r

~

(III)

12.2

sub-

the same method that was used in II.I).

a 6 ~ ~

It

%he

the same. ~

.

and

Q

there

with the f o l l o w i n g properties

vP(n)A(P (n))

Par.(vP(n)A(P (n) )) ~ Pari(F~) l

is a formula of

then the f o l l o w i n g

R two

conditions are equivalent:

(i)

vP(n)A(P (n))

(ii)

E F

VB~E~VT[F ~ i F B

and

term such that

Pari(T ) c Pari(F8)

T

PROOF.

The only interesting

Let

be an

Q

is an

n-ary elementary a b s t r a c t i o n

~>

case is (ii) ~ >

n-ary predicate

parameter

A(T) (i).

such that

E F8]. Thus assume Q ~ Par2(F a)

(ii).

221

E.G.K. Lopez-Escobar, W. Veldman and then let 12.1.

Then

A

constructed

be the subfan of

(ii) specializes

according

to Lemma

to

V 8 8 E A ( A ( I X l . . . X n Q X l . . . x n) E FS) , which in turn leads to

VSBEABP(A(Q) Proceeding

E F(Sp)).

as in the proof of Theorem

11.2 we arrive at

r~i-mA(Q). Then using the assumption

that

Q ~ Par2(F e)

we conclude

r I-~vPA(P) and then that

II

w

Deri~

in S e c t i o n

VPA(P)

II

E F

AND THE SPREAD

~,

Of the conditions

7.1-7.7

listed

7 the following one is the only one that remains to be

proven. 13.1

THEOREM.

For any sentence

S

of

]~

the f o l l o w i n g

two condi-

tions are equivalent:

(i)

Der~R(S )

VCCc~E;~(S E Fa).

(ii) PROOF.

Again the only interesting

case is

ter is proven with the help of the subfan (I)

SEA ~ >

(it) = >

~

of

(i)- and the lat~

such that

S ( r

(II),

(III)

Analogous

to (III) and

w

THE COMPLETENESS OF ~,

(IV) respectively,

of Lemma ii.i.

So far we have shown that for sen-

222

E.G.K.

tences

A

of

Leoez-Escobar,

W. V e l d m a n

I~:

14.1

Der]9(A) ~ >

Va~(A)

(see Theorem

14.2

Va~(A)

nerl~(A)

(see Corollary

However

=>

in order to have an honest

must show that it is plausible

14.3 where

completeness

"The sentence

A

~

theorem

for

~

we

Va~(A)

is an abbreviation of

8.3).

that

Valid~(A)--> "Valid(A)"

5.1)

for the

(informal)

statement

is logically valid from the inruitionistic

viewpoint" In order to avoid multitude

of realizations

the following 14.4

some of the problems associated

schematic

Given that then

~

for

~

with the

it is better to consider

14.3 in

form

is a r e a l i z a t i o n for

~

and that

Valid(A)

~I=A.

One way to show 14.4 would be to give the exact conditions under which:

a sentence

A

of our r e s t r i c t e d s e c o n d - o r d e r

language

i8 logically valid from the i n t u i t i o n i s t i c viewpoint.

Fortunately would Vali~

suffice

we do not need to know the exact conditions.

for our purposes

which would allow us to conclude

show that such properties In V a l i d ( A ) (2)

to specify

characterize

two important

Logical validity

The intuitionistic

viewpoint.

of

14.4, we do not need to ValidR(A).

concepts

and (~)

enough properties

are involved:

It

223

E.G.K. Lopez-Escobar, W. Veldman The e s s e n t i a l

(C1)

characteristics

of

(*) are that:

The validity of a compound sentence be reducible to the validity of simpler sentences,

(C2)

The validity of a sentence be independent of the interpretation of the non-logical symbols. On the o t h e r hand

to be in c o n f l i c t the t r a d i t i o n a l mathematical

with

view

"truth"

fied with

eventually

A possible

the

finding

compromise

as m e n t i o n e d

intuitionism

a proof

in the

is solely (and,

statement

of

(~)

concerned

in fact,

with

the intui-

is sometimes

identi-

of it).

is to satisfy

(**) as far as the

logical

consider

of an intuitionist.

approximation

is a sequence

(both by p r o d u c i n g The

seems

introduction,

In o r d e r to do the latter we must

world"

what he is doing).

characteristic

and t h e i r proofs

could be d e s c r i b e d

"There working

is that

are concerned.

As a first

for,

of a m a t h e m a t i c a l

"mathematical

tionist

(C2)

statements

tionistic

symbols

the e s s e n t i a l

the

"mathematical

world"

of an intui-

as follows: of m a t h e m a t i c a l

statements

new constructions

sequence

itself

on w h i c h he is

and by r e f l e c t i n g

can be thought

on

of as extend-

ing indefinitely". It w o u l d p r o b a b l y arranged

as an

flection

is an important

m-sequence

not a finitist) and

secondly,

experience m -seq u e n c e , person.

be a m i s t a k e

to c o n s i d e r

for two reasons:

part of i n t u i t i o n i s m

and as we all know h u m a n

even t h o u g h

of a given

it does not f o l l o w

Now we are

interested

that

may a p p e a r it w o u l d

in logical

sequence

Firstly,

because

to be re-

(the i n t u i t i o n i s t

reflection

it could be argued

intuitionist

the

that

is very erratic

the

subjective

to him as appear

validity,

is

(part of)

an

so to a n o t h e r that

is we wish

224

E.G.K.

characterize aZZ

those

sentences

intuitionists,

particular appears

creative

subjects

(i.e.,

are not

CI,

@

must

as true by

consider

proofs

any

the way

it

is progressing. for example:

(a) think

iff they

that

correctly,

do have

reduce @

and

Similar interest

both

considerations

assert

proofs

of

A,

struct

later on,

proofs)

the

(b) are and

(c)

@

ac-

of Divine

would

madness,

apply @

to

(AvB)

would

the c r e a t i v e

of

in the

[and

appear,

to us,

If we wish

but also

that he might

intuitionist

not as a

on w h i c h

sequence

(AnB).

subject,

iff

we

Of more

In view

of

see that he

of c o n v e r t i n g

the

that he might

con-

of an i n t u i t i o n i s t

statements

consider

he is w o r k i n g

to be m y s t i c i s m

include

should, c o n s i d e r

world

to u n d e r s t a n d

he a c t u a l l y

(linear)

3xA(x)].

or even those

of m a t h e m a t i c a l

statements

that we

Thus

B.

then we must

it is b e t t e r

accepted.

statements

assert

the m a t h e m a t i c a l

not only those those

was

be a c c e p t e d

in it.

into proofs

sequence

would

had also been

(A^B)

constructed

inspiration).

(apparent)

(A^B)

of m a t h e m a t i c a l

when

if we c o n s i d e r

actions

B

intuitionist

in view of a s s u m p t i o n

only when he had a m e t h o d

already

to be just the

and

concerning

(AnB)

(which

sentences).

see that

is to c o n s i d e r

Thus,

A

sequence

were

our i d e a l i z a t i o n s

then his

the

we w o u l d B

statements

to simpler

only when

was w o r k i n g A

h o w our p a r a d i g m a t i c a l

of c o m p o u n d

if we c o n s i d e r e d

would

intuitionist~

with

by

forgetful.

the truth

both

(i.e.,

that we imagine

world

as valid

too i n v o l v e d

idealizations,

c l a i m to have

Now we must cepts

It suffices

some

be r e c o g n i z e d

not become

to us that his m a t h e m a t i c a l to make

W. V e l d m a n

which w o u l d

so we should

intuitionist.

We do have

honest

Lopez-Escobar,

in his

considers later

on.

(or the result

the m e t h o d

in the

"mathematical

world"

at a given moment, Or in o t h e r words,

the m a t h e m a t i c a l

sequence

on,

but r a t h e r

world

of a

as a p a r t i a l

E.G.K. Lopez-Escobar,

W. Veldman

225

ordering. Our notion of r e a l i z a t i o n

for

called a "temporal" record of the

~

corresponds

to what might be

(possible) results of a creative

subject, w h e r e for simplicity we "record" only atomic acts or statements.

That is, suppose given

~

= .

corresponds to the structure of e v i d e n t i a l and e n v i s i o n e d by some creative subject. Dl(e) up

~;

D2(a)

Then given an

D2(a)

lection of rather simple species)

~EK,

objects e o n s t r u c t e d

is the c o l l e c t i o n of atomic

since species are properties,



situations e n c o u n t e r e d

gives us the c o l l e c t i o n of m a t h e m a t i c a l

to stage

Then

statements

(or

could be c o n s i d e r e d as a col-

c o n s i d e r e d up to stage

specifies those atomic statements v e r i f i e d by stage Now, granted such a reading for a r e a l i z a t i o n

a.

M(a)

a.

~

of our res-

tricted s e c o n d - o r d e r language, we obtain, by a simple induction on the logical c o m p l e x i t y of the sentences of

{A : A

is a sentence of

~

~,

and

that given

~EK:

~]=aA}

coincides with the c o l l e c t i o n of sentences c o n s i d e r e d as true sentences by the creative subject at stage In p a r t i c u l a r we obtain that ~I=A}

{A : A

~. is a sentence of

~

and

is the c o l l e c t i o n of sentences which are true for the given

creative subject at all stages of his m a t h e m a t i c a l world. Now we can return to 14.4. is a r e a l i z a t i o n of

~

Thus suppose that we are given that

and that V a l i d ( A ) .

Then

ered as true by one and by all of the intuitionist.

A

is consid-

Thus

appear in all the m a t h e m a t i c a l worlds of the intuitionist, A

is logically valid it would be in all the stages.

can be i n t e r p r e t e d as t h e ' ~ a t h e m a t i e a l world" then

A

would

and since

Thus if

~

of some i n t u i t i o n i s t

~I=A. Thus the truth of 14.4 is reduced to the question:

226

E.G.K.

Lopez-Escobar,

Can be an arbitrary r e a l i z a t i o n

W. V e l d m a n

~

of

~

be i n t e r p r e t e d as the

(possible) m a t h e m a t i c a l world of some i n t u i t i o n i s t ?

Actually,

in view

pleteness t h e o r e m tion.

answer

(?)

it suffices

Thus w h e t h e r

in fact

of Corollary

an honest

to c o n s i d e r

or not the

theorem

w

correctly

thinking

intuitionistic

restricted

second

realizations Furthermore of the

gives since

does

R

of

~

is

upon a p o s i t i v e

(see Section 8) be in-

the

intuitionists

and not forgetful) of them

(i.e.,

it seems

so as to have

AND KRIPKE MODELS,

predicate

order

logic

calculus ~,

us a (formal)

the notion

the r e a l i z a t i o n s

Kripke models

relationship for the

or not rests

shown

to be

not u n r e a s o n -

a positive

an-

(?).

R E A L I Z A T I O N S OF

of the

~

idealized

able to a l l o w our i d e a l i z a t i o n swer to

result we have

(possible) m a t h e m a t i c a l world of some i n t u i t i o n i s t ?

Since we have a l r e a d y honest,

realiza-

question:

Can the u n i v e r s a l r e a l i z a t i o n

terpreted as the

of the com-

just the u n i v e r s a l

completeness

completeness

to the f o l l o w i n g

8.3, for the p u r p o s e s

between

for

Since

has an a n a l o g u e

(formal)

of v a l i d i t y ~

specially

IPC via Kripke m o d e l s

leads

since

in the

for

~

vla

for the IPC.

are n a t u r a l

modifications

to c o n s i d e r

a completeness

to M a r k o v ' s

A

formula A~

validity

of IPC it may be of interest them;

every

principle

the

theorem and ours

not. It is now common

is e q u i v a l e n t "absurdity"

to s e c o n d - o r d e r

may be defined

that by further to all

knowledge

intents

restricting and purposes

that

second-order

intuitionistic

by

V(2)PP. ~2

minimal

logic

and that

What we have

we obtain

a conservative

logic

extension

in

observed

a calculus

~

~2 ~2 is

which

of IPC and

is in

E.G.K. w hich n e g a t i o n Because

tions even

(nor absurdity)

~

in view of our

of

~,

~J=l

verified)

ments

are

the v i e w p o i n t

15.1

such that

~ J=l.

treatment

The

(informal)

but r a t h e r

to the

considered

in

~

DEFINITIONS.

of

I

Or put

we obtain

appear

interpretation

for the r e a l i z a true

too few e l e m e n t a r y

in more

some

strange.

to the False being

fact that

~ .

227

rSle.

latter may

does not c o r r e s p o n d

from

W. Veldman

has no p r i v i l e g e d

of such n o n - p r e f e r e n t i a l

realizations However,

Lopez-Escobar,

picturesque

(or state-

terms,

is not very d i s c r i m i n a t i n g .

Let

~ =

be a r e a l i z a t i o n

and

e~K.

(i)

~

is a credulous

(2)

~

is a trivial r e a l i z a t i o n

iff every node

(3)

~

is a natural

iff there

for

Then node

of

realization

~

iff

~J=~ I. is credulous.

is a node

of

~

which

is

not credulous. (4)

is an ideal r e a l i z a t i o n

~

iff every node

of

~

is not credu-

lous.

The

ideal r e a l i z a t i o n s

that any

ideal r e a l i z a t i o n s

(as far as f i r s t - o r d e r versa.

15.2 (I)

tion

to Kripke

can be t r a n s f o r m e d

formulae

The t r a n s f o r m a t i o n s

are

concerned)

models

in the

sense

into an e q u i v a l e n t Kripke model

and vice

are as follows.

DEFINITIONS. Given

a realization

the Kripke m o d e l (2)

correspond

Given

a Kripke

predicate

let

Q

=

for

~

let

(~)i

be

. model

First

~

for

K = ~

obtained

Then define

(K) 2

be the r e a l i z a -

as follows:

be a new p r o p o s i t i o n a l

parameter).

let

parameter

for all

~(K:

(i.e.,

a

0-ary

228

E.G.K.

The r e m a r k

If

~

:

D(~)

D2(~)

=

{Q} U M(a).

the e q u i v a l e n c e

is an ideal r e a l i z a t i o n such that

K

for

is a Kripke

~

itf

model

A

A

iff

f

is a Kripke

model

15.6

If

~

is an ideal

realization

node

(i.e.,

if we simply then, Let

if

~

above

then

all the

anything

is a

of IPC

is an ideal

realization

((K)2) 1 : K.

for

and

~,

then

for any

A

a credulous

node

~ s 8

credulous

is left,

be the result

A

(~)i

of IPC

formula

is credulous

delete

provided Red(~]

first-order

every node

then

(K)21= ~ A*.

If

Because

~

(K) 2

15.5

essentially

ideal r e a l i z a t i o n s

~l=e A*.

then

and for any formula K~

for

for any formula

(~)iI= ~ A

If

between

as follows:

Kripke m o d e l

15.4

W. V e l d m a n

DI(~)

concerning

can now be stated

15.3

Lopez-Escobar,

nodes

we obtain

of d e l e t i n g

the

is also

then

8

a credulous is credulous)

of a r e a l i z a t i o n an ideal

realization.

credulous

nodes

of

~.

Thus

15.7

If

~

is a n a t u r a l

realization

then

Red(~)

is an ideal

realization.

It is easily if

~

natural tence

proven

is a credulous realization, A:

by induction

node

of

~,

on the c o m p l e x i t y

then

~ = ,

~ I= A.

we obtain

Then that

of if

A ~

for any

that is a sen-

E.G.K. Lopez-Escobar, W. Veldman 15.8

~ I=A = >

15.9

Red(~)l=A ~ >

229

Red(~) I=A,

Ve ( K ( 7 7 ~ I = a A ) .

In the case of the universal spread

~

: <~,S,...>,

Lemma 8.2

allows us to t r a n s f o r m 15.9 into:

15.10

Let

Red(e) l=A : >

MA

Ve (~ 7-13m(A ( F ( ~ m ) ) .

be an a b b r e v i a t i o n for the following:

Er(~m)) : >

MA : Va ( ~ 7 7 3 m ( A

Ve ( ~ 3 m ( A

E r(~m)).

Then from 15.10 and the p r o p e r t y of the u n i v e r s a l r e a l i z a t i o n we obtain:

MA =>

15.11

Since

Red(e)

15.3 - 1 5 . 6

Der~(A)).

is an ideal realization, we can apply the results

to conclude that for f i r s t - o r d e r sentences

MA, = >

(A

is valid in all Kripke models = >

Or in other words, recursive predicates

(~)

(Red(e)l=A = >

A

I--IPcA).

under the a s s u m p t i o n that for all p r i m i t i v e A(n):

V~ E~ u 7 3 n A ( a n )

~>

Va 6 ~ 3 n A ( ~ n )

we have r e c o v e r e d the completeness t h e o r e m for IPC w i t h respect to Kripke models which, as shown in Kreisel 1961, implies

Ve (B q 7 3 n A ( n , a )

=>

for primitive r e c u r s i v e predicates

V~ (B3nA(n,~).

A(n,~).

Since the nodes above a credulous node are also credulous~ we see that the r e a l i z a t i o n s up from ideal r e a l i z a t i o n s

for

~

can be c o n s i d e r e d as being built

(which c o r r e s p o n d to Kripke models

for

230

E.G.K.

Lopez-Escobar,

IPC) and trivial realizations

W. V e l d m a n

(in which every sentence

would appear that the reason why Markov's our completeness

theorem

a node in a realization

principle

is true).

is avoided

It

in

is because we do not have to decide w h e t h e r is credulous

or not.

Further evidence

to

this last remark is given by the following: 15.12

THEOREM.

If

~

is a natural realization for

=

and (D)

then for every sentence

S

of

l=S PROOF.

Let

complexity

K~ = {~K

iff

:7 ~ I =

Red(~)l=S.

I}.

Then we prove by induction

of the formula

(i)

for all

aEK*,

~ I = a A ~ Red(~)I: aA.

Let us consider the case when

A = (BnC).

Red(~)I= e
is straightforward.

(BnC)

and let

BEK

the assumption

that

@~K*,

then

B

B ~ a

Red(~)l= ~ (BnC)

by the induction hypothesis is a credulous

~

that

[ ~ I = 8B = >

That is ~ I=~ (BnC).

(D), that for all sentences Va E K[~I=aS]

S iff

~I=@B.

and

we conclude

node of v B~K*.

(i) has been established

thus that

gives us that

more we are assuming ~I= BC].

BEK*

That ~l=a (BnC)

Suppose

be such that

hence,

Once

on the

Red(~)I= ~ If

B(K*

Red(~)l= BC

then and

that

~ I=BC.

If

and thus

~ I = BC.

Further-

Thus we have that

we obtain,

VB {

again with the help of

of Va ( K*[Red(~)I=~S].

E.G.K. w 16. HISTORICAL The above lings

theorem

his book "The Foundations can be compared

structed

in w

231

seems to confirm E.W. Beth's

the way in which his theorem

One gets this impression

there

W. Veldman

REMARK

completeness

concerning

Lopez-Escobar,

from reading Section of Mathematics".

with the spread

The semi-models

could be saved. 145, last paragraph,

(The semi-model

~ of explicit

theories,

theories,

occuring

was right in thinking that defining a fan of models, some inproper

points,

and then applying

in

M mentioned con-

in M which are not proper models

be compared with the overeomplete

taining

own fee-

can

in ~ ). So Beth necessarily

the fan theorem,

conwould

give the desired result. It is a pity that he did not carry out this program and distorted Brouwer's

argument

for the fan theorem.

seen the crucial rSle

of negation.

paper are a continuation

of Beth's

He also does not seem to have

Nevertheless,

the ideas in this

and it is a matter

of historical

justice to mention his name here. (We are indebted

to Prof.

Troelstra

for asking our attention

for this.)

232

E.G.K.

Lopez-Escobar,

W. Veldman

REFERENCES BETH, E,W, 1949 Semantical considerations on intuitionistie m~thematics, Math., vol. 9.

Indag.

1955

Semantic entailment and formal derivability, Mededelingen der

1956

Semantic construction of Intuitionistic Logic, ibid., vol. 19,

Kon. Ned. Akad. v. Wet., new series, vol.

18, no. 13.

no. ii.

DYSON, V,H, AND KREISEL, G, 1961 Analysis of Beth's semantic construction of Intuitionistic Logic, Technical Report No. 3, Applied Mathematics and Statistics Laboratories, Stanford. GIRARD, J-Y, 1971 Une extension de l'interpretation de G~del ~ l'analyse, et son application ~ i'elimination des coupures dans l'analyse et la theorie des types, Proceedings of the Second Scandinavian Logic Symposium, (Fenstad, Editor) pp. 63-92. HEYTING~ A, 1966 Intuitionism, An Introduction. Amsterdam.

North-Holland

Publishing Co.,

KREISEL, G, 1962 On weak completeness of intuitionistic predicate Journal of Symbolic Logic, vol. 27, pp. 139-158.

logic, The

KRIPKE} S, 1965 Semantical analysis of Intuitionistic Logic, Formal Systems and Recursive Functions, (Crossley and Dummett, Editors), pp. 92-129. PRAWITZ, D, 1965 Natural Deduction, A Proof Theoretical Wiksell, Stockholm. 1971

Study.

Almquist and

Ideas and results of proof theory, Proceedings of the Second Scandinavian Symposium,

(Fenstad,

Editor) pp. 235-308.

TROELSTRA, A,S, 1973 Metamathematieal Investigation of Intuitionistic Arithmetic and Analysis. Lecture Notes in Mathematics, vol. 344. E.G.K. L6pez-Escobar Department of Mathematics University of Maryland College Park, Maryland 20742

W. Veldman Math. Instituut Katholieke Univ. Nijmegen

THE REAL E L E M E N T S

IN A C O N S I S T E N C Y TYPE T H E O R Y

Dedicated

to Kurt

SchHtte

PROOF FOR SIMPLE

I

on the o c c a s i o n

of his 65 th b i r t h d a y

Horst

Luckhardt

Abstract

This

is the first of two papers

of simple tion,

type theory mean,

the s t r o n g e s t

question

recursive plus

type

theory

functionals

results

one c o m p r e h e n s i o n

method

today.

by i n t e r p r e t a -

Concerning

to a r i t h m e t i c

the q u a n t i f i e r f r e e

over

consistency

the

first

are proved.

can be r e d u c e d plus

What does

and hew can it be a c h i e v e d

constructive

the f o l l o w i n g

(I) Simple

on the questions:

a special

over

the p r i m i t i v e

rule of e x t e n s i o n a l i t y

A-property

on objects

of type

O.

(2) C o n s i s t e n c y cator", types

is fully d e s c r i b a b l e

a generalization

and e x t e n s i o n a l i t y .

(3) A n e c e s s a r y primitive

recursive

types O(Oe)

based

it rather

instances

ency proof

has

of c o n t i n u i t y

for c o n s i s t e n c y contains

indi-

with r e s p e c t

to

is that

the theory T of the

no c o n t i n u i t y

indicator

for

points.

gives

to the negations

and to c o n t i n u i t y

to make

"continuity

Especially:

on c o n t i n u i t y

leads

prehension

of the m o d u l u s

functionals

at d e f i n a b l e

The ~ - o p e r a t o r ening;

condition

by the n o t i o n

very

no p r o o f - t h e o r e t i c

strength-

of the above m e n t i o n e d indicators.

fine distinctions.

Therefore

com-

a consist-

234

H.

(4) w - r u l e s

for

to t h e o r i e s

types

greater

of c l a s s i c a l

strength.

Moreover

AemiVa.e

= a imply

recursive

constructive

Markov's

incompatible

with

higher

second

paper makes

question

by

and

By G ~ d e ~ s w e l l - k n o w n creasing

strength

consistency

concepts

} O are

the

this

it f o l l o w s

intuitionistically

functional

functionals

which

Significant

by p u r s u i n g

which

was

interpretation have

only

results

Hilbert's

developed

in a p r e v i o u s

using [33.

than on the

But

this

proof

more

this

on

of

a descripon the

fundamental

in f u l l

paper

for

justified

for a r i t h -

the b a s i s

ordinal

segments

and

intuitionistic

means

properties

to the b a r - r e c u r s i v e

functionals.

of t h e s e

is shown.

For

today

in the

concentrates

to a q u a n t i f i e r f r e e

functionals

concept.

we have

more

notions

there

of

calculus

is v i a

their

- in the c a s e step

on

constructive

functional

first

con-

one way

inter-

ordinal

o n the c o n s t r u c t i v e The

in-

inductions

f o r m of v a r i a b l e

second

analysis

is o n l y

So the o n l y

of

and relations

into

transfinite

of p r o o f s .

In the

systems

by an i n s i g h t that

good

for

abstract

not mean

analysis

This method

and more

proofs

does

is a v e r y

combinatorial

in a r e d u c t i o n

consistency

can only be

say b y f i x e d

- although

pretation

From

of a n o n -

survey

require

meaning.

to do this, them

of

obtained

results

them which

structive

the e x i s t e n c e

absurd.

type

of a c e r t a i n

and K r e i s e l - T r o e l s t r a ' s

that

- explicit.

the a u t h o r

analysis

consistently

interpretation.

Introduction

between

are

be a d d e d

: a.

failure

means

above

and analysis

of

and A~Va.~

the

cannot

(MP)

is n o t

objects

of t h e r e a l e l e m e n t s ,

functional

O.

principle

- even with

intuitionistic

concept metic

(MP)

comprehensions

tion by

zero

intuitionistic

function

fo~ lawlike

second

than

intuitionistically

that w-rules

The

and

Luckhardt

step

content consists

of a n a l y s i s ,

the c o m p u t a b i l i t y

the n e w a b s t r a c t

notions

H.

used

The

are of

first

mainly

principle

is an i n d u c t i o n

computation the author

t i o n of B r o u w e r ' s

The

second

tool

computation ciple

according

consistency

constructively

of

to r e a l i z e

matical bert's

real

vantage

and

that

it g i v e s

the q u e s t i o n

The

answer

no!

classically

directly method

Can

in t h e

and

gives

of

of

us

this

the

opinion

- although

connexion

it c o n c e r n s

with

which

notions

the

are

prin-

for the theorem;

which

domains

occasion method

to s i m p l e simple

enable

of m a t h e on H i l -

has

a constructive

the adcontent.

type

type

theory?

theory

- the higher

is

types

cannot

be g r a s p e d

forced

to set u p a n e w

analysis.

treatment

the

that we dispose

analysis

So h e r e w e

that

way.

fact

theorem

objects

says

is n e e d e d

at a n o t h e r

of

species.

underivability

be e x t e n d e d

the p r i n c i p l e s

The

is the a d d i t i o n a l

interpretation

is t h a t

manner.

from

which

and uniform

classical

its p r o o f - t h e o r e t i c a l

author's

on the

explained

idealized

up

but which

second

-

is the g e n e r a l i z a -

such fundamental

this m e t h o d

The reason

new

rests

abstract

elements,

in an e f f e c t i v e

which

analysis

ics.

is:

is:

a generalization

produce

proof

to e a c h

built

the m e t a t h e o r y

as a n a l y s i s .

this

in a c o n s t r u c t i v e

the a u t h o r

ideal

Now

only

As

strong

This

the s y s t e m

the c o m p u t a b i l i t y

activity.

types.

to G ~ d e l ' s

sufficient

makes

principle,

in all

directly

as

to s p r e a d s

is n o t w i t h i n

it c a n b e v i s u a l i z e d

us

works

which

for a t t a i n i n g

bar-induction

proof

the

- locally

introduced

procedure

consistency

theory

principle

is a u n i f o r m i z a t i o n

here which

In short,

235

two k i n d s :

tool

the

Luckhardt

in E33

fundamental

The

significance

is s t r e s s e d domain

of

because

of m a t h e m a t -

H.

236

The

aim of

these

will

look

for n e w

to us

a further

two p a p e r s

and

notations

direction

type

theory.

be t r a c e d

the

this

properties.

situation.

These

At

first

then w i l l

we

indicate

of r e s e a r c h .

axiom

theory

systems

over

the

considered

functional

here

for a r i t h m e t i c ,

language

as w e l l

as

the

used.

w 2 is d e v o t e d type

is to a n a l y s e

characteristic

In w I we d e s c r i b e analysis

Luckhardt

to the p r o o f - t h e o r e t i c

By the m e t h o d s

back

o~ G ~ d e l

to c o m p r e h e n s i o n s .

ducible

from

one

simple

Finally

also

extensionality

reduction and C o h e n

Further,

comprehension

and the

each

(C)~

axioms

of

a pure

of

choice

comprehension

of

can be e l i m i n a t e d

simplification

can

is de-

A-property.

by r e l a t i v i z a t i o n

as

in [3].

Because

~((C)~

sistency

of type

represents

a generalization and

theory

is e x p r e s s e d

of the

"modulus

extensionality.

theory

T of

indicator that

for

the

types

the a d d i t i o n

strengthening

of

and

functionals

based

because

distinctions.

w 4 it is p r o v e d to c l a s s i c a l

the

~-operators of

consistency

definition.)

that

w-rules

or i n t u i t i o n i s t i c

points.

respect is,

that

the

no c o n t i n u i t y

does

the

not

shown

alter

formation

the

of c o n t i hand

- is a p r o o f - t h e o r e t i c

the p r e c e d i n g

level

a consistency

~ 0 cannot

theories

to type

(On the o t h e r

assumed

types

-

- It is f u r t h e r

~((C)~

of

con-

indicator"

contains

allow

Therefore

for

with

on c o n t i n u i t y

(C) a, ~ ~ 0 - c o n s i s t e n c y

make

ently

Such

in w 3 the

for e x a m p l e

at d e f i n a b l e

the d e r i v a t i o n

by t r u t h

In

of c o n t i n u i t y " condition

of ~ - o p e r a t o r s

be p r o v e d fine

O(0~)

strength.

indicators

the a d d i t i o n

A necessary

continuity,

by a " c o n t i n u i t y

the p r i m i t i v e - r e c u r s i v e

proof-theoretic nuity

a certain

which

can

proof

has

be a d d e d

contain

the

then to

consist-

H.

primitive-recursive theorem

and w h i c h

Moreover tive

principle

compute

(MP)

O with

of the e x i s t e n c e

the failure

comprehensions

means

elements,

and analysis,

is studied

on the c o n s t r u c t i o n

are c o n c r e t e

m e n t of type

hints

theory

Functional

the n o t a t i o n s

there

in this

statements

indicating

Axioms

language

of type

which

treat-

to formulate

Notations.

the theory of simple

only e q u a l i t y the n o t i o n

types

of type o. Besides "type degree"

~: m a x ( g s l , . . . , g ~ m ) + 1 , the c o r r e s p o n d i n g

an e x t e n s i o n a l i t y

rule

(ER)-qf for q u a n t i f i e r - f r e e

to that in [33,

for f i n i t e l y m a n y e l , . . . , ~ m for gs > g6.

be realized.

proof-theoretic

type ~ can be coded

~ 6 is short =

re-

proceed.

of type theory.

the type O s l . . . ~ m ( m ~ O)

>

to the case of

can a c t u a l l y

Each

>

for arith-

characteristic

set hierarchy.

analogous

of

concept

in [43

Contrary

case,

might

in [33 here we still use

for

Then H i l b e r t ' s

how a natural

E3J using

g ( O ~ 1 . . . s m) denotes

are,

it is s u i t a b l e

in the f u n c t i o n a l

holds

objects

the

can only be describ-

by the author

for type theory.

by i n t e r p r e t a t i o n

language.

For our p u r p o s e s

a manner

this

interpretation

which

explicit.

w h i c h was r e a l i z e d

strictions

T plus

construc-

with M a r k o v ' s

= a. From

for lawlike

functionals

- is m a d e

and analysis,

the power

together

of the f u n c t i o n a l

- even with

arithmetic

w hich

recursively.

of a n o n r e c u r s i v e

As~Va.s

of e-rules

and the fan-

(MP) and A s ~ l Va.s = a follows.

of the real

I.

arithmetic

constants

intuitionistically

incompatibility

ed by i n t u i t i o n i s t i c

These

function

and K r e i s e l - T r o e l s t r a ' s

In the second part

metic

237

intuitionistic

all their

can be p r o v e d

intuitionistic

higher

functionals,

the n o n a b s u r d i t y

function

Luckhardt

pp. and

24-26 6 with

level

in each type

provided

6 within

formulas,

g6 > ge;

in

in

the same

g6 ~ m a x ( g s l , . . . , g S m ) .

238

H. L u c k h a r d t

E s t i m a t i o n s of p r o o f - t h e o r e t i c

strengths - by this we u n d e r s t a n d the

imitation of proofs of one system by proofs of another system in a fixed manner -

(The insight in such facts has its own p r o o f - t h e o r e t i c

are e x p r e s s e d by <, _<, =. [] denotes a quantifier,

strength.)

and Ao,Bo,...

are

q u a n t i f i e r - f r e e expressions.

The following systems are given in the form e x p e d i e n t for functional i n t e r p r e t a t i o n [3~. These a x i o m a t i z a t i o n s differ from the usual ones but are p r o o f - t h e o r e t i c a l l y e q u i v a l e n t to them.

Arithmetic A C l a s s i c a l p r e d i c a t e logic E q u a l i t y axioms for type O Peano axioms Primitive-recursive

functionals T of all finite types

Q u a n t i f i e r - f r e e e x t e n s i o n a l i t y rule

(ER)-qf

Analysis A plus e-choice

(~AC) ~

Q u a n t i f i e r - f r e e choice

(AC)~'B-qf

In [3] it is shown: A $ A r i t h m e t i c ~ T and

Simple type theory A plus Extensionality Comprehension

(ER) (C) ~ for all

Q u a n t i f i e r - f r e e choice

(AC)a'B-qf

~

$ Analysis ^ = T U BR.

H.

is a g e n u i n e (mAC) e comes

part

of

7.

Luckhardt

(C) ~ and

289

(AC)~'~-qf

give

-->

9

~o~

-->

VF~~176

vz 1

(AC) c~,c~

0

AX , y a A ( x , y , z l y x ) ' ))

Gz~~

~

Gzn'

= z(Gzn)n

namely

F = Gz I

(Here and b e l o w

functional

It f o l l o w s

the r e s u l t s

~

from

contractions

in [3]

~ is r e d u c e d (c)O~-A

suppressed.)

and b e l o w

proof-theoretically

: vyO(O~)AxO~[yx

2.

Reduction

to

Our

first

is to s i m p l i f y

aim

the a c t u a l

a model

that

(C) ~ is not

provable

of

essence

of

(see [3],

side

chapter

(functionally

and

the

ing

the old

(ER),

extensionality II).

But

~ O].

of

observation

By the m e t h o d s

that

(C) e

the

(ER)

and

to b e c o m e of G ~ d e l

aware

and C o h e n

(C)~:

can be e l i m i n a t e d flexibility

(ER)-qf

comprehension

~ in o r d e r

(C) ~

for m o r e

interpretable)

instances

AZ~'XZ

to r e d u c e

in A plus

7 ~ A plus Now

~ 0 <-->

the p r o b l e m .

~ can be d e f i n e d

the r i g h t

to A plus

(C)~

~ A plus On

are

for ~ > O.

In w 2

of

from which

as f o l l o w s :

Ax~

in

(AC) ~'~

and for

are

thus

we

by r e l a t i v i z a t i o n add

again

the

simple

obtain:

all simplified.

introduction

We do this

of h i g h e r

types

by f o l l o w increases

240

H.

the s t r e n g t h pressed:

situations

of h i g h e r

Theorem all A

o

of p r i n c i p l e s

Luckhardt

well-known

can be s i m p l i f i e d

in lower

types or o t h e r w i s e

in a u n i f o r m m a n n e r

with

ex-

the use

types.

1(a)

F o r all types

B, 6 i, ~ ~ B, u

L 6i

(i = I ..... n) and

:

. ~ A o ( ~ - l y ~ ' ~-I Y I ' ' ' ' ' A (C)~_[]ly ~ "'" []nZn

~ lyn)

<-->

(C)B-DIX~I ... [ 3 n X ~ A o ( X B , X I ..... x n) where

~, ~-I

The p r o o f

are c o d e s and t h e i r

is o n l y an e x e r c i s e

[3x6B[x] f o l l o w s

in coding.

the r i g h t

the left side by s u b s t i t u t i n g

Theorem

I (b)

Proof:

By t h e o r e m

~-(C)~ A

r

<-->

from w h i c h

one gets back

to

for x.

for all B >

I (a) w i t h ~ - B, n = I

= o-->

Icl~

h e r e can also be w r i t t e n

But this f o l l o w s

I(c)

Because DyYB[~-ly]

side,

(C)~

Icl~

Theorem

types.

= O <--> D 1 X l . . . [ ] n X n A o (%-ly,xl ..... Xn)]

y by #x B y i e l d s

The p r e m i s e

of a p p r o p r i a t e

f r o m ~ - 1 ~ x = x the left side states

VzAya[zy Replacing

inverses

directly

from

F o r e a c h A there

(C)~

-->

as

= o nvB (IY'X(~-Iy))Yl (C) oe ---~I

= O.

(C)~

is an ~ such that for all

B >

(C)-A

A

Proof:

It s u f f i c e s

to p r o v e

the a s s e r t i o n

for p r e n e x

A ~ []iXl...~mXmAo[X I ,. . . , X m , X , U I ,. .. ,u n], w h e r e variables

of A. We use i n d u c t i o n

with respect

u I ,. ..,Un are all free

to m.

H.

I.

m=

Let

~ be

Luckhardt

241

O: the

characteristic

functional

of A

in T

([3],

p.

62) :

O

~ U l . . . U n u = O <--> In t h i s

case

take

e H: 0 a n d

A o [ U , U I ..... u n] ~ U l . . . u n as

the

desired

comprehension

functional.

II.

m+

O:

Case

1:

A

E Ax] D2x2...DmXmAo[Xl

= By

induction

Ax'~

B [ x 1 , x , u I ..... u n]

hypothesis

(C)~

-->

..... X m , X , U I ..... u n]

there

VyAx,x]

is a 6 w i t h

{yxx I = O < - - >

B[Xl,X,Ul,...,Un]}

Thus (c)O6-A^(c)OY-A

-->

VyAX

{AxI.yxx I = 0 <-->

VYIAuOY

(+)

-->

namely

z = l x . y I (yx).

~-:

gives

~ > 6,y

and

(C)~

{zx = 0 < - - >

A X ~ ' U X I = O}

A[X]}

6

if

Y

otherwise

all

g6

> gy

B >

--> (C)~

,, (c)~

Theorem

(+)

--> (C)-A Case

2:

According

A

-= Vx] to c a s e

(C)~

A

Putting

{

for

VZAX

{yl u = 0 < - - >

A[X]}

B[Xl,X,u1,...,u ] there

n]

is e s u c h

Vzmx

{zx = O < - - >

-->

VzAx

{~(zx)

-->

VyAx

{yx

that ~A[x]}

= 0 <-->

= O <-->

for

A[X]}

i[x]}

all

@ >

I (b)

H.

242

namely

y = Ix.s-g(zx).

Remark:

Similarly

For e a c h A there

one

theorem

paragraph

Theorem

I(c)

can be

prehension

can g i v e

is ~ such

(C)~

Using

Luckhardt

^^

(c)~

continued.

figure

is c a l l e d

for

all

-->

(C)-A.

there

reduced

of

Because

in a p r o o f

~ =^ ~r w h e r e

in a p r o o f

that

the r e d u c t i o n

instances

2

an i n t u i t i o n i s t i c

there

type

in the b e g i n n i n g

are o n l y

of A plus

is at m o s t

namely:

B >

~ given

~r H: A plus

reduction,

this

many

com-

(C) ~, we n o w h a v e

(C)~

one

finitely

of

with

(C)~

the r e s t r i c t i o n

that

axiom.

theory.

r

Remark:

From

equivalent

the n e g a t i v e

to its

"intuitionistic"

The

significance

the

impredicativity

meaningful the

special

asserts in our type. tivity

of this into

Thus

reduction two of

comprehension

(C)~

the the

normally

obstacle

considered

lies

in the the

the p r i m i t i v e which

- from we have

but

this

consequences

for

at once

that

~r is

version.

of a f u n c t i o n a l

zero-functional actual

it f o l l o w s

components:

impredicativity

the e x i s t e n c e case

translation

fact

natural

to o v e r c o m e

and

predicative

a fixed

the o t h e r

classically

functionals

formally

separating

it a n a l y s e s

and c o n s t r u c t i v e l y

recursive

looks

all

that

functional

objects

is not

accepted

the

of

the

and same

impredica-

separation

principle.

N o w we d i s c u s s

the

consistency,

which

can be e x p r e s s e d

H.

with

the use

of

Z

243

Luckhardt

as f o l l o w s :

r r

<=>

There

is an ~ w i t h :

~ v x ~176

Ay

{xy : O <-->

Az-yz

= O}

A (Deduction

<~>

There

is a n e w i t h :

~-~ V x O(O~) A + (ER) In this w a y w e a continuity

see

~-~

Here

o n the r i g h t

described

3.

<~>

Ax

For

all

by

Definition:

of

Ay(xy

= O-->

a separation

~: ~ - ~ Ax~176 A + (ER)

Az.yz

principle

indicator,

Consider '''''

u 6 <=>:

= O) }

involves

In g e n e r a l

the

continuity

such a situation

is

p-operator.

9n

%x is a n o t h e r

= O}.

of

^ yz#O)}

indicator".

8 ~ O8n...61

A x e6

Vy,z(xy=O

the u n d e r i v a b i l i t y

Vy ~ ~ . x y

a "continuity

'

In short:

the n e g a t i o n

side we have

{x~ = O -->

Continuity

the place

that

{xC~ = O ^

statement.

(~)

property

theorem)

(n ~ O).

is a c o n t i n u i t y {x(%x)~XU

argument

indicator

for

A % x ( ~ i x ) ... ( g n X ) ~

effectively

different

type

~6 a t

9 .. (~n x) }

from u with

the

same value.

The

"continuity

continuity".

indicator"

Every

modulus

is a g e n e r a l i z a t i o n

of

of

a continuity

continuity

gives

the

"modulus

of

indicator

244

H.

but not vice

versa.

Luckhardt

As an e x a m p l e

for

type

O(OO)

the m o d u l u s

of c o n t i n u -

ity X w i t h Ax ~ < • 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 gives

the

~ o,

following

B ~ o0,

Remark:

Of

indicator;

The

here

clarity

from

the

these

continuity

the p l a c e

the

above

of the n o t i o n

immediate

situations

insight

are

them

in g e n e r a l

[13,

71).

The

possible portant

to t r e a t example

N o w we r e t u r n indicator tained

r

in T,

"modulus into

continuity

intensional,

Ix~

of

~ (o (o~))

, 41

not

for

comes

there [23,

only

statements

be g i v e n

consistency

and

to a r b i t r a r y

continuity

into

the

continuity

constructively

of c o m p u t a t i o n s .

(see K r e i s e l

indicator"

this w i l l

to the

1 ~176~:

suffices.

of c o n t i n u i t y "

the

sequences

certain

(o (o~))

at p l a c e

incorporated

notion

extensional

from

for

u B can be

weaker

"continuity

lus of c o n t i n u i t y "

indicator

essential

to m a k e p.

uv s uw

Cu ~176176-.: i, X(u,I) , @i u ~: X(u,I)

B I ~ 0;

course

: w~176

is no h i n t p.

of h o w

154 and K l e e n e

generalizes

types,

However

the

but m a k e s

"moduit also

extensionally.

An

im-

below.

~ and

line

type O(O~)

(*).

If a c o n t i n u i t y

at the p l a c e

~o~

is con-

i.e. uO(~

(@u)

~ u~^

~U(@lU)

~ 0 ~ ~(@lU),

T then

for

be p r o v e d

this

~ the

continuity

statement

on the r i g h t

in A:

u~ = O--> -->

Consequently

u(~u)

= u~ = O ^

Vy,z(uy

Cu(r

: O A yz + O)

@ O

side

of

(*) c o u l d

245

H. L u c k h a r d t

Theorem (a)

3

Z ~+ ~

=>

There

for type O(O~) (b)

~r

~

<=>

is no c o n t i n u i t y

at the p l a c e ~o~

The e x i s t e n c e

at the p l a c e ~o~ form AxV~x,

(b) f o l l o w s equivalent VxAy

p l a c e ~o~. finable

of a c o n t i n u i t y

in A+(ER)

to

(*),

is, in A,

(ER), f i n a l l y

~ contains

afortiori

~o~ h e r e c h o s e n

no m o d u l u s

of c o n t i n u i t y

can be r e p l a c e d

at the

by any o t h e r de-

p l a c e of the same type.

o

(~I) u(l,~u)

~z o [u(i,z)

can be a d d e d

D-operator

(with l B -: lXnB" ...x~ 4 .I for

} ul

to a r i t h m e t i c ,

T and a n a l y s i s ,

the p r o o f - t h e o r e t i c

continuity

for type O(~0)

indicators

the f o l l o w i n g

T U BR c o n s i s t e n t l y

strength.

In these

can e a s i l y be e x h i b i t e d .

extensional

continuity

so

- : I,UU

u(@u)

o u(1,~u)

@U(@lU)

o

, @i u -: Wu

, further

So p p r o d u c e s

indicator

@i are a r b i t r a r y ,

= ul

= 1,~u(uu)

= s $ i = l(@lU)

(BI)

and

"inessential"

I: Cu o (ao)

B---OBI...B n)

by the a x i o m s

u(1,v)

changing

extensions

o ul]

extensional

o ul

v ~ < ~u-->

because

to

A z . y z = O}.

is c h a r a c t e r i z e d

without

(C)~

{xy = x ~ <--> A z - y z = O) and, w i t h

_ The p l a c e

puO(~O)

(~2)

for type O(O~)

or in ~ in the l o g i c a l

(a) from the fact that

In the n e x t t h e o r e m we s h o w t h a t the f o l l o w i n g

which

indicator

~i x.

to V x A y ~

Thus a c o n s i s t e n t

%oe(o(o~)) , ~I~(~176

in T for all ~.

c a n n o t be p r o v e d

analogous

{xy = x ~ -->

indicator

at p l a c e

246

H.

In c o n n e c t i o n for

simple

ries

of

with

type

equal

theorem

theory

3

Luckhardt

(e ~ O)

requires

proof-theoretic

this

fine

shows

that

distinctions,

strength

with

a consistency

namely

respect

between

to their

proof theo-

continuity

properties.

Remark:

The

place;

fixed

the c h o i c e

Theorem

place of

i is t e c h n i c a l l y

p ,

arithmetic

T U BR = T U BR p l u s

the

can be r e p l a c e d

by any o t h e r

definable

motivated.

4(a)

T $ T plus

Proof:

I above

It s u f f i c e s

statements

for

Z

,

~ arithmetic analysis

to p r o v e

plus

= analysis

the a s s e r t i o n s

arithmetic

and

analysis

plus

for then

the

functional

follow

domains;

by the r e s u l t s

in [3].

The

computability

can be g i v e n treatment local

by

given

there.

(~

continuity

arithmetic

Theorem ~(C)~

Proof:

p of

(T U BR p l u s

~) of

[3],

X, X I I I

chapter

Formalizing

proof

(see [3],

this p.

a closed

as in [3],

137)

for

term

analogous chapter

T plus

p

of

type

to the

XIV

statements strong

over

do not

theory.

T plus

For

increase deductive

the c o m b i n a t o r i a l

a p)

consequences

strength

we c o n s i d e r

p.

4(b) has

p-

gives

(T U BR plus

O

^ = T U BR).

a sufficiently

point

the m e t h o d

consistency

A in A = T

Thus

in T p l u s

an i n t u i t i o n i s t i c

By t h e o r e m

1(b)

is the c o n t i n u i t y

proof

it s u f f i c e s indicator

in A+~.

to s h o w

previously

~(C)~176 formed

The

with

~.

starting-

of the

H.

u~176176 = O-->

1,Bu(uu)

Ax ~176176 {Xl = 0 - - >

~Vx

{xy = O <-->

Hence

= 0 A Vz~

= 0 -->

Az.yz

ly~176176

Thus

Continuity

sions

(in the

consistently presence

Az.yz

trary

sential

~ I)}

= I)}

= I}

properties

form

of Z).

(C)).

And

does

to the

= ul = 0

Av.uv

= O} w i t h

~v~

u =:

I)-

on every

~(C)~

247

1 ^ u(1,Uu)

n V z ~ 1 7 6 1 7 6 ~176{zu = 0 < ~ >

x =:

of

vyOO(xy

{xl = 0 ^ A y ( x y

nVxAy

= O @

Luckhardt

Therefore

together

not

alter

with

imply

classical

constructive

situation

extension

constructively

type

functional theorem

negations

domain

4(a)

(C)~

as w e k n o w f r o m

is a d d e d theorem

I(c)

and

does

not work

not

that

strength; which

comprehen-

[e.g.

we have

the p r o o f - t h e o r e t i c

where

theory

of

in the

the addition

this

is c o n -

constitutes the

theory

a n es-

of

truth

definitions.

In c o n c l u s i o n

Theorem

the

"continuity

perspective"

c a n be

summarized

as f o l l o w s .

4 (c)

A

T = A plus

~

,

~(C)~

^

T U BR = ~plus All

systems

lar

they are

Proof:

~

, BR,

between

have

consistent

Arithmetic

~(C)~ the

same proof-theoretic

strength;

in p a r t i c u -

[3].

and analysis

are

treated

in p a r a l l e l .

A

T

(T U BR)

= A

(~

)

[3]

< A

(~

plus

BR)

plus

Z,

= A

(~

plus

BR)

plus

~

A

7 (c)Oe-A Theorem

4(b)

H.

248

4.

Invalidity

Also

T

(T U BR)

T

(T U BR)

of h i g h e r

connected

with

plus

Luckhardt

~

functional Theorem

observations

all

for

(total)

4[ u s ] taken

classically

(eR) ~ e x p r e s s e s "natural" also valid O(OO). ~

equals

initial the

0 or not.

segment

This

~

of

has

vy~176

functional

in c o n t r a d i c t i o n

VX O(OO)

follows

~-rules

f u n c t i o n a l c o n s t a n t s ~i

for

Take

case we - say of

length

n - which

so t h a t ~ ( ~ , n

Vy(xy

(~R) O(~

= 0 A Vz.yz

+ O)}

= 0 ~>

(C)~176

Az-yz

= O) }

whether

a finite

contains

all

* i) = 0 l i k e -

~ O)

c ~.

Is t h i s

~ of t y p e

steps

(~,n * l)n = I + 0

o(oo)

from

of

determine

to

immediately

constant

in ~:

= O ^ Vz~

{X~ = 0 ^ Ay(xy

and e v e r y

consistently.

number

can also

computation,

* i) = O ^

{x~ = 0-->

this rule

numbers,

in ~ a f u n c t i o n a l

expressed

constants

the n a t u r a l

in a f i n i t e

first

the

of

to a d m i t

the a r g u m e n t

needed

Ax O(OO)

which

In t h e

~(~,n

pO = 0 -->

for a l l

concept

types?

can be fully

= 0-->

are higher

(u v a r i a b l e )

it c a n be e s t a b l i s h e d

information

wise.

therefore

for h i g h e r

Then

4(a)

or c o n s t r u c t i v e l y .

a standard

theory

[33

e-rules

our previous

(~R) ~ A [ ~ O ] . A . ~ .I ] ,.

interpretation

now gives

H. L u c k h a r d t

249

Thus we have proved that an w-rule for type O(00)

cannot be added con-

sistently to theories over c o n s t r u c t i v e f u n c t i o n a l domains w h i c h contain i n t u i t i o n i s t i c logic and

(C)~176

This result can easily be extended to

all higher types because we have

T h e o r e m 5(a)

Proof:

(wR)~ ~ -- (wR) B T

Let AIr, I, A [ ~ ]

for 8 <

.... be p r o v a b l e for all functional constants

B Then also A [ ~ - I ~ ] A[~-I~] are p r o v a b l e for all f u n c t i o n a l ~i" ' '''" constants ~ Now

of type ~. Here % is a code for B in e and ~-I its inverse.

(wR) ~ gives A[~-lue];

so in p a r t i c u l a r we have A [ % - 1 r B] from w h i c h

A[v B] follows with the coding r e l a t i o n ~-1%u8 = u by

(ER)-qf.

Consequently

T h e o r e m 5(b) (wR) ~ cannot be added c o n s i s t e n t l y for ~ > 0(00)

to theories over con-

structive f u n c t i o n a l domains w h i c h contain T, i n t u i t i o n i s t i c

logic and

(C)~176

The case of n u m b e r - t h e o r e t i c functions

is not covered by this method.

But the result holds here too and is also valid for m u c h weaker theories.

T h e o r e m 5(c) w-rules for types d i f f e r e n t from zero cannot be added c o n s i s t e n t l y to theories in which all f u n c t i o n constants can be computed r e c u r s i v e l y and w h i c h contain functionals,

(relative to their language)

the p r i m i t i v e - r e c u r s i v e

i n t u i t i o n i s t i c a r i t h m e t i c and the fan-theorem.

For instance all

systems of i n t u i t i o n i s t i c

analysis

(with ~ i n

(wR) ~

250

ranging

over

(because

[I],

Proof:

By

leads

freely

the

Kleene

growing

fan-theorem p.

115)

theorem

tree w h i c h

all

Where

choice

objects)

come

5(a)

under

this

it s u f f i c e s This

([I],

to s h o w

for

p.

by

according

that

by u s i n g

the a d d i t i o n Kleene's

all r e c u r s i v e

W1(x~176

-: T1((X)o,X,y)

Az < Y ~ T 1 ( ( x )

, (u~

I are

the

R is d e c i d a b l e

projections

-: V t < x

and has

Vy<x

A y < w ~ " u y : v ~176 y-->

Let o o

be a r e c u r s i v e

consideration. so we

Then

of

can m a k e

I:

1,x,Z)

a primitive

=

[R(u;w)

~ is also

I <-->

have

<-->

recursive

pairing

constant

informally numbers

R(V;W) ]

of

fo,fl

VyT1(fo,t,y),


o

the

following

sg(~t)

,fl > , sgu ~

sg(~f)

= I

=:

derivation

Ix.sg

(ux) .

the

consistent

a well-defined

(Kleene

Case

but n o t

t-W sg (ut) (t,y)

function

effectively

sg(~t)

f -:

of a

the p r o p e r t y :

(i)

Put

(~R) ~176


R (LlO O ;x ~

(2)

of

112).

Az < y ~ T 1 ( ( x ) o, X t Z )

^

to

example

sequences

~: T1((x) 1,x t y) ^

function

N o w we

(AC) ~

of

theorem.

is done

is w e l l - f o u n d e d

sequences

here

as all e x t e n s i o n s

Wo(x~176

(U~

rithm;

as w e l l

can be p r o v e d

to a c o n t r a d i c t i o n .

binary for

Luckhardt

H.

theory

under

recursive

algo-

with

= O <-->

[O],

within

p.

the

VyTI (fl ,t,y )

281)

theory.

H. L u c k h a r d t

By

(2) ,I t h e r e f o r e

(2),2 gives

we have ~ with:

(4) From

to K l e e n e

Az < y (3) , (4)

A ...

[03,

Case

p.

197

W 1 (f,y)

and w i t h

then

also

x -: f+7+I

VxOR(sgg,x;x)

2:

sg(~f)

(I)

= 0

in the same way

Thus

function

for each

Ay~

Hence w i t h

< I ~>

oo

I.

of the t h e o r y

Ay.~y =

(sg~)y

Vx~

(5) , (I

(wR) OO

Az ~ and with

as in case

constant

-->

This

e166a

f. Wsg((sgq~)f ) (f,y)

(5) is p r o v e d

(6)

^ 7TI ((f)1,f,~)

7TI ((f)1,f,z)

Vf<x Vy<x (5)

T1((f)o,f,~)

in this case

7TI ((f)1,f,O) According

(3)

251

{Ay. zy < I -->

Vx~

the f a n - t h e o r e m Vm~

~176{Ay. zy < I -->

is n o w refuted.

Take

Vx < m . R ( z , x ; x ) }

~ as the f o l l o w i n g

primitive

tion: I I (<m~ t ~ ) -:

if t<m A V y < m

0

if t<m ^ V y < m -

0

otherwise

" t-W O(t,y) t.W I (t,y)

recurslve

func-

252

H.

Luckhardt

Then %0(<m,t>)

(7)

< I

x --< m A R ( l y . w ( < m , y > ) , x ; x )

-->

Vt<mVy<m-'t

-->

With

(7) , (8)

< m

Am~

Further,of

by

sequences

(MP)

[~(<m,t>)=O

-->

Wo(t,y) A %0(<m,t>):1] }

~ 1 7 6{ A y . z y

to

< I ^

A

can

the rule

that

variable in

the

([5],

last

{Ay.ay

< m

317).

is

lawlike

proof,

we

< I -->

the

choice

objects.

If w e

obtain

Vx~

For

gives

transformed

by

has

objects apply

the

same

is n o w this

re-

re-

only

}

of

this

(~R) s w h i c h

u s for

a result

But

this

can be

Kreisel-Troelstra's

repaired

theory

of

choice

Vx~

intuitionistic

into < I -->

rule

fan-theorem.

< I -->

{Ay.sy

the

the variable

As ~ { A y . s y

As

9 ~ R(z,x;x)}

interest

of

= a, p.

Ax

a s for

the premise

As ~ V a . s

be

,x;x)

(6).

intuitionistic

is n o t

using

which

W I (t,y) A ~ ( < m , t > ) : O ]

(t,y)

i t is p r o v e d

Aa OO

by

-->

(~R) s e x c e p t

stricted

which

{[~(<m,t>)=1

~ R(ly.~(<m,y>)

in c o n t r a d i c t i o n

placed

--< mVy<x'-t _< m-'t. W s g g ( < m , t > )

(7)

Ax

as

Vt<x

~

(8)

form

-->

Vx~

logic

and

Markov's

principle

H.

With

this

Theorem For

supplement

the a b o v e

proof

theories

in w h i c h

contain

functionals,

all

function

(relative

intuitionistic

incompatibility

holds

situation

the basic

thus

ideas.

Theorem

which

in f o r m of

(see K l e e n e

[O], Aa O O

(Ch T) O

p.

and

recursively

the p r i m i t i v e - r e c u r s i v e

the

fan-theorem

the

following

5(b)

(MP) ~

will

0 =

perhaps

is b a s e d

idea behind

become

clearer

o n the c o n t i n u i t y

theorem

the f o l l o w i n g

I

5(c), (d)

consequence

if w e r e v i e w of c o n s t r u c t i v e

is r e c u r s i v e n e s s

of C h u r c h ' s

thesis

281)

9 ar

with

(Ch T) ~ is c o n n e c t e d

with

~e~

~: V e A x

{~x=O

<-->

(~R) ~ 1 7 6in the f o l l o w i n g

VyT1(e,x,y)}

way.

5(e)

theories

contain

case.

c a n be c o m p u t e d

language)

arithmetic

explained

The basic

For

in this

for ~ ~ O:

functionals. enters

constants

to t h e i r

(~R) ~, As ~ 7 V a ( ~ = a),

Theorem

also works

5(d)

and which

The

253

Luckhardt

in w h i c h

the relevant

all function part

of T a n d

constants

are recursive

intuitionistic

logic

and which

the f o l l o w i n g

holds: (~R) ~

Proof:

A~ O O

By t h e o r e m

gether

with

within

the

From

~

this

5(a) :

theorem theory

9 ~e~

,

(~R) ~176 ~

(~R) OO ~

IV in K l e e n e

VeAx

the a s s e r t i o n

{px=O

[O],

<-->

follows

(~R) O p.

I

Aa O O

(~R) ~176~ 281

used

VyT1(e,x,y) } for

with

(~R) ~ 1 7 6resp.

9 ae~

(~R) O

(~R) O to-

informally every

gives

constant

(~R) ~176

oo

2~

H.

Remark: For

The

same

theories

which

necessarily tion

partial,

= {e}.

= {e} ~

A[{I}] ....

We n o w

system

tionistic the

with

we

arithmetic

and

AS

only

A[a]

to s h o w

(~R) ~

with

as far

Then

AaVe.a

the

first

=

thesis.

within

of T, part

form

AE{O}],

Church's

part

situa-

in the

as p o s s i b l e

the r e l e v a n t

fan-theorem.

thesis

-

now

to this

in p a r t i c u l a r

follows

5(a)

to c o n t a i n

the

(~R) ~176e x t e n d e d

A[~I] .... ; then

of t h e o r e m

ax = Uy}.

{i} of their,

(~R) ~ to C h u r c h ' s

we have

(~R) ~

suppose

(~R)~176

(9)

A[~o],

the p r o o f

which

over

{T1(e,x,y)^

enumeration

constants,

foregoing

and A [ { u ~

AaVeAxVy

a recursive

equivalent

Assume

formalize

(Ch T) :

function

By the

(~R) OO.

for

contain

is d e d u c t i v e l y

AaVe.a

the

holds

Luckhardt

intui(up to

gives

{~c~ ^

Ay-~y

< I -->

Vx~

Consequently

I

Ae.~c~-->

Ac

{Ay-~y ~

I -->

Vx~

Aa.ae~-->

Aa

{Ay.ay

< I -->

Vx~

(10)

The

second

part

(beginning

with

the a p p l i c a t i o n

of

the f a n - t h e o r e m )

becomes

(11)

As

{Ay.~y

N o w we

see

that

(I0),I

and

(11).

Applying

the

< I -->

theorem

additional

Vx~

5(c)

is a d i r e c t

argument

for

consequence

theorem

5(d)

to

of

theorem

(10).,2 and

5(e) ,

(11)

gives

(12)

Again

As n~ Va-~

theorem

5(d)

= a ,

follows

(MP)

~

~Aa~176

immediately

from

this w i t h

theorem

5(e).

H.

(12) of

can be i m p r o v e d

Luckhardt

by s t a r t i n g

255

a similar

argument

with

(11)

instead

(10) ,2:

~A~

77{Ay-~y

77V~

Together

~

{Ay.~y

with

(9)

I -->

Vx~

}

< I ^ ~Vx~

this

(11) , (MP)

}

implies

~V~o~ From

this

the n o n a b s u r d i t y

tive

function

of

the

existence

of a n o n r e c u r s i v e

construc-

7 7 Va-a#~

follows

with

Theorem

5(f)

For

theories

tionistic

As77Va'e

which

contain

arithmetic

(MP) ,

This

concludes

this

be g r a s p e d

constructive

part

= a

and

!

the p r i m i t i v e - r e c u r s i v e

intui-

the f a n - t h e o r e m :

A~77Va.~

I; p a r t

= a

II d e a l s

proof-theoretically

method

functions,

available

today?

~

77Va.a#C~

with

the p r o b l e m :

by i n t e r p r e t a t i o n ,

In w h a t the

way

can

strongest

256

H. Luckhardt

References

[03

Kleene,

S.C.:

Amsterdam [13

Kleene,

Introduction

S.C.

and Vesley,

Mathematics. [2]

Kreisel, logic.

[33

[43

Amsterdam

R.E.:

The F o u n d a t i o n s

Luckhardt,

logic 27

of intuitionistic

(1962),

H.: E x t e n s i o n a l

G~del Functional

Proof of Classical Analysis.

Lecture Notes

in Mathematics

Kreisel,

306,

H.: Uber Hilbert's

fur m a t h e m a t i s c h e

Annals math.

logic

Springer

Interpretation. 1973.

reale and ideale Elemente.

Logik und Grundlagenforschung,

G. und Troelstra,

of intuitionistic

predicate

139-158.

A Consistency

Luckhardt,

of Intuitionistic

1965.

G.: On weak completeness

J. symb.

Archiv [5]

to Metamathematics.

1962.

A.S.:

Formal

systems

to appear.

for some branches

analysis. I (1970),

229-387.

Postscript I If partial

functionals

are admitted

in w 4 then the connection between

the higher types and type O must be revised, 5(b), (c), (d) and Postscript

?

The m e t h o d

of w

(f) carry over under appropriate

also gives a nice f u n c t i o n a l

metical

(C) ~ : This reduces

pretable

via

p. 46, 74,

but our main theorems

(AC)O,O_Az o

88-91).

,

to

(C) ~ - Ax ~

(4 A 4)0

-

conditions.

interpretation

of a r i t h -

w h i c h is f u n c t i o n a l

VyOAz o

by

T u BR ~

(see

inter[3],

CHURCH ROSSER THEORE~ UNENDLICH Herrn Professor

iWOR A - K A L K U L E

MIT

LANGEN TER~EN Dr. Kurt SchGtte

seinem 65. Geburtstag

zu

gewidmet

W. Maa~

In dieser Arbeit wird mittels

Transfiniter

Theorem fdr einen typenfreien bewiesen.

Der Beweis

vorliegenden

l~Bt sich unmittelbar

~ -KalkGle

Induktion

mit unendlich

halten.

Wir geben am Schlu8 an, wie die auftretenden Anwendungen

und Schwichtenberg zahlen bewiesen Martin-LSf

zur Termbildung

dieser Art mit Typen hat Girard ~

~

an

angegeben. entstehenden

das Church Rosser (unver~ffentlicht).

der Terme

I)

0 , S

und die Variablen

2)

Sind

a

3)

Sind

ai

und

b

fGr alle

k-KalkG1

haben Barendregt

entwickelten

Methoden

i)

lob = IsL : J il :~

2)

l~xal

3)

ll

lal+1

,

sind Terme.

so sind auoh Terme,

der L~n~e eines Terms

=

KalkG1

9

(~xa)

so ist auch

:

labl = max(lal,lbl)+1

= sup(Jail+l)

der

Wit benutzen beim Beweis die von

Xl,X2,..

i ~ ~

bei

in [I] einen Beweis mit

F~ir den durch Weglassen

:

Terme,

ent-

Ordinalzahlen

[~ ).

Definition

Definition

und Reduktion

Theorem mit Hilfe von Ordinal-

und Tait fur den endlichen

(siehe Stenlund

auf die meisten

genGgend klein gehalten werden kSnnen.

Hilfe von Fundierungspr~dikaten Reduktionsregel

langen Termen

(mit oder ohne

die noch zus~tzliche

Ftir A -KalkGle

Regeln

ausdehnen

langen Termen

Typen),

beweistheoretischen

das Church Rosser

l-Kalktil mit unendlich

und

(ab)



Terme.

ein Term.

W. IVIaaB

258

Mitteilun~szeichen

:

i,j,k,m,n fur natGrliche Zahlen) zahlen; a,b,c,d,e fGr Terme; n mit n-maligem Auftreten yon

a,~,~,6 ffir abzihlbare Ordinalfir Terme der Gestalt (S(S..(SO)..)

S 9

Unser Ziel ist, das Church Rosser Theorem fir den folgenden Reduktionskalk~l ~ zu beweisen :

I)

a~

a

2)

(Ixa) b ~

3)

~

4)

( b) o ~

5)

a ~

6)

ai ~

7)

a ~

ax[bU

~

an

a'

,

a!l b

,

b

b ~

b'

=>

ab ~

fir alle

i e ~

b ~

=>

a'

a'b'

=>

a ~

, ~xa



~

~ Xxa,



a'

Wir geben einen zu diesem Kalkil iquivalenten ReduktionskalkG1 an, bei dem den einzelnen Reduktionen Ordinalzahlen als "Reduktionsordnungen" zugeordnet sind. Das Church Rosser Theorem fir den KalkG1 list sich dann dutch Induktion iber diese Reduktionsordnungen beweisen.

Reduktionen mit Reduktionsordnun~en (wir schreiben

~

anstatt

~ a,1

(I)

a

~

a

fir alle

a

(2)

a

~

a'

und

~

(3)



~



(4)



~



( b) c (5)

a

~

a'

(6)

a

~

a'

=> ,

~ ,

=>

b

: )

b'

-nb

~

b'

=>

(~xa) b

~ ,

~

b'

~xa

~

=>

Xxa'

ab

a'x[b']

a'n c

~

c'

b' b

~

~

a'b'

=>

Maa~

W.

(7)

ai

(8)

a

>

fGr alle

a!1

~

b

259

i e ~

=>

nach elner der R e g e l n



~

(I),..,(7)



und

at

b ~,n

mit (9)

=>

~ <

a

~

jeweils

und

b

Bemerkun~en:

Lemma I.

b

Anstatt

(j)

a

~

~ ~,n

a'

a'

J ~ a' a,n

=>

a

~

a'

a,n+l

der Regel

eine anschlie~ende

Wir schreiben Regel

a

(8)

kSnnte man bei den Regeln

Reduktion

~ ~,n

um mitzuteilen,

mit dab

~ < ~ a

(I)-(7)

zulassen.

~ ~,n

a'

zuletzt nach

erschlossen wurde.

I :

c

> ~,n

c'

8~

d

2.

c

3.

kxa

4.



=>

c

~

c'

f~ir

~ ~,n

c'

g >~

a und

~ a,n

d

~xa'

~

=>



a

=>

mit

~ a,n

~ < a

C

=>

[~

C'

a'

ai

>

a!1

C

8~

c'

ffir alle

i E

Beweis: I.

C

I~

C

>

2.

c

8~

d

=>

Dann gilt nach

I.

=>

C' c

J~

e

e

c'

~ ~ ,m

d

mit

j < 8

mit

8 := max(~,N)

und < a

~ < ~ . und

k c

8,k =>

c

i--

~

>

e

3.,4.

c'

Induktion nach

Lemma 2 : 0rdinalzahl

=>

c

8>

c'

6,k

G.

a

~ a

a

~m+n

<=>

a

mit einer a b z ~ h l b a r e n

260

W.

Beweis:

"=>"

Zur B e h a n d l u n g benutzt ai

~

Induktion

nach

der Regel

6)

(Gber Lemma a~

I. I.)

fGr alle

Maa~

der D e f i n i t i o n wird

von

die Regel

a

(8)

~

a'

des KalkGls

:

i e ~

=>

(I.V.)

a~

ai

mit abz~hlbare

ai 0rdinalzahlen

~i

fGr alle

i E ~

=>

a~

ai

fGr

a

mit

a a > ai

fGr alle

"<="

Induktion

Lemma

3 :

Beweis:

c

i e nach

~ ~,n

also

Nebeninduktion

(Regel

(7))



~



a

c'

und

d

~

d'

=>

Cx[d]

~

Induktion

I) n = I ,

=>

nach

~+n

J~

c'

c

nach

Ic[

~ ~,n

C'x[d']

. mit

j e {I,..,8).

. Die F~lle

j = I,..,7

machen

keine

Schwierigkeiten. Sei j = 8 :

c

8~

o'

=>

c

k~

e

~

und

~ < a . Da die B e h a u p t u n g

hat man d'

>

ergibt 2~ F~ir

Cx[d] d'

:

sich n ~ I

Church Rosser es gibt

Beweis:

einen

~

ex[d'] Ox[d]

C'x[d']

c'

k~

e

mit

k ~ (I,..,73

schon b e w i e s e n

erh~lt

. Mit Regel

:

c

nach

c

mit

~ a,n c'

a ~ ~

Nebeninduktion

unmittelbar

(8)

und Lemma

aus der I.V.

c'

~,m ~ ~,m

8 ~,n

. nach

Icl

wurde,

man w e g e n

C'x[d']

die B e h a u p t u n g

Theorem Term

c

Aus der I.V.

~ ~ ,m

~

folgt

Induktion

11 n = m = I

exCd']

f~r

~ ~,m

1.2.

W. MaaS Zun~chst

wird

die Behauptung

j,k ~ {I,..,7}

mit Hilfe

Jeder einzelne

261

c J>

f~r die Fille

c'

,

der N e b e n i n d u k t i o n s v o r a u s s e t z u n g

Fall ist trivial,

soda~

einige Beispiele

k>

o

bewiesen.

gen~gen

:

~ = 2 ~ k = 5 : (Xxa) b

2>

a,x[b, ]

mit

a

(kxa) b

5>

(Xx~) ~

mit

(nach Lemma

a'

>.

~

~

. Aus der N.I.V.

folgt,

und

sowie

~

~

0

~

a'

und

b

1.3.)

> a

da~ es Terme

~

b' >

und

und

gibt mit . Es

a

ergibt sieh

(~xg) ~ 2~

ax[b ]

und nach Lemma 3

a'x[b']

&x[6]

9

~ = 3 r k = 5 : _n

3~

an'

mit

n

5~

<~i > n

mit

>





~

<~i >

0

Aus der N.I.V.

folgt:

es gibt

einen Term

, sodaZ



b

,

^

also nach Lemma

1.4. auch

al,o

~

~i =

<~i > ~ 4

,

k

=

5

4>

c

und

~

Zun~chst

~

k' = 8

&

n

, und





,

an

b' b

( b) c

und

k' ~

, dabei

e

kann man die Behauptung

der N.I.V. Ftir

~

:

( b) c ~

a' n

ist

ftir die F~lle

5>

mit

e

k' = 1,3,5,8 k' = 1,3,5

m~glich. sofort mit

beweisen. hat man dann

b

J'~

eI

,~,

e

mit

j' < 8

und

P < ~

. Aus der Gestalt

yon

b

folgt

j' ~ ~1,3,5}

9 Es ergibt

262

W. M a a ~

s i c h das

Diagramm

(

b)

wobei

c

(+)

aus

dem vor-

her B e w i e s e n e n (++) b'

~

Nachdem kSnnen

C

8>

~

:

=>

k !

C

der

<

I.V.

d

F~lle

~ k = 8

m @ ~

aus

folgt.

fur die F A l l e

die r e s t l i c h e n

E {Ir..r7}

eI

-

die B e h a u p t u n g

wegen

~ ~

und

~

e

j,k

behandelt

n

s {I,..,7}

bewiesen

ist,

werden.

~

mit

k'

~ {I,..,7]

und

~ <

P Es e r g i b t

sich

wobei

c

(+)

F~llen

~ =k=

aus

und

den v o r h e r

(++)

behandelten

aus d e r

I.V.

folgt

8

~an beweist

diesen

Fall

in a n a l o g e r

Weise

unter

Verwendung

des v o r i g e n

Falles. 2~ n >~ I ~ m = I

Nebeninduktion

nach

n

mit H i l f e

yon

3) n >~ I , m >~ I

Nebeninduktion

nach

m

mit

yon 2

Hilfe

I

W. Maa2 Fiir die Theorie formuliert angegebenen dadurch

der primitiv

mit unendlich

lisation

der Terme

duktionen Gber nur Terme mit

rekursiven

Funktionale

langen Termen nach Tait

Reduktionsregeln

sich ergebende

263

nur Ordinalzahlen

Reduktionsbegriff

durchf~hren

endlicher

[4] < ~o

zugelassen.

ist stark genug,

zu K~nnen.

um die Norma-

zu k~nnen,

feste Ordinalzahl

beschr~nkten

zu. Es ist dann beim Church Rosser Theorem mitzubeweisen,

L~nge

der zur Erg~nzung

nicht ~berschreitet. finite

Induktionen

Mit dieser Methode wurde bis

system mit unendlich lysis

interpretiert

benStigten

~o

Terme

in Maa2

das Church Rosser Theorem

langen Termen bewiesen,

In-

l~2t man

L~nge

der Diagramme

Der

Um auoh die verw~ndeten

die L~nge yon Termen einschr~nken einer durch eine geeignete

Typen,

, werden bei den

da2 die

diese Schranke [21 durch Trans-

f~r ein Funktional-

in dem die pr~dikative

Ana-

werden kann.

LITERATUR [I]

J.Y.Girard,

Theoreme

de Church-Rosser

infinis. [2]

W.Maa2,

pour un systeme

de termes

(unverSffentlicht)

Eine Funktionalinterpretation Analysis.

Dissertation

der pr~dikativen

an der Universit~t

MGnchen

(1974) [3]

S.Stenlund,

Combinators, D.Reidel

[4]

W.W.Tait,

k-Terms

Publishing

and Proof Theory. Company,

Dordrecht

Infinitely

long terms of transfinite

In: Formal

systems

and recursive

(ed. Crossley/Dummett),

(1972)

type.

functions

S. 176-185.

U B E R S K O L E M E R W E I T E R U N G E N IN DER I N T U I T I O N I S T I S C H E N L O G I K MIT G L E I C H H E I T H e r r n Professor Dr. Kurt SchGtte

zum

65. G e b u r t s t a g gewldmet H. O s s w a l d

Als erster hat G. Eo Mints in

(7) gezeigt,

da6 im G e g e n s a t z zur

k l a s s l s c h e n L o g l k mit oder ohne G l e l c h h e i t und im G e g e n s a t z

zur

i n t u l t l o n l s t l s c h e n Logik ohne G l e i c h h e l t in der i n t u l t l o n i s t i s c h e n L o g i k mit G l e i c h h e l t

(abgekGrzt ILG) S k o l e m e r w e i t e r u n g e n im a l l g e m e i -

h e n keine k o n s e r v a t l v e n E r w e l t e r u n g e n slnd. G. E. Mints hat eine Formel A angegeben,

die die e l n s t e l l i g e

F u n k t l o n s v a r l a b l e F nicht enth~lt und fGr die gilt: ILG + V x R x F x ~ A aber ILG + V x 3 y R x y ~ A. Mints definierte: Rxy

: = (Gy ^ x = a -~p) --,Mx,

w o b e l G, M b e l i e b l g e e l n s t e l l i g e R e l a t l o n s v a r l a b l e n und a, b O - s t e l l i g e Funktionsvarlablen A

sind

und p elne A u s s a g e n v a r l a b l e

ist, und setzte

: = 3 z ( ( G Z -~ p) --~Ma A Mb).

H i e r soll eln etwas e l n f a c h e r e s B e l s p l e l mit d e n s e l b e n Eigens c h a f t e n a n g e g e b e n werden. Sind @, I, QXy und

B

r O - s t e l l l g e F u n k t i o n s v a r i a b l e n und d e f i n i e r t m a n

: = (x = @ ^ y = ~) v (x = 9 ^ y = ~) v (x = e A y = c) : = 4@ = ~ ~ c = @ v ~c = @,

so gilt e b e n f a l l s (I) ILG + V x Q x F x ~ B aber

(x) ILG + Vx~yQxy ~ B, wie m a n durch eln K r l p k e - M o d e l l n a c h w e l s e n kann. D i e s e s Modell ~ l~Bt slch g r a p h l s c h so darstellen:

H. 0sswald

Q

265

9

|

0 Die Situatlonsmenge yon ~ hat 2 Elemente ~,8

mlt a ~ 8. Die

belden klasslschen Strukturen ~a' ~8 haben Jeweils 3 Elemente, die gleichzeitig O-stellige Funktionen sind, wobei in ~8 @ u n d c

gleich

sind. Die Pfeile beschreiben die Relationen Q ~ und Q~ . Also: %

= C(~,~),(~,~),(~,~)~,

In ~ ist Vx3yQxy g~ltig, B aber nlcht gGltlg. Die Aussage

(I) kann man ebenfalls semantisch zeigen:

Gilt ILG + VxQxFx ~ B, dann glbt es elne Kripke-Struktur B mlt Situationen ~, 8,~ r 8, so dad gilt:

(I) a ~ VxQxFx (2) ~ ~ ~

=

(3) a ~ =~ (~) ~ = @ . Aus (I), (~) folgt a~F@

=4

~Fr

= r

Hieraus folgt :

8~Fr Hieraus und aus (4) folgt

Widerspruchl Die Aussage ILG ~ Q|

Denn aus (2) folgt 6 W ~ @

= ~.

(I) bewelst man syntaktisch so: Zuerst zeigt man ^ QcF~ -~B.

266

H. 0 s s w a l d

D u r c h V - Q u a n t i f i z i e r u n g und S t r u k t u r s c h l u B erh~lt man die Behauptung. A.

So T r o e l s t r a machte darauf aufmerksam,

das u n a b h ~ n g i g yon der A r b e i t

dab dieses Beispiel,

(I) e n t s t a n d e n ist, in e n g e m Z u s a m m e n h a n g

mit a n a l o g e n B e i s p i e l e n in der i n t u i t i o n i s t i s c h e n A n a l y s i s Es sei @ die C a u c h y f o l g e der O, ~ die der folge yon r a t i o n a l e n Zahlen,

Iund

steht:

~ eine Cauchy-

ffir die c = @ nicht e n t s c h e i d b a r ist.

D a n n gilt zwar ffir [@,~,~] Vx3yQxy, w e n n man Q wie oben deflniert, gibt aber keine A u s w a h l f u n k t i o n F mit VxQxFx.

es

D e n n well man ~ = @

nicht v o r a u s s e t z e n darf, muB F@ = ~ und F~ = c sein,

well m a n aber

auch ~c = @ nicht v o r a u s s e t z e n darf, k a n n man nlcht zeigen, dab F eine F u n k t i o n isto

Literatur (I) Mints,

G. E.: Skolemts m e t h o d of e l i m i n a t i o n of positive q u a n t i f i e r s in sequential calculi, DOklo Akad. N a u k SSSR, 861 - 864, Bd

169,

1966.

EINE VARIANTE

DES BEZEICHNUNGSSYSTEMS

Herrn Prof.

W(X)

FOR O R D I N A L Z A H L E N

Dr. K. Schfitte zum 65. Geburtstag

gewidmet

H. Pfeiffer

In dem in [I I b e s c h r i e b e n e n der dort e i n g e f a h r t e n az = bz W(X)

sein.

System

Bezeichnung

W(X)

far Ordinalzahlen

fur verschiedene

Dadurch wird die Definition

und damit auch der Wohlordnungsbeweis

genden wird ein System Vergleich

zu W(X)

W~(X)

angegeben,

so eingeschr~nkt

W(X) wird in W*(X)

ordnungstreu

W*(X)

Ordinalzahlen

ebensoviele

Bei V e r a l l g e m e i n e r u n g e n ante W*(X)

Ordinalterme

a, b

der O r d n u n g s r e l a t i o n

< auf

recht kompliziert.

Im fol-

bei dem die Termbildung

wird,

dab aus

eingebettet,

a + b

womit

im

az # b z

gezeigt

folgt.

ist, dab in

b e s c h r i e b e n werden kSnnen wie in W(X).

yon W(X)

auszugehen,

kann mit

darfte

es natzlich

um den erforderlichen

sein, yon der Vari-

Wohlordnungsbeweis

zu ver-

einfachen. I. Definition Sei

X

und E i g e n s c h a f t e n

eine durch

kleinstes

Element.

eventuell

mit

<

wohlgeordnete

Als V a r i a b l e n

simultan

a ist Term yon

Wt(X),

Ga

(1.1)

Jedes

Element

Sind

al,

und a2 ein

S(al,a2)

:= Sa 2

Sind

sei

O

ihr bezaglich verwenden wir

x,y,z,

Induktion:

yon

X

sei

es

sei

Sx

:= x

yon

aus

Term yon

al,...,a

# ...

x

y

(al,a2)

aI

durch

a ist Hauptterm,

Terme

(ao,Y) , wobei

(1.5)

Menge und

fur ihre Elemente

die Stufe

Sa

yon a und den

yon a.

Hauptterm, (1.2)

Wt(X)

Indizes.

Wir definieren Grad

yon

~ ap

S(a I # ...

p

und

ist

# ap)

und und

W*(X),

und

G(al,a2)

Term yon := max

Term yon

und

W*(X) X

Hauptterme ein

ein

W*(X),

Gx := 0 hat

a I nicht

Sa 2 < y zwar

ein

ist

die

< S~ ~

p > I,

W*(X),

aber

kein

{Sajll

~ j

~ p}

zwar

ein

so

Form

gilt,

Hauptterm,

:= Ga I + Ga 2 + I

und

und

.

so und

sei es

sei

. sei

Hauptterm, und

und

es

268

H. P f e i f f e r

G(a I # ...# ap) Wir b e n O t z e n yon

W*(X).

Sind

al,

:= Ga I + ... + Gap + I

a, b, c , e v e n t u e l l Die M e n g e

...,

ap, bl,

..., bq

a := a I # ...

# ap

a I # ... ~ ap

# b I # ...

Jeder

T e r m yon

#

assoziativ

und

wird

mit

die D e f i n i t i o n und der M e n g e n

der

als V a r i a b l e n bezeichnen

for E l e m e n t e

wit mit

H*.

p, q a I ,

~ bq

, so b e z e i c h n e n

wir den T e r m

a # b

ist auch T e r m yon

for Terme

X E X

Hauptterme,

# bq

und k o m m u t a t i v

W*(X)

Indizes, yon W*(X)

b := b I # ...

W*(X)

Die G l e i c h h e i t

mit

der H a u p t t e r m e

aus

W(X).

W*(X)

ist, wie

so e i n g e f O h r t ,

in

W(X).

Ebenso

x-Koeffizientenmengen

daG die O p e r a t i o n kann

K* a

for

a

des Terms

aus i

for

X

Obernehmen bei

A

erkl~rte az

weiterhin

und

B

Yea aus

endliche

Relation

nur e r k l ~ r t

Die R e l a t i o n

auf

in

[I] for

W(X)

gegeben

[I] die S c h r e i b w e i s e n

A <* b

Teilmengen

und

W*(X)

sein mGge,

<*

wie

sei wie

ist,

yon

W*(X)

sowie

x~

w e n n nicht folgt

(2.1)

{a, b} c H * A Sa < Sb ~ a

(2.2)

Sb = x i x <* (a, b)

(2.3)

a = (al,a2)

[a z1 ~ * b z1 ^ Kx~a l

(2)

a 1 = b 1 A a 2 <* b 2

(3)

~ b zI <. a zI ^ a ~* K xbl

und

<*

a ~* B , wo-

die n a c h s t e h e n d

a z , wobei

a = (a1,Sa)

Wit

allerdings

^ z < Sa < Sa I

gilt.

erkl~rt. b

+ ^ x := Sa = Sb ^ z := xal,b I A

^ b = (bl,b2)

(1)

und

werden.

U {a2}

<* b

V

V U {b2}

]

a <* b

(2.4)

{ao,...,ap,b ~ .... ,bq} c H * A (I) Vi(i

E p + I A j E p + I A i + j ^ fi = fj m a i <* bfi)

(p = q ~ Sk(k = a ~ ~ ...

Bemerkungen: we(x)

und

+ I) - (p + I) A

E p + I = a i ~* bfi ) A

(2) V i V j ( i (3)

p + q > O ^ ~f(f:(p

E p + I A a k <* bfk))

~ ap

I. Zu ( 2 . 3 ) :

<* b ~ # . . . Sind

2. Zu (2.5):

Aus

Alternative

a T = b~

folgt dar,

~ bq

a = ( a l , a 2)

z = x: I ,b I j so sind

vollst~ndige

A

a zI

und

b~

und

at= b I ; daher

und es gilt

der

b = ( b l , b 2)

definiert

Terme yon

*

stellen

(I) bis

(3) eine

H.

Satz

1.

W*(X)

Bemerkung <

auf

sprechenden

<

dutch

<*

linear

3. Da die Definition yon

W(X)

Satz I u n d gen,

wird

Pfeiffer

269

geordnet.

<~

auf

Satzes ffir W(X). Dasselbe der Wohlordnungsbeweis

einfacher als die von

gilt ffir den Wohlordnungsbeweis.

ffir

(W~(X), <~)

in [I] bewiesenen S~tzen ffir W(X)

auf

W~(X)

ist, wird der Beweis yon Satz I einfacher als der des ent-

W~(X)

, weil

folgen aus den analo-

<*

die Beschr~nkung yon

ist.

Wie in [I] Lemma 7 zeigt man dutch Induktion fiber Ga Lemma I. Sb 2 ~ x ^ a ~* K*x bl U {b2} m a <* ( b l , b 2 )

2. Ordnungstreue

Einbettung yon

(W(X), <)

Um zu zeigen, dab das Teilsystem (W(X),<)

~hnlich in

(W*(X) ,
(W~(X),< t)

ist, definieren wir eine Funktion sie

in

r

ebenso stark wie

(W~(X),
abbildet.

(3.1)

x E X ~ ~ x := x

(3.2)

Sa 2 < y < Sa o ~ ~((ao,Y)

(3.3)

~[a I = (ao,Y) + n ^ Sa 2 < y < Suo] = ~(al,a2)

(3.4)

p > I ^ {al,

Dabei sei

a + n a

+ 0

+ n, a2)

:= (~(ao,Y)

..., ap} c H * ~ ~(a I # ... # ap)

ffir n E ~ :=

a

;

(W(X),<)

yon W(X) in W@(X) und zeigen, dab

+ n + I, r := (~ al, ~ a2) := ~ a I # ... # ~ ap

induktiv fiber n dutch a

+

( n

erkl~rt. Man zeigt leicht:induktiv,

*

1 )

dag ~

:=

( a

W(X)

+ n

)

# 0

injektiv in

W*(X) ab-

bildet. Dutch

Induktion

Lemma 2 . 1 .

fiber

Ga

Sa=

Scpa

2.

r

) c K~ r a

3.

y a -- y~cpa

4.

x a

+

5.

b

e

w(x)

c

man f f i r a l l e

r

a E W(X)

) U {O}

+

=

xcpa

-~(a = ( a o , Y )

Mit diesen Hilfsmitteln a,

beweist

+ n #% z < y < S a o )

,* r

(a z)

-- (cp a) z

ergibt sich durch Induktion fiber

Ga + Gb

ffir

270

H. P f e i f f e r

Satz

2.

a < b = r a <* r b

Wir ffihren den B e w e i s b = (bl,b2) (5.3)

und

hier nur

gebildet werden,

~(al,a2)

for den Fall

Sa = Sb =: x

= (~(ao,Y)

d.h.

ist,

daft

~ a

durch,

nach

a I = (ao,Y)

+ m + I, ~ a2)

und

da~

a = (al,a2)

(3.2) und

+ m

A

~(bl,b2)

r b

,

nach

x < y < Sa o ,

= (~ bl, ~ b2)

gilt.

+

Sei

z := x a l , b I

Wit s c h l i e ~ e n

9

Wegen

dutch

I. a < b

nach

[I],

^ Kxa 1 U {a2]

x < y

Induktion

Ya I U Yb I , die g r 6 ~ e r

=

als

Sa I

ist

x < z .

fiber die A n z a h l

x

sind.

Wegen

Z

(7.3-I):(a

n

der E l e m e n t e

x < y

ist

Z

Z

yon

n ~ I

< b I V a I = b I ^ Ga I < Gbl)

^

< b.

Aus a~ = b~ ^ Ga 1 < Gb 1 , d . h . b 1 = (al,z) A z < Sa 1 f o i g t e n t g e g e n der Voraussetzung, da$ ~ b nach (3.2) gebildet sein mfgte. Also gilt alZ < blZ

und nach

n

r

gilt

1.1.

Ist

Induktionsvoraussetzung

<* r

Nach Lemma

y ~ z , so gilt

a I + I
sein,

bezfglich

r

~ b

(~ bl) z = ~(ao,Y)

w~re

entgegen

Ga + Gb

oder

<* ( ~ bl)Z-

~ a I = ~ ( a o , Y ) + m <~ (r bl)Z.

, denn w ~ r e und

(I.V.)

2.5 ist

Dann

ist auch

+ m + I , dann m f ~ t e

der V o r a u s s e t z u n g

nach

(5.2)

gebildet.

1.2.

Ist

z < y , so gilt

Also ergibt (*)

sich in b e i d e n

(r

Ferner

~ (a~)

= (r

+ m + 1) z

F~llen

+ m § I) z <* (r bl)Z

gilt

nach

I.V.

~ (Kxal)

U {~ a2}

<* ~ b

und nach

Lemma 2 . 2

.

Kx(ep a 1 + 1) = K hauptung

liefert.

2.

~ a

Well

sind, 3.

ist

Gilt

I.V.

ep a 1 O {O] <* tp b , was m i t

und

~ b

a 1 + b 1 , also a < b

nach

~ a ~* ~(Kxbl)

a at K~

nicht kann

(7.3-3) O [~ b2]

b 1 O {~ b 2 ]

nach derselben

aus

a < b

nicht

[1],

so i s t

(*)nach

(2.3-1)

Bildungsregel nach

(7.3-2)

die

Be-

entstanden aus

a ~ Kxb 1 U {b2]

[1]

gelten.

und nach

, u n d w e g e n Lemma 2 . 2

<~ ~ b .

Literatur: [1] P f e i f f e r , H . :

Archiv

f. matn.

L o g i k u. Grundl.

1__3_3, 74 - 90

(1970)

AN UPPER BOUND FOR THE PROVABILITY OF TRANSFINITE INDUCTION IN SYSTEMS WITH N-TIMES ITERATED INDUCTIVE DEFINITIONS Dedicated to Kurt Sch~tte on occasion of his 65 th birthday. W. Pohlers Several people conjectured the ordinal of the system ID~ (i.e. the intultionistic theory of N-times iterated inductive definitions) to be -@)S~N+10 in the notation of [I][2], c.f. [5]. In this paper we will show that in fact this ordinal is an upper~ bound for IDN( i.e. the classical version) a ~ therefore also for IDa. In a paper in preparation we will show that vice versa all instances of transfinite induction up to ordi@

nals < @~2N+I0 are provable in IDa. The result of this paper here is es~ablished by a technically rather complicated method. We first introduce systems H N and prove their consistency using transfinite induction up to our ordinal by a combination

of methods due to Gentzen and Takeuti

(the

complicated part). H N corresponds to a system of 2nd order arithmetic I with N-times iterated HI-comprehension with no set-parameters in the comprehension formula and without induction. However H~j, i.e. H N + axiom of complete induction,

is relatively consistent to H N what is proved in

IIl by an inner model argument (following the ideas in [9]). The inter-" pretation of ID N in HNJ' done in IV, is a trivial consequence of 4the fact that an inductive definition corresponds to one application o~ H$-com~rehension. So consistency of ID N is proved by transfinite induction up to ~N+I0,

whence it follows that this induction is not provable in ID N.

Some people asked me to write this paper in English. As I have not much practice in doing so and had no chance to get the paper revised by somebody who does, I want to apologize for all the grammatical- and orthographical mistakes probably occuring in the typescript.

I. The systems H N I. Syntactical notations 1.1. Basic symbols are variables for individuals (denoted by x,y,z,...) and sets ( denoted by X,Y,Z,...), sive functions and -predicates

constants representing primitive-recur-

(among them 0 for zero, S for successor,

= for equality) and the logical symbols -~,V . 1.2. A n-place notational-form ~ is a string of symbols containing besides basic symbols at most the symbols * 1 ' ' ' " * n " strings ~[Ul,...

If Ul,...,u n are symbol-

Un] is obtained from U by substituting Ul,...,u n for

272

W. Pohlers

"I" " " "'*n respectively. I .3. Terms and formulae are defined as usual. A formula is called atomic if it is X(t) or R(tl,...,tn) with X a set-variable or R a n-place predicate constant. 1.4. By n w e

denote the numeral

S(S(...(S(O))...)). A term is called n-tlmes

numerical, if it does not contain any free variable. If all constants are interpreted in the intended manner each numerical term can be evaluated primitive-recursively. We call two numerical terms equivalent if their evaluations give the same value. Two formulae FI,F2 are called numerical equivalent if Fz is obtained from F2 by replacing numerical terms by equivalent terms. A formula containing neither free nor bound variablel is called numerical. The truth value of a numerical formula can be computed primitive-recursively~

We call a formula ~[x I, ...,x k] verifiable

(falsifi~ble) if for every sequence n I , . . . , ~ is a true (false) numerical formula. 1.5. We define ~F to be F - * 0 = I

of numerals U[n I , ...,~k ]

. In order not to have to deal with s t r u c

turals properties and difficulties arising by them

we introduce positive.

and negatlve-parts, due to SchGtte, which generalize the concept of anteand succedens in usual sequent calculus. To make this precise we define posltive- and negative-forms 1.5.1. *z is a posltive-form

as l-place notational-forms inductively by (p-form)

I .5.2. For each formula F ('i-*F) is a negative-form

(n-form)

1.5.3. If G is a p-form (n-form) (F-*G) and (~G-*F) are p-forms

(n-forms)

Every antecedent A~ of a sequent AI,..~ ~ is a n-part of the interpreting formtLla A~-*...-*A ~ B 4 ..~.~ B~. 0theTwlse if a negatlo~ appears in the antecedent the negated f6rmula is'"already a p-part of the interpreting formula. So for instance an inference 'negation left' is superflous etc. As syntactical variables we use ~ for p- and ~ for n-forms. A negative~positlve-form is a 2-place notatlonal-form of the shape ('i-~['2]), ( ~ - * ~ [ * 2 ] ) or (~['2]-*~). We denote n-p-forms by ~. A formula A is said to be a positive-part (p-part) or negative-part (n-part) of a formula F if there is a p-form $ resp. n-form ~ s.t. F is ~[A] or ~[A] (i.e. a ppart of F is always understood together with its place in F. We refer to p-parts and n-parts as parts. A part is called minimal if it does not contain any part except itself. By F = ~ t r G ( G is a structural consequence of F) we mean that every minimal part of F is a minimal part of G. A formula whose minimal parts are all atomic is said to be primitive. 1.6. By a n-place predicate

( n > 0 ) we understand a n-place notational-

form ~oB.t. ~[Xl,...,Xn] is a formula. For a n-place predicate

G and term~

tl,..,,t n we define @(tl,...,tn) to be @[tl,...,tn]. So if S[X] is a

2'73

W. Pohlers

formula U[@] is the formula where each denoted occurrence of X(t) is replaced by @(t) ioeo @[t]. I .7. A formula not containing bound set-variables is called arithmetical. I A formula VX~[X] with U[X] arithmetical is said to be H I . A formula F i s called arithmetical in a finite set of predicates

[@1,...,@n)if there is

an arithmetical formula ~[XI,...,X n] s.t. ~[@I,...,@ n] is F. We define collections T N of formulae inductively by I .7.1. T o contains all arithmetical formulae. I 1.7.2. T~+ I contains all formulae H I in (a finite subset of) T N. I .7.3. TN+ I contains all formulae arithmetical in ~N+I ~ The rank r(F) of a formula F is the least N s.t. F is in T N if such a N exists, otherwise it might oe ~. 2. Axioms and inference-schemata 2. I. Axioms 2.1 .I. Each formula ~[AI,Aa] with AI,Aa numerical-equivalent atomic formulae, is a logical axiom. 2.1.2. Each formula s =t-~@[~[s],U[t]]

, with ~[s]

atomic is an equality-

axiom. 2oi .3. Each formula ~[P]

(~[P]) where P is a verifiable

(falsifiable)

primitive formula is a mathematical axiom. 2.2. Inference-schemata (S~) ($2,o)

~[~A], ~[B] ~[~[x]]

II-

~[(A-~B)] ~[Yx~[x]] ~ if the eigenvariable x resp.X does

($2,1)

~[E[X]]

I-

@[VX~EX] ] J not occur in the conclusion

(s3,o) ~[t]-~[vx~[x] ] I-~[vx~[x] ] (S3,1)

~[@]-~[VX~[X]]~-~[VX~[X]]

~[X]is called the eigenformula of an

instance of (S3,1) ($4)

~[C], C-~F

I-

G

if ~ [ F ] = ~ str G. An instance of ($4) is said

to be a cut with cut-formula C. The underlined parts in the conclusions are called main-parts of the inferences. All main-parts are minimal. 2.3. Corresponding parts in an instance of an inference Let M be a minimal part in an instance J of an inference schema. We define the parts corresponding to M by 2.3.1. if J is a cut, each minimal part of t%~e premise which has the shape of M corresponds to M, 2.3.2. if J is an inference with main-part M, every minimal-part of the formulae denoted in the premises in our list ($I) - ($4) cogresponds to

M, 2.3.3. if J is no cut and M not the main-part of J M occurs in the pre-

274

W. Pohlers

mise(s) of J and this occurrence is to correspond to M. 2.4. By a derivation

we understand a finite tree with axioms at its top-

nodes being locally correct with respect to ($1) - ($4). We denote by %1-F that F is the endformula of the derivation %. 2.5. The fibre of a formula F in a derivation is defined inductively by 2.5.1. Each minimal-part of F belongs to the fibre of F. 2.5.2.

If a minimal-part M belongs to the fibre,

each minimal-part cor-

responding to M belongs to the fibre. 2.6. A minimal-par t is said to be explicit in a derivation % if it belong to the fibre of the endformula,

it is said to be implicit in % if it be-

longs to the fibre of a cut-formula in %. We call an inference explicit (implicit) in % if its main-part is explicit

(implicit)

in %.

2.7. Inductive definition of the endpiece 5 of a derivation %. 2.7.1. The endformula of ~ belongs to 5. 2.7.2.

If the conclusion of an inference which is not implicit in % be-

longs to 5its premise(s) 2.7.3. The premise(s)

also shall belong to 5.

of an inference implicit in % do not belong to 5.

We say that an inference is on the boundary of % if it is implicit and its conclusion belongs to the endpieee. 2.8. A formula is called pure if it contains no free set-variable. dicate U is called pure,

A pre-

if ~[t] is pure or a formula X(t).

2.9. A derivation is said to be a derivation of H N if each main-part of an implicit inference

(S3,1) is pure and of rank ! N.

II. Consistency of H N I. We introduce two new inference schemata:

(S) U[X]

I- ~[@]

if X does not occur in the conclusion. An instance of

(S) is called a substitution with eigenvariable X. The corresponding part of a minimal-part EK@] in the conclusion of a substitution is the minimal9 [X] at the same place in the premise. (Str) F I- G if F = ~ str G called structural rule. If M is a minimal-part in the conclusion each minimal-part of the premise which has the same shape as M corresponds to M.

(Both schemata are actually superflous in

H N but they are needed to make the reduction-proceedure work). 2. A modified derivation is defined as a derivation but with respect to the schemata

($I) ..... ($4), (S) and (Str). In a m.d. a minimal-part may

belong to more than one fibre. We call it implicit if it only belongs to fibres of cut-formulae,

explicit otherwise. A m.d. is said to be a m.d.

of H N if moreover holds: 2.1. Each main-part of an implicit

($3,1) is pure and of rank ~ N.

W. Pohlers

275

2.2. All substituted predicates are pure. If M is an implicit minimalpart in the premise of a substitution then r(M)< N. 3. Ordinals We assume familiarity with the ordinal notational systems of [8]. Z*(N) is obtained from Z(N) by the restriction Sa ~S~+I add 2 o to the terms of Z*(N), denote ~

O

for terms (m,~). We

by o, (o,o) by I etc. First we

define some special ordinal functions and list their properties. 3.1. 2[a,n]is defined inductively as 3.1oi. 2[~,n] = a if S ~ < n . 3.1 .2. ~[a,n] : (~[~,n+1],2n) if n < S a . Proposition I.

a) ~ cZ*(N) =~

~[a,n] ~Z*(N)

for all n.

b) S(2[a,n]) = min[Sa,n}. c) k < n = ~ Ek(~[~,n] ) = ~ . d) ~ < ~ , K k ~ < 2 [ ~ , k ] for n ! k ! S ~

=~[~,n]

<~[8,n].

3.2. We use a ~ 8 as an abbreviation for m < 8 and K k ~ < 2 [ G . k ] For s = o we omit the subscript o.

for s < k < S a .

3.3. We define ~[s,a] inductively by 3.3.1. ~[o,~] = (o,~) 3.3.2. *[s+1,~] = (o,~[S,~]). Proposition 2.

a) ~ ~Z*(N) =~ b)

~[n,~] ~Z*(N) for all n.

S(~[~,a])=Sa.

c) ~(~[n,~]) d) m < n

= ~

=~[m,~]

e) m ~ n ,

for k<S~. <~[n,~].

~<m[m-n,~]

=~

m[n,~] <m[m,~].

f) ~ c ( 1 , 2 N) = O m [ n , a ] c(1,2N)

for all n.

4. Height of a formula in a m.d. The k-de~ree of a formula ~[XI,...,X n] with XI,...,X n the only set-variables oceurringfree in it, is the length of the formula ~[@,...,@] where @ is a predicate of length k. In the following we assume a fixed m.d. We choose k s.t. all predicates used in inferences

($3,1) and (S) are of

length < k. We omit k and write degree instead of k-degree *) 4.1. By the de~ree of a cut we understand the degree of its cut-formula. 4.2.

The height of a formula F in a m.d. is the maximal degree of all

cuts below F in the m.d. 4.3. The potentialdifference of a cut in a m.d. is the difference bet*) At this place I wish to thank Prof. Sch~tte for discovering an error in a first version of this paper.

276

W. Pohlers

ween the height of its premises and its conclusion. 5. Order o f a m.d. of H N To each node of a m.d. 9 we assign an ordinal of Z*(N) by induction on the length of the tree above the node. 5.1. Every axiom in % gets the ordinal QN" 5.2. If a and 6 are the ordinals of the premises of an inference

($I)

(o,a@ 6) is the ordinal of its conclusion. 5.3. If a is the ordinal of the premise of an inference

($2) or ($3, o)

(o,a) is the ordinal of its conclusion. 5.4. If a is the ordinal of the premise of an inference eigenformula of degree n, the ordinal of its

(S3,z) with an

conclusion is ~[n+1,a].

5.5. If a and 6 are the ordinals of the premises of a cut with potentialdifference n, the ordinal of its conclusion is w [ n , a @ 6]. 5.6. Let a be the ordinal of the premise F of a substitution in ~. Then there is a i < N s.t. r ( M ) < i i

a rank

for all implicit minimal-parts of F. We call

of the substitution and assign the ordinal ~ [ a , i ] @ ~ N to its

conclusion. 5.7. The conclusion of an inference

(Str) gets the same ordinal as its

premis e. The order o(~) of a m.d. is the ordinal of its endformula.

It is clear

that for every m.d. ~ o ( % ) ~ (I,~N) holds. Proposition 3. Suppose al,...,a m to be a string of ordinals in a m.d. with no substitution of degree < i between a I and a m. If there is a k < i

s.t. 61 ~ a I for

an ordinal 61 and 61, ...,6 m is the string built up analogously starting with 61 instead of a I 6 m ~ a m holds. The proof is by induction on m. W e just indicate the only remarkable case that the last inference is a substitution. By induction hypothesis we have 6m_ I ~ am_ I, hence by proposition I d) 6 m = ~ [ 6 m _ I , j ] ~ N Further it is K ~ m =

< ~[am-1" j ] # ~ N = a m

[~[6m_1,j]}<~[am, r] for J < r

~[am_1,r ] <~[~m,r] for k < r <

j. So ~ m %

since k < j

is provided

and K r 6 m = K r 6 m _ 1

<

am"

6. Reducible modified derivations A m.d. %I-F is ca~led reducible if % is cut-free or there are m.ds. ~il- F i satisfying

~.I. o(~i) ~o(~), 6.2. if there are cut-free derivations of each F i there is a cut-free derivation of F. I~a

I.

If E is a numerical formula and % a m.d. E ~ E

~ is reducible.

W. Pohlers

277

Fromm lemma I we get b y induction on ~[o(%),o] lemma 2. If a numerical formula is provable in H N it is cut-free provable. So we have THEOREM I H N is consistent established b y a transfinite induction up to ~[(1,2N),O] which is, by work of Buchholz and Sch~tte[2],

equal to @ ~

+10. If we succeed in pro-

ving lemma I elementary enough we may conclud~ by (a modification of) G8del's second theorem that transfinite induction up to @ ~ provable in ~ .

+i0 is not

(This result also can be obtained without t~e use of

G6dels theorem by methods used by Gentzen for first order arithmetic.) Until otherwise mentioned let % and E be as in lemma I. We may assume that all individual-variables

of % which are not used as eigenvariables

are replaced by ~. Proposition 4. If % contains an explicit inference in its endpiece ~ is reducible. Proof. As E is numerical it contains no quantifier. So there are at most explicit inferences

($I) in the endpiece. Suppose % to be

m[~A]

re[B] Y ~[(A-*B)]

where

6 E (A-*B) is a n-part of E. We construct %~ and %2 as

[,A]

and

~[B]

(Str) ~ A-*~[ (A-*B) ]

6"

6' and ~ c y

resp.

B-*~[(A~B)]

IA-*E As ~ y

(str)

hold we get by prop.3

6, c 6

B-*E and 6" c8.

If there are

cut-free derivations of I A-*E and B-~E we get a cut-free derivation of (A-*B)-~E by an inference

($I). Condition 6.2. is then established by

Proposition 5. If G = ~ str E and %1-G is a aut-free derivation then there is a cut-free derivation %'I-E.

278

W. Pohlers

The proof is obtained easily by induction on the length of %. Proposition 6. If % contains a cut-formula A-~B in its endpiece Proof. Assume % to be

it is reducible.

n2

J~ Y G 6 GI

8

G2

n2

J2 GO

nl

E where J2 is J1 if J1 has a potentialdifference ~ O. Otherwise J2 is the first cut with potentialdifference + 0 below J1. First we remark that from

(A_,B~_~ F

we get m.ds. IA-~F ~i \I/

We define %' as

and B ~ F

(str) ~l

__B F (Sir)

~ G-~A-~B

--~(Str

)

62 ~

i ~-~GI

G2

B-*G s

~ GI-~A'-~B '

G2

iG

AS-*Bt

n2

~i

n2

s

B-~GI

G2 GO

n2

n4

~2

A--~G~ G

9 63

B 9

A~-*Go

with ~ i ~

n3 nl

0

E We want to show ~' c~. It is ~' =~[n 3- n ~ , ~ @ ~2] < ~ [ n 2 - n l , 6 @ s] = ~ if ~I, S 2 < ~ [ n a - n 3 , 6 @ s ] . Since degree(K) < d e g r e e ( A - ~ B ) < n 2 it is n2-n 3 > 0. ~ =w[n2-n3,6~@s] <~[na-n3,6@e]because 6~ s<6~s by prop.3 forit is ~ ~Y. ~m =~[n4-n3, e~@ o2] < ~ [ n ~ - n 3 , 6 @ s ] if c~,o~ < ~ [ n ~ - n 4 , 6 @ s ] s i n c e n~-n4 > 0 because degree (B') < n~. From 6 i < 6 (i =2,3) we get o~ =~[n~-n4,6~+ ~] <~[n~-n4,6+ s] and c~ =~[n~-n4,63# e] <~[n~-n4,6~ s]. Moreover Kk~, =Kk6 ~ UKk6 ~ U ~ 6 3

U~ks so ~' ~ .

W. Pohlers

If B is 0 =I

the proof is similar

279

(but %' is somewhat

Proposition 7. If 9 contains a logical axiom in its endpiece

9

simpler).

is reducible.

Proof. Assume % to be

[A~, A2 ] q/

j

~[A]

A-~F Y G

6 E cut below Q[AI,A2]. If neither Az nor Aa is in the for the derivation of ~[A] and as well we fibre of A A is inessential

with J the uppermost may derive

C~

~[F] (Str)

G 8' E Trivially 8' cSj

hence ~ is reducible.

of A A is an implication

If Al or A2 belongs

atomic and obtained from Aa by some substitutions. shape ~' [AI*,A] where A and ~~ are numerical

6 or A is

Then ~[A] is of the

equivalent.

@'[Aa,F] = ~ s t r G we have AI*-+F ==>str G. From A construct ~' as AI -~F G

to the fibre

and hence ~ reducible by proposition Since

F we get A

P and we

(Str)

8' E

Clearly 6' c 8 .

The proof is similar if A-~F is below the axiom.

Proposition 8. If ~ contains a mathematical Proof. Assume % to be

axiom in its endpiece

9

is reducible.

W. Pohlers

280

~' [P' ]

P' -~F Y G 6 E

where

J is the u p p e r m o s t

cut below ~[P]

. If P is not in the fibre of P'

we do as in the same case of the proof of p r o p o s i t i o n

7. Assume

P to be

in the fibre of P'. By p r o p o s i t i o n 4 we m a y assume that there only are substitutions

between ~[P] and ~'[P'].

that P' is atomic.

Since no variable

By p r o p o s i t i o n

P' too does not contain free variables. merical atomic formula.

6 we m a y provide

of P can be used as eigenvariable So P' m a y be regarded as a nu-

If P' is false we construct

E' as

q/ ' [P' ]

and as

P' -*F

(str)

(str) 7P' -~G

P' -~G

8' ~P' -+E

6' P' -*E

if P' is true. milary).

In both cases 6' c 8 .

(The remainig

cases are treated si-

Condition 6.2. is established by

Proposition 9. Suppose E to be a quantifierfree derivation 9

I-P-~E

closed formula.

(~ I - 1 P - ~ E ) w i t h

formula then there is a cut-free

P a true

derivation

If there is a cut-free

(false)numerical

atomic

9 ~E.

The p r o o f is by induction on the length of %. The only interesting

cases are that P-~E is a logical-

All other cases are trivial tical axiom because ference because numerical PI as a

If P-~E is a l o g i c a l - a x i o m

to P what means

of an in-

Q[P, PI] Px is

that PI is true too. But E contains

positive part and is therefore a m a t h e m a t i c a l

is an e q u a l i t y axiom

for a m a t h e m a -

it is true and it can not be the main-part

it is atomic.

equivalent

or e q u a l i t y axiom.

since P can not be essential

s =t-~Q[~[s],~[t]]

axiom.

we distinguish

If P-~E

the subcases~

P occurs in ~. Then E is also an equality axiom. -

P is s = t .

Then U[s] and ~[t] are n u m e r i c a l

equivalent

and E is a lo-

gical axiom. -

P is ~[s].

If s = t

is true E[s]

and ~[t] are numerical

equivalent.

Hence ~[t] is true. Since it is a p-part of E, E is a m a t h e m a t i c a l If s = t

is false E is a m a t h e m a t i c a l

axiom,

because

s =t

axiom.

is a n-part of E.

W. Pohlers Proposition

281

1o.

If the endpiece

of 9 contains

an equality axiom ~ is reducible.

Proof. Again we may assume that no free individual variable piece of ~. So if s = t - ~ [ U [ s ~ [ t ] ] merical formula.

s = t is a nu-

If s = t is true we have a logical axiom and ~ i s

ble by proposition

7. If s = t

is reducible by proposition

is false we have a mathematical

reduci-

axiom and

8.

6. A part in the conclusion of an inference weakening part

occurs in the end-

is in the endpiece

if none of its minlmal-parts

($4) or (Str) is called a has a corresponding

part

in the premise(s).

A cut in a modo 9 is called suitable

of its cut-formula

contain either a weakening part or the main-part

an inference Proposition

if both fibres of

on the boundary of ~. 11.

A m. d. without axioms in its endpiece

contains

a suitable

cut.

The proof is obtained easily by induction on the number of cuts in the endpiece. Corollary. contains

a suitable

cut or is reducible.

Lemma I. (repeated) is reducible. Proof. By the corollary above we may assume that 9 contains

a suitable

I. Suppose that one of the fibres of the cut-formula

contains

part. Say 9 is of the shape ~i Fi A T --~r !

9[A]

A~F Y G 8 E.

But then we may derive as well

(i=1,2

resp. i = I

)

cut.

a weakening

282

W.

Pohlers

F. l

F'

(str)

G

8' E with 6' c8. The proof is analogous if the weakening part is above Z[A]. 2. Now we assume that both fibres contain an inference on the boundary. If the cut-formula is an implication we are done by proposition 6. So we only have to consider the cases where the cut-formula is Vx~[x] or V xu Ix ] . 2.1. Suppose the cut-formula is Vx~[x].

Then Z is

C~ 0

~o

~l ' [~l IX]] (Z1

VxU2 [x ] -*F~

~ ' [vxU~ [x] ]

Vx~[x]-'F J1

n2

7 5 GI

G2

n2

Ja Go

nl

E

where J1 and J2 are defined as in the proof of proposition 6. We construct a m.d. ~' as pointed out on the next page. We only have to show that ~' c b holds.

Since s o c u l

and ~o C ~i we get 6 1 Q 8

and 82 c 6 by

proposition 3. It is ~'= ~[n3-ns,~1@ ~ 2 ] < ~ [ n 2 - n 1 , 6 5 e] if ~i and ~2 are smaller than ~[n2-n3,6@ e], because degree (~[t]')< n2 implies n2-n 3 > O. But it is ~i =~[n2-n 3,81~ s] < ~ [ n 2 - n 3,6~ e] and ~2 is equal to ~[n2-n3,62~ e] < ~ [ n e - n 3 , 6 ~ e]. Since Kk~' = Kk6z UKkS2 UKke we get

W. Pohlers

~83

(z

o S,'[UI It] ] 9

~

8o

(str)

[t]-~[

Vx~

[x]]

!

6'

B ~ ~ [t]-*~ [VxU[x] ]

[~xU [x ] ]

V x ~ [x]-*F

[t ] -~Vx~ [x] -~F

7 ~[t]-'G

I ~ [t ]'-*GI

G2

62 ~[t ]'-~i

n2

Om

n2

b2

U [t ]'-*0o

[t ]'-~G~

n3 nl

Go E 2.2. Suppose

the cut-formula is

VX~[X].

Then % is

6o

0

~i [~l [x ] ] Z~ [ V X ~ [X ] ]

V X~2 IX ] "~F~

n3

6 [V X~ [X ] ]

V X~ IX ] -~F

n2

$ G2

n2

Y G

6 GI v G

~o

n~

G' E

Suppose r ( V X ~ [ X ] )

=i+I. Then G'Is defined to be the premise of the

first substitution below G whose

rank

is < i, if such a substitution

exists. Otherwise G' is E. (The case where G' is between G and G~ is treated like the case we are going to show). By proposition 4 ~

may

assume that there are no inferences with maln-part below 61. So VXU2[X]

284

W. Pohlers

only could be a l t e r e d by a substitution. set-variable

occurs free in V X ~ 2 [ X ]

Since it is p r o v i d e d that no

we m a y conclude

that ~2 is ~. We now

construct ~'. 0

~J. [~:t Ix] ] (Str)

~I [x] ~ i [vx~ [x] ] C~W

Vx~ [x ] --~F

~[x] ~ $ [vx~ [x] ]

n4

7 G"~U IX]

n4 ~t

8'

G2

7 G1 "~U IX ]

n4

V !

~Oo~U[X]

n4

~r -~G, ~ [ x ]

J

~ , - . . . P~ [Q ]

[@ ] "~ VX~~ ] "~'F1

n4

~l t

Vx~ [x ] ---.G' --'-F~

n3

VX~ Ix ] -~TG' "~F

n2

C~

IVX~ IX ] ]

ne I G' -'G

n2

6" I G' "+GI

G2

n2

V t!

7G' -~G

W/

nl

o

IG' -~G' G !

E Before we start to show that ~' is a reductum of ~ (I) If m>n

We mention:

~ is a part of a m.d. and we replace the height n of F b y n the ordinal of F becomes G' c ~ .

The p r o o f is an easy i n d u c t i o n

on the length of the s u b d e r i v a t i o n

above

F. First we have to show that ~' again is a m.d. ference

of H N. ~' contains

(S3,1) which is not in ~. So 2.1. is fullfilled. Because

no inof

W. Pohlers

285

r(~[X]) < i %' is a m.d. of H N . It is an easy consideration that it is possible to assign to each substitution, different from J, the same rank as in %. Since all implicit minimal-parts of G' are provided to be of rank < i < N we may give the rank i to J. It remains to show ~"c ~. It is nl < n2 < n 3 < n 4. By (I) we get m~c ml, hence by proposition 3

(2) ~ ' ~ . It is ~i' = ~[n4-n3, P@ ~o'] and ~l =~[s+1,~ o] with s the degree of ~[X]. So s < n 4. If s < n 4 it is n 4 = n 3. Hence n4-n 3 < s + I . By (I) we have ~o 'c_ ~o < (O,~o). It is p = ~ [ ~ ' , i ] @ ~ N < (O,~o) since S ~ o = N . So we have ~i' <~l. For i < k it is ~ i ' = [~[~',i]}UEk~ o' < ~[~1,k]. Hence

(3) ~l'c ~i. i+I Every substitution between ~i' and ~" contains VXU[X] as implicit min~mal-part and therefore is of rank > i+I o By definition there is no substitution of rank _
(~) ~"~

i+1 ~

.

Suppose k < i+I and ~ e Eke". It is easy to realize that (in the set theoretical sense). Jf ~ ~ ' it is ~ e ~

KkG~' c Kk~L" holds < ~[~,k]. So

assume ~ e ~ ' = KkPU~ o' . If ~ eKk8 o' < ~ [ ~ , k ] < ~[~,k] we are done. For k < i it is KkP = Kk~' < G[~,k] by (3) and KiP = [~[~',i]] < ~[~,i] also by (3). So we have (~) ~ " < ~[~,~] f o r k < i + ~ . By (4) and (5) we get ~"c ~ and the proof of lemma I is completed. III. Consistency of HN} I. Let ~V be the l-place notational-form VX(X(O)-~Vy(X(y)-*X(Sy)) -~X(*I)). The axiom ~ of complete induction then is expressed by the formula V x ~ ( x ) . HN$ is H N with ~ as additional axiom. We shall show that if ~N is consistent H N ~ is consistent too. By F ~ we denote the formula F with all first order quantifiers restricted to J~. Proposition 1 2 . The following formulae are provable in H N for N > I . (I) ~ ( 0 ) (2) J~(t) -~ J~(St)

(3) s = t (4) ~ ( t I ~

(5) } d

Y(s)~N(t) ... -+~(tn) -~ ~/(f(tl,. ..,tn) ) for each n-place function constant

286

W. Pohlers

Proof. (I), (2) and (3) are trivial. (4) is proved in the usual way by induction on the definition of primitive-recursive functions. We shall give the formalized proof of (5) in order to show that ($3,1) is used in the right manner. For X a set-variable we define 8[X] to be q (~(*x)-+IX(*x)) i.e. ~ ( * z ) ^X(*z). ~[X] is an abbreviation for X(0) ^Vx(X(x)-+X(Sx)). In H N we get: I) Vy(~/(y)-+X(y)-+X(Sy))-+ ~(x)-+X(x)-~X(Sx). From I) and (2) we get 2) " -~ 8[X](x)-~8[X](Sx). Since we have 3) X(0)-+8[X](0) by (I), we get 4) B[X]~-+B[8[X]]. By 4) and the axiom 8[X](x)-~X(x) it follows 5) (B[~[X]]-*~[X](x))-+~[X]~-+X(x). From 5) we get 6) g~(x)-+ B[X]Mr-+X(x) with an inference (S3, z) whose main-part is~/(x). ~/(x) is pure and has rank I. From 6) we get ~ ~ provable in H N (N>I). Lemma 3. If U[x I ..... Xn] is a formula resp. ~[xl,...,Xn]a term with only the indicated individual variables occuring in it it holds

b) I-IN9 I-

=:>

HN

I- ~/~(XI )

"'~ "~d'['/(Y~s " ~[XI-''~ ~ n]

The proofs are straight foreward by induction on the length of ~[Xl, .... Xn] resp~ the length of a derivation of U[x I .... ,Xn]. Lemma 4. If H N is consistent so is H N ~ . THEOREM 2 H N ~ i s consistent.

IV. Embeddin 5 of ID N int O HN%. I. Derivation properties of HN~. Let ~ N be the collection of formulae arithmetical in a finite set of pure

formulae of rank

i N.

Propositon 13.

a) b) c)

HN~ I-~[F,F] i f F ~ N HN~ I- VxU[x]-+U[t] i f HN~ I- VXU[X]-+~[ @] i f

VxU[x] c ~ N. VXU[X] and ~[6] are i n ~-N"

Proof. I t is easy to prove a) by i n d u c t i o n on the l e n g t h of F. b) and c) are immediate consequences of a ) . Corollary.

d)

HN~ ~ ~X~/x(X(x) -*U [ x ] )

for U[x]G TN, i . e .

HN~-- (H;-CA)N ~

"

287

W. Pohlers

[ by (H~-CA)N ~ we mean the theory of N-times iterated HI comprehensiona x i o m n o t containing free set-variables with axiom of complete induction~

e)

IIN~ I- ~[0], U[x]-*U[Sx]

I-~[t]

i f U[O]

~.

Rema rk. In [6] it is proved that the ordinal of (HII-CA)~ i.e. H11 comprehension with axiom of complete induction is ~ 0 . By d) and our result below it follows that the ordinal of (HI-CA)~ is ~ 0 too. I conjecture that this does not hold for the theories with induction schema~ There the ordinal of (HI-CA) - might be ~ 0 whereas the ordinal of (]11-CA) seems to be @(@~(~+I ))0 i.e. the first ~-critical orinal of the ~-th number class collapsed to the 2nd number class. 2. Let IDN(~ I , ...,~ ) be the theory of N-tlmes iterated inductive definitions, i.e. first order arithmetic augmented by the set constants Qi and their 'defining axioms' Qi.1)

~i[Q1,...,Qi,x] -+ Qi(x)

Qi.2)

u176

~ i=1,..o,N,

-* Vy(Qi(y)-+~(y)) 3

where ~i is a i+1-place notational form, s.t.~i[X1,...,Xi,x] is an arithmetical formula satisfying the 'monotonicity condition' for X i. For details see [3] [12]. To each formula F o f the language of IDN (the union of all IDN(~ I,...,~N )) we define a formula ~ in the language of H N ~ by substituting for all occurrences of Qi the predicate ~i~WX (Vx (~i [~I "''' ~i-I' X, x] -* X (x))-+X ('i)) Proposition I4. If F is a formula in the language of IDN ~

eTN holds.

Proof. Clearly each formula in the language of IDN is arithmetical in QI"" "' QN so we only have to show that all formulae ~i(x) are pure and of rank i which is immediate by induction on i. THEOREM 3 ID N

I-

F

=~

HN~

I-

F@o

Proof. The proof is by induction on the length of a derivation of F. By propositions 13 and 14 all cases concerning first order arithmetic are clear. We are done if we succeed in proving Qi.1 )* and Qi.2)* in HN%. First we prove Qi.2)*. 1) Vx(~i[~ 1,...,~i_1,~,x]-~(x)) -+ Vx(~i[~ 1,...,~i_1,~,x]-~(x)) and 2)

S(y)-~(y)

are provable by prop.13 for ~(x) ~ N "

288

W. Pohlers

3) Vx(~ i [~I,''. ,~i-I ,~,x] - ~ (x)) - (Vx(~ i [~I " ' " ~ i-I ,~,x] -~(x) ) -~(y) ) - ~ (y) follows by I ) and 2). From 3) we get by an inference (S3,z) 4)

Vx(~i[ ~1,..,~i_1,~,x]-*~(x))-,~i(y)-~(y).

Since ~i(y ) is pure and of rank i this clause is permissible in HN}. From 4) Qi.2)* is immediate. Similar to 4) we get: 5) Vx(~i[~ I , .... ~i-I ,X,x] -+X(x)) -~Vy(~i(y ) -*X(y) ). By monotonieity condition (which is provable without using (S3,z)) we get 6) Vx(~i[~1,...,~i_1,X,x]-+X(x))-+~i[~1,...~i,y]-~i[~1,..., Hence

~i_1,X,y].

7)

~i[~I ..... ~i,Y]-+ Vx(~i[~ I .... ,~i_I,X,x]-+X(x))-+X(y) and from 7)

8)

~i[~1,...,~i,y]-+ ~i(y)

by an inference (S2,z)o

Conclusion. The consistency of ID N is provable in ID N + transfinite induction up to ~e~N+10., hence this induction is not provable in ID N ( and therefore also not provable in IDNZ, the intuitionistic version. ) This follows from Theorems 1,2 and 3 since besides transfinite induction (and the provability predicate of H N which is formalisable in ID N and even replaceable by a primitive recursive one) we only used finitary methods in our consistency proof. Remark: @ ~ N + I 0 is the least upper bound for the derivability of transfinite induction. In a paper in preparation we will show that induction up to any ordinal smaller than ~ N + I 0 is provable in IDiN which implies that the ordinals of the intuitionistical - and classical theories of iterated inductive definitions coincide. Lit e rature K1]

Buchholz~W.: Normalfunktionen und konstruktive Systeme yon Ordinalzahlen. This volume.

[2]

BuchholztW. Sch~tteIK.: Die Beziehungen zwischen den Ordinalzahlbezeichnungssytemen 7 und @ ( ~ ) . To appear in Archiv f~r mathemati~ sche Logik und Grundlagenforschung. Feferman, S. Forms/ theories for transfinite iterations of gernerallzed inductive definitions and some subsystems of analysis. Intuitionism and proof-theory. Proceedings of the summer conference at Buffalo N.Y. 1968. A.Kino, J.Myhill, R.E.Vesley (editors)Amsterdam-

[3]

London (North. Holland Publ.Co)197o.

W. Pohlers [4] [5]

[6]

[7] [8]

[9] [lo] [11] [12]

289

Gentzen~G. : Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen 112 (1936) Martin-L8ftP. : Hauptsatz for the intuitionistic theory of iterated inductive definitions. Proceedings of the Second Scandinavian Logic Symposium. J.Fenstad (editor). Amsterdam-London (North Holl. Publ. Co) 1971 Pohlers~W.: Eine Grenze f~r die Herleitbarkeit der transfiniten InI duktion in einem schwachen [II Fragment der klassischen Analysis. Dissertation Mttnchen 1973 Schdtte,K.: Beweistheorieo Berlin-G~ttingen-Heidelberg (Springer) 196o SchGtte~K.: Ein konstruktives System yon Ordinalzahlen I u n d II. Archiv f~r mathematische Logik und Grundlagenforschung 11 (1968) und 12 (1969). Takeuti~G.: On a generalized logic calculus. The Japanese Journal of Mathematics 23 (1953) Takeuti~G.: On the fundamental conjecture of GLC V. The Journal of the Mathematical Society of Japan Io (1958) Takeuti~G.: Consistency proofs of subsystems of classical analysis. Annals of Mathematics 86 (1967) ZuckertJ. : Iterated inductive definitions, trees and ordinals. Metamathematical investigation of intuitionistlc arithmetic and analysis. A.S.Troelstra (editor). Berlin-Heldelberg-New York (Springer) 1973

COMMENTS

ON GENTZEN-TYPE PROCEDURES

AND THE CLASSICAL NOTION OF TRUTH Dedicated

to Kurt Sch~tte

on occasion

of his 65 t birthday. Dag Prawitz

Gentzen probably tion to Hilbert's cal work

thought

program.

(Gentzen 1934-35)

of his work as essentially

But it appears

that he also understood

tion to the general understanding argued elsewhere

(Prawitz

this latter aspect Hauptsatz

of Gentzen's

is formulated,

did himself,

of the structure

1965 and 1971)

a contribu-

from his first main logiit as a contribuof proofs.

I have

that the significance

work is better brought

not for the calculus

but for the system of natural

out when his

of sequents

deduction

of

as Gentzen

or equivalently

for a typed lambda calculus. However,

there is a third aspect

missed by this reformulation

of Gentzen's work which is

and which concerns

ence between the rules of his calculus the semantical

definition of truth.

with this third aspect

the close correspond-

of sequents

this aspect himself.

to stress the correspondence

between Gentzen-type

semantical

rules of truth was perhaps Beth

1957, and SchGtte completeness

has much contributed logic

(SchGtte

(see also Kreisel

to obtain very natural

theorem for first order logic.

Gentzen-procedures

The first logician inference

was utilized by Beth 1955, Hintikka 1956,

of

of Gentzen's work.

Gentzen never mentioned

The correspondence

and the clauses

The present paper is concerned

proofs

rules and 1958).

1955, Kanger of G~del's

In particular,

SchGtte

to the theme with which I am here concerned.

were used by him in connection with infinitary

1951) and higher

latter case the correspondence

order logic

(SchGtte

1960);

in the

mentioned was the subject of special

attention. I shall here try to give a systematic of this third aspect

of the kind of procedures

into logic by Gentzen. briefly mentioning

exposition

of some features

that were introduced

I shall start by treating first order logic,

also infinitary

propositional

logic,

and shall

D. Prawitz

then show how this treatment second order logic. in which Gentzen's natural

Among

can be extended

other

calculus

In contrast recursion

logical

Valuations

seml-formal

in some respects

1.

as the

truths.

and related notions of truth which

of formulas,

is by a

the rules of a formal

or

system may in general be understood as defining certain inductively.

An obvious way of approaching

notion of truth by the use of a Gentzen-like notion

to

I want to emphasize a sense

to the usual definition

over the complexity

predicates

things,

of sequents may be understood

system for generating

I.

291

of inductive valuation

the classical

procedure

is via the

as defined below.

Definitions 1.1

Bipartitions

of sentences.

I shall call bipartitions

of sentences written

is the set of all (individual) individual and

~

constants

an~ parameters

terms belong to

~

of as a set of true sentences

.

and that

~

(e,~,~)

over

O

~

belongs

to

of atomic

sentences.

9

or

, i.e.

A

sentences

(|

where

of

@

and

is said to be included to be an extension ~'~. A bipartition the inductive

~

over

~'

and

(symbolically

@

$

(|

~'

$

and

and

$

is meant

are sets the bi-

are the sets of atomic A bipartition ~ ~'

= (@,$,*) ~ ~)

if

~' or

= (~,@',~') ~

is said

@' ~ ~

and

is called a base when it is used as the base for

said to be atomically

Let

A E ~ 9

(in the given

~ , and atomic when

in a b i p a r t i t i o n

definition

over

is to be thought

|

of another bipartition.

for bipartitions may also be used about bases. sentences

sentences whose

@

is said to be consistent when

respectively.

of ~ '

and

is true in the bipartition when

By the atomic ~art o f a b i p a r t i t i o n partition

where

as a set of false sentences,

total when every sentence

language)

,

symbols,

is false in the bipartition when

A bipartition are disjoint,

and function

Intuitively,

and

and I shall say that a sentence A E ~

(8,@,~)

terms that can be built up from some

are two sets of sentences

individual

We shall be dealing with what

complete when

~ U ~

The notions A base

defined

(|

is

is the set of atomic

| .

be the set of all terms that can be formed from numerals

292

D. P r a w i t z

and s y m b o l s for p r i m i t i v e

recursive

set of all true s e n t e n c e s ences

t = u

where

is an e x a m p l e

u

belongs

and

complete.

W h e n there |

and

of a c o n s i s t e n t

more a t o m i c a l l y

terms

t

writing (~,!)

individual

parameters

or in some s e n t e n c e

.

Then,

~=

sent-

(8,~,~)

the a r i t h m e t i c a l

I may

suppress

of

base.

the set of

it simply as a pair

(~,~)

over

~

;

|

or

I

.

8s

A

,

or

is

terms that can be b u i l t up from the together with

the s y m b o l s

occurring

to be first

order

s , respectively.

the s e n t e n c e s

s e n t e n c e s u s i n g the l o g i c a l symbols

~

as a b i p a r t i t i o n

as, a 2 ...

In this section,

range

to

w h i c h m a y be w r i t t e n

the set of i n d i v i d u a l

A

be the

terms d e t e r m i n e d by a s e n t e n c e

understood

in

~

and atomic b i p a r t i t i o n w h i c h is f u r t h e r -

By the set of (individual) of sentences,

and let

the set of all f a l s e

It w i l l be called

may then also speak about

s

~

is no risk of c o n f u s i o n ,

in a b i p a r t i t i o n ,

by a set

functions,

t = u

are s u p p o s e d

constants

from some g i v e n language.

~, ^ ,

and

V

A

and

The letters

and p r e d i c a t e B

are to

over sentences. 1.2.

Inductive

b_~ a base (T, P)

~

valuations.

= (~,~,$)

, also w r i t t e n

generalized

The i n d u c t i v e

, sometimes written (T~, F~)

V~

valuation

V

induced

, is the b i p a r t i t i o n

, d e f i n e d by the f o l l o w i n g

simultaneous

induction:

(I)

@ c T

(2a)

If

A E T , then ~ A E F .

(2b)

If

A E F , then ~ A E T .

(3a)

If

A E T

and

(3b)

If

A E F

or

(4a)

If

At

E T

for each

t

in

|

, then

YxAx

E T .

(4b)

If

At E F

for some

t

in

8

, then

~xAx

E F .

A, B, and The

and

At

converses

by r e c u r s i o n

$ ~ ~

.

B E T , then B E F , then

are here s u p p o s e d of (2) - (4)

over the c o m p l e x i t y

A ^ B E T . A ^ B E F .

to be s e n t e n c e s

(used in the u s u a l of f o r m u l a s )

over

@

definition

are taken notice

. of truth of in

the f o l l o w i n g d e f i n i t i o n . 1.3.

Semi-valuations.

ing

($,,))

and)

the c o n v e r s e s

valuation.

if

V

V

is a s e m ~ v a l u a t i o n

is a b i p a r t i t i o n of clauses

(T, F)

over

satisfying

(2) - (4) in the d e f i n i t i o n

8

(contain-

(clause

(I)

of i n d u c t i v e

D. Prawitz

1.4.

!Total) valuations.

1960, a (total) valuation consistent b i p a r t i t i o n (4) in the definition 2.

293

Following

over a set

(T, F)

~

over

the terminology

~

satisfying

the clauses

(2) -

1.2. and their converses.

Remarks 2.1.

While

is defined

in the usual definition

simply as the complement

of truth by recursion

the consistency

is in a way the prototype

Hauptsatz when formulated for a classical The clauses be understood

in the definition

and

be "from formal

A ET

B

is in

and

A E T

infer

~A

E F" ~

etc.

and

~

valuations may also

and

of the fact that

A E T

or

clearly in many respects by Gentzen.

B E F

in the rule corresponding (e.g~

respectively.

the subformula property

initiated of the rules

of any cut rule. also some of the questions

one immediately

or the corresponding

system are typical for questions

this kind.

There are three such natural questions

about Gentzen system of (with w e l l - k n o w n

that I shall consider next in section 3: one concerns

the relationship

and semivaluations

between inductive valuations

(invertibility

a second one concerns valuations

of the Gentzen rules);

the consistency

of the inductive

(Gentzen's Hauptsatz);

(iii) a third one concerns

their completeness.

The inductive valuations

system may be understood depending

valuation,

in tree form)

The system is

semi-formal

2.2.

is

The system is of course semi-

to ask about the inductive valuations

(il)

A

in the base are decidable)

similar to the kind of calculi

As already remarked,

(1)

where

and the inference rules would

a derivation

A E F

Note in particular

and the absence

answers)

the expres-

of the system

With each true and false formula in an inductive

we can in an obvious way associate

wants

of sequentso

As axioms

A E T

of the infinitely many premisses

to (4a).

calculus

A E F .

~ , respectively,

(provided the sets

because

of the proof of a Gentzen

of inductive

we would thus have all expressions ~

The proof of

as the rules of a semi-formal system, where

sions are of the form

falsity

of truth, we have here to prove

the consistency and totality of inductive valuations.

in

of SchGtte

is defined as a (total and)

or the corresponding

either classically

on how we read the various

clauses

semi-formal

or intuitionistically in 1.2.

It is to be noted that the problem of proving an expression may not be the problem of finding or effectively

describing a configura-

294

D. P r a w i t z

tion that

is a d e r i v a t i o n

a derivation

or not.

but

to decide w h e t h e r

For instance,

we can easily and e f f e c t i v e l y such that

if

A

arithmetical

is true

base,

then

is to know w h e t h e r

T

if

describe

A

a tree

in the inductive ~

T

of

last

induced

A ~ T .

or not.

is

theorem,

ending w i t h

valuation

is a d e r i v a t i o n

is a d e r i v a t i o n

the c o n f i g u r a t i o n

is F e r m a t ' s

A E T

by the

The problem

See f u r t h e r

section

3.3 and II.3.b. 2.3.

While

tive v a l u a t i o n we accept

the e x t e n s i o n s

as sound,

tion induced

of the sets

may be open to d i s p u t e s the existence

by a base

of the

is an i m m e d i a t e

of g e n e r a l i z e d

inductive

definitions

of. t r a n s f i n i t e

induction

over the

to extend

the results

far as possible, bility first have

complexity

order less

case

just because

clause

The e x i s t e n c e

system.

containing

(~,~)

I. sequence (i.e.

is c o n s i d e r e d

Va

case

are

to yield

of all

footnote

Since order

I want

logic

as

use of the possi-

by an ordinary

induction

to be p o s s i b l e

in the clauses

in the

(2) - (4)

class)

in the case

in section

in h i g h e r where

to be s a t i s f i e d

Va+1)

and Vwl

Va

IV.

3.1.b.

rules

of

of a

semi-valuations

in ~S

logic)

V 0 = (~,i), Va

an induc-

to a t r a n s f i n i t e

the clauses

is then an i n d u c t i v e

to section

of the premiss

order

rise

for limit

(where

of its

4.

of course

by a p p l y i n g

a bipartition

the c o n c l u s i o n

of the p o s s i b l e

(and e.g.

V B, e < a .

containing

Since

be taken as inference

1.2 give

obtained

the premisses

in the third n u m b e r Cf.

viz.

The g e n e r a t i o n

of the kind

of

sions are added the u n i o n

matter.

V0, VI, V2,... , V , V +1,...

the e x t e n s i o n Va

I shall not make

of s e m i - v a l u a t i o n s

they cannot

In the g e n e r a l

tive d e f i n i t i o n

valua-

of the p r i n c i p l e

class.

to second

which happens

disjunctive,

(4b),

semi-formal

inductive

of the a c c e p t a n c e

or e q u i v a l e n t l y

the p r e m i s s e s

different

is s o m e t i m e s (3b) and

(one unique)

induction

in an induc-

than the c o n c l u s i o n I.

is a s l i g h t l y

clauses

transfinite

F

on what r e a s o n i n g

consequence

order logic

of formulas,

complexity

2.4. (@,i)

this

and

second n u m b e r

w h e n not n e c e s s a r y

to replace

over the

for first

T

depending

V~+ I

is

(2) - (4) to

and the concluordinals

is the first valuation.

a

is

ordinal

D. P r a w i t z

3.

Immediate 3.1.a.

results Inversion

by an a t o m i c base

principle.

the t r u t h and f a l s i t y the c o n c l u s i o n

by an i n d u c t i o n

of the f o r m u l a s .

of the c l a u s e s

A semi-valuation but

3.1.b.

valuation

over

.

Then

The p r o o f

of the c l a u s e s and

B E F

V = (T, F)

A E T

and

(I) - (4),

.

proof

hence

u

E T

of G e n t z e n ' s T h e n by 3 . 1 . a and

there are s h o r t e r

derivations

B

valuation

i n d u c e d by an

is total. over the c o m p l e x i t y

t E |

, either

of the i n d u c t i v e At

E T

or

c l a s s i c a l l y but not i n t u i t i o n i s t i c a l l y

, At

of f o r m u l a s .

i n d u c e d by a con-

A E ~ 9

The i n d u c t i v e

complete base

on a c l a s s i c a l u n d e r s t a n d i n g

t E 8

be a

in the i n d u c t i v e

from the i n v e r s i o n p r i n c i p l e

for some s e n t e n c e

Completeness.

G i v e n that for each

and hence

Y xAx E T

of f o r m u l a s and valuations:

At E F

, we can

that e i t h e r for each

or for some

t E e

, At

E F

and

E F .

This

situation

is a c l a s s i c a l

introduction

l o g i c a l matters:

of formal

while

to see that s o m e t h i n g bound

(@,V)

of the u s u a l

The proof goes by i n d u c t i o n

Frege's

Let

valuation

immediately

that b o t h

inspection

3.3.

to be an i n d u c t i v e

is c o n t a i n e d

part of

a prototype

of

atomically

is

in the p r e m i s s . )

The i n d u c t i v e

is o b t a i n e d

Assume

B E T

in

is c o n s i s t e n t .

and is as m e n t i o n e d Hauptsatz:

required

of s e m i - v a l u a t i o n

on an i n d u c t i o n over the c o m p l e x i t y

Consistency.

of

to such a v a l u a t i o n :

V

i n d u c e d by the atomic

3~2.

conclude

(The d e r i v a t i o n

of s e m i - v a l u a t i o n s .

|

The p r o o f d e p e n d s

sistent base

induced

over the d e r i v a t i o n s

does o b v i o u s l y not n e e d

it can be e x t e n d e d

Embedding

semi-valuation

valuation

of the d e f i n i t i o n

s h o r t e r t h a n the one a s s u m e d

valuation,

depends

An inductive

is a l s o a s e m i - v a l u a t i o n .

The p r o o f is i m m e d i a t e

of course

295

troversies

in a s e m i - f o r m a l

instance,

of r e a s o n i n g

sound.

For

example

in the proof above)

by c l a s s i c a l

c o m p l e t e but this r e a s o n i n g

that

a reduction

logical reasoning

in a f o r m a l

of the r e a s o n i n g

over what k i n d

of the sense in w h i c h

constitutes

only r u d i m e n t a r y is a d e r i v a t i o n

in the c o m p l e x i t y

t h i n g is a d e r i v a t i o n

illustration

systems

required

system,

of

is r e q u i r e d there is no

to see that some-

system with accompanying

con-

that is to be a c c e p t e d as

reasoning, ~(VxAx

we see

^ ~WxAx)

is not a c c e p t a b l e

(continuing

E T ~ if ~

the

is atomica~y

intuitionistically.

296

D. Prawitz

It was

one of G e n t z e n ' s

we get a b e t t e r truth

if we replace

finition vely.

A E F

A E T

understood

II, we can c o n s t r u c t i v e l y

A E T

or

conscious

A E F , although

or not)

of the classical and

of such e x p r e s s i o n s

In such a d i s j u n c t i v e l y

that

(whether

approximation

the e x p r e s s i o n s

1.2 by a sequence

in section

A E F

represent

of

in the de-

understood

sequence,

that

notion

disjuncti-

to be c o n s i d e r e d

the c l a s s i c a l

we can prove n e i t h e r

fact

A E T

nor

constructively. But from a s t r i c t l y

that

insights

constructive

the i n d u c t i v e

complete

base

constitute

of truth:

Each

such that

A E T

and only if

A

is false

simple

3.4.

V'

part

because

The inductive

complete

base

V = (T, F)

induced of 1.4.).

it is clear

total

and c o n s i s t e n t

that

V' c V

and

A E F

each model

M

if

determines

the form of the sup-

about

generation

of total

in total valuations: valuation

i n d u c e d by a

is a total valuation. over

~

is i d e n t i c a l

by the atomic The second

by the d e f i n i t i o n s

ation

M

has

of 3.4 is just a summary

the t e r m i n o l o g y

diately

in

notion

of course a m o d e l

of G e n t z e n - t y p e .

of s e m i - v a l u a t i o n s

total v a l u a t i o n

The first (using

is true

system

conclude

and a t o m i c a l l y

of the c l a s s i c a l

section by a result

and a t o m i c a l l y valuation

A

we must

determines

M; and conversely,

Total valuations.

each

inductive

if

semi-formal

and e m b e d d i n g

consistent more,

in

(T, F)

this r e p r e s e n t a t i o n

We may sum up this valuations

of view,

by a c o n s i s t e n t

a representation

if and only And

point

induced

such v a l u a t i o n

such a valuation. p o s e d l y most

classical

valuations

part

of

of v a l u a t i o n

to the

(e,T,F)

of 3.1.a, part

Further-

3.2 and 3.3

then follows

and i n d u c t i v e

, but then since both

.

V

and

immevaluV'

are

B

3.5.

Embedding

consistent total lemma

extending

Lemma. if 3.6. valuations

i.e.

~

and

Each and

to a total valuation. first

3.1.b,

to a c o n s i s t e n t

~'

be two bases with

V~ _c V~,

truth.

and truth

valuations.

to a c o n s i s t e n t

then the f o l l o w i n g and a t o m i c a l l y

com-

3.4.

, also

Lpgical

each e q u i v a l e n t

V',

the atomic base

Let

in total

can be extended

by a p p l y i n g

and f i n a l l y

~ _ c ~'

V

valuation

is i m m e d i a t e

plete base,

Then,

of s e m i - v a l u a t i o n s

semi-valuation

inductive This

V = V'

In view of the e q u i v a l e n c e

in models,

to logical

the same set of terms.

.

the f o l l o w i n g

truth:

three

between

conditions

total are

D. P r a w i t z

is true

(i)

A

consistent

and

complete

base

~=

(ii) (iii)

A

is true

in each total v a l u a t i o n

over

|

A

is false

The e q u i v a l e n c e equivalence

between

total v a l u a t i o n

4.

Generation

set

in view

~

in the i n d u c t i v e

between

(i) and

(ii) and

(iii)

of finding

a valuation

this n o t i o n

(i.e.,

them to

applications

and so on.

is the e x t e n s i o n

of

(Ti,Fi)

(Ti,F i)

in the case

disjunctive.

If we want

containing

(|

of (3.b)

can

a consistent

to g e n e r a t e , we have

apply

of clauses

I shall r e p r e s e n t of the form

and

by one a p p l i c a t i o n

the

(2) - (4) obtained

(Ti+1,Fi+ I)

of the

2.3), m a k i n g

(4.b) where

some arbi-

the c o n c l u s i o n

systematically

to c o n s i d e r

(T,F)~

of the sentences

successively

, F0 = ~

I to sect.

of pairs

I may write

of

are false

, then to the results

TO = ~

and

The c o n s t r u c t i o n

sequences

~

of f i n d i n g

the c o n v e r s e s

obtained

choices.

by an e x p r e s s i o n

the formulas

set

I.e. we form a sequence

possible i.e.

"

from 3.4 and the

in w h i c h

we may

(@,~)

where

(cf. footnote

choice

tions

@A

(~,~)

We first a p p l y

to

over

(and the fact that a

of some

such a s e m i - v a l u a t i o n

(To,Fo) , (TI,FI) , (T2,F2),...

trary

from 3.5

.

semi-valuation).

to the problem

containing

defining

clauses

(~A,~,~) "

semi-valuation

(ii) follows

are true and the formulas

To g e n e r a t e

from these

induced by any

of s e m i - v a l u a t i o n s

semi-valuation

in 1.2).

valuation

in no c o n s i s t e n t

of 3.5 be r e d u c e d

clauses

atomically

is a c o n s i s t e n t

The p r o b l e m some

297

instead

all these

may then be r e p r e s e n t e d the pairs r => A

in the sets

T

(T,F)

where and

is

all s e m i - v a l u a -

r

as a tree

by sequents, and

A

are

F , respectively.

e.g. E,F=>A,A

to denote

a sequent w h o s e

of the s e n t e n c e s is the sequence In the figures sentences. finite able

in

E

consisting below

I shall

E

@~d~

"

of the

is the sequence

sentences

is always

restrict myself

sets and shall

set

antecedent

and then by the ones

consider

to denote

in in

F A

consisting and whose

followed

a sequence

to the case when

only s e m i - v a l u a t i o n s

~

first

succedent

by

A .

of atomic and

~

are

over the d e n u m e r -

298

D. P r a w i t z 4.1.

and let

Definition. ~

valuation sequent

over

~

and

order)

the

(a)- and

(b)-rule

by w r i t i n g of the

whose

(b)-rules

immediately

(b)-rule.

applicable applicable,

below

If there

(a)-rule. F => A

upwards

obtained

F => a

with the

in

a sequent

~

any

(b)-rule, (b)-rule

is to be a top sequent

and

In other F => &

above

we apply

nor any

in

then we apply

the same form as

the sequent(s)

is no such

If n e i t h e r

semi-

by a l t e r n a t i v e l y

of an (a)-rule,

(a)-rules

F => A

the line

instead

(a)-rule

in the final

the

is

tree.

(b)-rules (b2)

~,A,F

=>

A

E,F

=>

4,mA

=>

F

~,A,A

"IA,F =>

(a3)

E,A

(b3) E,A,B,F

=> A

E,AAB,F

=> i

F

=>

=,A,A F

(a4)

=>

1~

=> E,B,A

E,AAB,A

(b4) E,At,r,YxAx E,YxAx,r

=> A

(taken in some fixed antecedent

in some

tl, t2,.., 4.2.

order)

sequent

The m i n i m a l T = UD. i I

a (minimal) be g e n e r a t e d V

is a

t

is to be the first

such that below

by

and

~

.

of

over of)

generated

occur in the

@

F0 => AO '

Let

containing

the tree.

containing

(~,~)

over

In (b4),

|

As is easily

is any s e m i - v a l u a t i o n V'

does not

of the s e m i - v a l u a t i o n

F = ~Ai

(the branch over

At

...

term in

the one to be constructed.

semi-valuations.

of a b r a n c h

semi-valuation

seml-valuations if

of (a4),

are to be all the terms

be the sequents and let

F => ~,At I,A 1~ => ~ , A t 2 , A r => E , Y x A x , A

=>

In an a p p l i c a t i o n

there

The

of sequents

the line has

(a2)

i.e.

of sentences

below as far as possible.

stage have

above

sets

~ U ~ .

the sentences

is c o n s t r u c t e d

sequent

by

is the tree

the top of the tree by the a p p l i c a t i o n the

be finite

(i.e. we take

which

if we at a certain

~

determined

~ = (|

as origin

fixed

applying words,

tree @ => ~

in some

Let

be the set of terms

e

tree

seen, (~,~)

Furthermore,

by the s e m i - v a l u a t i o n

=> AI' (@,~,~)

(T,F)

is then

"''

and is said to all m i n i m a l

are g e n e r a t e d containing

s

over

this way;

(~,i)

tree over

, then

(~,~,$)

299

D. P r a w i t z

such that

V' ~ V

4.3.

.

The s i m p l i f i e d

tree c o n s t r u c t e d

semi-valuation

tree

in the same w a y as in 4.1

over

except

(|

is the

that rule

(b4)

is

r e p l a c e d by: (b4') r => ~,Aa,A r => ~ x A x , where

a

is the first

A

parameter

ai

in

|

that does not

o c c u r in

s e q u e n t s b e l o w the one to be c o n s t r u c t e d . It is e a s i l y seen that if the s e m i - v a l u a t i o n V2

tree over

V1

(~,~,~)

g e n e r a t e d by the s i m p l i f i e d

such that

VI

cance

|

V2

branches

never

by some

for the c o n s t a n t s

of the s w i t c h to s i m p l i f i e d split in m o r e

g e n e r a t e d by

, then there is a s e m i - v a l u a t i o n

semi-valuation

is o b t a i n e d f r o m

tion of terms in

is a s e m i - v a l u e d

as,

trees

tree

over

(|

(simultaneous) a2,

...

The s i g n i f i -

is of course

than two b r a n c h e s

substitu-

that their

at one and the same

point. 4.4.

Consistent

semi-valuations.

It is e a s i l y

the s e m i - v a l u a t i o n g e n e r a t e d by the b r a n c h valuation where

tree is i n c o n s i s t e n t ,

some a t o m i c

succedent.

sentence

semi-valuations.

semi-valuation truncated

closed.

semi-valuation

ends in a s e q u e n t s

If all the b r a n c h e s

4.5. over

said above

for f i n i t e

only if the t r u n c a t e d

to g e n e r a t e

of the

exists

sets

@

(simplified)

and

A branch

of a t r u n c a t e d

tree are closed,

also

and f r o m w h a t

conclude:

a consistent ~

(simplified)

I shall speak about a

W i t h this t e r m i n o l o g y

There

and the

in q u e s t i o n is said to be

of a t r u n c a t e d

in this s e c t i o n we may

C l o s e d trees.

(e,~,~)

tree.

of the k i n d

the tree is said to be closed. has b e e n

if we only w a n t

in this way,

semi-

a sequent

of a b r a n c h w h e n we

W h e n the c o n s t r u c t i o n

tree is m o d i f i e d

(simplified)

tree w h i c h

of this kind,

contains

in the a n t e c e d e n t

We can b r e a k off the c o n s t r u c t i o n

have r e a c h e d a s e q u e n t consistent

t h e n the b r a n c h

occurs b o t h

seen that if

of a ( s i m p l i f i e d )

where

semi-valuation

semi-valuation

e = ~0~

if and

tree over

(~,~,*)

is not closed.

4.6. base

Semi-valuations

(~,~,~)

c o n s i s t e n t w i t h a base

I shall say that a s e m i - v a l u a t i o n

~.

Let

~

be a

tree is t r u n c a t e d

S00

D. P r a w i t z

with respect to

~

when

is m o d i f i e d by b r e a k i n g a sequent to

$

F => a

or some

the c o n s t r u c t i o n

is r e a c h e d w h e r e

either

belongs

to

~

said to be c l o s e d w i t h r e s p e c t

to

~

branch

A E a

is stopped

The g e n e r a l

Calculi

the f o r m u l a s

of f o r m u l a s , remarked

was

these

completeness

F

belongs tree is of e a c h

of s e q u ~ o t ~

in the i n f e r e n c e

important

sequents make

of the i n d u c t i v e

As a l r e a d y

to r e p r e s e n t

valuations

constructive

thus

or s e q u e n t s

inventions.

it p o s s i b l e

rules,

the

constructively

approximation

of the c l a s s i c a l

of truth.

Let a c o n d i t i o n or

in

in the d e r i v a t i o n s by s e q u e n c e s

and thus gives us a b e t t e r notion

A

w h e n the c o n s t r u c t i o n

side-formulas

one of G e n t z e n ' s

in 1.3.4,

classical

some

A semi-valuation

idea of s i d e - f o r m u l a s

The idea to i n t r o d u c e replacing

.

tree

of a b r a n c h as soon as

in that way.

II.

I.

of the s e m i - v a l u a t i o n

off the c o n s t r u c t i o n

A E F .

on

V = (T, F)

in the d e f i n i t i o n I.I.2 clauses

(2) - ( 4 )

(2a')

If e i t h e r

then e i t h e r

~A

side-formulas

of i n d u c t i v e v a l u a t i o n s

A E T

can then be e f f e c t e d by r e p l a c i n g

the

in the f o l l o w i n g way:

~ F

A ~ T or

An so on for the other

V

on the scope

or

V

satisfies

the c o n d i t i o n

X

,

satisfies

clauses.

From a constructive depends

be a c o n d i t i o n of the form

The idea to i n t r o d u c e

point

of the

of view,

(informal)

the i m p o r t a n c e quantifier

of this change

in (4a').

This

clause n o w reads: (4a')

If for each

the c o n d i t i o n Given

•

that for each

n o w c o n c l u d e by YxAx

t E @

, then e i t h e r

(4b')

t E |

, either YxAx

At E T

E T

or

V

, either

At

E T

that for e a c h

E F ,

from which

f o l l o w s by

VxAx

E F

Furthermore

get

YxAx ^ ~ YxAx E F , which

when

the i n d u c t i v e v a l u a t i o n s were

t E 8

(4a')

or

V

satisfies

satisfies or

, either

that e i t h e r

by (2a') and two a p p l i c a t i o n s

At At

E F E T

~xAx of

X

E T

, we can or or

(3b'), we

could not be c o n c l u d e d c o n s t r u c t i v e l y d e f i n e d as in s e c t i o n I (cf.I.3.3

and 1.3.4).

2.

The n o t a t i o n

of s e q u e n t s and the i n f i n i t e

The new c l a u s e s

calculi

(2') - (4') o b t a i n e d by i n t r o d u c i n g

side-formula8

D. P r a w i t z

301

in the d e f i n i t i o n of i n d u c t i v e v a l u a t i o n s rules

in a s e m i - f o r m a l

read c o n s t r u c t i v e l y , occurring

is p o i n t l e s s

elements

of this s e m i - f o r m a l

understood

sentences

A

and the s e n t e n c e s

occurring

in a s e q u e n c e

as in s e c t i o n

ence

F

~=

1.4.

But

is false (e,~,~)

(R I) of

~

of s i d e - f o r m u l a s

B

A E F

i.e.

understood

obtains.

of the form

in c o n t r a s t

S

A 6 F

the r u l e s

in

A

the

in a seof the form

as a s e q u e n t

to the s i t u a t i o n there,

(with a c l a s s i c a l

sequences as a s s e r t -

If we c o l l e c t

o c c u r r i n g in c o n d i t i o n s

or some s e n t e n c e

F => A this

"or"):

e i t h e r some sentI is true.

of the s e m i - f o r m a l

system with a

become:

For all

or some

of c o n d i t i o n s ,

and

A , we can r e p r e s e n t

In this n o t a t i o n , base

sequences A E T

in c o n d i t i o n s

is now to be r e a d

in

the "or"

disjunction;

s y s t e m can now be taken to

one of these c o n d i t i o n s

quence

sequent

tension between

( 2 ' ) - (4') and c o n s t r u c t i v e

of the form

ing that at least

B E T

a strange

definition

(see s e c t i o n 3 below).

be " d i s j u n c t i v e l y "

F

taken as i n f e r e n c e

point of v i e w the i n t r o d u c t i o n

The e x p r e s s i o n s

with

is b e t t e r

U n d e r s t o o d as an i n d u c t i v e

there a r i s e s

in the c l a u s e s

and f r o m a c l a s s i c a l

S

system.

F, A

B E A

such that e i t h e r some

is e l e m e n t

of

~

A 6 F

, the sequent

is e l e m e n t F => A

is to

be an axiom.

(R 2a)

r => ~,A

(R2b)

A, r =>

-I A , F => A (R 3a)

F => A,A

F => &, -IA

F => A, B

(R3b)

A, B, F => A

F => A, A A B

(R ~a)

F => A,At I

AAB,

F => A, At~

(R 4 b )

...

At,

F => A, Y x A x

to d i f f e r e n t

in the a n t e c e d e n t

the ones u s e d in 1.4.1

I.

Alternatively,

the i n f e r e n c e r u l e s s h o w n by $chGtte. main

idea m o r e

orderings

and succedent.

to operate

As seen,

the rules

that we pay

are the same as

semi-valuation

sequents

on c e r t a i n parts

of the

of s e q u e n t s

F => A

of the f o r m u l a s

of i n t r o d u c i n g

Since the n o t a t i o n

clearly,

the c o n v e n t i o n

or r e p e t i t i o n s

for c o n s t r u c t i n g

instead

F => A

YxAx,

The rules are to be u n d e r s t o o d w i t h no a t t e n t i o n

F => A

trees except

one may a l l o w sentences

as

seems to show the

I shall use this n o t a t i o n here.

302

D. Prawitz

that we there the risk these

prescribed

of a c e r t a i n

different

rules

coincide,

The calculi w i t h

the rules

calculus

of sequents

sequent

S

calculus

determined

is provable differences) 1951

3.1

F =7 A B

in

~

When

I shall write base

the infinite

= (| ~

infinite

in L o r e n z e n

between

the inductive

~S

.

is (except induction

the

The for some

introduced

1951.

calculi

A

belongs

valuations

and the

of sequents:

to

some

A in

r

belongs

to

F~

T~ .

is obtained

immediately

by an i n d u c t i o n

over the

of derivations.

Thus, calculus

from a s t r i c t l y

of sequents

inductive

valuation

3.2. sequents

induced

determined

above:

point ~

of view,

yields

new above

the

~.

The Hauptsatz for the ~

either

, or both

for some A E T@

by 1.3.2,

infinite

is a c c o r d i n g l y

~9 F => A , A

is e x c l u d e d

the infinite

nothing

and

calculus

a triviality

~& A, F =7 A ,

B E r , B E F@

and

A E F~ .

we have

of

or for

Since

the

again by the result

~& r => A .

3.3.

Restrictions

derivations

on the rule

in the infinite

are to be e f f e c t i v e l y complete

by

If

above,

B E A , B E T~

last p o s s i b i l i t y

by

by a c o n s i s t e n t

of view:

then by the result

classical

determined

The Hauptsatz.

from this point some

, I shall call

contained

if and only if either

This result length

system,

the system with

The e q u i v a l e n c e

or some

(For

the fact that

result

infinite ~

(R I)- (R4)

in this

and also

the formulas.

concerning

by the base

by the a r i t h m e t i c a l

by SchGtte

Classical

order b e t w e e n

see 6.3).

determined

notational

3.

a certain

oversimplification

calculus

described,

bases ~ = ( @ , ~ , ~ )

(R 4 a ) .

with

If we require

of sequents

we obtain

determined

for c o n s i s t e n t

denumerable

|

that

the

by

and atomically

and d e c i d a b l e

~

and

the result: If

A E T~

by the f o l l o w i n g (|

],[A})

scribed,

~

=7 A

observations.

truncated

The

semi-valuation

tree

over

with respect

to

~

can be e f f e c t i v e l y

and if closed w i t h r e s p e c t

to

~

, it is a d e r i v a t i o n

In the c a l c u l u s respect

, then

to

determined

~ ) is not

by

~

.

closed with

If the tree respect

to

(truncated @

deof

--~A

with

, the tree g e n e r a t e s

D. Prawitz a seml-valuation and

$

V'

with an atomic

are disjoint

atomically

complete

and

~'

base,

and

then

V'

, A

cannot

(using 1.3.2). of theorems

4.

and

effectively,

obtain two notions

" ~

is an

A

is false

is consistent

of the fact that the set that the derivations

are to

1959.

pointed

=> A"

the interest

out above:

definition

and

" ~

the equivalence

to left,

the questions in 1.3. have

3.1 holds

about inductive

The answers

T~

base)

F ~ , we

that constitute

than obtained

valuations

by

a

T ~ and

P~.

only from right

raised

in 1.2 and

anew for the infinite

(first given by SchGtte

is the arithmetical

and

constructively

to be considered

of the present

By the introduction

of

A =7"

of truth and falsity

answered ~

~

proved by Shoenfield

in the inductive

better approximation

when

~'

if ~

~ . Hence by 1.3.1.b 3ince

provided

point of view,

is the fact already

of side-formulas

of sequente.

V~

by requiring

first

such that Hence

results

From a constructive

Since

~'~

V' ~ V~ .

This gives a simple proof

Constructive

approach

(@,~',~')

are disjoint.

@'~

then be true in

is not affected

be described

part

~

and the lemma in the proof of 1.3.5, in

303

1951

follow the general

calculi

for the case pattern

of 1.3

with some slight modifications. 4.1. rules

Inversion

(R2),

miss(es). provable,

principle.

If the conclusion

(R3) and (R4b) is provable, If the conclusion

~xAx,

r => A

then there exists a sequence

Atl,r I => At; r',r => A,A,

At2,r 2 =7 A2;

...

can be obtained

r',r 2 =7 A2,A';

of any of the

then so is (are) of the rule

of provable

the pre(R4a)

such that a derivation

from derivations

of

is

sequents of

r',r I => AI,A';

...

The proof is immediate

by induction

over the length

of deriva-

tions. 4.2. ~

Hauptsatz.

The Hauptsatz result

yields

1.3.2 and follows

principle

~

r => A,A

and

~A,

F => A , then

and is a generalization like this result

now using induction

is a negation ~

If

r => A .

r => A ~

~ B , then if

A

~

directly

over the complexity B,r => A

is a conjunction

and

~

of the consistency from the inversion of formulas:

F =7 A,B

B ^ C, then

~

If

and hence

r =7 A,B

and

A

304 ~

D. P r a w i t z F => A,C

and

~

B,C,F => A

is a d e r i v e d

rule and)

~

If

F => A .

for every and ~

t

~Ai} i

A

is a u n i v e r s a l

and

~

...

Constructively, that either section

5.

A

that

~

A => A

are true

but

introduced

(i)

complete

,

F,F i => Ai,A;

If

~

is complete,

basis

indicated

in

Since

for

not to describe

the

truths of sequents valuation

induced by some

truths,

i.e.

valuations

induced

by c o n s i s t e n t

purpose,

different

the sentences

which

and

A,F

=> A,A

individual not

changes

section: about

the base

is its

are axioms.

terms now have

occurring

of a term

two obvious

(R I) by:

given a d e r i v a t i o n

is a p a r a m e t e r F => A,At

thing known

we replace

sequents

substitution

there are

(R I)- (R4) of the last

in the axioms,

t

for

for every

t .

(R4b) is now e q u i v a l e n t

(R4b')

then it holds

the logical

this latter

(RI') All

rule

~

for complete

But as a l r e a d y

Since the only positive

of

F => A,Bt

bases.

in the rules

a

prove

.

in the inductive

completeness,

(ii)

~

[ti]i,~Fi)i

F =~ A .

in g e n e r a l

his calculus

to generate

When we have to be made

thinning hypothesis

.

in all inductive

atomically

Hence

A =>

of the logical

and f a l s i t i e s ~

~

~

then

sequence

in 4.1.

also

that

we have:

The g e n e r a t i o n

truths

or

T e r t i u m non datur.

Gentzen base

and hence

VxBx,

for some

stated

we cannot

=> A

I above,

4.3. every

~

(by the fact

of the i n d u c t i o n

sentence

B t i , F i => A i

with the p r o p e r t y

F,F 2 = > A 2 , ~ ;

and hence

by two a p p l i c a t i o n s

in

of

the same

F => A,Aa r

or

status

, where

A , we can by

a , obtain d e r i v a t i o n s In other words,

to the finite

the infinite

rule:

F ~-> A,Aa F --~-> A,VxAx

where 6.

Completeness 6.1.

logically

PFoof true

a

does not

occur

of the calculus of c o m p l e t e n e s s .

if and only if

in

F

or

A .

of sequents The proof

~ => A

of the fact that

is now immediate

A

from the

is

D. Prawit z

equivalences simplified of

=> A

1.3.6 and 1.4.5 and the fact that a closed truncated

semi-valuation

tree over

(|

in the calculus of sequents.

sequents

305

in general

~ },~A))

is a derivation

The same proof holds for

if we define logical

truth for them in the

obvious way. 6.2.

Remarks.

predicate plete.

There is nothing

calculus which makes

In contrast,

sequents,

truth I.

this calculus generate

one expect

that the calculus

from the very construction

it is immediately

to logical

in the usual formulation

obvious

the first order logical

of a system intended to

truths. of construction

culus of sequents which makes the calculus

(i)

of

that it is complete with respect

is the natural formulation

to logical

is com-

of the calculus

One seems thus to be justified in saying that

To state in summary this principle respect

of

obviously

of the cal-

complete with

truths, we may recall the following

facts:

Studying an inductively defined notion of truth, we saw that the problem of finding a total valuation a sentence

A

is true is equivalent

sistent semi-valuation

in which

that the non-existence

of a consistent

which

A

is false,

A

is equivalent

in which

to finding a conis true, and hence semi-valuation

to logical

in

truth of

A

(section 1.3). (ii)

Furthermore,

we found a construction

semi-valuations consistent itself

that generates

all

in such a way that the non-existence

semi-valuation

in which

A

of

is false shows

in the fact that the construction becomes

closed

in a certain way (section 1.4). (iii)

Hence,

we just take these closed constructions

as deriva-

tions and are sure to derive exactly all the logical 6.3. calculus

I.

Additional of sequents

remark.

Although

is sufficient

truths.

this way of describing the

to account for its completeness,

This point has also been stressed by Kreisel and I am grate-

ful to him for much stimulating See also his contribution

of the theme treated h e r e . _

tc the present volume which became known

to me only after I had completed manuscript.

discussions

the present revision of my lecture

306

D. Prawitz

it does not explain why the inference the converses just the rules side-formulas

of inductive valuations

after the introduction after all,

existence

of certain semi-valuations, sentences

of those defining

the (a)-rules

in the construction

of the (a)-clauses

defining

of sequents;

valuations.

semi-valuations of these

are the

On the contrary,

of the semi-valuation

trees applied

satisfy the converses

the inductive valuations

coincide with the generalization side-formulas,

may here be appro-

the semi-valuations

the inductive

in order to make the generated

introducing

induced

does not simply depend on the defi-

fact that the clauses defining

converses

the truth

valuations

complete bases.

The colncidence mentioned

nitional

In other

(simplified)

but also as asserting

in all inductive

A warning against a certain oversimplification priate:

of

trees can be looked upon not only as stating the non-

and falsity of certain by atomically

are

the calculus

in the latter way.

it should be explained why the closed truncated

semi-valuation

i.e.

the semi-valuations,

and some obvious modifications;

of sequents was first described above words,

rules of the calculus,

of the rules for generating

(a)-clauses

i.e. with the (a)-rules

do not at all obtained by

of the calculus

instead they coincide with the g e n e r a l i z a t i o n

of the

(b)-clauses. It is to be recalled are interpreted valuation

that the sequents

trees.

When the upwards

tree is broken off because read downwards

in the calculus

as just the negation of the sequents construction

of inconsistencies

instead with the opposite

of sequents

used in the semi-

of the semi-valuation

in all branches and is

interpretation

of the se-

quents, we are in effect replacing

the semi-valuation

clauses by the

transpositions

replacing

and

by

" EF"

and

of their converses, " E T"

.

Thus,

wanted to explain depends (with the replacement

the coincidence

on the essential

just described)

defining the inductive valuations (b)-clause

Infinite

The notions and results sentential

in question

. ~ F" that we

fact that the transposition

of the converse of an (a)-clause

is identical

to the corresponding

and vice versa.

~!~

to infinite

" ~ T"

sentential parameters,

logic.

sentential

of sections

I and II are easily extended

Given an infinite

we consider sentences

of negation and conjunction

logic

~ Ai iEI

denumerable

set of

formed by the operations

of sentences

~Ai)iE I

with a

D. Prawitz

denumerable

index

set

I ; when

modifications

are necessary.

I.

valuations

Inductive Leaving

the clauses

The

has a h i g h e r

out the set of c o n s t a n t s

from the bases,

(3) - (4) in the d e f i n i t i o n

of i n d u c t i v e

we now replace

Ai 6 T

for all

i E I , then

~ Ai 6 T . iEI

(3b')

If

Ai E F

for some i E I , then

~ Ai A F . iEI

other n o t i o n s

of section

and the r e s u l t s

are then defined

in s e c t i o n

in the same way

1.3 then i m m e d i a t e l y

extend

to

case.

Generation

of s e m i - v a l u a t i o n s

In the c o n s t r u c t i o n for finite

I.I

some

v a l u a t i o n by

If

the present

conjunction

of s e m i - v a l u a t i o n

and u n i v e r s a l

trees,

we replace

quantification

the rules

by the rules

(b3')

(a3') ~,Aj,r, ~ A i => A iEI

r => ~,A 1,A

In the a p p l i c a t i o n s that

Aj does not occur

to be constructed,

At, A2,

...

trees,

containing

valuation

tree

over

semi-valuation determined

~

r => A,A I

.

below

are g e n e r a t e d

of a c o n s i s t e n t

to the t r u n c a t e d

closed.

the

of (b3'),

by semi-

semi-

But we now make no use of

trees.

as in s e c t i o n

we may now i n t r o d u c e

m i n e d by bases

sequent

such

[Ai)iE I .

semi-valuations

is e q u i v a l e n t being

j E I

by a base

For the same r e a s o n s way as there,

of

and the n o n - e x i s t e n c e

(~,~) (~,~)

of some

in the a p p l i c a t i o n s

I, all m i n i m a l

valuation

(R3a')

and

are to be all the s e n t e n c e s

As in section

Calculi

is to be the first

in the a n t e c e d e n t

sequent

simplified

Aj

of (a3'),

the s e m i - v a l u a t i o n

r => ~,A2,A

r => ~, ~ Ai,A iEI

~, ~ A i , F => A iEI

3.

cardinality,

(3a')

as there

2.

I

807

In place

of the rules

r => A,A 2 ...

r => A, ~ A. iEI i

II and in the same general

infinite

calculi

of sequents

(R3)-(R4), (R3b')

Aj,

deter-

we now have r => A

~ Ai,r=> iEI

A

308

D. Prawitz

where and

AI, A2, j

...

in (R3a') are to be all the sentences

in (R3b') is to belong

The results

Calculus

to generate

To generate

sentential

extend to

logic.

the logical t r u t h s

the logically

logic, we replace

~Ai}iE I

I

of section II.3 and II.4 now immediately

these calculi for infinite 4.

to

of

true sentences

the axioms in the calculi

in infinite

sentential

determined by a base

by the axioms

(El') as in first order logic but leave the other rules

(R2a),

(R3a'),

remains

(R2b),

infinite.

and (R3b')

as they are.

Except for notational

The calculus

differences,

thus

this calculus

is the one studied by Tait 1968. Since the rules of the calculus in the construction

that a closed truncated tion of logical

F => A

are the same as the ones used

of the semi-valuation semi-valuation

in the calculus,

truth is immediate

IV.

tree over

(r,A)

is a deriva-

the completeness with respect

in the same way as in section III.6

that we now make use of semi-valuation semi-valuation

trees and it is thus clear

trees instead

to (except

of simplified

trees).

Second

order valuations

and related notions

The notion of truth for second order sentences

that we could

hope to approach by extending the notions and results I and II is the notion of truth in generalized

of sections

second order models

in the sense of Henkin. Let a domain sequence be a sequence D0

is a non-empty

empty set of in

~

set of individuals

n-ary relations

of the descriptive

language,

in

~0

and "

constants

~

= ~0' ~1' ~ 2 '

~n'

n > 0 t

and parameters

individual variables variables

over

~n

"

order domain sequence

of formulas

range over

T

in

(~,I)

(~,I)

and the n-ary ~

to ~ n

(il,i2,...,in)

such that

I'

except for assigning

is like

I

by recur-

(n > 0)

predicate

is said to be a (normal)

if for each second order term

belongs

I

of a second order

G

~ x l x 2 . . . X n A ( X l , X 2 , . . . , X n ) and for each interpretation tion of

is a non-

in the usual way, letting the

~0

The sequence

where

Given an interpretation

we can then define the notion of truth in

sion over the complexity

"'~

' (i.e.

A(al,a2,...,an) ij

of the form I, the denota-

the set of all n-tuples

is true in to

second

aj

(9,I')

belongs

where

to ~ n )

9

D. P r a w i t z

It is truth second

in second

order d o m a i n

in all such m o d e l s

I.

sequence

(~,I)

and logical

where

~

is a (normal)

truth in the sense

that shall be c o n s i d e r e d

of truth

here.

Definitions 1.1.

Second

is u n d e r s t o o d 80' ~I' 82' that

order m o d e l s

309

order b i p a r t i t i o n s .

a triple

"'"

(8,q0,$)

such that

By a second

where

80

|

is a sequence

dicate

symbols

constants

of second to

The logical also

to

order)

80

'

constants,

over

case

, i.e.

are s u p p o s e d quantifier

order v a r i a b l e s

the term is atomic,

formula

at most

containing

case

the term is molecular.

and

G

is an

obtained

n-ary

second

by s u b s t i t u t i n g

and then,

If order

G

~

to use

if the term was m o l e c u l a r ,

or c o n s t a n t s and

now being a l l o w e d

A(G)

eliminating

(second

in

8n ,

is a second free,

is a second

then

as

of the form

..., x n

occurrences

u

to bind

n-ary

is either a symbol

x I, x2,

term,

two sets

4, ^, An

pre-

individual

parameters

A(Xl,X2,...,Xn)

for free

n-ary

are

whose

(or xn)~

~ x n A ( x n)

terms

and p a r a m e t e r s

and

or an e x p r e s s i o n

where

e

X

8 of ranges

Xxl,x2...XnA(Xl,X2,...,Xn ) over

@

sentences

n-ary predicate

the u n i v e r s a l

second

constants

n > 0 , is a set of

and where 8

sentences

term over a sequence

in w h i c h

for

and whose

8 n " second order

(n-ary)

Cn

and parameters,

order s e n t e n c e s

terms b e l o n g belong

and

of ranges

is the set of all i n d i v i d u a l

can be built up from some i n d i v i d u a l

and f u n c t i o n

order b i p a r t i t i o n

order

in w h i c h

order sentence

is the sentence of

Xn

the

in

A ( X n)

X-symbols

in

the usual way by conversion. The sequence of s e n t e n c e s such that and

en

s

~0

of ranges d e t e r m i n e d written

or

and the p r e d i c a t e

or in s e n t e n c e s

of

The t e r m i n o l o g y also used for second 1.2. order base

The ~=

(4) by

9s

bE a sentence

of the

terms n-ary

parameters

or by a set 80' ~I' 82'

determined predicate

and c o n s t a n t s

by

A

"'"

or

s

parameters occurring

in

A

s , respectively. introduced

for b i p a r t i t i o n s

in section

I.I.1

is

order b i p a r t i t i o n s .

(second

order)

(| | )

A

is the sequence

is the set of i n d i v i d u a l

is the set c o n s i s t i n g

P~, P~,...

clauses

~A

inductive

is defined with

valuation

as in I.I.2

the a d d i t i o n a l

induced by a second

(replacing

clauses:

|

in the

310

D. Prawitz

(5a) If

A(G) E T

for all n-ary terms

G

over

0, then

YX~A(X n) E T.

(5b) If

A(G) E F

for some n-ary terms G

over

|

yxnA(x n) 6 P.

1.3.

Quasi-valuations.

then

I shall also consider the quasi-valuation

induced by a second order base

~

, which is the pair

as in second order inductive valuations

except

(T,F)

defined

that the clauses

(5)

are replaced by: (5a') If

A(P)6 T

for all predicates

P

in

~n

'

then

yXnA(xn) 6 T.

(5b') If

A(P) 6 F

for some predicate

P

in

|

'

then

y X n A ( x n ) 6 F.

1.4.

Semi-valuations

order semi-valuations

and

order inductive valuations corresponding

(total)

in a way analogous (T,F)

over

|

We define

valuations

second

from second

to the one in which the

first order notions were defined.

sider semi-quasi-valuations verses

(total) valuations.

and second order

In addition,

we con-

that satisfy the con-

of the clauses defining the quasi-valuations,

i.e.

(2)- (4)

and (5'). 2.

Remarks

and further definitions

The notions

of truth and falsity in second

all the properties showing

defining the second

that these valuations

can be generated

the one obtained for first order logic thus have an approach similar

and the consistency

satisfy

in a way analogous

(result I.I.3.4),

for first order sentences.

we find that the inversion

Investiga-

principle

3.1.a

3.2 in section I are proved in the second order

valuations

3.1.b and in particular

valuations

3.3 now fail, also when classical

valuation

the completeness

of semi-

of inductive

reasoning

is accepted.

it is easily seen that no second order inductive

induced by an atomic base

showing e.g.

to

one would

case in the same way as in section I, but that the embedding

Indeed,

By

to the notion of truth of second order sentences

to the one established

ting this possibility,

order models

order total valuations.

that

is total;

VX I~ (xlt A ~ X ~ )

belongs

Any derivation to

T

or

F

already have to contain such a derivation as a proper part.

woul~ It can

be shown that only a quite special kind of second order sentences get a value in second order inductive that these valuations for second

valuations.

cannot be used to represent

order formulas

It is thus clear the notion of truth

that we are concerned with here.

In the case of quasi-valuatlons ever, all the basic results

and semi-quasi-valuations,

of section 1.3.

how-

- i.e. 3 . 1 - 3.3 - imme-

D. Prawitz diately

carry over without

valuation converse

change.

311

But it is clear that a quasi-

is not n e c e s s a r i l y a valuation of clause

since clause

(5a) in 1.1 will not be satisfied

(5b) and the in the general

case. We want the second order variables finable

to range

by second order terms as expressed

definition

of inductive

one hand, we cannot

valuations,

in the inductive

definition

valuations

of quanti-

because

the exten-

sions of the second order terms may depend on the meaning

of quanti-

fication

of the

(a fact formally reflected

inductive

valuations),

de-

(5) in the

but the dilemma is that on the

take this as an inductive

fication as attempted

over the relations

in the clauses

in the incompleteness

and on the other hand,

the quasi-valuations,

which are inductively defined and are total when the base is atomically complete,

give the variables

with our original

a range that is too small to accord

intention.

Any straightforward

extension

of the Gentzen-like

procedures

of first order logic to second order logic is therefore However,

by paying attention

be valuations,

to the quasi-valuations

we can get some solution

with in the end of section 1.3: tions and that of embedding In fact,

that happen to

of the two problems

that of generating

given semi-valuations

dealt

the total valuain total valuations.

given a total valuation V, it is easily seen that by an

appropriate

choice of an atomic base

quasi-valuation

induced by

~ .

also a given semi-valuation end, I make 2.1. and let

the following

be an G

~

,

V

can be embedded

Less easily,

can be embedded

in this way.

Let

~

n-tuples

To this

be a second order base

n-ary second order term over

relative

in the

it can be seen that

definitions.

Possible values. G

value of of all

excluded.

to

~

is a partition

(t1,T2,...,tn)

~

.

A possible

R = (RI,R 2)

of terms in

e0

(e,~,~)

of the set

such that

(1)

if

Gtlt2...t n E ~ , then

(tl,t2,...,t n) E R I ; and

(il)

if

Gtlt2...t n E ~ , then

(tl,t2,...,t n) E R 2 9

2.2.

By a representation

= (e,~,~)

is understood

for each molecular

n-ary

and for each possible unique contain

predicate

a sequence (n > 0)

value

symbol

of possible values relative

R

of

PG,R E e*n

|

= ~,

~* e2,

to

... such that

second order term

G

G

there is one

relative

to

and such that

Just these symbols and no others.

~

@~' |

over @

"'"

e ,

312

D. Prawitz 2.3.

Atomic

closure

of a bipartition.

Let

05

...

be a r e p r e s e n t a t i o n

* 82,

order b i p a r t i t i o n

(~,s

and let

|

of possible

relative

to

By the atomic

usin~

values

~*

is understood

i)

@0 = |

ii)

|

the atomic

and

|

(t 1,t 2 ,. .~ ~'

(using

A second

the atomic some

in

second

if the b i p a r t i t i o n ~

is closed

of

(~,v)

2.5.

order

and

where

written

~5 = (|

of possible

then already

order base

(|

over

of the form

as in 2.3,

in

~ = (8,%o,~)

where

V

, in other words,

atomic

values

occur

and

R = (R I,R2).

is closed

is a consistent

|

sentences

sentences

relative

05

of

to

~

,

| is said to be normal

is the q u a s i - v a l u a t i o n if

if

closure

is an atomic

induced

closure

and total,

term relative

symbol

that when

there to

68 , and

n-ary

P E 8n

a bipartition

is exactly term

~ G

~

one possible is then over

@

= (e,T,F) value

closed there

if and

Gtlt2...t n E T ,

then

Ptlt2...t n E T ; and

(ii)

if

Gtlt2...t n E F ,

then

Ptlt2...t n E F .

is consistent

possible

the atomic

closure

but not

total,

values

of a second

of

may a c c o r d i n g l y

~

there may be n o n - d e n u m e r -

order

term relative

have

record

the immediate

results

to

~

and

to be non-denumerable.

Results I first

n-ary

such that

if

~

of a

is an

(i)

ably many

~0

(tl,t2,...,t n) E R 2 ; |

~

It is to be noted

When

3.

in

of

only if to each m o l e c u l a r predicate

in

.

is consistent second

all atomic

order b i p a r t i t i o n part

such that

sentences

PG,Rtlt2"''tn

and all

where

of which must

An atomic by

~

representation

the symbols

all atomic

is the set containing

are the symbols

2.4. already

of

E R I ; and

PG,Rtlt2...t n PG,R

closure

~3' = (~',~0',~')

0 O*n ;

of the form

n)

that are not

here

= |

base

is the set containing

all sentences

lii)

~.

be a second

mentioned

above.

D. Prawit z

3.1.a and 3.2. results order"

Inversion

principle

3.1.a and 3.2 in section before

"inductive

3.1'-3.3'.

for q u a s i - v a l u a t i o n s .

as in section

3.1.b. valuation

Embedding

logic,

a semi-valuation

the second

order

trivially

V

by the 3.2

valuation,

(a)

Every

consistent

semi-

I fails

for second

order

e

if

to an i n d u c t i v e

base

V

that

can of course

(@,T,F)

V'

valuation

valuation .

Since

is and by 3.1.a

V'

since

induced

furthermore

V'

is a v a l u a t i o n

also in

by

is always

if

a semiI is consistent.

V

Total v a l u a t i o n s . The q u a s i - v a l u a t i o n base

i n d u c e d by a c o n s i s t e n t

is a total v a l u a t i o n

the d e f i n i t i o n (b)

over

in the i n d u c t i v e

non-atomic)

it follows

3.4.

in section

V = (T,F)

case be e x t e n d e d

is c o n s i s t e n t

- 3.3

to a v a l u a t i o n . 3.1.b

is i n c l u d e d

(possibly

V'

3.1

" i n d u c t i v e valuation" by by " s e m i - q u a s i - v a l u a t i o n " .

of s e m i - v a l u a t i o n s .

the result

The results

I.

can be e x t e n d e d

Although

The "second

and " s e m i - v a l u a t l"o n " .

in section I hold also when we r e p l a c e " q u a s i - v a l u a tlon " and " s e m i - v a l u a t i o n " Proofs

and consistency.

I hold also w h e n we insert

valuation"

Results

313

A closed

1.1

of inductive

quasi-valuation

quasi-valuation

if it s a t i s f i e s

and a complete

clause

(5b)

in

valuation.

is a total valuation.

Hence

the

induced by a normal

atomic

base

is a total

be a total v a l u a t i o n

over

~ .

Then

valuation. (c)

Let

V = (T,F)

included closure Hence,

in the q u a s i - v a l u a t i o n of

if

(~,T,F) (@,T,F)

quasi-valuation Proof. a consistent complete, tion

1.2 if it s a t i s f i e s

I. there

satisfy

3.1.b

complete

is also a s s e r t e d

is i n s u f f i c i e n t ;

i n d u c e d by an atomic

then

V

is i d e n t i c a l

(5b)

base

is itself

of clause

1960 but

to sect.

1.2.3.

of

to the (~,T,F).

i n d u c e d by

consistent

(5a)

and

in the d e f i n i -

in that definition.

by SchGtte

cf. f o o t n o t e

part

the q u a s i - v a l u a t i o n

the converse clause

V'

is

is also a total valuation.

i n d u c e d by the atomic

and 3.3'

and a t o m i c a l l y

V'

is closed,

V'

Since by 3.2'

it must

and

V

Hence, by

the p r o o f given

314

D. P r a w i t z

the d e f i n i t i o n

of total valuation,

To prove easily

Let

atomic base

Then

over

V = (T,F)

(@,@,$),

term over

~

assertion

(b), we note

proved by i n d u c t i o n

Lemma.

n-ary

the part

(a) follows.

the f o l l o w i n g

lemma,

P

be a p r e d i c a t e

such that for all

tl, t2,

induced

in

|

in

Gtlt2.~

n E T , then

P t l t 2 . . . t n E T , and

(ii)

if

Gtlt2.~

n E F , then

Ptlt2...t n E F .

for every

(ill)

if

A(G)

E T, then

A(P)

E T , and

(iv)

if

A(G)

E F, then

A(P)

E F .

To use

the lemma

quasi-valuation

there

P E |

lemma hold

(see s e c t i o n

inductive

because closed

The second

trivially

is a total v a l u a t i o n

V

Embedding

it follows

over

semi-valuation let

I.

3~

V = (T,F)

V'

3.5, we make use

second

order

logic

1960.

of

an

n-ary

in the c o n c l u s i o n 1.2 of

by part

case

(a).

of the next

from the first

one

that

is

(|

(|

and hence

V ~ V'

and that

V'

in total v a l u a t i o n s I.

can be extended

to a total valuation.

be a c o n s i s t e n t induced

1968.

by an atomic

(see V.2) with respect

lemma,

1967.

calculus

That

of

.

which

is proved

technique

used

the result

of sequents

to l o g i c a l

over

closure

V ~ V'

The e s s e n t i a l

in T a k a h a s h i

of the c u t - f r e e

semi-valuation

such that

of the f o l l o w i n g

is proved by P r a w i t z

the c o m p l e t e n e s s

by SchGtte

closure

is a total v a l u a t i o n

in the proof was also present implies

(iv)

~

(ii) of the

in the d e f i n i t i o n

that also

be the q u a s i - v a l u a t i o n

To prove

over

in (c) follows

of s e m i - v a l u a t i o n s

consistent

Then

is a closed

~ .

precisely, V'

G

V

(i) and

in (c) is only a special

Each

.

if

is a total v a l u a t i o n

assertion

More

(e,T,F)

term

Hence by (5a)

part is an atomic

by the first a s s e r t i o n

and let

that

V' ~ V; and by the a s s u m p t i o n

its atomic

3.5.

clause

Thus,

first a s s e r t i o n

3.5.

n-ary

2.5 above).

satisfies

valuation.

The result

(b) we note

such that the c o n d i t i o n s

V

an :

that

is to each

predicate

of the lemma,

A(X n)

G @O

if

it follows

by an

and

..., t n

(i)

to prove

is

A:

be the q u a s i - v a l u a t i o n

and let

which

for

truth was proved

D. Prawitz by induction over the length of Lemmao tion over

As in 3.5, let |

and let

by the atomic closure tion

8"

A :

V = (T,F)

be a consistent semi-valua-

V' = (T',F')

be the quasi-valuation induced

(~',~,$)

(8,T,F)

of

of possible values relative to

formula

A(XI,X2,...,Xn)

G I, G 2, ~

Gn

respectively, GI, G2,

315

,

n ~ 0 ,

using some representa-

(8,T,F)

~

of the same number of arguments as

and for all possible values

.~., G n

relative to

Then, for each

for all second order terms

(8,T,F)

XI, X2,

RI, R2,.o. , R n

..., X n,

of

it holds:

(i)

If

A ( G 1 , G 2 ~ . . , G n) E T, then

A(PGI,RI,PG2,R2,...,PGn,R n) E T';

(ii)

If

A(GI,G2,o..,G n) E F, then

A(PGI,R1,PG2,R2,...,PGn,Rn)E

When

n = 0, the lemma asserts that

To see that

V'

V ~ V' ~

is also a total valuation,

using the notation

of the lemma it suffices by (b) of 3.4 to show that closed, over

i.e. by the observation 2.5, that to each

8'

fled.

there is a

Let

R

occur in

m

(|

is

m-ary term

G

such that (i) and (li) of 2.5 is satis-

be the partition

([(t 1,t2,.,~ and let

P E |

F'.

Gtlt2...tm~ T'}, ((t I,t2, .... tm): G t l t 2 . . . t m E P'])

PGI,RI, PG2,R2,~

be the predicates from

8*

that

G , which therefore may be written G(PGI,RI'PG2,R2'~

Applying the lemma to

n)

G(Xi,X2,...,Xn)tlt2..~

A(XI,X2,~

n) , it follows that

G(GI,G2,~

n)

relative to

R

(8,T,F)

m

in place of

is a possible value to Hence, we can take the

predicate PG(G1,G2,...,Gn),R which belongs to 3~

8*m

and thus to

Logical truth.

~m

as the

P

required in 2.5~

By the results above, the following three

conditions are equivalent to logical truth in the sense of truth in all second order models:

316

D. P r a w i t z

(i)

A

is true

atomic

order base

(ii)

A

is true in all total

(iii)

A

is false

over

|

The e q u i v a l e n c e tional

fact

order

valuation second

struction

logic

(iii)

second

follows

follows

valuation

order

over

@A"

semi-valuation

from 3.4.c and 3.4.b. 3.5

(and the defini-

is a c o n s i s t e n t

semi-valuation).

of s e m i - v a l u a t i o n s (8,~,~)

@ U ~

,

from

sentences,

~

in s e c t i o n

1.4 is e x t e n d e d

In the d e f i n i t i o n

where

and where

@ ~

are now finite two rules

to

of a semi-

is now the sequence

and

we add the f o l l o w i n g

of ranges

sets of

for the con-

of the tree:

(a5)

E,

(ii)

straightforwardly.

over

by

order

order v a l u a t i o n s

of s e m i - v a l u a t i o n s

tree

determined

of (i) and

of (ii) and

The g e n e r a t i o n second

second

in no c o n s i s t e n t

that a total

Generation

i n d u c e d by any normal

(@A,@,~)

"

The e q u i v a l e n c e

4.

in the q u a s i - v a l u a t i o n

second

(b5)

VXnA(X n)

A(G), r,

VXnA(xn), r

~,

In a p p l i c a t i o n term over occur

~

=>

=> A

of (a5),

G

fixed

in some

of (b5),

order)

sequent

GI,G2,...

r

A

=>

~, A(G2),

E, y X n A ( x n ) ,

=>

is to be the first

(taken in some

In a p p l i c a t i o n over

r

a

in the a n t e c e d e n t

terms

r => ~ , A(GI),

n-ary

such that

below

a

A

second A(G)

order

does not

the one to be constructed.

are to be all

n-ary

second

order

~ .

As before,

all m i n i m a l

semi-valuations

are g e n e r a t e d

by such

trees. The s i m p l i f i e d 1.4.3,

semi-valuation

now also r e p l a c i n g

trees are defined

analogously

to

(b5) by

(b5 ' ) r => ~, A(P),

A

r => z, ~ X n A ( x n ) , in the a p p l i c a t i o n parameter

among

of w h i c h

PI' n p~,

one to be constructed.

..

P

a

is to be the first not

As before,

occurring

n-ary

predicate

in the sequents

the m i n i m a l

below

semi-valuations

can

the

D. Prawitz be obtained

from the ones generated

trees by substitutions, second

by simplified

now also substituting

the result

definition

second

Calculi

Infinite

and closed

of sequents

order case and the

trees need no change.

for second

order log!~

of the infinite

order logic

calculi

of sequents

is obtained by adding

(R5b)

F => A, A(GI)

F => A, A(G 2)

A(G),

r => A, ~XnA(X n) in (RSb)

G

are to be all the However,

is to be any n-ary

n-ary

term and in (R5a)

terms over the sequence

not only from a classical

II.3 still holds,

or false

~

=> A

or

in the second

upon classically). are never complete. (also when

~

A =>

Furthermore,

true sentences

point

valuation

also

~

complete

A => A

The finite

(looked

fails

in general

base). A, F = ,

would be provable

A,A

as axioms,

then

(as will be seen in

that are not logically

The finite

A

these valuations

in such a calculus

2.

of view,

induced by ~

IV.2,

but it seems that the set of provable

to warrant

...

in question.

The only sentences

the section below) interesting

GI~ G2,

point of view for which the

And as we saw in section

is an atomically

r => A

can hold are the ones that are true

order inductive

If we should add all the sequents all logically

of terms

but also from a constructive ~

F => A

yXnA(xn),

there seems to be no point in such calculi. for which

in section II

the two rules

(R5a)

result

order terms for

calculi

An extension to second

1.4.4 hold in the second

of truncated

V.

where

semi-valuation

order parameters.

Also

I.

317

sentences

true is not sufficiently

this kind of calculi.

calculus calculus

of sequents

for second

order logic is ob-

tained from the one for first order logic by adding rule the preceding section and the rule

(R5a') r => A, A(P) F => A, VXnA(X n)

(R5b)

of

318

D. P r a w i t z

where or

P

A .

is to be an The rules

truncated

the calculus. in the first is p r o v a b l e

predicate

of the c a l c u l u s

ones for c o n s t r u c t i n g a closed

n-ary

simplified

simplified

Hence

p a r a m e t e r not

are thus a g a i n

semi-valuation

semi-valuation

o c c u r r i n g in

identical

trees.

A

in the calculus.

is l o g i c a l l y

to the

In p a r t i c u l a r ,

tree is a d e r i v a t i o n

from IV.3.6 and IV.4, we i m m e d i a t e l y

order case:

r

in

o b t a i n as

true if and only if

=~ A

D. Prawitz Bibliographical

319

references

Beth 1955, E.W., Semantic entailment and formal derivability, Mededelingen der Kon. Nederlandes Akademie van Wetenschappen, Afd. letterkunde, n.s., 18, 309-542, Amsterdam. Gentzen 1934, Gerhard, Untersuchungen ~ber das logische Schliessen, Mathematische Zeitschrift, 39, 176-210. Hintlkka 1955, Jaakko, Form and content in quantification theory, Two papers in symbolic logic, Acta Philosophica ?ennica, no. 8, 7-55, Helsinki. Kanger,

1957, Stig,

Provability in logic, Stockholm.

Kreisel 1958, Georg, Review of Beth, La crise de la raison et la logique, J. Symbolic Logic, 23, 35-37. Lorenzen 1951, Paul, Algebraische und logische Untersuchungen Gber frei Verb~nde, J. Symbolic Logic, 16, 81-106. Prawitz 1965, Dag, Natural deduction, A proof-theoretical Stockholm.

study,

1968,

-

Hauptsatz for higher order logic, J. Symbolic Logic, 33, 452-457.

1971,

- Ideas and results in proof theory, in: Proceedings of the Second Scandinavian Logic Symposium (ed. J.E. Fenstad), 235-307, Amsterdam.

Sch~tte 1951, Kurt, Beweistheoretische Erfassung der unendlichen Indukticn in der Zahlentheorie, Mathematische Annalen 122, 369-389. 1956,

1960,

Shoenfield

-

-

Ein System des verknGpfenden Schliessens, Archiv fGr mathematische Logik und Grundlagenforschung, 2, 55-67. Syntactical and semantical properties of simple type theory, the Journal of Symbolic Logic, 25, 305-326.

1959, Joseph, On a restricted ~-rule, Bulletin de l~cademie Polonaise des Sciences, 7, 405-407.

Tait 1968, William, Normal derivability in classical logic, in: The syntax and semantics of Infinitary languages, Lecture notes in mathematics (ed. J. Barwise), 72, 204-236. Takahashi 1967, Moto-o, A proof of cut-elimlnation in simple type theory, Journal of the Mathematical Society of Japan, 19, 399-410.

BEMERIiUNGEN

ZU

B.

REGEL

UND

SCHEMA

Scarpellini

Einleitung In

dieser

f~lle

Arbeit

yon

(konsistente) zwei

mit

schaft

:

E)

t ein

Die

Frage

Ist

T zudem

T

(VX)

+

Wir und

setzen

durch

~ uns

der

An

: ~)

Fragen

haben

es

i)

allgemeine

als

deren Die

kann

2

wird +

S[tze

es

konsistent : 2)

~ B(x)

beantworten

durch

B(t)

.

?

Besitzt

?

wo

A(x)

sein, ) eine

befasse]

eine

positive

Frage

2)

zu

er-

konsistente,(eVtlo

?

nat~rlich

Teil

T ~

Theorien

Fall,

Primerweiterung

zum

)

fragen

zweckm~ssig

(yx)(A(x)

sind

gilt

intuitionistischen dem

B(x) Eigen-

so

Eige.schaften

) mit

A(x),

nur

die

sehr

zur

spezielle

Verf~gung

For-

stehende

wordeno Beispiele

nichttriviale Der Form

Grund

der

hat

gibt ist

S~tze

auch

Grundstock

igkeit

&

noch

B(x)

eine

zu (u.a.

ausser

ist,

die

den

S~tzen

2,

3 weggelassen,

versehiedene

Platzmangel yon

einem

zeigen,

dass

Formen

der,

dass

gewissen

yon

ob-

Refle-

es d o c h

Interesse

eher ist~

Anwendung.

Arbeit

gewissen

T

illustrierende

durehaus

man

T

folgender

gilt, ~

Sei und

mit

A(t)

Spezial-

lautet.

verf~gt,

(Vx)(A(x)

mit 1

und

T ~

so w i r d

disjunktive

- 3)

xionsprinzipien)o die

T +

Stellen

Besitzt

Terme

disjunktiven

Anhang

einigen

suggeriert

Wir

Ist

Methoden

folgt

Variablen,

den

ausschliesslich yon

zu b e w e i s e n d e n

Technik

f~r

) die

wie

konstante freien

Term,

B(x)

aufz~Lhlbare)

Die

~ber

: i)

beweistheoretischen typische

einziger

damn

Ausnahme

rekursiv

die als

mit

eine

intuitionistiseh,

werden

ist.

wir

denen

konstanter

lautet

Formel

wohl

x

(A(x)

(mit

men

yon

Theorie,

Formeln

Ist

behandeln

Fragen,

Resultate

an Hber

den

Zweck

zu

beweistheoretiseher die

Struktur

man,

Technik,

ausgehend mit

intuitionistischer

yon

relativer Theorien

einem Leicht-

erzielen

o 0hne

nern Formeln, dutch

Zweifel

. Hingegen n~mlich

lassen

sich

beschreiben die

Primer~veiterun~en

, dass

die

bier

sie

vermutlich

sich

erzwingen

die

bewiesenen eine

S~tze

disjunktiven

lassen

(Frage

stark

Eigent[imlichkeit Eigenschaften 3).

verallgemeipositivel schon

B.

I. S p e z i a l f ~ l l e , (a) Sei

Regel

und

in welchen

S ein k o n s i s t e n t e s , A, B zwei

dass

folgende

man

Sch~tte,[ o)

Dann

S

~

trivialerweise

S , A~B

Uber

Ferner

Funktionen

Funktionen. primitiv

Fall

S ~

seien

gibt

Sei n u n G d i e

lelchte

sich

Gehen wir

der Begriffe

aus

o) g e s n h l o s s e n

S klasslsch,

S ~A

Situation

und

so ist d i e

S ~B

weniger

(, d i e m i t

viele

ebenso

~i

folgen,

einfaoh,

wie

fol--

Zahlvariablen

' ~i

Konstanten

die

Formel

Insbesondere

zugehSri~en

' ... bezeich--

fLtr p r i m i t i v

re-

definierenden

fur n i c h t - p r i m l t i v - r e k u r s i v e Prim(T ) , d i e

ausdr~ckt:

ist darn% fttr j e d e n k o n s t a n t e n

~

ist

~h/nktor

~

Prim(F)

.

Formel

Rechnung

~

flndet

G ~ ~(~ ~ ) q P r l m ( ( )

ist also

ist,

die ~blichen

wen~

dass

(S~)(Prim(~)A-Prim(~))

offenbar

.

konsistent

in Z i + ~G

die

Rege~

gilt

~ (~)~Prlm(~)

,

z i + ~G

~

^P=i=(~)

aber yon der Theorie

Im Verlaufe

dieser

(3~)(~Prim(~)

Regel

zum

Schema

Uber,

Z.x + w G + G . Arbeit werden wir uns mit

Problem

besch~ftigen.

Gegeben

welches

d i e disju/iktive

ist

Eigenschaft

u n d hat t d a

Eigenschaften.

dlsjunktiven

Zi + nG

konsistente

.

man

Z i + IG

sofort~

2)

Ist

wUrden

Konstanten

Zi ~

~ibt

: Kann

?

verfUgt

(~$)qPrim(~)

Die Theorie

Fallp

B

ist

gen~gend

es eine

Zi

Harrop-Formel

(wegen

der

:

l)

Durch

oder klassisches es h H u f i g

Zahlentheorie t die nebst

vorhanden~

rekursiv.

F beweisbar

hat

Frage

ist d i e

aber keinerlei Dann

ist

zeigt.

Funktionsvariabeln

Glelchungen~

impliziert.

).

intuitionistische

net w e r d e n ) . kursive

, so

ja (, sonst

zu o)

Belspiel

auch

Schlussregel

konsistent

Im intuitionistischen

Zi die

Formeln.Dann

sich die naheliegende

Antwort

Sei

Schema

40 ) :

A

dass

im W i d e r s p r u c h

das

formales,intuitionistisches

zul~ssige

werden,

gendes

Regel

geschlossene

S S , pg.

wenn

stellt

die

Schema

System t

siehe

321

Scarpellini

er-

:

so ) .

so e r h a l t e n

Varianten

und

wit die

yon

ein intuitionistisches besitzt,

~ G eine Daraus

Formeln

in-

s

System A

~ B

S~ , fur

322

B.

welche

die Regel

dingungen

gilt

gefunden

: " wenn

werden,

ist u n d

die disjunktive

gewisse

einfache,

oretischen heit

mlt

] , ~S

3 und

Beispiel

IndlviduenDefinition

ist

Formel

2

Satz

1

: Seien ~(x

mit keinen

andern

schwach

positiv.

liebige

Terme

El)

eine

werden,dass

der beweisthe-

gewisse

Vertraut-

yon

positiv,

wenn

positiv,

A ~ V

, ~

M heisst

solchen

S kan~

enthalten.

schwach

vem~ge

Pr~dikatenkalkGl;

sie w e d e r

wenn

, V

7

sie aus

alleine

noch

aufgebaut

P r i m b a s i s t werLn sie n u r

enth~it.

Wir nennen

D

Primformeln ist. Prim-

M konslstent,

I ... x n k ) , B k ( X 1 ... X n k ) , k = 1 , 2 , . . . freien

Variabeln

als d e n

Sei M e i n e k o n s i s t e n t e

t I , ...

S~MvE

, tnk

gelte

endliche

~ ( t

S ~ ~ B k ( t

die

Es w i r d

F~lle

konslstent

ist.

ist E e i n e

I s t d a n n M'

soll gezeigt

triviale

intuitionistische

Formelmenge

konsistent

Es

es s o l l e n B e -

S + A mB

[7 ~vorausgesetzt.

heisst

heisst

Primformeln : Eine

;

dass

Pr~dikatenkalk~l

S der

t[nd N e g a t i o n e n

SuM

hat.

v~llig

L 6 ~oder

1 : Eine Formel

Eine

Definition formeln

nicht

und FunktionskonstaLnten

und negierten

wenn

garantieren,

Eigenschaft

aber doch

(b) D e r i n t u i t i o n i s t l s c h e

enth~it.

S ~ A ~ so S ~ B "

welche

Behandltuag zug~Lnglich sind.

~4

Im ersten

Scarpellini

angegebenen

Primbasis.

FUr

Formeln

| die ~'s

seien

Jedes k und be-

:

Primbasis,

I ...

t

) gilt

1 ...

t

) .

eine maximalkonsistente

sodass , d~n~

Primbasis,

gilt

auch

die M umfasst,

so ist

Theorie T = SuM,

u U (V ~)(Ak(~) ~Bk(~) ) k und hat die Ubllchen disjunktiven Eigenschaften.

konslstent Beweis:

Der Einfachheit

ger nk'S behandelt

man

halber analog.

theorie

intuitionlstischer

wlesen.

Wit

I ) S e i M'

nehmen

Primformel,

Sequenzenkalk~l

wie

SequenzenkalkUls

----~ p

~ falls

kUl nennen

wlr

; den Fall beliebi-

die Terminologie

fGr Details

Primbasis,

sel z.B.

auf

die M umfasst.

so p 9 M t o d e r ~ p e M ' . W i r

folgt:

schen

= 1 an

der Beweis[5]

ver-

in Schritten.

elne maximalkonsistente

Ist p e l n e

nk

Wir verwenden

Systeme;

fiihren d e n B e w e i s

wit

zu den Regeln

(GS) a d d i e r e n

wit

Gbertragen

und Axiomen fur jede

des

Wit bemerken: S~M

t in den

intuitlonisti-

Primformel

p das Axiom

p & Mttttnd ~p-----@ t f a l l s n p ~ M t . D e n r e s u l t l e r e n d e n G S M I . Zt~m K a l k ~ l

GSM'

addleren

wlr nun

alle Regeln

Kal-

B.

der

Scarpellini

Form

,

~(t),V WO

t eln b e l l e b i g e r

Formelfolge,

zu zelgen,

dass

den Theorien

Term

GST ~ Der

A

ist

Formel

bezelchnen

bzw. ---~A

und

also bewiesen,

Wit

GST ~

----* A

wit

disJunktiven

Wenn

GST~A

II)

AvB

9 so

Im f o l ~ e n d e n

diese

die

Ublichen

Konsistenzbeweis Begriffe:

usw~

Da nun ist,

noch,

f~r alle

k~nnen

eine H~he

wit

h(s)

H~hensprungs III)

eine

schlUsse

die

jeder

Sequenz

im E n d s t U c k

ordnen

wir

zu.

Schnitte) Ausnahmeo

~

Es b l e i b t

~ -Schlusses

~(S)

= ~(Sl)

ncch der Fall

setzen

: 0(S)

~d(~)

haben

=~d

(s

wir

die

Ordinalzahl

IV)

Fttr B e w e i s e

duktionsschritte

P aus

st~ck,

3)Elimination

SI,S 2 / S gleich

eines

Schlusses

B k ( t ) in d e r

Pr~mlsse~

gleich

0 ist~

eingeftihrt

worden

P aus

GST in bekannter

Weise

dann

auch der Begriff

des

werden.

S in P induktiv SchlUsse

sich alles

glelch

SI/S

' wo d = h ( S l ) Als

yon

und

oben nach

Struktur-

wie bei

Gentzen~

zu d i s k u % i e r e n .

- h(S)

0rdinalzahl

ist; ~ o ~(P)

yon

Wir

' @ ' P nehmen

SE yon P .

wir nun die bekannten

Gentzenschen

Re-

als d a sind

, 2) E l i m i n a t i o m

eines

in d e r E n d s e q u e n z ,

Primformel

eingeftthrt

der Endsequenz

GST kSnnen

i) V e r d ~ h l n u n ~ s r e d u k t i o n

die

eines

SI/S 2 ordnen wit jeder Konklusion S -I (statt 0(SI) ~ 1 w i e b e i G e n t z e n ) .

Bedeutung~

definieren

kann

zwei~en

also

Hauptformel

"Komplexit~t"

~-Schlusses

@ O(SI))

O(SE)

einer

fiir d e n

~ 2 zu

eines

die ~bllche

Formel

Sequenz

blelbt

:

ftihren 2qlr

Wir haben

Schnittes

F~r Axiome,logische

(inklusive

eines

jeder

GST konsistent

ein 9 d i e

slnd~

Hauptformel

S im B e w e i s

Insbesondere

f~Ir S c h n l t t e

0rdinalzahl

der

bzw.

als d a s i n d

GST und

Komplexit~t

Komplexit~t

A

usw.

c sines

der Begriff

hat,

im E n d s t ~ r

~

sind mit

GSM v ~

k~nnen t dass

Begriffe

solchen

die Komplexit~t dass

eine

schwierig

~quivalen% kurz

P im S y s t e m

Schluss

~ch~ittformel

zuordnen.

0rdinalzahl

Beweise

einer

auf eine kleine

Im Falle

GST~B

Kcmplexit~t

Sohl~sse

Als n ~ c h s t e s

unten

die

der

per Definition

Man beachte

Bild

ist d i e

~

.

eharakteristisch

kritiseher

Schlusses,

Wie ~blich,

der Komplexit~t ist

Gentzen

, und

Es ist n i c h t

schreiben

zeigen

beweistheoretischen

yon

EndstUek,

kritischen

wir

GST.

Eigenschaften

oder

betrachten

ist

GST vollkommen

Satz.

wenn

, Bk(X)

enth~It.

w l r mit

GSM Iund

Tim

ist u n d d i e ~ b l i c h e n GST ~

.o.

fiir x in ~ ( x )

eine

die Kalk~le

f~tr G S M t ~

Satz

frwi

Kalk~l

SuM'

k = 1,2,

~

die h~ohstens

Den resultlerenden

bls

323

logischen

logischer

Zeichens

Axiome

aus d e m E n d -

aus d e m E n d s t ~ c k .

324

B.

Reduktiensschritte

y o n d e r Art

Reduktionsschritte

bezeichnen.

Wir mUssen

nun noch

~-SchlGsse

aus d e m

Sei a l s o

ein kritischer

A)

EndstGck

P wie

wir kurz

einfUhren,

die

als v o r b e r e i t e n d e

es

uns g e s t a t t e n ~

zu e l i m i n i e r e n .

,~ in P ; s e i n e

sei

ein B e w e i s

~hldern w i r

2) w o l l e n

"~

~-Schluss

Voraussetzung

Es l i e g e

Dann

i),

Reduktionsschri~e

Bk(t),U Ak(t),r

Folgende

Scarpellini

Konklusion

liegt

also

im E n d s t U c k o

erf~llt:

Po y o n folgt

~ B k ( t ) im S y s t e m

ab

G S M ! vet.

:

p

9 o

__---~Bk(t)

,

Bk(t),

r

9

, Schnitt,

Verdiinnung

e

Wir

sagen,der

resultierende

Beweis

P'

folge

aus

P durch

einen

~-Reduk--

iogischer

Zeichen

wollen

tionsschritt. ~-Reduktionsschritte s~mmenfassend

als

haben

einen Beweis

schritte

Reduktionsschritte

folgende

P k~nnen

angewandt

2) v o r b e r e i t e n d e

Elimination

eigentliche

Reduktionsschritte i) A u f

und

nur

werden

fundamentale endlich

viele

wir

zu-

bezeichnen. Eigenschaften

:

vorbereitende

Reduktions-

,

Reduktionsschritte

vergr~ssern

die

Ordinalzahl

yon

P

nicht, 3)

eigentliche

Im F a l l e in

eines

~d(~O

eines

~-Schlusses

dies

Form

ersetzt

Po aus GSM'

einen ~

wird

; die

Beweis

oder

a.wendbar ( [ 6 ]

~7])

BL)

eine

Ist

folgt

die

Ordinalzahl

aus d e r T a t s a c h e , dureh

Ubrigen

die

F~lle

dass

y o n P. das

~o

Ordinalzahl

~ < ~o aus d e n U e b e r -

folgen

yon Gentzen.

y) W i r n e n n e n die

verkleinern

@ O(SI) ) jetzt

Beweises

legungen

Reduktionsschritte

A,r---*~

ein Beweis

P aus G S T N o r m a l b e w e i s ,

~ F

hat.

oberste

Po aus G S T y o n

---~A

wenn

Auf Normalbeweise

Sequenz

des

Endst~ckes

, fur w e l c h e n

seine

ist das

~(Po ) ~

yon

P,

~(p)

Endsequenz

Basislemma

so e x i s t i e r t gilt.

B.

VI)

Wir

beweisen

nun

durch

Scarpellini

transfinite

325

Induktion

Gber

O(P)

folgende

Be-

hauptung: B)

Sei

P ein

wo

F schwach

ein

Beweis

Ist

~(P) in

Beweis

aus

positivist P

<

aus

o ~o

' so

P vorkommt

(und

wir

Einschr~nkung

bereitenden

dann

ein

kommt

GSM'

der

existiert

~(P1)

< O(P). M'

Da

~ndererseits

eine

yon

: Die

ist

anwendbar.

9 yon

und

ein

J

M'

Axiom

yon

Also

yon

Dann

P ist

~

transformiert gleicher

Dann

P vorhanden

Satzes GSM'.

halten wieder mMss

muss

und

F.

der

kein

Schluss

O(P)

~ GO

Ferner P keine

sich

folgende

Form

aus

GST

muss

yon

~ und

Gem~ss

, fQr

darns a b e t

k~nnen vor-

p

F~lle

~ ,

yon

das

~ p

: GSM' ~

impliziert ~p

~ M'

Axiom

mit

p ,

dies

sein

p

p ~

, was

M~

der

~ , p prim,

somit

ein

Beweis

Sequenzens Wir

nun k~nnen Beweis

im

der )

Existenz einen

P' y o n

9

F 3 notwendigerweise

kleinerer einen

ist

~-Schluss

existiert

dann

ein

Beweis

Induktionsvoraussetzung

und

eines

aus

und

GSM'.

wegen

Beweises

P

o

a(P') Die

die

< 0(P)

Endsequenz Form

>

Wegen

der

der

Hypothese

yon

~

~-Reduktionsschritt mit

aber

Reduktionssehritt

kritischer

~(t)

Pr~dikatenkalkUl

die

P1 yon

ergibt

P

o

gilt.

und

somit

. Das

logischer

(y)

~(P)

Pl y o n

Reduktionsschrit~

Beweis

9

Basislemma

Induktionsvoraussetzung. bis

kein ein

Hypothesen

loglscher einen

~

, ~

den O(Po)<

alle

ein

P in

notwendigerweise

sein.

~(t)

einen

F I

existiert

dass

der

ist,

kein

Endsequenz

Bk(t)

aus

P1

GST,

sich

Beweis

des

also

bedeutet

kann

ein

yon

es

zusammen.

Axiom

Primbasis

ist

widerspricht.

aber

: P ist

existiert

dann

lassen

Bowels

F

valenz

Nun

ein

Induktionsvoraussetzung.

erf~llt

~ F,

Pvor.

zur

Po

Sei

annehmen,

zul~sst.

Widerspruch

-

ist).

da

EndstGck

Allgemeinheit

mazimalkonsistente

Endsequenz

Ordnungszahl,

in

odor

P vorkommen.

F 2

yon

9

9 % Fund

C bewiesen,

seinem

Baslslemma

p

yon

anwendbar.

Form

Nach Induktionsvoraussetzung

aber

~

Form

.

konsistent

mit

yon

Endst~ck

gem~ss

Da

Konsistenz

Beweis

im

Dann

EndstQck

da

S die

F

Behauptun

P nicht

S der

.

: P ist

p prim,

die

Sequenz

hat

9~

Reduktionsschrltte

unterscheiden F i

yon

ist

f~llt

einer

; dann

GSM'

speziell ohne

GST

ausf~hren . Das yon

B

Aequi-

P muss mit

E i)

(t) k und er-

widerspricht

F haben,

es

also

aber ge-

positivem

326

B.

F ~

: Die

Endsequenz

wendigerweise

und

P in

ein

einen

Dann

gilt

F ~

: Die

aber

Beweis

Pl y o n

GSM'

p

der

~

Endsequenz

nicht

P hat

die

eigentlicher

Reduktionsschritt Wenn

yon

Scarpellini

, so m u s s

9

nach

yon

auf

Form

w

p

, p prim 9

Reduktionsschritt

p mit

auf

~(Pl)

<

Dann

muss

P anwendbar

O(P)

not-

sein,

transformieren.

Induktionsvoraussetzung.

P hat

die

Form

P anwendbar,

so

~

~p

. Ist

schliessen

notwendigerweise

ein

wir

ein

wie

eigentlicher

unter

kritischer

)

F 4

.

~-Schluss

Form

p,P C--*~p in

P vorkommen p

Wir

ein

(,den

Bild

behaupten

W~re

dem

~

folgt

ab~Ludern

kurz

mit

Sl/S

in der

Endsequenz

hat.

: p

~

Axiom

nicht

weise

wir

so,

p ein

ist

ein

so m G s s t e Axiom

wegen

yon

GSM'

bezeichnen),

yon

der

sein.

GSM'

dessen

, und

somlt

Maximalit~t

yon

Wlr

dann

kSnnten

Hauptformel

GSM'

~

mp

.

M I notwendigerden

Beweis

P wie

:

5

p

, p

, ~

P

Das

Resultat

hat

S die

w~re

die

Komplexit~t

ein

Beweis

Ordinalzahl

~(S1)

0 hat

benutzt

raten

somit

Also F

GSM'

6

: Die

nicht

yon

der

Form

F

(3~)

=

wird

~

in

der

Form

A v

B

L~sst

P einen

unter

F

geben

:

in

P1

~(P1)

hingegen

<

O(P)

0(S1)

nur

: denn

in

@ 1

Da p

9

P

h(S)

= 0

Axiome

einen

die

, also

O~d(O(S1)

Ordinalzahl

Widerspruch

mit

~

1)

= O(S1)

1 erhalten

der

.)

~ Wir

1

,

ge-

Induktionsvoraussetzung 9

.

Endsequenz

A(~)

4

~p

-

, dass

wieder

,

, ist

d = h(S1) wobei

,it

P1 y o n

@ 2

yon p oder

, A A B an

P hat

Form

, p prim 9

, (~x)

A(x)

~

F

Dann

, (Vx)

, F

muss

A(x)

schwach

positiv

und

F notwendigerweise sein

; wir

nehmen

yon z.B.

9

eigentlichen

. Wenn

-~p

die

nicht,

Reduktionsschritt

so m u s s

es

einen

zu,

so

kritischen

schliesst ~

~

man wie - Schluss

B.

>

P

dessen

nauptformel

(~x)A(x)

Schluss

so

diesen 0(PI

)<

Analog

Sx)A(x)

verf~hrt

man

Behauptung Liegt

Form

)

A v B

Basislemma,

dass

kommt,

nun

einen

k~nnen.

und

- (~x) Indem von hat

der

Satz Es

M" Ak

Diese

wir

B

P aus

von

Lassen

)

also

, A V

GST

vor,

A(t)

: GSM'

~

B

wit mit

A(t)

, ~x)A(x)

mit

so

~

,

.

Endsequenz

aus

- Schluss

h~chstens

Reduktionsschritten

B)

bei

finden t)

einem mehr

wir

einen

. Analoges

gilt

Eigenschaften

und

aus

in

P vor-

Beweis

eigentP1 y o n

anwendbar

im

de.

anwenden

vielen

~eweis

ist

der

und

, der

endlich

Reduktionsschritt

gewisses

elner

folgt

Rk-Reduktionsschritt yon

aber

, die = 1

, 2

wird

naheliegende ersetzt

in

beantwortet

: FGr

seien

Voraussetzungen

die

ist.

P

Fall

aus A v

GST

B

konsistent,

.

womit

Satz

~ ~)

konsistent

.

die

Primbasis

l)

p ~ M1

genau

2)

-~ p e M I g e n a u

die

wenn

damn

vielleicht

ist,

Formeln

1 erfGllt.

M l wie

dann

1

eine

die

maximalkon-

vorausgestzt

aufz~thlbarer

(V~)(Ak(~)~

k

Satz

Weise

kleinere dass

gegeben

M

,

sind.

die

M und

yon

in

dutch

aufz~hlbar

durch

Primbasis

ob

kann

rekursiv

To = Su~

: Definiere

Frage,

werden

rekursiv

, ...

FolKerung

Bewels

hat.

ist.

M")

und

P1

gilt

= A A

A(x)

AusUben

kein

sin

M'

disjunktiv

Endsequenz

Beweis

kritischen

disjunktiven

die

die

(~x)

F 6 schllessen,

sich

Frage

F

Beweis

jeden

den

(fUr die

der

einen

F~llen

nach

Primbasis

, k

in

bewiesen.

~

auf

, auf

bewiesen

(M ~ , Bk

wir

A(t)

stellt

wir

beliebiger

unter

somit

sistente

den

somit

vorbereitenden

wie

)

GST

in

odor

enden

A(x)

wit

erhalten

Bild

Ordinalzahl-verkleinernden

Also

lichen

ein

ein

t

(~x)A(x)

9

.

B) i s t

VII)

A(t)

Induktionsvoraussetzung

. Nach

GSM, ~

also

Die

weg,

0(P)

327

Scarpellini

folgt

T

~

, Bk ist

, k die

= i

, 2

, ...

Theorie

Bk(~ ) )

:

T O ~- p

wenn

~ Dann

,

Ip

.

O

Die

Theorie

gleichen

T1 = S u

The0reme

Wir

k~nnen

den

entsprechenden

dlsjunktlven

nun

wie

dutch

MI v

k.J ( ~ x ) ( ~ ( ~ ) ~ Bk(~ ) ) hat dann dis k T o und ist semit konsistent nach Satz i .

Repetition

des

SequenzenkalkUl

Eigenschaften

hat

Beweises

yon

GST 1 beweisen,

; dabel

vereinfacht

Satz

i

dass slch

, angewandt schon der

auf

T 1 die Beweis

etwas,

328

B.

well

die

wsrden Satz bei

Konsistenz

muss

1 und

nur

Formeln

die

noch

verlangt

mit

die

Platzgr~nden Belspiel

konstanten sistente die

man

wird~

Primbasis,

die

und

nicht

mehr

bewiesen

so g i l t

~ts

hat

man

positiv wie

der

der

nur

endlich

viele

Funktionskonstanten

jeder

endlichen

aus

Harropim Zu -

erweiterten

implikatlven

Theorie

Axiome

erh~it,

nicht

eingehen.

Individuen-

, Ist

Erweiterung

zu t

werden~und

meistens

Verallgemeinerungen

S enthalte

in

zugelassen

sich dann,

Konsistenz

HinzufUgen

keine

Verallgemeinerungen

Axiome

die

auf diese

yon

~ cn und

M

eine

S v M

kon-

u E

:

so folgt

aus

der

S V M u E ~ (Jr)

konsistent freien

und

Folgerung

disjunktiv

Variabeln

A(x,y)

s ~ M ~ E ~ A(~,~ 1) v zu

Satz

S U M U ~,/ ( V x ) ( ( ~ y ) A ( x , y )

zwei

wird

verschiedene

zus~tzliche dass

durch wit

Sprache

~ ...

R) w e n n

Also

lassen als

Allerdings

kSnnen

: Die cI

Regel

1 garantiert

Harrop-Formeln,

vorauszusetzen~ Aus

Folgerung

Harrop-Formeln

zusammensetzen.

sammenhang

Satz

,

welchen

wobei

durch

Scarpellini

ist,

wo

durchl~uft.

...

i, d a s s

D A(X,r A(x,y)

alle

v A(~,o n)

die

~ A(~,~).

Theorie

V ... Formeln

V A(X,C n) V A ( x , x ) ) yon

S mit

genau

B.

II.

Die

(a)

intxrltionistische

Zahlentheorie

Bemerkun~en

Versucht

man,

die

Methode~

nlsfiische

Zahlent~eorie

lich

die

nur

oder

zu

nicht-konstruktiv) man

Schwierigkeiten,

herrfihreno

Zus~tzliche

auf

der

finitiren

etwas

dem

auf

rekursiv

die

nStigen

(b)

Der

Erg~nzungen

Hinblick

D~

auf

fur

ste~ige

die

Analysis

~

wie

bei

hilt;

(kurz [

wit

Funktor

. I s.t

~e~

f(i)

..~

,Xp

heisst sine

= ai

eine die

Folge

Belegung

uns

mit

vielleicht

Zusatzbetrachtung

den

ist.

fn'gl .....

wir

' i ~

yon mit

sine

zuordnen~mit

: Sind

u 9

~ f

und

N

....

2,

, an_l~

Zeiehen

freien

yon

"'"

Funktoren nehmen

fULhrt

wit

an

,fUr

vorhanden

Abbildungen

er k e i n e

n

~ , x t(f,g,~)

; u ~ v , so

sin

.

. Bin

Variabeln

gm'Ul .....

ist

. Jedem

% 0

, so

die

1

ent-

St

und

yon ist

Up

m

(fi,gj

Jedem

t(~

yon f

'

, ~l ....

e NN

so , U k G N)

, ~ ,x)

und

sin

Zahl-

Weiss Term

Sirule

gleicher t(f,g,~)

u ~

Zahlvariabeln,

Term

(im

Verkettung wit

. . . . .

in nat~rlicher

bezeichnen.

F01genzahlen

~(v,w,~)

'

n Punktions~

gilt

ist

schreiben n

' ~

. . . .

Ststigkeitsfunktion v,w

,

' i = i, und

seien

Zahle~.Ferner

=
halber

f~r

vorkommen.

Z s t ~

fl

~

~ Ui

stetige

wenn

~l WP-Konstanten,

[ ~ ])

g

rekursive

f e NN

sioh

, w ~

>

Zahlentheorie

Konstanten

= nat~rliche

Terms

u,v,w

fl ......

induktiv

Nt

.

formalen

( N

Einfachheit

Zahlen-

aus

) und

Folge

und

f,g,u

der

Folge

~ an_l>

lisst

v

sich,werna wlr

aufz~hlbaren

einer

.... N

mit

n-i)

yon

in

wir

,wenn u i ~

zugeordnet,

wert

sich

intuitionistische

endliche

. Der

d~rfen

yon

Folge

Belegung

jeder

~

'~

abzihlbare

auf

aber

(i

N t

konstant,

bezeiehnen

u. = .< a o ,

X

~ :(NN) s x

uns

WF-Konstanten

Folgenzahlen u,v

ergeben woli~n

Funktionen

WF-Konstanten).

] sin

heisst

die •

' ~i

jede

sine

Abbildung

sehr~nken bzw.

fur

Zahlen

Kleene

stetige

Spiter

ergeben

rekursiv

in

Zeichen

' ~2

(NN)s

und

Wahlfolgenkonstanten

Ublichen

(~I yon

wir

Kons~anten

den

vorhanden,

nat~rlichen

Term

zul~sst.

mathematisch

den

eigent-

.

statten

und

Funktionen

Abbildungen

das

jede

M~thede

Art

diese

erst

, dass

(konstruktiv

, so

Trotzdem

wir

, und

intuitio-

Schnittelimination

technischer

wollen

lassen

sich

der

einer

, da

anbringen

Z.m mit Variabeln wir haben neben

Variabeln

man

Fehlen

verharrt.

Zuerst

es

~-Regel

Zahlentheorie

vom

befassen

Acht

zeigt

einer

die

Formalismus

theorie

yon

finit~re die

auf

Uebertragung

Komplikationen

ist.

ausser

mit

direkte

die

Zahlentheorie

vSllig

Abschnittes , so

versehen

Aufz~hlbaren

interessanter

Asp@kt

ist

sich

letzten

~bertragen

sine

technische

man

des

Zahlentheorie

Beschr~nkt

Im

329

Scarpellini

yon

L~nge,

t(~,~,x) [ 6 ] , gilt

:~t(v,w,u)

- i.

330

B.

Sind nur

Gt

rekursive

rekursiv

f~r

Konstanten

w~thlen.

jede Bele~ung

beim

Primformeln f,g,~

gilt

Ein konstanter

Term

t(~

) heisst

( ~ ~ f , ~ ~ g) g i l t

(c)

der

Form

Systeme

0

der

Z.m bezeichnen theorie, welches

~

f~r

wir

0"

das

neben

saturiert,

= t(g)

,an_l>

sich wenn

wenn

f~r alle Belegungen

f,g

numerische

Terme~

System

der

WF-Konstanten

nur

Funktionale seien

intuitionistischen Zeichen

(und

enth~it.

zudem

Zahlen-

F~r

die Axiome ~

quantorenfreie jede

(k)

WF-Konstante

= ak

f~r

zu Z l a l l e w a h r e n P r i m f o r m e l n (im S i n n e d e s l e t z t e n A b s e h n i t t e s ) ~ so e r h ~ i t m a n d a s S y s t e m Z ; mit Z bezeichlI 12 n e t m a n d a s U n t e r s y s t e m y o n Z. , in w e l c h e m n u r r e k u r s i v e K o n s t a n t e n 11 beim Aufbau yon Termen und Funktoren verwendet werden. Uebertr> man die

Systeme

Zi,Zil,Zi2 dabei

Induktionsaxiome

delnd),

so e r h ~ i t

noch mit

Addiert

wahr,

Zahlentheorie

rekursive

...

vorhanden.

so l ~ s s t

,~,x) h e i s s e n

.

, ..o h e i s s e n

formale

den

primitiv

wo XI = ~ao,

k ~ n-i

,

t beteiligt,

= q(~

:

intuitionistischen

Mit

Axiome)

O'

,

vom

= q(f,g,~)

t(f)

Terme

Aufbau

t(~n,~,x) :

t(f,g,~)

yon ~

Scarpellini

man

man

in bekannter naeh

dem

Weise

Vorbild

Kalk~le

in den

yon Gentzen

GZi,GZiI,GZi2

einer Umsetzungsregel

Sequenzenkalk~l

versehen~

die

I l l als R e ~ e l

. Jeder es

( die

dieser

behan-

Kalk~le

ist

gestattet,numerisehe

T e r m e d u r c h a n d e r e y o n g l e i c h e m Weft zu e r s e t z e n (siehe ~ I , C ~ J )" F ~ r das F o l g e n d e b e n ~ t i g e n w i r e i n e n H i l f s s a t z f~r p o s i t i v e F o r m e l n : Hilfssatz

: F~r

A* i n p r ~ n e x e r ist,

angeben,

jede positive Normalform,

Formel

deren

A lisst

sich

effektiv Tell

quantorenfreier

eine

eine

Form~l

Primformel

sodass Z.

I-- A ~ A *

(k

= ~,1,2)

1k gilt.

Hinweis:

Man benutze,

~quivalent

ist

zu

dass

jede

(~x)(~)((x-i

Formel

A v

(~)B(~)

V A)A

(x%l ~ B ( ~ ) )

intuitionistisch

)

.

B.

(d)

Wahre

Im

pr~nexe

Fulgenden

@

Ist

~

und

"wahr"

ist

A

(~y)

,~,x,f,g

durch

@

) ist

wahr,

wer~n

"Stetlg

und

wahr"

oder

auch

definiert. so

heisst

wahr

wird~

A wahr,

A(~t~,~,x,y

wernq

) wahr

A

f~ir

jede

iSto

mit ahr,

@

(~)

A(~t~,~,x,~

dass

A(~

Lassen

wir

nur wahr"~ ist

Da

H

4 unten

unter

sieh

1 haben 4

Die

:

]

A(~u) ist

und

(

zu~

so

stetigen

stetigen

sprechen

wir

Furmktionale

von

vorhan-

Funktional-lnterpretation

d o c h m ~ s s e n wir die

1

ohne

freie V a r i a b l e n

f~r ~

(v) @ 0

den

= leere

Beweise

yon

H

lem-FurLktionen"~ , so

) pr~nex

, ~ elne

0( u 's verschieden Folge)

sagen

dann

konstruktive nur

e

sie erf~lle

Gegenstiicke eines

: Es

, H

3

deren lassen

als

existiert

Sei einer

e(v)

A(~u)

und ~

A(~u,v)

Zi2~- A.

eine

w a h r ist.

wahr.

A(~u,~,x

yon

Formel

, sodass

und A rekursiv wahr,so

Folge

sind

yon

und

WF-Konstanten,

alle

den

4

verlaufen

Existenz sich

eine

durch durch

A.

Mit

zu

H

Illustration

eine

die

unteren

Index

o

A(Cgu,~,x ) ist wahr, g e n a u d a n n w e n n A ( ~ u , ~ , n ) w ~ h r

kursiv

existiert,so-

so Z

eine p r ~ n e x e

ist

Darm

~

alle

zur

F

sind l e i c h t b e w e i s b a r ,

Stetigkeitsfunktion

wir

Sprache

aequivalent

Hilfss~tze

H ~ : Sei A(~u)

wir

Furmktoren

~- A . II : Sind alle K o n s t a n t e n in A r e k u r s i v ,

H 2

FgOgl~,~,x

aus Platzgr~inden w e g l a s s e n .

H i : Ist A wahr,

In

Funktor

sod ss

ist.

t und

unserer

exi,tiert,

].

Die n ~ c h s t e n Beweise

in

ein

) wahr Terme

"wahr"

yon Kreisel ~

Term

wahr,we~m

rekursive

"rekursiv sind~

) ist

,~,x,F[~u,~,x]

den

H

bis

aussagenlogisch

A(~i~,~,x,y

WF-Konstanten,Furaktions-

Furh
Q

@

H

yon

) quantorenfrei,

f~g~

@

Folgen

Zahlentheoretische darn~

= A(~t~,~,x

Belegung

331

Formeln

sind

Zahlvariabeln kurz

Scarpellini

~hnlich

wie

in

~

~

garantiert

eine Hilfe

3,

H

4

f~ir

einzige dieses

Stetigkeitsfur~ktion~

- i die F o r m e l A ( ~ u . v )

Begriffes

sp~teren

pr~nexe sodass

.Dann

mlt

Formel gilt

Sind

die

"Sko-

werden,

re-

e kodifizieren; lassen

(und

Gebraueh ~

].

GSdelnummer

formulieren

Partialrekursive konstante

bis

E~

f~r alle n iSto

sich

leicht

beweisen),von

denen

erw~en.

folgenden

Eigenschafteno

und

OSdelnummer

: Ist

erf~llt

e die e(v)

~ 0

~ so

erf~llt

~(e) die F o r m e l A(~u).

B.

332

(e)

Regel

und

Schema

in

Scarpellini

Z l1

Satz als

2

: Seien

einzigen

liebige

%(~,x)

freien

nume~ische

Terme

E) g i l t

Z.11~ % ( F , n )

Dam i s t

die Theorie

und

, Bk(~,x ) Formeln

Variabeln;

ohne

die % ' s

n und k o n s t a n t e

, so g i l t

~

positiv.

Funktoren

Zil ~

T = ZilU

WF-Konstanten,

seien

F~r

mit alle

F gelte

~ , x k

, be-

:

Bk(F,n ) .

(V~,~)(%(~,~)

~ ~k(~,~)

) kon~ist~.t

disjunktiv.

Bemerkun~en

:

i) Bei

in Z. m U s s e n die V a r i a b l e n in d e n P r i m a x i o m e n n i c h t lI g e h a l t e n W e r d e n ; i n s b e s o n d e r e sind die u n i v e r s e l l e n Abschl~s--

Herleitungen

konstant se d e r 2)

Primaxiome

beweisbar.

Unter

"disjurLktiven

Eigenschaften"

etwas

Allgemeineres

als

in I)~

einer

Theorie

Wir verlangen

T verstehen

jetzt

z~

wit

nun

:

Gilt T ~ ( ~ ) A(~u,~) (mit (~)A(mu,~) konstant), so gibt es e i ~ e Stetigkeitsfunktion ff , s o d a s s f u r ~ ( v ) @ 0 e i n k o n s t a n t e r Funktor Fv e x i s t i e r t m i t T ~ A v( ~ u) , v. , FAehnliche Formulierungen gelten in den Fillen m A(~ u) V B(~ ) , T ~

~d

yon der

Satz

2

fUhren

Wir

: Auf

wit

Zuerst

den

Teil

eine

Beweis

im

also

( s i e h e z . ~ . [ ~ ] ~heorem ~ yon

annehmen,

iibersetzen

addieren

,

Grund

Allgemeinheit

quantorenfreier

I)

A(~ , ~ )

i~ rekur~i.e~ Fall

Beweis kung

(~)

wir zu

Primformel

Theorie

alle

Regeln

1 kSnnen

%'s

seien

ist.

Wie

SequenzenkalkUl die

GZil

Hilfssatz die

und

~

Bk(r, t ) , P

beim

Satzes der

~

F,t

frei

fur

mit

GT.

II)

Fiir Beweise

griffe aus delt wird. Die

[l]

Komplexit~t

eines

~,x

P

aus

%,

GT

ein,wobei

eines

Beweis

in

Bk

fiihrem die

. Das

wir

den

Induktionsschlusses

die

Induktionsregel

Strukturschlusses,

definieren

System

Satz

in

1

Schritte.

SequenzenkalkNl.

wie

...

bezeichnen

beweistheoretischen gleich

eines

wir

deren

%

resultierende

wieder

Einschr~hl-

yon

ihn

k:l,2~

in

ohne Formeln~

Form

%

wo

wir

pr~h~exe

zerlegen

T des

[ ~ ] ).

~

wie

logischen

in

in

[i

Be] behan-

Schlusses

[ I ]~ w ~ h r e n d

wir

die

oder

Kom-

B. Scarpellini plexit~t

eines

~-Schlusses

Die B e g r i f f e H~he, wie

per Definition

Endst~ck~

kritiseher

333

die

Komplexit~t

Schluss

usw.

y o n B k iSto

sind w i e d e r

gleieh

in [ ~ ].

III)

Jeder

Sequenz

oben nach unten verfahren setzen

wir

gleieh

setzen wieder

IV)

F~r Beweise

Beweises

P aus

GT o r d n e n

0rdinalzahl 0(S)

zu 9 Mit

wie

[ 5 ]. F ~ r

wir wieder 0(S)

Wir

[ I J:

Seines

eine

0(P)

in [ I

],bzw~

wir

Ausnahme einen

induktiv

~-Sehluss

=~d(~o

~ ~ ( S I ) ) , wo d = h ( S l )

- h(S)

= r

wo

P ist.

S die

Endsequenz

yon

yon ~-Schl~ssen

yon

SI/S ist~

GT d e f i n i e r e n w i r n u n R e d u k t i o n s s e h r i t t e w i e

P aus

2) E l i m i n a t i o n

in

Reduktionsschritte,

i) V o r b e r e i t e n d e

logischer

Zeiehen,

~) I n d u k t i o n s r e d u k t i o n . Hinzu

kommt

In P liege

die

~-Reduktion~

ein k r i t i s c h e r

die wie

%-Sehluss

Der Funktor

saturiert,

mit

GZil yon

Fund

der Term

P

B k ( F , n ) vor.

#

t seien

Wert n

numerischem ~

Dann

ist.

Form

Bk(;,t) , V %(F,t) vor.

definiert

folgt der

konstant;

zudem

. Schliesslich ~ndere

man

liege

P wie

sei d e r T e r m ein B e w e i s

folgt

P

o

t aus

ab:

o

Bk(F,n)

Umsetzung

Bk( ,t)

Bk(F,t)

Schnitt~ Verd%innung

Ak(F,t) ,~ a

Reduktionsschritte eigentliche liche

der

Sorte

Reduktionsschritte

Abschnitt V) V o n n u n

IV)

des

deren

freien

Variabeln

haben

Beweises

an w o l l e n

weise,

9

die

enthilt.

gesagte

Ein Normalbeweis

wieder Satz

hat,

Form

Alles

gilt

P heisst F

yon

, 3) u n d genannte

wit nut noeh

Endsequenz

und Basislemma

oder

2)

Reduktionssehritte

%-Reduktionsschritte werden.

die

Eigensehaften

und

i) bis

eigent-

2) im

1 .

Normalbeweise m

oder

im B e w e i s

P betrachten~ 9

yon

Satz

F

hat,

d.ho

Be-

wo F k e i n e

1 ~ber Normalbeweise

auch hier.

Standardbeweis, wo

sollen

Vorbereitende

F eine

pr~nexe

wenn

seine

Formel

Endsequenz

ist,

deren

die F o r m

quantoren-

B.

334

freier VI)

Teil

eine

Primformel

FUr Normalbeweise

sei

saturiert,

yon

P satnriert

wit

~

)

~mit + 0

der Eigenschaft

, so

ist

P

VII)

Wir beweisen

nun durch

0hne

mit

seinem

trifft F 1

also

:P ist

zusammen. Es

und

P

tim

A(~u)

P

~hldst~ck

. Erse~zen

einen Normalbeweis eine man

Kontraktion. eine

folge

v

gilt

so h a t

kSnnen

zu,

Auf

ergeben

saturiert

sagen,

Stetig-

0(Pv) ~ber

seine

a~s

=

P d~roh

O(P).

O(P)

:

Lhudsequenz d i e

Form

ist w a h r .

der Allgemeinheit

B) zu.

sagen,

Induktion

i n GT,

Reduktionsschritte Endst~ck

Wit

transfinite

) und A(~U)

Einsehr~nkung

vorbereitende

Term

~

so f i n d e t

0ffensichtlich

P ein Standardbeweis

Wir

:

s~turiert.

v Kontraktion.

A(~

yon

Normalbeweis,

saturierende

>

konstante)

, so e r h a l t e n wit u~ v s a g e n ~ P v f o l g e aus P d u t c h

Wit

.

Begriffe.

dutch ~

eine

B) Xst

zwei

Sei P e i n N o r m a l b e w e i s

A(~u~v)

~(~)

wir noch

jeder(notwendigerweise

ist.

P ein nichtsaturierter

keitsfunktion Ist

ist.

P benStigen

i m gs_nzen B e w e i s

Pv y o n Ist n u n

wenn

Scarpellini

2)

wit

0(P)

arunehmen: ~

~

.

o Standardbeweise

alle

s i e h darn% f o l g e n d e

enth~it

keinen

Dar~n m u s s

P notwendi~erweise

einen

oder

Induktionsreduktion

i) P l ~ s s t

P f~llt

kein(

dann nicht

P' m i t O ( P ' )

< 0(P I

Fille:

kritischen

logischen

Schluss

in der Endsequenz.

eine

logischen

zulassen

oder

Reduktionsschritt einen kritischen

Sehluss

%(F,t)

,~

;

enthalten.

In

den

ersten

logischen einen

Beweis

Pl

A u f Pl t r i f f t B)

beiden

F~llen

transformiert

Reduktionsschrittes

bzw~

derselben

somit

Endsequenz,

die

sich

einer

P vermSge

eines

lnduktionsreduktion

aber

mit

Induktionsvoraussetzung

in

~(Pl)

~

zu~

also

~(P)

.

trifft

a u f P zu.

Xm letzten

Fall

~(F,t) Term f~gen 9

gibt

es g e m ~ s s

mit 0(PI) <

t saturiert

und hat

sit

erhalten

O(P2) < O(P)

tionsvoraussetzung

zu,

Zil ~

%(F,n)

. Auf

Zil ~

Bk(F,n ) , also

also

Grund GZ

. Auf

wit P2

ist ~ ( F , n )

der Bedingung ~- B k ( F , n ) .

mI

einen Beweis

o Da P saturiert

einen numerischen

einer Umsetzungsregel %(F,n)

Basislemma

~(P)

Wert

wahr.

ist d e r

no D u r c h

einen Beweis trifft

P1 y o n

ist,

somit

P

Hinzuyon

2 die Induk-

Es g i l t

somit

Satzes

folgt

dann

somit

einen

~-

E) d e s

Wir kSnnen

B.

Reduktionsschritt einen Beweis P' F 2

trifft

Dann

saturiert

mit

Bild

existiert

und

in der

sine

enth~it

zu~

keinen

Stetigkeitsfunktion

yon

F 1 ~ nach

Endsequenz Also

9

A(~u)

, P1 y o n 0 ( P l) < 0 ( P ) wahr 9 also

Auf

folgt

~unktiven

~

einen

a u f Pl zu

analog wie

D)

in ~6 ]~pg.

B)

# 0 , so

somit

den Vor-

auf P

zu. v f~r ~(v)

, und

Somit ~ 0 ist

mit

Bild

erh~it

gswis~en

in der End-

. Unterl~sst

man

einen Beweis

Funktor

F[~11 ]_

o B ( ~ u , F [ ~ u ] ) is% F~lle

o Da somit

aufgezihlt

sind,

B) hat.

folgt~

enth~it~

: Ist 5 (v)

(~)B(~u,~)

) . Da alle m6glichen

Eigenschaft

Eigenschaft

a u f P zu.

logischen

gen~gt

y

also

Schluss,so

B)

P

Schluss

A(~u)

ist

~-Schluss

B)

Auf

~ach H 3 .

B(~u,F[
~ mit

A(~)

io~ischen z.B.

logischen

P die

GIu/nd y o n

einen kritischen

v

A(~ u) ~ r

. Sei

aush A~u

dass

trifft

kritischen

saturiert.

Form ~

einen kritischen

man den kritischen

folgt,

der

ist

also

P in

< ~(P).

~ndsequenz.

F 1 trifft

P sine

sich dann

aber mit O(P')

aussetzungen

sequenz

(f)

transformiert

P

: P enth~it

Daraus

Es

Endsequenz,

die Induktionsvoraussetzung

A W u . ~) w ~ r .

VIII)

335

ist d e r k o n t r a h i e r t e B e w e i s

hat

F ~

auf P aus~ben.

P' d e r s e l b e n

: P ist n i c h t Schluss

Scarpellini

dass

sin Normalbeweis

P, d e r

einen %-Reduktionsschritt ll0

, dass

GT und

somit

zulisst.

T die dis-

Ei~enschaften haben.

Regel und

Schema

i n Z.•

D a Z. n u r T e r m s u n d F u n k t o r e n e n t h ~ i t , d i e aus r e k u r s i v e n K o n s t a n t e n a u f 12 ~ e b a u t sind, ! ~ s s t s i c h j e d e r B e w e i s aus Z. durch sine einzi~e G6delnum12 m e t e k o d i f i z i e r e n . S c h l i e s s l i e h w o l l e n w i r sine T h e o r i e T k u r z r e k u r s i v aufz/hlbar

nerunen~

T rekursiv

aufz~hlbar

Dann

gilt:

Satz

2

meln

falls

Z. ~ i ist.

: Sei % ( ~ , x )

T ~ Z. und m2

falls

, B k ( ~ , x ) k = 1,2 . . . .

die Menge

der Axioms

sine rekursive

Men~e

yon

yon For-

ten

aus Z , m i t ~ ~ x als e i n z i g e n f r e i e n V a r i a b e l n u n d o h n e W F - K o n s t a n 1 ~ die %'s s e i e n p o s i t i V o F e r n e r sei ~ sine p a r t i a l r e k u r s i v e Funktion

mit

der Eigenschaft ER)

Ist

e die

G6delnummer

sin konstanter die Dann hat

sines

Funktor

G~delnummer sines

die Theorie

aufz~hlbare

: aus

Zi2 y o n % ( F , n )

und n sin numerischer Beweises

T = ZiU

Primerweiterung

Beweises

ist,

so ist ~ ( e )

y o n B k ( F , n ).

(~,x)(%(~,x)

T', w e l c h e

Term

, wo F

=

Bk(~,x ) )

die disjunktiven

sine

rekursiv

Ei~enschaften hat.

336

B.

Aus

Platzgriinden

weitert Form

man

sind

F,t

Hilfe

eine

aus

der

Flhnktionen

einem

Zi2

uns

Kalk~l

und

mit

einem

GT durch

Hinweis

begn~gen.

Hinzuf~gen

aller

Zuerst

Re~eln

er-

~

der

in den

die

f~llt ~(e)

die

Die

Definition

mit

Hilfe

nannten

> ~

frei

ffir ~,x ~

eines

Formel

ist

Menge

Satz

%

mit

der

, Bk

3 und

Gegenst~cken ~

" . Darau

der

definiert

man

partialrekursiven

zu H 3

~ H

4

(z.B. ~ i n H

5)

Eigenschaft: P yon

~

A aus

GT,

so e r -

A. unter

Fixpunktsatzes,

von~

in

in

Standardbeweises

erfolgt

Eigensehaften

aufz/hlbare

%(F,t),r

Funktion

von~

des

Hilfe

~

konstruktiven

GSdelnlnnmer

r

~k(F,t),r

partialrekursiven

partialrekursive

Ist'e

Mit

mfissen w i r

zu

:

Dabei mit

GZi2

Scarpellini

und

Bezugnahme der

auf

Beweis,

die

dass

Reduktionssehritte

~ wirklieh

hat,

erfolgt

durch

transfinite

Induktion

es d a m n

nieht

sehwer

, ausgehend

yon

M yon wahren

Primformeln

aus

Z

zu

die

genannt

~ber

T eine

erzeugen,

~(P).

rekursiv welche

12 die

Eigenschaften i)

Ist

A

Z.

1

2) Auf

Qilt

T U

es

zeigt Die

sich

konstant M

U

~

z.u i

M

karhn sich

klassischen

nicht

und

2,3,4

pr~nex

k

Analysis bietet

trivial,

zwar und

gilt

T U

(~ konst~t)

(F,n)

nun

den man

Beweis hie

lassen

fHr

und

M

~

A

, so

gilt

schon

.

M~A man

:

yon

genStigt

sich

intuitionistische

Einschr~nkungen

Systeme

A,

, dass

S~tze auf

hat

%'s

hat) keine f%thrt

und . Die

2

ist

Hber

der Bk'S

den

Rahmen

fast M

Z.U

M~k(F,n)

1

wSrtlich

, falls der

~bertragen das

fibertragen

&

lassen

sie

(mit

System

S~tze

Sehwierigkeiten, dieser

.

hinauszugehen~

. Insbesondere

Analysis

Ausdehnung

prinzipiellen Hber

Satz

verallgemeinern

Systeme die

, so ~ilt

Arbeit

die

2,3,4

gewissen

St~rke auf

ist hinaus.

der

diese aber

doch

B.

Scarpellini

337

Anhan 6 1 Es

gibt

noch

kationen Wir

weitere

M~glichkeiten,disjunktive

zu untersuchen.

begn~gen

uns

der

Eine

meln

. Verallgemeinerungen

sind

( unter

geeigneten

Satz

: Seien

A~

beweisbar. Beweis

: Wir

GZ. die 1

auf

sei

kurz

halber den

Fall

~eschlossene

dem

yon

Formeln

Z. + A m l

B

Impli-

geschlossener mit

und

(A D

freien

For-

Variabeln

schwierig. B)

~

A in

Z. n i c h t 1

disjunktiv.

Zi + A ~

Hbersetzen

Fall

) nicht

Formeln

yon

besprocheno

mit

Zusatzvoraussetzungen

B zwei

Darn% i s t

davon

Einfachheit

Eigenschaften

B in den

SequenzeD~alkNl

, indem

wir

zu

Regel

R hinzuf~gen. kann

Ein

dann

mit

Hilfe

Das

w/irde

spruch

des

F

~

Z. + A ~ 1 2)

Sei

Modell

B

~

einen

und yon

Beispiele:l)

so

+

A ~

B

kSnnen

~

die

und

Ist

ZK

eine

so

sind

die

somit

P

dem

yon

m

kSnnte

man

in

Muster

yon

klassische

GZ.R

Andernfalls

GZ.R yon 9 A finden~ o 1 , Mithin Z. ~ (A ~ B) ~ A ~ im Widerm auf Beweise P in GZ.R mit einer Endm klassische Gentzensche Reduktions-

A

nach

Theorie

enthalteno

somit

daraus

GZ.R 1

ernaltenen

Beweis

uneingeschr~nkt

anwenden,

A

der

R-Schluss

: Z. 1 Wir

Annahme.

Eigenschaften

~

in

Basis-Lemmas

9

theorie

P

kritischen

bedeuten

zur

sequenz

ZK

Beweis

keinen

Z. 1

+

yon A

~

B

[

Zahlentheorie~

Voraussetzungen

des

] die

disjunktiven

herleiten~ nicht

Satzes

ZK

erf~llt

~

B~

abet

~ mithin

B disjunktiv.

~)Z(~) hat

eine

. Setze

B

Formel,die

ausdr~ekt,

=

, A

~I

)Z(~)

=~

B

dass

Zermelo-Fraenkel

. Nach

Beispiel

ein

iist

dann

Z. + A D B d i s j u n k t i v . Addiert man zu dieser Theorie die Harrop-Formel l , so b l e i b t d i e ~ r w e i t e r u n g konsistent, verliert aber die disjunktive

A

Ei~enschaft. ~)Sei die

wieder

B

Theorie

Formel

T

B nicht

" werun T ~ ~ B T

+ IB

D B

=

(~)Z(~)

= Z

+

, A hin~egen

(~B ~

B)m

1 entscheidbar

so T ~

B

ist

" gilt

, so b l e i b t

diese

sei IB

D

B disjurLktiv. in

ZK

also

, gilt

B

Da

o Naeh T ~

nicht

trivialerweise.

Beispiel

ZK

, und

T ~ nB Bilden

konsistent, v e r l i e r t

aber

1 ist

da

die

. Die wir

die

Re~el

die

Theorie

disjunktiven

Eigenschaften. Es

ist

Z. ~ I ~) A

A

uns so

nicht Z. ~ 1

Sehliesslieh =

(~)qPrim(~)

sicher

nicht

B

gelungen, , 2) sei

Z

i

ein + A D

nochmals

, B

beweisbar

=

Formelpaar B

das

ist

Beispiel

(~)(qPrim(~)9 in

ZK

, also

A~B

zu

konsistent in

~(a)

Prim(~)) nicht

in

Z

finden

mit

, aber

nieht

. Somit

ist ist

Wenn

disjunktiv.

betrachtet

. Dann i

: i)

Setze (A D

B)D

Z. + A D 1

A D

338

B.

disjunktiv

nach

es

nicht

zu

beweiseno

Anhang

Satz

mSglich,

konsistent).

leicht

lassen

Realisierbarkeitsbegriff

spiel

kurz

theorie

skizziert.

ohne

yon

verstanden.

Satz

: A(x), E)

Sei

Dann

B(x)

Ist

T

man

leicht

Zi + A ~

verifiziert~

B mit

Hilfe

ist

yon

Satz

Beweis

gew~hnliehe

T

~

beruht

auf

werun e i n

Dies

sei

an

intuitionistische sei

im

passen-

einem

Bei-

Zahlen-

Sinne

yon

Kleene

Satz

Formeln

rekursiv

gilt

erzielen,

steht.

"Realisierbar"

der

mit

folgender

aufzihlbare, A(n)

(Vx)(A(x) = B(x)

i~t

Resultate

Verf~gung

ZI d i e

gilt

seien

eine

ZI u n d Da~

sich zur

Funktionsvariabeln.

~]

Der

Wie

Disju~tivit~t

2

verh~Itnismissig der

(und

die

Scarpellini

, so

Eigenschaft

disjunktive

gilt

T

m

: Erweiterung

yon

B(n)

realisierbar

einem

technisehen

Lemma,

das

wlr

ohne

Beweis

an-

geben. Lemma: A).

: Sei

Dann

terung

A

eine

l~sst

T yon

gesehlossene

sieh

ZI +

effektiv A

Formel

eine

angeben

und

gelte

rekursiv

mit

der

e / A

(d.h.

aufz/hlbare,

Eigensehaft

e realisiere

disjunkte

: Jedes

Axiom

Erweiyon

T ist

realisierbar. Beweisskizze effektiv terung T

T n B(n).

~

n alles

(~x)(A(x) Erweiterung auch

schon

eine

sich

B(x) : Aus

=

B(x)

+ A(n) es

ist

die

folgt

[Z]

die

rekursiv

mit

der

des

gilt

yon

folgt

ohne

im

funktioniert

E) f/

dass

Partialre-

ZI

+

aus

dem

Lemma

positiven T

A(x) yon

Verfeinerungen

kombinieren, erzielen,

aueh

:

disju~tiv~

automatisch.

Kombinationen

schliesslich

auch

. Da

yon

eines

fast

zu

Satz

f vermittelt

weiteres

ihnliehe

diese

im

aufz/hlbare,

und

man Erwei-

B(n)

eine

sofort,

Methode

Satzes

finder

Primerweiterun~

Realisierbarkeit

auf

e zu

Falle

aufz~hlbare

Hinweise

nach

f mit

Realisierbarkeit

so

Diese

Lemma

realisierbare

sehwierig

~ rekursiv

dass

letzten

und

ein

nieht

nieht

) existiert.

selbst

A(n)

in

Lemma

beweistheoretischen auch

Beweistheorie

dem

Beweis),

Versehirfungen

~--

realisierbare

seinem

. Gemiss

Ueberg~ig

damn

Hingegen

B(x)

der

den

sich

und

Satz

eine

n 62

prinzipiell

dem

disjunktive~ D

es

P

Satz

.

)

e / A(n)

disjunktive

. Wegen

) sofort

aus

bei

etwa

naeh

ergibt

T hat. nicht

Platzgr~nden Die

: Sei

aufzihlbare~

finden,

Beweistheorie

sentliche aus

ist~

(~x)(A(x)

geben die

gibt

~

B emerkun~en

+

ZI

Also

~ Daraus

(Vx)(A(x)

ZI

yon

9(x,y) zu

) = f

(und

Satzes

rekursiv

effektiv

kursive ~(nTe

des

eine

dann

um

doeh

Man

erkann

ganz

we-

m~ssen

wir

weglassen. noch,

wenn

kein

B.

passender

Realisierbarkeitsbegriff

St~rke

der

Die der

Scarpellini

Analysis

beiden

und

Methoden

3~

verhanden

fur

Theorien

mit

~berlappen

sich

ist

: fur

Funktionalen also,

Theorien

yon

hSheren

aber

sie

der

Typs.

implizieren

einan-

nicht.

Anhan~ In d i e s e r A r b e i t w u r d e unter

Benutzung

von

Allgemein Lemma

so

man

dass

die

eine

durchgef~hrt

in dass

Schliessens

die An

Teile

dafOr sich

dieser

urn

'starke

Stelle

sei

in

[$

und

] dank

etwa

das

Schema. nicht

Systemen

des

Basis-

dem fur

Trotzdem vom

dem

nach

Reduktionstheorie

Arbeit

in

heurisdie

ist

Regel~

anzunehmen

2

beweistheoretischen

nat~rlichen

Schliessens

kSnnen. in

dieser

yon

Richtung

aber

sich Das

ein

auf

gesteigerter

R.

vQn

Herrn getan.

Systeme

Basis-Lemma

Normalisation" Herrn

wird

Dissertation

~6~

lassen. ist

Methoden

eine

befindliehen

Obertragen

Erscheinung, dig,

Schritt

Vorbereitung weite

dieser

.

nahestehen

man f~r

durchaus werden

] die

recht Hat

in

und

wichtiger

zeigt,

Problem

Untersuchungen

h~tten Ein

~6

dass

Reduktionstheorie

abh~ngen

einer

aus

sagen,

funktionieren:

Rahmen

in

sich

Regel-Schema Motto

hat

Techniken

l~sSt

dem

tischen

a u s s c h l i e s s l i c h mit dem Sequenzenkalkfil g e a r b e i t e t I

des tritt

gruppieren

Haberth~r

ffir

HaberthNr wird

ge-

natNrlichen dabei

Aufwand

R. Deft

an

nicht

mehr

Begriffen

in

notwen-

. wertvolle

Diskussionen

gedankt. Insbesondere tippte,

und

sei ohne

auch

dessen

Herrn Hilfe

Ch.

0ppliger

diese

Arbeit

gedankt, nicht

der zustande

dieses gekommen

Manuskript w~re.

B.

340

i] G . G e n t z e n

:

Neue

Scarpellini

Fassung

die reine

des W i d e r s p r u c h f r e i h e i t s b e w e i s e s

Forschungen

zur Logik

Wissenschaften, [2]

S.C.Kleene

:

Introduction

Neue

S.C.Kleene-R.E.Vesley

: The

(1962)

:

(1965)

Interpretation Functionals in

[6]

[7]

B.Searpellini:

"

Proof

. Ann.

181

[8]

in M a t h .

Notes

K. S c h ~ t t e

:

Second

Consistency

o

Intuitionistic 212

and T r a n s f i n i t e

Systems

, Springer Induction

.

(1971)

.

in Intuitionis-

.

of M a t h . L o g i c ,

Vol.

of Intuitionistic

Commentarii [4

(1969) and

A Model

"

of G e n t z e n s

Lecture

tie S y s t e m s

of Constructive

in Mathematics"

Proof Theory

Annals

by means

Type

(1959)o

Some Applications

Induction

,,

of Analysis

"Constructivity

Math.

of I n t u i t i o n i s t i e

.

of Finite

North-Holland ~]

exakten

.

North-Holland G.Kreisel

der

(1938).

o

Foundations

Mathematics

~4]

& Grundlegung Folge

to M e t a m a t h e m a t i c s .

North-Holland, 3]

fGr

Zahlentheorie.

Math.

Beweistheori~

.

Helv.

4 , No.2

, (1972).

Analysis. ,Vol.

Springer

45

(1960)

, Fasc. o

4

(1970)o

Infinite

Terms and Recursion

H.

Schwichtenberg

Systems of infinite

in Higher

and S.S.

Wainer

terms defining functionals

were first considered by Tait [10] and further Feferman

[3] initially

unpublished

constants

in a proof-theoretic

notes Feferman

terms inductively

introduced

generated

abstraction

context. o

Later in of infinite

of all finite

recursive

and autonomous

of finite type

developed by

the system T

from variables

for the ordinary primitive

application,

Types

types and

functions by

enumeration:

if for each

n, f(n) codes a term tn~ T o and f is itself defined by a term of T O then the t e r m < t n > n ~

N is in T o .

to an arbitrary f u n c t i o n a l ~ a n d denoted by To(~)

.

the resulting

Feferman proved

functions

definable

in~(This

also follows

This definition

that i f ~ i s

in To(~) are precisely

This immediately poses used to characterize

can be relativized

system of terms is of type 2 then the

the functions

recursive

from our results here together with [11]) . the problem of whether

full Kleene recursion

specifically whether, f o r ~ o f

infinite

terms can be

in higher types and more

type n+2, To(T) gives a characterization

of the n+1 - section of ~. We show in w

that for arbitrary ~ o f

of types ~ n+1 definable appearing

in T O (~) are just those functionals

in a naturally- constructed

based on ~, which generalizes recursively "jump").

type n+2 the functionals

[11].

though not necessarily

K l e e n e - type hierarchy (This hierarchy

recursively

The proof of this equivalence

expands primitive

since~may

not be a

uses normalization

for To(~)-

342

H. S c h w i c h t e n b e r g ,

S.S. W a i n e r

As a consequence we obtain a n e g a t i v e answer to the second p r o b l e m above as follows . are p r e c i s e l y

The type n+l

the functionals

Ha n+1 , a E 0 U + 1 [ 5 ] . Ha , aE

0

functionals d e f i n a b l e

in To(n+2E)

o b t a i n e d in K l e e n e ' s hierarchy

But M o s c h o v a k i s

[7] has shovm

that the h i e r a r c h y

does not exhaust the 2 - s e c t i o n of 3E .

In c o n n e c t i o n w i t h the first p r o b l e m m e n t i o n e d above F e ~ e r m a n [4] has r e c e n t l y o b t a i n e d a new d e f i n i t i o n of full r e c u r s i o n in higher

types w h i c h , a l t h o u g h not f o r m u l a t e d as a system of terms, is

n e v e r t h e l e s s m o t i v a t e d by the idea of autonomous we investigate ways of g e n e r a l i z i n g

the autonomous

so as to obtain complete c h a r a c t e r i z a t i o n s (The obvious idea is first to allow d e f i n a b l e functionals f u n c t i o n s as in T o.

In w

sequencing scheme,

of h i g h e r - type r e c u r s i o n

"long" s e q u e n c e s , e n u m e r a t e d b y

of a r b i t r a r y pure But

enumeration.

type, rather

than just

this is i n s u f f i c i e n t as it stands, and n e e d s

to be m o d i f i e d further.)

This leads to a h i e r a r c h y of systems of

terms To,TI,T 2 and Long Partial

Terms,

the last one of which turns

out to be nothing other than a r e f o r m u l a t i o n of F e f e r m a n ' s d e f i n i t i o n

[4] 9 w

The System To(~) of Infinite

Terms.

Type symbols are 0 and w i t h o-,T also ~

T for

(o-1-*(o-2-*...(0-n~V)..)).

symbols are denoted b y ~ , ~ defined)

Let M

A

M

= M

(T -~ ~"

, i.e.

be

the class of all

Mo = N ,

(set-theoretic)

the natural n u m b e r s , and

M , the set of all mappings from M T

into M

0-

U Mqr are denoted b y ~, F , G, H, ~,# and finite

. E l e m e n t s of q-

sequences of them

etc.

We fix a f u n c t i o n a l ~ o f will

a (canonically

T

~-

F,G,~

As usual we write

Finite s e q u e n c e s of type

etc. and we let ~ I b e

code n u m b e r of T .

f u n c t i o n a l s of type T

(0--~7) 9

be b u i l t

up

a r b i t r a r y type ~ .

from v a r i a b l e s

'

x% ,

The t e r m s ......

.

for

of

To(~)

each

H.

type ~ , the s y m b o l ~ , k-th primitive abstraction

Schwichtenberg,

~3

Wainer

and for each k ~ 0 a constant

recurslve

f u n c t i o n , by means

and autonomous

the introduction.

S.S.

formation

Pk for the

of application,

of sequences

as described

Each term will have only finitely-many

in

free

variables.

We define

inductively

(i) a set C ~

N of codes,

term ta denoted by the code a ~ C ~ , (iii) a function for each a g C ~ , Typ a sequence

of variables

for each a ~ C ~ t h e assignment by Typ

(a) determines containing

value

all variables

in t a, (iv)

a type-preserving

of variables

determined

(a).

type of t a and ~--TI,...,T n is to be

the form ~ , ~ w h e r e

thought

sequence

~ = Xl,...,x n of free variables

variable

x~i if T i is the J-th occurrence

this ~ we also write

be clear

that ~ contains

variables

~=~-1,...,~-n

of as determining

in t a (i.e.

ta(~) for t a .

~ is t h e the

x i is to be the

of that type symbol

in ~ ) .

From the definition

it will

all of the free and none of the bound

of t a .

I (Variables)

a = e

.

II (Application)

Typ (a) : ~ , ~ - ~

C ~ if I ~< i ~< n and

' ta : t a ( ~ )

Let a 1,a 2 ~ C % where Typ

Typ(a2 ) =~[,or ~ . Then a = ~2,a I,a2~ ~ C ~ , Typ and

free

of t a under

to the sequence

For each a ~ C ~ Typ (a) will have

With

Typ such that

the type of t a and furthermore

[a] E (in ~ ~ )

of F = F I , F 2 , . . . , F n

(ii) the

[a] ~F = [al]-F[a2] -F "

= ~ i and [ a ] ~ = F i

(a I ) = ~ ~ ,0--~~ and (a)= ~,p~, t a = (tal

H. Schwichtenberg,

344

S.S. Wainer

III (Abstraction) Let a I E C~ and Typ(a I) =r~=m,p~

.

Then a = < 3 , a I ~ C ~ , ta(X)=~y0 ta1(X,y) jSyp Ca).= r~,~_.p~ and [ a ] E G =

[al]E'G for all G ~ M

.

IV (Autonomous Sequences) Let a I E C$, Typ(a I )=r0,0~and for all n, [ a l ] n = b n E C ~ a n d a=

,al>

C%

Typ (bn)=CT,0n.

ta = <

,

Typ

Then (a)=r,r,O -~ -

0"1

and

[aSFn = [bnSE for all n e ~ . V (Primitive Recurslon) Let al,...,anE 0~ where n ~ 0 is the number of arguments of the k-th. primitive recurslve function Pk and for I ( i ( n Typ (ai)=r~,0~~

Then a = < 5 , k , ~ ,

al,...,an> E C~,

ta=Pk(tas,...,tan) , Typ (a)=T,0~and

[a]E=pk([a 11E,..t.Ian] ~)

VI (The Constant ~) a=<6,r1~ e C~ , ta= the symbolS, Typ(a) = rZ, ~ and [alE= ~.

Obviously Typ can be chosen as a primitive recurslve func ti on. To(Y) is the set of all terms ta, a~ C$ .

We want to

normalize the terms of To(~), that is eliminate all subterms of the form (kxt)s.

For a system of nonconstructive infinite

terms (in the sense that no restriction is imposed on the formation of infinite sequences) this was done by Tait [10], extending earlier work of Lorenzen, Novikov and ~ch{Jtte concerning infinite proofs. Tait's paper.

We assume here some knowledge of

Now it is moreor less standard how such operations

on nonconstructive infinite terms can be paralleled by operations on their constructive counterparts such as terms in To(~ ) or, more precisely, codes in C ~ (see e.g. Feferman [41, Lopez-Escobar [6]~Schwichtenberg [9])

9

Hence we do not give proofs but

merely state the proper lemmata, following mainly Feferman [4].

H. Schwichtenberg, S.S. Wainer

345

Most of them (Lemmas I-4) are proved using the primitive recursion theorem.

The type level L T of a type symbol T is defined by L o = 0 , L(~ ~ T ) = max (Lr+ I, L r),

The rank of a code

aE C~is defined as the supremum of the type levels of all subcodes of the form a1=<3,.,.> occurring in a context ~2,al,a2>.

Mo~e

precisely we inductively define Ra for a ~ C ~ a s follows R =

0

R <2,ala2> = m a x (RaI,Ra2,La I) if a I has the form <3,.,.>. = max (Ra1,Ra 2) otherwise. R <3,aI>

= Ra I

R <4,~u, al>= max (Ra I, s~p R[a 1]n) R < 5 , k ~ l, al,...,a n > =

max (Ral,..,,Ra n)

R <6,F~>= 0 . -~,~. Here La I = L ~ where Typ(al)-C

C!early we have Ra ~ ~.

A code a e C~ (and the corresponding term ta) is called irreducible or normal if and only if R a = 0 . Lemma I (Extension) There is a primitive recursive function Ext such that for all a E C}and all types ~ the following holds. Let Typ(a)=~ ,p~. r~,~p~

Then Ext (a,r~~) g C~, Typ(Ext(a,r~)) =

R Ext(a#~)= Ra and for all G,~ of the appropriate

types, [a]E= [Ext(a,ro~] G'E .

Lemma 2 (Interchange)

There are primitive recursive functions PiJ such that for all a E 0 ~ the following holds. Let Typ(a~)=r~ , ~-~ . Then Pij(a) e G~ , Typ (plj(a))= r~lj(~) ,o-q , R P l j ( a ) = R a and

346

H. Schwichtenberg,

for all ~ of the appropriate where ~lJ interchanges

S.S. Wainer

types ~ [a]~= [Pij(a)]~lJ(~),

the i - t ~ a n d

J-th. components in the

respective n-tuple.

Lemma ~

(Substitution)

There is a primitive recursive function Sub such that for all a, b E C~ with Typ (a) =r~ , X ' PN and Typ (b) =r~, ~w the following holds. Sub ( a , b ) E C ~, Typ (Sub(a,b)) = Z~ ' P .I, R 8ub(a,b) ~< max(Ra,Rb,Lb) for all Z of the appropriate

L emma 4

types,

[a][b]F'~= [Sub(a,b)] E 9

(Reduction)

There is a primitive recurslve function Red such that for all m and all a g C~wlth Ra ~< m+l the following holds~ Red ( a , m ) = a ~

~, Typ(a')= Typ (a), Ra ~ ~< m and for a l l F

the appropriate

types, [a I ]F=

~orma~ization

[a]E

of

.

Theorem I

There is a primitive

recursive function N such that for

all a g C~the following holds , N(a) = a ~

C~, Typ(a ~) = Typ(a) ,

a ~ is in normal form, i.e. R a * = 0, and for all ~ of the appropriate

types, [a~]E= [a] -F .

Each term ta in To(~) defines a functional, namely kF.[a] -F, whose arguments occurring in the term. characterization

correspond

to the free variables

We wish to give a recursion-theoretic

of the functionals

definable in To(~), and

since arbitrary finite types can be canonically coded into pure types it will henceforth be more convenient restrict attention ~=a1'''''am

for us to

to those functionals h whose arguments

are of pure types ~< n and whose values are of

type 0. ~ is now assumed

to be an arbitrary but fixed type

and

H. Schwichtenberg,

347

S.S. Wainer

n+2 object.

If h(~1,...,~ m) is definable in To(~) then it is defined by a normal term of type O.

Such a term can only be either

a variable of type 0 or a term of the form Pk(Sl,...,sr) where sl,...s r are normal terms of type 0, or else a term of the form st where s and t are normal.

In this latter case s

cannot be of the form ((SoSl)...)s k with k ~ I since s o would then have to be a variable of impure type, so s must be either ~ o r

a variable of pure type ~ I or a term of the form

where kx.a x is defined by a normal term. Hence t must be either of type 0 or else of the form ky.t' where t ~ is of type 0 (If t were of the form then we could replace it by ky. Y) 9

Thu~ it

is clear that each of the functionals h(a1,...,~ m) definable in To(~) can be generated by means of the schemes 4,...,7 below. The converse,

that the functlonals generated by schemas I,~..,7

are all definable in To(~), should be clear and can easily be proved by a simple application of the primitive recursio~ theorem. Each scheme defines a functional h e where the index e codes up (in the usual way) all relevant details of th~ particular being applied.

We now let ~ = xl,...,x k denote variables of

type 0, ~ = ~ 1 , . . . , ~ m

variables of pure types ~ n and # a variable

of the appropriate pure type ~ n 9

1.

he(~,~)= Pk(~)

2.

he(~)

=r

3.

he(S)_

= ~ j ( ~ . h e l (~,~))_ where type of ~ j > 1,

4. he(g) 5.

scheme

where type ~i = I .

= }(X~.he1(a,~)) )

provided that for each x, he1(X) is an index for a functional with arguments ~ .

~8

H. Schwichtenberg,

S.S. Wainer

6.

he(s):~ he1(he2(g), 5)

7.

he(~)= he1(~' ) where ~' is some permutation of ~ . To be precise,

the above schemes should be interpreted as a

simultaneous inductive definition of a set of indices e, and for each index e a functional h

We believe however that the

e

intention is clear. w

The~-hierarchy. We now develop a recursion-theoretic hierarchy based on a

fixed but completely arbitrary type n+2 object ~, and prove that the functionals of type ~ n+1 appearing in the hierarchy are precisely those functionals definable in To(~) 9

The hierarchy is

just a generalization of [11] to higher types. Let lelF(~) , e < ~ , be a standard enumeration of all functionals (with arguments ~ of type ~ n) primitive recumsive in a type n+1 object F (in the sense of Kleene [5]).

We assume ~elF(~)= 0 if e

is not an index for a functional of the appropriate string of variables. We associate w i t h ~ a n ~(F)

(<x,~>)=

operator~defined

as follows

<[x~F(~,on),~(X#.[x~F(~,#))>

The ~-hierarchy is then obtained by iterating ~ o v e r a simultaneously generated set of ordinal notations.

Note however that the word

"hierarchy" is used in a rather broad sense here, since ~ m a y not be a jump operator in the usual sense (and a l t h o u g h ~ r a i s e s recursive degree" it need not raise "degree").

"primitive

As a result of this

our hierarchies will not in general have the uniqueness property. Definition.

I 1

and

for a c O ' a r e

inductively defined as

follows, where ~,# are variables of type n. will usually drop the superscript ~ )

(Since ~ i s

fixed we

H. Schwichtenberg, S . S . (1) I E 0 ,

"/(b
Wainer

~9

9

(ll) If a E O t h e n

2 a E o , b ) =<~x]a(=, on),~(kp.~x| a(~,#))~

where On here denotes the zero type n object. (Ill) If a E 0 a n d , = ~e~ Fa is a function such that , ( 0 ) = a , , ( m ) e and $(m) ) = F~ (x) (=) " Clearly if a
Examples (I)

I f ~ i s of type 2 then the above hierarchy exhausts the

l-section of ~(see (2)

[11]

)9

If ~ is the functional n+2E which introduces quantification

over type n then the above definition gives an alternative version of Kleene's proposed hierarchy of hT"per-order n+1 predicates ~ ] . Our definition differs from Kleene's particularly in the formation of limit levels, where we insist that fundamental sequences ~ b~ primitive recursive (rather than Just recursive) in previous levels ~ However standard methods show that the two definitions give rise to the same class of predicates and functlonals (and coincide at limit stages).

~9schovakls KT~ has shown that~ for n = I , the

hierarchy does not exhaust the 2-sectlon of 3E (nor the l-section of 3E). (3)

If~Is

the superJump functional we obtain an alternative

version of Platek's hierarchy [82 but again, Aczel and Hinman 11~ have shown that this does not exhaust the 1-sectlon of the super Jump.

0

350

H. Schwichtenberg, S.S. Wainer

Limit Property. There are primitive recursive functions M and N such that if for each m,k~.G(m,g)= ~$(m)~ =r F F , , = ~e~ and a=r ~o r 3asM(e)E0, r

where
then


Proof Choosa

F F M so that [M(e)~ a(0)=a, [M(e) 1 a(m+1)= 2r

.

Let <~a>n denote a standard primitive recursive coding of a sequence Z as a single type n object, and lets o be a primitive recursive function such that ~So(J)IF(n, on ) = (F2r (m)(<8o (~ (m)) '<~>n>))o = (F3asM(e)(<m+1,<So(~(m)),>)) 0 ~ NOw let m n denote the type n object with constant value m and let S I be a primitive recursive function such that IS1(J)JF(mn,o n)= ~J]F(m) for any type n+1 F object F. Then ~(m)= li~ a(m)= ~S1(i)~Fa(mn,0 n) = (F2a(<Sl(1),mn>))o = (F3asM(e)(<1,<S1(i),m~>>)) o.

We therefore have

S(m,~) = (F3asM(e) (<m+ I ,<8 o (F3aSM (e ) (> ) )o'n>> ) )o and it remains to choose N so that N(i) is an index of this expression as a function of m and ~, primitive recumsive in FSasM(e)" Lemma ~. There are primitive recursive functions I and C such that if e is an index of a functional h e defined by schemes I,...,7 then for any b E 0 ,

C(e,b)g0,b
Proof First note that the arbitrary bg0appears because in order to deal with scheme 5 we need to locate k~.he(X+l,m) above

H. Schwichtenberg, k~.he(X, ~) i n O s o

S.S. Wainer

351

that the Limit Property can then be used

to piece together the whole functional kx,~.he(X,~).

Also in

dealing with scheme 6 we will need to locate h

above h . eI e2 These complications arise because there is no corresponding

Uniqueness

Property for an arbitrary~-hierarchy,

since

Uniqueness requires quantification and we do not in general have 2E recursive i n ~ .

I and C will be defined simultaneously by the primitive recursion theorem, with induction on the definition of h e by schemes I,...,7. Suppose h e is defined by I, so from e we can find k so that he(X,~) = pk(~) . Clearly there is a primitive rgcursive function fl such that fop any F of type n+1 , ~f1(k)]~(x,~)=pk(~).

Thus

we only need to put I(e,b)= f1(k) and C(e,b)= 2b in this case. Suppose h e is defined from hel by 2,3, or 7.

By induction

hypothesis we can assume b
If h e ( ~ ) = ~ k ~ . h e 1 ( ~ , ~ ) ) by scheme 4 then again by hypothesis we can assume b n,~) . Therefore he( ~ ) = (k~. If2( e, I(e I ,b) ) ~FC(el ,b) (<~>n,~)) = (F2C(e I ,b) (
352

H. Schwichtenberg, S~

Now put C(e,b)= 2

C(el,b)

Wainer

and h e is clearly primitive recursive

in FC(e,b) with index I(e,b~ primitive recursively computable from e and I(el,b).

Next suppose h e is defined by scheme 5 9 Then he(X,~)--hhe~ ) where, by the induction hypothesis, I

b
Define ,(0)=C(el,b)

and , (m+1) -- C(hel (m) ,, (m)) , and define ~(0)= O, ~(m+1)= I(h e (m),,(m)) . Then for each IF m,ka.he(m,~) = [~(m+1)~ ,(m+1) where , and ~ a~re primitive recursive in FC(el,b) with indices z and i primitive recurslvely computable from C(el,b), I(el,b) and primitive recursive indices of C and I.

Also ~(0)= C(el,b) ~
every m, by hypothesis.

Therefore by the Limit Property, h e

is primitive recursive in FC(e,b) with Index I(e,b) where C(e1'b?sM(Z ) C(e,b)= 3 and I(e,b) is given by a simple primitive recurslve function of N(i).

Finally suppose he(~)= hel(he2(~),~) by scheme 6. By induction hypothesis we can assume b < o C(e2,b), he2= |l(e2,b)l

C(e2'b) and for all d~O, d
he1= ~I(el,d)~C(el 'd) .

Define ,(0)=C(e2,b),r

and $(m+2)= 2~(m+1) . Then ~ is primitive recurslve (and hence primitive recu~sive in FC(e2,b ) with an index u primitive

H. Schwichtenberg,

S.S. Wainer

353

recursively computable from e,b, and a primitive recur sive index of C 3C(e2'b)su

Also r 0"

<

o

r

c(

Put C(e,b) = 3

by hypothesis and so e2'b)5u.

Then b < o C(e2,b)< o C(e,b)

and since FC(e2,b )=k~.FC(e,b)(<0,~ >) it follows that he2 is primitive recursive in FC(e,b) with an index primitive recursively computable from I(e2,b).

Now for some fixed

primitive recursive function f3 we have

hot(X,~) = II(el,C(e2,~))~F~(1)(x,~) = ~f3(l(e1,C(e2,b)))~Fr

n, O n )

= (F,(2)(n>)) o = (FC(e,b) (<2'n>>) )o Thus hel is also primitive recursive in FC(e,b) with an index primitive recursively computable from e,b and primitive recursive indices of I and C.

Hence h e is primitive recursive

in FC(e,b) by Kleene's scheme S&, with index I(e,b) given as a primitive recurslve function of I(e2,b), e , b , and primitive recurslve indices of I and C.

We give I and C the value 0 if none of the above cases applies. Inspection of the above cases shows that C(e,b) and I(e,b) are defined simultaneously from C(el,b) C(e2,b), I(el,b) I(e2,b),e,b and primitive recursive indices of C and I.

Since el,e 2 < e the

simultaneous definition is a primitive recursion on e.

Therefore

by the simultaneous primitive recursion theorem (e.g, Lemma 2.1 of [2]) we can indeed find primitive recurslve indices of C and I which satisfy this definition ~ This completes the proof.

354

H. Schwichtenberg,

S.S. Wainer

Next we show that every functional G(~) , with arguments ~ of pure types ~ n and with values of type O, which appears in the~-hierarehy, is definable by a term of To(~) 9 Lemma 6 There are primitive recursive functions p and Pl such that if the type n+1 functional F is defined by a term t c of To(~) thenle] F is defined by the term tp(c,e) of To(~) and x,~.

Ix~F(~) is defined by the term tp1(c) of To(~) .

proof We first define p by the primitive recursion theorem with cases corresponding

to the schemes 8o,...,$8 by which ~e~ F

is defined. In this proof and the next, u,v will be used to denote ~ariables of To(~) of the appropriate types ~ If ~e~ F is defined by $I,$2,$3 then [e~ F is Just a primitive recursive function of its numerical

arguments and

so p(c.e) is given explicitly as a function of e .

If ~elF=k~.~eIDF(le2~F(~),~) assume inductively that tp(c,el ) defines ~e2 ~F.

through S~ then we can defines le1~F~and tp(c,e2 )

Therefore ~e~ F is defined by the term

~ . tp(c,el ) (tp(c,e2)~)~ and we can clearly compute p(c,e) as a primitive recursive function of p(c,e I) , p(c,e 2) and e.

If ~el F is defined by $5 then lelF(0,~)= ~e1~F(~) and lelF(x+1,g)= le2~F(le~F(x,~),x,~) inductively that tp(c,el )

where again we can assume

defines ~e1~F and tp(c,e2 ) defines

le2 ~F . Now let r(O)=p(c,e I) and r(x+1)= the code for the term ~ .

tp(c,e2 ) (tr(x) ~) x ~ ~

Then for each x, tr(x)

H. Schwichtenberg,

S.S. Wainer

defines k~.~e~F(x,~) and therefore < t r ( x ~ x ~

355

defines le~ F .

But r is primitive recursive, with index i primitive recursively computable from p(c,el)

p(c,e2) and e.

Hence we can primitive

recursively compute from i, first a code for the term defining r, and then the code p(c,e) for the term < t r ( x ) > x E g

which defines ~e~ F.

The cases where ~e~ F is defined by S6 and $7, corresponding to permutation of arguments and function application, are trivial.

If lelF(g)=~i(k#.~e1~F(~,#))

through $8 then it is easy

to define p(c,e) primitive recursively from e and p(c,e 1) such that t p ( c , e ) = k ~ ,

ui(kV.tp(c,el)~V)

9 The case S0 is treated

similarly, replacing ~i by F and u i by t c 9 It is clear from the above cases that p is primitive recursive, as required.

To define Pl simply note that k x~. defined by the term <tp(c,x)> x E g ,

Ix~F(~) can now be

whose code is given as

a primitive recursive function of c.

Lemma

7

There is a primitive recursive function q such that if a~ O ~ t h e n

q ( a ) E C ~ and tq(a) defines F ~ a

q

Proof Again by the primitive recurslon theorem. so that t q ( 1 ) = k t ~ O

.

Define q(1)

Now assume tq(a) defines F a.

Since x = < x , ~ > o (0) and ~ = < x , ~ > 1

there are terms tk and t$

356

H. Schwichtenberg, S.S. Wainer

which define the decoding functions k~.~o(O) and k~.a I 9 Fa P But F 2 a = k a . < l ~ o ( O ) ~ (~I,0 n),~(k#.I~o(O) ~ a(a1,#))> and so F2a is defined by the term ku. <tpl (q(a))(tku)(~u)0n' ~(kv. t l

(q(a))(tku)(t~u)v)> whose code q(2a) is clearly given as

a primitive recursive function of q(a~. ks. F e~Fa(~o(O)) (~i) , so if r inductively that Fr

If 3a5 e g O t h e n F3ase=

~e~ a we can assume

is defined by tq(r

) for each x and

therefore F3ase is defined by the term ku.
~ (tkU)(~u) 9

Now r is defined by the term tp(q(a),e ) and so kx.q(r

is

defined by a term whose code is primitive recurslvely computable from q(a), e and a primitive recurslve index of q .

Thus we can

compute q(3a5 e) primitive recursively from q(a), e, and a primitive recursive index of q, so that tq(3ase ) is the term ku.
which defines F3a5e ~

The

primitive recursion theorem then provides an index of q satisfying the above definltlon, and this completes the proof.

Putting the above results together we have Theorem 2 A functional with arguments of pure types ~ n and values of type 0 is definable in To(B) if and only if it is primitive recursive in F~a

for some a ~ 0 ~

Corollary If~Is

of type ~ 2 then the functions definable in T0(~)

are precisely the functions recursive i n ~ .

But for ~ o f type >i 3 the functions definable in TO(~) do not, in general, exhaust the 1-sectlon of ~ .

H. Schwichtenberg, @3.

Extensions

S.S. Wainer

of To(~)

The reason why To(~) f o r ~ o f Kleene recursion

in'seems

type level ~ 3 does not give full

to be that sequences used to build up

terms in To(~) are indexed

by natural numbers and so each term

can be regarded as a countable well-fotmded Kleene-computations Thus it is tempting

in types ~

3

tree , whereas

are in general uncountable.

to allow sequences

objects and to consider defined

$57

indexed

by h i g h e r - t y p e

a system T I(B) of infinite

just as To(~) in w

except

terms which is

that clause IV is now generalized

to read as follows IV ~ (Long autonomous

sequences)

r~ 01 and for all ~F~ M Then a - ~ 4 , a l " ~

, [a I ]F = b F ~

But if t a = ~ t b ~ F ~ M ranges over M of b F

and Typ(D F)=Z,r 0 u

F~ G types, [a]~F=

[bF]~G

is a term formed by IV * then as

there can still only be countably- many different .

Thus the "depths"

terms in TI(~) remain countable, adequate

c~

C~ , t - a = ~ t b ~ p ~ M - ~ T ~ p ( a ) = ~ , ~ ~ 01 and f o r

all F,G of the appropriate

values

Assume a I ~ C ~ , Typ(a I ) =

of the trees corresponding so we cannot expect TI(~)

to define all functions recursive

in~.

to

to be

In fact for the

case ~ = 3E we have : Theorem The functionals those definable

of type ~ 2 definable

in TI(3E ) are just

in To(3E).

Proof For i = 0,1 we let C ~ b e and for each a ~ C~ we denote XE.

[a]~ .

the set of codes for terms in Ti(~), the corresoonding

functional by

We show that there is a primitive recursive function p 3E such that if a ~ C I is normal and Typ(a)= r ~ , ~ where ~ is a 3E sequence of types 0 or I, then p ( a ) ~ C o and for all

~8 ~e~

H. Schwichtenbe~g, , [a]~ : [p(a)]~ .

The only non-trivial

a = 4 4 , a l b , Typ(a)= r~,z-, 0 ~.

E

E

S.S. Wainer case is when

Then for all

9

3E

roceeding by induction on

we can then assume that [a]~ ~ = [p[p(al)]~] I .~ow using the function-quantifier

3E we can primitive

p(a I) and a primitive

~n.[b] n

enumerates

[c]~ = # n

recursive

all the values of }~

= [~4,b~]~[c]~

compute,

p([P(al)]o~)

and

Then for all ~,# ,

, and so from b and c we can primitive

recursively compute p a5 such that We finally obtain

Eal

the required p by the primitive

Clearly this Theorem will hold for a n y ~ s u c h definable

in To(~) , and it will also generalize

when relativized

from

index of p, codes b and c such that

([hi n = p([P(al)]~o)).

[a]~_

recursively

recursion

theorem.

that 3E is to higher

types

to 4E,SE etc.

-~tb~ F M

The depth of a term t~4,ai~-

the obvious way by depth(t~4,al>)=

formed by IV ~ is given in

supF(depth(tal)+l,depth(tbF)+1)

and since each b F = [a I ]~g C ~ we are here only taking the supr~mum of c o u n t a b l y - m a n y

(countable)

terms with uncountable

ordinals.

depth is to allow the ~F's to be used as

constants

in t bF, i.e. to let bF~_ C }'-F .

suggestion

of Feferman,

give new systems

we further

extend our systems

types,

define,

simultaneously

fixed ~ = ~ , . . . , F n .

F 1,...,Fn)

for all F of the

[a;F] ~ defined by that term.

[a;~] ~ in order to make explicit

and VI as before

of terms to

a set C ~ of codes and for each a ~ C ~ a term

ta ~ T2(F) and a total functional ~ . We write

Thus, following a

T2(~) as follows.

This time we inductively appropriate

Now a natural way to get

The clauses

in the definition

(but with VI introducing

together with

the relativization

to the

are I,II,III,V

each of the constants

H. Schwichtenberg,

IV ** (Long relativized autonomous

sequence s )

type Z , Type (a I)= ro-,0~ and for all G~ ~ , and Typ(b G)=

oI .

r p_,

Then a = ~ 4 , a l > ~

~9

S.S. Wainer

C -F, ta

Assume a I ~ C-F, F of [a I;F] ~ = b G ~ C~F'~

:
,

Typ(a)= rp~,~ _. O~and for all G e M , HeMp~ [a;~]~

: [ [ a ~ ; F ] -G ; ~,G] ~

.

With F the empty sequence we thus obtain C and T2 , so if we denote the depth of a term ta in T2(~) by laI ~ then the depth lal of a term ta= ~

Ial where

in T 2 is given by

~E~

= sup

(la~l

+ ~ ,

IbGI ~ may now, of course, have uncountably many different

values,

and so lal will in general be uncountable

1,2 in Moschovakis

(cs

definitions

[7]).

We shall show (Theorems 4 and 5) that for arbitrary a with Typ(a)=~Uthe

partial functionals k~.[a;~]

are just the Kleene

partial recursive functionals k~.lel(~) by the lemma below,

It then follows

that the total functionals

exhaust the functionals recursive

[a;~] with a ~ C~ ,

in~.

Lemma For each ~,~

there is a primitive recursive function f such

that (with F,~ ranging over ~ , ~ , (i) ~ 2 ( a ~ C ~'g)

respectively)

~-~ f(a)e C ~

(ii) ~G(a~ C~'~)

~ [a;~,G]~= [f(a);F] ~'~ & lal~'~ < If(a)1z

Proof Given ~,~ , we can easily find a primitive recursive function q such that for a l l ~ , [a;F,G] ~

=

~Mo~,[q(a);F]~= a .

Hence

[[q(a);~]~; F,~]~ [~4,q(a)> ; ~ ] ~ [f(a);Z]~'~

by IV *~

360

H. Schwichtenberg,

with f(a) lemma

depending

primitive

S.S. Wainer

recursively

on q(a).

The proof

of the

is now obvious.

Theorem 4 There

is a primitive

(i) (ii)

recursive

function

g such that

Iet(~)~ ~ g(e)~ C~ Ie](~)$ ~

[g(e);~]=

tel(g)

Proof We shall define the primitive is by cases

g from its own primitive

recursion

depending

The implication proved by induction

theorem

in the usual manner.

for left to right on lel(~)~ w .

to left in (i) is by induction

after

the definition

Case

function

(ii) are

The proof of the implication on Ig(e)I ~ and will

to the cases

= le11(~21(~), ~)

that as in w

Sub such

in (i) together with

from

be clear

$4, $8 and $9, the other

or similar.

84; lel(~)

First note

The definition

is completed.

ourselves

cases being obvious

index using

on the form of e.

right

We restrict

recursive

we can easily

9 obtain a primitive

recursive

that b ~ C ~ implies

(i) [a;[b;2]~;F] ~ ~ [Sub(a,b) ; 2 ~

(il)

lal[b:= ]e '=< [Sub(a,b)lZ and Ibis< ISub(a,b)~.

(However, usual ~2,~2,

note

that if SUbo(a,b)

to t e r m - s u b s t i t u t i o n

is the function

we have to put

corresponding

Sub(a,b)

=

Co, SUbo(a,b)~ , b ~ with C o ~ C -F such that [ C o ; F ] G H I H f = H I )

We now obtain

tel(c~)= [ g ( e l ) ; [ g ( e 2) ; ~],_~] = [Sub(g(el),g(e2)) Hence

it suffices

by ind.hyp.

; ~]

to put g ( e ) = Sub(g(es),g(ef) )

9

as

9

H. Schwichtenberg, Case

88:

S.S.

~el(~) ~- ~j(X~e11(_~,~))

9

g ( e l ) E C ~'~ and le11(~,/~)= [g(el);Z,#] lemma~le 1 ~ ( ~ , # ) = from g(e I).

By ind. hyp.

recursively

It remains

to set g(e) 89:

= [a2;~_]

9

compute

= [a};~]

= a3 9

lel(x,~)

= [g(x);z]

can easily

-~ Ixl(~)

9

By ind.

Now from a primitive

hyp. we can assume

recursive

But then an application

index of g we

of IV m* yields < 4 , a ~

; ~]=

Ixl(~)

C x'~ such that

x , ~ ] = [[a2;x,G] ; x,~ ] = Ixl(_~) and it then remains

to put g(e)=~/4,a2~2

simply

9

~.

There is a primitive (i)

that

a code a I e C x'~ such that [ a l ; x , ~ ] = g(x) and then

a code a 2 E C x'~ such that [[a2;x,z] ; x , Z ] = [[al;x, ~]

Theorem

computable

from a I such that

~j(k#le11(a,E) )

[~4,a2>;

we have

for all # , and hence by the

[a lIZ] p with a I primitive

X~eI~(Z;~)

Ixl(~)

361

It is now easy to obtain a2,a } also primitive

recursively

Case

Wainer

recursive

function

h such that

a e C~ ~* l h ( a ) l ( ~ ) ~

(ii)

a e C~

-* ~h(a)l(~)

= [a;~]

It is fairly straightforward primitive

recursion

Since functionals method terms.

the treatment anyway,

scheme

involved

codes as values.

the

first

under which

of partial

for a more direct

in the context

this is to return

functional

a discussion

to look

recursion

the conditions

such anh using

the details.

IV * may be applied,

that the enumerating

general be partial.

of T2(~)

partial

One way of doing

sequencing

we omit

it seems natural

of introducing

T I (~) and then relax

defined

theorem;

to define

of infinite

to the system the autonomous

by not requiring

any longer

given by a I has only previously

The functionals

so defined

But not only [a] ~ as a function

will now in of G will be

9

362

H. Schwichtenberg.

partial

S.S. Wainer

(as we would like) but also the values [a]~ for certain

fixed ~ may be partial functionals and as such will not even be objects of our underlying domain U T ~ we instead let ta be the term<%2F

To avoid this difficulty M

with

a sequence

of

variables of type ~ , so that the values of [a] ~ , when defined, are natural numbers

(i.e. total objects of type 0) .

This leads to a

system of infinite "partial" terms ta,aE C defined by I, II, III, V and IV *** below (We no longer relativize t o , s i n c e necessary here.

it is not really

One can easily show, for this new system,

is a primitive recursive function ka.a'

that there

such that if a E C~then

a ' ~ C and for all F [a'] ~'~ = [a] ~) . IV ~**

(Long partial autonomous sequences) Assume a I ~ C and

Typ(al)=

fT.,On.

Then a = < 4 , a 1 > ~

C and ta = < t b ~ F ~

b F = [a I]~ and tbF is undefined if bF r C.

where

Furthermore T y p ( a ) = } ,

and [a] E is defi~ed with value m if and only if (i) [al]~ is defined, (ii) [al]~ : b F ~ C

with T y p ( b F ) = } , 0 u, and (iii) [bF] ~ is defined

with value m. Now in what sense do I,II,III,IV ***, V constitute a definition of the concepts a ~ C,t a and [a]~?

The formerly critical point in

the inductive definition of C was the use of quantification over M

in IV (with ~ = 0) and IV*,IV ** (with T arbitrary), which meant T I that C was "at least" a complete H 1 set. But this clause has now

been removed to give IV * ~

and so the new C can be defined independently

of ta and [a] ~ , and is simply primitive recursive of indices for partial recursive functionals).

(as is the set

Incidentally the

primitive recursive function Typ also needs to be redefined so that Typ(<4,a1> )= Typ(al).

We next consider [a] ~.

Since [a] ~ may

now be undefined we need to give a definition of the relation [a] ~ ~ G, to be read "[a] ~ is defined with value G".

This relation

is clearly analogous to Kleene's lel~(~) ~ z and is given by the

H. Schwichtenberg, following

induction

(I)

Variables.

(2)

Application. G2~ M

(3)

S.S.

Wainer

S~

:

r ~ , F: FI,. ..FnE M

[a] E -~ F i if a = ~ 1 , i ,

[a2 IF -~ G 2 where

If [a I]F ~ GI and _

and

_~p and

G I~M

then [a] ~ -" GIG 2 where a = ~ 2 , a l , a 2 ~ .

Abstraction.

If [a I]F'G _~ HG for all G ~ M

then [a] ~-~ H where

a = ~ , a1"~ .

(4)

Long partial

autonomous

then [a] ~ -- m where (5)

Primitive

sequences.

a--~4,a1~.

Recursion.

If [ai]~F -~ m i for I ~ i ~ n k then

[a] ~ -~Pk(ml,...,mnk ) where

a = ~ 5 , k , r ~ ~, al,...,a_~,nk _F~M~

Pk is the k - the. primitive For details.

the "partial"

Notice

a=~4'a1~and

quite arbitrary in particular may have

$9).

the problems

x where b F - [al]~

which

t a starting

(where

Since a I is

the values bF

tree

;

the structure

occurs

of [a] -F from given a,F node.

In such

in the case of

in the case of sequencing

of t a in this case has an infinite branching).

at an inductive mentioned.

definition

functionals .

(I)...(5)

This definition

point of [4].

(as is done in [4]) or by reduction

functionals

to

of ta has only a l-fold branching),

occurs

and is the starting

that the partial

(analogous

ta

through Kleene's

from the outermost

the structure

terms are not explicitly

recursive

about

can arise

an infinite b r a n c h i n g

We have arrived

directly

can arise when

~

but only a 2-fold branching

Feferman,

which

One can think of a computation

a computation,

(whereas

C we omit corresponding

of a non-well-founded

computations

and

function.

t a for a ~

we do not know anything

through

abstraction

terms

F~M

the structure

as working

recursive

we may have bF = a for some F and so in general

the undefined scheme

however,

ta : ~

If [all -F -~ b and [b] -F- : m

X~.[a] -a exhaust

in which

is due to

One can show either to Theorems

4 and 5,

the Kleene partial

3~

H. Schwichtenberg,

S.S. Wainer

REFERENCES. [I]

P.Aczel and P.G. Hinman, Generalized Recursion North-Holland

[2]

S. Feferman,

"Recursion in the Superjump",

in

Theory (Eds. Fenstad and Hinman),

(1974). "Classifications

of Hierarchies",

of Recursive Functions by means

Trans. Amer. Math. Soc. vol 104 (1962)

pp. 101-122. [3]

S. Feferman,

"Ordinals and ~kmctionals

in Proof Theory", Proc.

of Int. Congress of Mathematicians Nice (1970), pp. 229-233. [4]

S. Feferman,

"Recursion in Total Functionals

of Finite Type",

to appear. [5]

S.C. Kleene,

"Recursive Functionals and Quantifiers

of Finite

Types I, II", Trans. Amer. Math. Soe. vol 91 (1959) pp. 1-52, vol 108 (1963) pp. 106-142. [6]

E.G.K. Lopez-Escobar,"Remarks Constructive

Formulas",

on an Infinitary Language with

Journ.

Symb. Logic vol 32 (1967)

PP. 305-319. [7]

Y.N. Moschovakis,

"Hyperanalytic

Predicates",

Trans. Amer Math.

Soc. vol 129 (1967) pp. 249-282. [8]

R.A. Platek,

"A Countable Hierarchy for the Superjump",

Logic Colloquium (1971)

[9]

'69 (Eds. Gandy and Yates) North-Holland

9

H. Schwichtenberg, "Elimination of Higher Type Levels in Definitions of Primitive Recursive Fnls.by Transfinite Recursion to appear in Proc. of Bristol Logic Colloquium 1973 (Eds. Rose and Shepherdson),

[lo]

North-Holland.

W.W. Tait, "Infinitely Long Terms of Transfinite Formal Systems and Recursive Functions Dummett) North-Holland

[11]

in

S.S. Wainer,

Type", in

(Eds. Crossley and

(1965).

"A Hierarchy for the l-Section of Any Type Two

Object", Journ.

Symb. Logic vol 39 (1972) pp. 88-94.

Consistency Dedicated

Proofs and Ordinals

to Kurt Sch~tte

on the occasion

of his 65th birthday Gaisl Takeuti

Proof theory began with Hilbert; Hilbert's program.

more precisely,

it began with

Much has been written about Hilbert's program,

by Hilbert and by others, and a variety of interpretations objectives

are possible.

school was doing, Hilbert's

Nevertheless,

from Hilbert's

reaction

influence

to the mathematical

its consistency

Hilbert's

on other people,

events of his day, it is my view

a +b

= b +a

finite standpoint.

sive operation.'

1 + 1 + l,

given before us, and

A consistency

proof,

For example,

we

form.

in the finite standpoint,

"Suppose a proof-figure

interpret

'finite operation'

usu-

to a

By a 'general

statement'

free deduction.'

to mean 'primitive

However,

his finite standpoint his standpoint

Such people

there is reason to believe to encompass Ackermann's

is beyond quantifier-free

and lies close to Gentzen's

standpoint.

this interesting historical

recur-

of a Gedaaken experiment,

finite standpoint with quantifler-free

investigate

Ne can finitely

for arbitrary numerals a and b from Hilbert's

they mean a 'quantifier

arithmetic.

by

is given before us."

Many people

Hilbert's

for mathematics

as a Gedanken experiment.

ally starts with the following contradiction

is the following:

figure like

infer a general statement can infer

and from

from his finite standpoint.

finite standpoint

operate on a concrete

of Hilbert's

Judging from what Hilbert's

that his main objective was to provide a foundation proving

both

then identify

primitive

recursive

that Hilbert

function.

primitive

intended

If so, then

recursive

arithmetic

I would hope that someone might question

concerning Hilbert's

366

G. Takeuti

standpoint

and Ackermann's

function.

From a modern viewpoint Hilbert's program sounds rather strange and there are at least two reasons for this: 1.

In Hilbert's

set theory. confidence

day people

really worried about contradictions

But today people have great confidence is based partly on their experience

in set theory.

in This

and partly on habit and

simply not thinking about the subject. 2.

GSdel's

incompleteness

Hilbert's program completely. proofs now require a method is nevertheless impossible

theorem has changed the meaning Because

that is finite

(or constructive)

very strong when formalized.

lies completely within Hilbert's of the accessibility

from transfinite

not transfinite

of Peano's arithmetic.

finite standpoint

induction,

sequences

of ordinals.

to extend that of Hilbert's.

standpoint

in the following way: figures,

experiment

Thus by Gentzen's

S,

is

it was necessary

for his

Let me formulate Gentzen's

We can finitely operate on a concrete

and

..-.

standpoint

The part

S

we can finitely operate

that is given before us.

be primitive

the order relation,

addition,

state-

... will be explained

further we need the following

Let a system of ordinals i.e. the set

Accessibility

given before us, and infer a general

initial part of a sequence standpoint

e 0.

that we need for consistency proofs.

standpoint

ment as a Gedanken

except for his proof

induction and it is usually accessibility,

Since Gentzen had to deal with sequences

sequence of concrete

His proof

of the ordinals we mean that there are no

strictly decreasing

Gentzen's

The situation

of the ordinals up to the first ~-number,

By the accessibility

finite

think this is

the continuum hypothesis.

Gentzen proved the consistency

later.

People

but which

similar to that of finding a new axiom that carries convic-

tion and decides

different

result consistency

or at least unlikely and extremely difficult.

is somewhat

infinite

of GSdel's

of

on any To discuss

definitions.

recursively presented, multiplication,

and

G. Takeuti exponentiation,

etc. are primitive

be a concretely given sequence Definition. al,a2 9

,

867

recursively

of elements

described.

of

Let

al,a 2, ....

S.

An element a n is a terminal point of the sequence

if and only if

l)

the sequence

2)

a n ~ an+ 1

terminals at

Definition.

A sequence

if and only if

~1,~2,...

an ,

or

~1,~2,...

is a fundamental

is a strictly

increasing

sequence

for

sequence and

is the limit of the sequence. By a p r o o f a method

M

of the a c c e s s i b i l i t y

of the ordinals If

F r o m the following less than

a I < ~,

Mn

Let

with

al < ~n"

then it is sufficient

to check

of

concretly

the a c c e s s i b i l i t y

a I +l

initial

al,a 2, ....

be a fundamental

sequence

for

~

and let

for finding a terminal point

for every sequence

Then we can easily construct

a method

terminal point for every sequence then we can easily find a fundamental

for every sequence

sequence

for

of

~0"

~l < ~2 < "''

be a method

we mean the c o n s t r u c t i o n

facts we can prove

terms in order to find a terminal point 2)

S

for finding a terminal point

g i v e n before us.

l)

of

~n ~.

al,a2 9

with

with

al < ~n

We then apply

to find a

a I < ~:

because Mn

M

to

al,a2,---

If

a I <~,

~1,~2,... al,a2,..,

is a and

find a terminal point. 3)

There

fundamental

is a very simple

sequence

systematic method

for any limit ordinal less than

A proof of the a c c e s s i b i l i t y as follows:

for c o n s t r u c t i n g

By l) we prove

that

of the ordinals ~

~0"

less than

is accessible.

r

Repeating

finite number of times 9 we prove

that

~, ~ + m, J ~ + ~ + m, ...

accessible.

that

m2

this

9

Then by 2) we prove

we prove

less than

e0

that

~2

is accessible.

a~, ..., ~ ~ ,...,~ ~ ,-..

is accessible.

and

a

goes l) a are each

Repeating

hence each ordinal

G. Takeuti

368

This proof is very clear and transparent the primitive theless,

recursive

structure

if one is familiar with

of the ordinals

~0"

Never-

there are many people who argue that Gentzen's proof does not

increase our confidence

in the consistency

of Peano arithmetic.

one believes from the outset that the consistency is obvious,

then of course this is the case.

one that I share for the following definition l, l + l ,

less than

of the natural numbers.

l+l+l,.--.

reason.

of Peano arithmetic

This view is however not There are two choices for the

One is a recursive

The other is a definition

definition

With the first definition we are lead naturally of quantities

problem of foundations

proof, which consists

of the elimination

bility proof for the ordinals does add to my confidence

difficulties.

of what the

Anyway since I a m a

am very familiar with the magic of quantiflers

less than

Gentzen's

of quantifiers

logician and consistency

and are accessi-

e0' is greatly reassuring.

in the consistency

Nith

to a conscientious

and here we become conscious

really entails.

i.e.

inside set theorem.

the latter definition we encounter many deep foundational

interpretation

If

It

and truth of Peano arithme-

tic. Although malization

the accessibility

is not simple.

The same method

and the number of repetitions becomes

larger and larger.

the 'furthermore' finitely operate operations

proof described

is repeated many, many times

becomes higher and higher as the ordinal

Analyzing

part of Gentzen's on concrete

above is clear its for-

this situation we can now complete

standpoint:

operations,

etc. and infer a general

"Furthermore

concrete

operations

we can on concrete

statement about them, as a Gedanken

experiment." About 25 years ago, I first sought to carry out Hilbert's program through cut elimination the following 1.

in type theory.

This effort was organized around

ideas:

I noted that combinatorial

been very successful

reductions

in all my examples.

to eliminate

Consequently

cuts had

I hoped to obtain

G. Takeuti a proof of the cut elimination

369

theorem that would be very combinatorial

and hence constructive. 2.

Cut elimination

interesting mathematical Sch~tte

implies the consistency problem

investigation

theorem that lead to nonconstructive

case.

of the cut elimination

solutions by Tait, for the second

by Takahashi

It should also be mentioned

and Prawltz

that Sch~tte

for the general

introduced

duction which makes the meaning of cut elimination cases.

and is an

in its own right.

started a semantical

order case, and independently,

of analysis

infinite

in-

clearer in several

Later Girard found an Ingeneous proof of the cut elimination

theorem and Martin-LSf

and Prawitz,

to prove a normalization

theorey,

independently,

used Girard's

idea

which is a sort of elegant variant

of cut elimination. Much earlier than this I introduced cut elimination of

theory for

1 ~l-comprehension

proof,

ordinal diagrams and proved a

~-comprehension

axioms and the consistency

axioms with generalized

like Gentzen's,

inductive

lies completely within Hilbert's

point except for the proof of the accessibility Systems of ordinal diagrams are complicated. have clarified

definitions. finltist

My

stand-

of ordinal diagrams.

Sch~tte and his school

several relations between ordinal diagrams and other

ordinal notations. The ideas involved in the proof of the accessibility diagrams are, at this time, much more abstract the proof of the accessibility and I are improving

of ordinals

than those involved

less than

the proof of the accessibility

but we have not yet succeeded

in reducing

that of the proof of the accessibility

of ordinal

eO"

Marlko Yasugl

of ordinal diagrams

its abstraction

of ordinals

in

to the level of

less than

e O.

MARKOV'S PRINCIPLE AND MARKOV'S RULE FOR THEORIES

OF

CHOICE SEQUENCES

Dedicated to Kurt Sch~tte on occasion of his 65 th birthday.

A. S. Troelstra

w I.

Introduction.

[K,T],

IT4].

We shall freely use notation and terminology from

The following forms of Markov's principle will be discussed:

MpR

Vxy(m ~ S z T ( x , y , z )

~ SzT(x,y,z))

~R

Vn(V~ ~ - S X R n ( ~ , x )

~

~

Vn(~y.1 ~XRn(~,~)-~i~.I

where of

R

V~XRn(~,x)) ,

and its weakening

~XRn(~,x)),

is a standard enumeration of all primitive recursive predicates

n

~,x

M

Va[~mZx(ax=O)

Mc

v~[~x(=x=

MI

V~[VS(A(~,8) V ~A(~,8))

c

,

-- ~ x ( a x = O ) ]

o) - ~ ( ~ x = o)]

,

,

& ~8A(~,8)

"

~8A(~,8)]

9

Assuming the lawlike functions to contain all primitive recursive functions, and all lawlike functions to be equal to choice sequences (i.e.

VaZ~(a= ~))

it follows immediately that

MI~ C

M

C

-~ M-~ Mp

as we shall show, in

R ~

c

C~S also

M

c

~ MI . c

The principal observations and results of this little note are as follows : I~ ) for any notion of choice sequence satisfying a few simple axioms relative to lawlike sequences,

Mc ~ MpR * - ~ -1 CT

A.S.

where

CT

(Church's

thesis for lawlike

CT

VaSxVySz(Txyz & ay = Uz)

(see

~ction

2).

Troelstra

As a corollary,

Mc

sequences)

2 ~) M c [K,T]

relative

can be extended to

in

as

C~S.

between completeness

logic, Markov's principle

is consistent

is formulated

is underivable

In section 6 we discuss the relationship istic predicate

371

of intuition-

and Church's thesis.

C~S; in fact, the elimination result of

C~S+ M c

on the one hand and

ID.~BI + M + K ~ = K

on

the other hand, where Koa ~def VbSx(a(~x) ~ 0 ) & V n m ( a n ~ O ~ a n = a(n.m)) . 3 ~ ) The underivability underivability

of

realizability".

of

MpR.

Mc

in

~

is of course a weaker result than the

To show the latter Kleene introduced his "special

We describe ~ simplified proof of this result in section 3.

4 ~ ) We use the elimination under Markov's

of choice sequences

rule

MR c

~V~x(A(~)

for

FIM,

v ~A(~,x)),

~V~X~A(~)

the system of intuitionistic

"choice-part"

to give a proof of closure

of

C S,

since

C~S

= bV~xA(~,x) analysis

of [K,V]

is conservative

over

(which is the

FIMM by [T3])

(section 4).

5 ~ ) We show

CS ~ M c -- M1 c

w 2. Incompatibility for

of

CT

(section

and

5).

M c , elimination

of choice sequences

CS + M c .

2.1. Let us assume lawlike and choice functions in", and let the

to be closed under "recursive

K - variables be supposed to range over a class

K*

of

neighbourhood

functions which is not assumed to coincide with the inductively

defined

Let

K.

KI

be defined by

K1a ~ Vb~zVc~b(a(~z) JO) & V n m ( a n J O ~ a n = a(n.m)) .

872

A.S.

Troelstra

We shall make the following assumptions about the properties of lawlike and choice sequences: (I)

V6Sx(a(~x) /0) & V n m ( a n / O ~ an= a(n*m)) ~ K*a

(2)

Va~(a= ~), ~ a ( ~ =

(3)

K*a ~

a)

K1a.

We start with two simple propositions with almost trivial proofs:

,, A. Proposition Proof.

Mc

Assume

aE

Ko

~

Ko,

Vb~x(a(~x) / 0 ) ,

c K* ~ KI" --

i.e.

V n m ( a n / 0 ~ an= a ( n . m ) ) .

By (2) V~~ ~ ~ ( ~ ( ~ x ) / o ) . Applying

Mc

with

kx.(1 : a(~x))

for

4,

it follows that

v~(~(~x) / o), hence with (1)

a 6 K*,

Proposition B. Proof.

~

and with (3)

a6 KI .

~CT .

Kleene constructed a binary tree with primitive recursive character-

istic function which is well-founded w.r.t, recursive sequences but not uniformly bounded (of. [K,V], lemma 9.8). a primitive recursive function

~

So, on assumption of

CT , there is

such that

Va~x(~(~x) / O) , Vnm(tm'l/O ~ C a = ~(n.m)) , but

~/K I .

On the other hand, as in the proof of proposition A, if

Va~tx(~(Tx) / O) , ~{

i t f o l l o w s by (2) t h a t

~V~Zx(~(~x)

~{K I Remark.

so

~x(~(~x) / 0), and t h e r e f o r e wi th

/0) ; but this is obviously false since

~V~Sx(~(~x) /0) . Note that in fact

Corollary.

V~ ~

C~S+ M c ~ ~ CT

proved in [K,T], 6.2.1).

~

Thus

K ~ K I , but

MpR ~ ~ C T .

for particular

n

refutes

CT.

(since (I), (2), (5) are provable in

CSS;

(2) is

A.S. 2.2. Theorem.

Let

for formulae

A

T(A) ~ A

(ii)

~

(iii)

CS + M c ~ A

Then,

for formulae of

ID~BI ,

~ T(A) ~-eA

As a corollary

* ID~BI+ ( K o = K ) + M ~ T ( A

Mc

M

c

T(Mc)

.

C~S but underivable in

C~+M.

IDBBI + (K ~ : K) + M

is derivable in

as

v~[~ v ~ ( ~ J o ) ~

:x(~=o)]

Below we shall regard sg(tx) = I.

)

is consistent relative

We first show that

We rewrite

(I)

be the translation as defined in [K,T], 7.1.

not containing choice parameters

(i)

Proof.

9

373

Troelstra

tx ~ 0

Application of

. as an abbreviation of the prime formula

T

to (I) yields

Ve[ Vf ~ VxVa( (e I (fl a) )x I O) S e ' V n ( e ' n % 0 ~ Va((el(nla))(e'n~1 ) = 0))] which is equivalent to (2)

V e K V f ~ V x V a ( ( e l ( f l a ) ) x ~ O ) ~ ~e'Va((ela)(e'(a)) =0)] .

Now note that Vf~VxVa((e1(fla))x~O ) ~ (For

~,

substitute

V b ~ V x ( ( e l b ) x ~0)

fb

satisfying

implies

Vf~a~Vx((el(fla))x ~0) .)

Ve[~Vx((elb)x~ which with (3) With

M

o) ~

~Vx(el(fl~y.1)x~O) Therefore

for

f,

for

*

note that

, hence

(2) is equivalent to

~e'Va((ela)(e'(a))

=

0)]

-- ~ e ' W ( ( e t b ) ( e ' ( b ) )

=0)] .

it follows that (3) is true.

On the other hand, sequences of

VC(fblc= c)

reduces to

Ve[~x((elb)x=O) K =K o

Vb~Vx((elb)xfO ) .

C~S

The consistency of

Mc

implies

M,

and in view of t ~

fact that the choice

satisfy (I), (2), (3) of Proposition A in 2.1, also Mc

relative

C~S then follows by the remark that

K o=K.

374

A.S.

(i)

FI~M+M

Troelstra

is consistent relative

B+M,

where

B

is Kleene's "basic

system" obtained by omission of the continuity axiom from

FIM;

this is

shown by realizability (see [K3, Part II), (ll)

in

FI~M one can prove the choice sequences to be a model for

IDB I + (Ko= KI) satisfying

(assuming the lawlike variables to range over choice sequences

FI~M) - this is implicit in [K,T~, 6.3, noting that

in the model indicated (ILi)

C~

Remark. C$.

FIM

and

IDB I .

Variants of this theorem hold for the weaker systems C$

but

K

is defined as

C So(C~SI)

and

IDB I + (Ko= K) + M

Also it follows that

Mc

by

w.r.t, arithmetical sentences for

CSS

E~L+AC-NF+M(E~L+AC-NF+

is underivable in

2.3. Corollar2 to the proof of 2.2.

C So, C S I

K. , and we add an axiom

V~Sx(e(~x) ~0) ; then the assertion of the theorem holds with by

and

K = K. o

is conservative over

is similar to

K~KI,

replaced (Ko=KI) +M) .

CS I + M .

H+M+CONT

I

is conservative ever

H -z EL, E L + ( K o = K 1 ) ,

ID B I + K o = K .

H+M Here

VaSbA(a,b) -~ Sc ~ KoVaA(a,cla )

CONT I

where of course Proof.

cla= b =def Vx~y(c(<x>*~y) = bx+1) .

Combine the proof of the preceding theorem with the method for

proving theorem I of ~T4~, where

H, CS H

in IT43, w 3 are to be replaced by

H+M,

C~H+M c

respectively.

w 3.

A new proof of the underivability of

3.1.

Kleene introduced in [K,V~ "special realizability" to show

underivable in

FI~M and

underivable in

ID~BI

MpR

i_n ~

and

CS.

C~S. To do so it is sufficient to show

since

C~S

is conservative over

Note first that the validity of the rule

IPR

IPR

~ (~A ~ SxB) = ~ ~x(~A ~ B)

not free in

for

ID~BI

(x

to be MpR

to be

ID~BI .

A)

is sufficient to yield the required underivability : for assume

A.S.

(I)

IDB 1 b V x ( ~ V ~ . T x x z

then with

Troelstra

375

~ ~ Txx~)

IPR

IDB I ~ V x ~ y ( ~ V ~ T x x z

~ Txxy)

and hence (2)

~DBBI ~ a V x ( . V z ~ T x x z

Now IDB I + CT IDB I + C T

~ T(x,x,ax)) .

is consistent relative

the recursiveness of

hence (I) underivable in

3.2. Theorem. Proof.

I DB I

IDB1,

~Txxy

and we can derive from (2) in

; this is obviously contradictory,

IDB I . So it remains to establish

is closed under IPR.

We use de Jongh's variant

5.1.13, [J], w 5 ~ read to a formalization of

"P" ID~BI

for with

E IA

of Kleene's

"D"

(see ITS],

in (i) of 5.1.15 of ITS]), relative

K-variables

(i.e. all prime formulae are of the form it is more convenient to axiomatize

F IC

IDB I

but without the constant

t = s).

Instead of the schema

K K5

with the equivalent schema of

induction over unsecured sequences :

(i)

W ( e n ~ o ~ Qn) ~ Vn(VyQ(n*~) -- Qn) ~ QO.

The definition of

E IA

is extended to

(vii)

E IVaAa m Va(E]Aa)

(viii)

EI~aA a E ~a(EIA a & ( E ~ A a ) )

(ix)

E Iveje

(x)

EI~eA e m ~e(E]Ae & (E--Ae)) .

(If

a, e

ID~BI

by

m Ve(EIAe )

do occur in

E

Now it is routine to show

we must rename IDB I + E

~A

a, e

on the right hand side.)

= IDB I + E I E

this by the verification for instances of

(I).

Assume (2)

ElVn(en~O

-~ Qn) & Vn(VyQ(n.#) -~ Qn)

,

~EIA.

Let us illustrate

378

A.S.

Troelstra

From (2) we obtain

V~(El(en/O ~ (E ~ e~/O)) -- EIQn) which is equivalent to V~((E ~ e n # O ) ~ EIQn ) which in turn implies (4)

~ n ( e n / 0 -- Elqn ) .

From (2) also Vn(Vy(EIQ(n.#) ) & (E ~ vyQ(n.~)) ~ EIQn ) which implies

(5)

~(~y(EIQ(n*~)) - EIQ~).

From (4), (5) with (I) ~pplied to

EIQ~

instead of

Q~,

EIQO. Then, as in [TS~, 3.1.14- 3.1.15, we obtain ~losure under IPR.

3.3. Remark.

The underivability of

ed from the underivability of

imply

IDB I

on the other hand is readily obtain-

M c : assume

it would already be derivable in C.~S form a model of

M

M

to be derivable in

C~S, then

IDB I ; but since the choice sequences of

(provably in

C~S, of. [K,T], 6.3), this would

C~S ~ M c , which we have refuted already.

A.S.

w

4.

Markov's rule for

4.1.

IDB 1

FI~

Troelstra

(from Markov's

377

rule for

I DB I

).

can be proved to be closed under Narkov's rule by extending the

Dialectica interpretation to

ID~BI ; this is done by first embedding

in a theory

WE-ID~B m

W~E-IDBW+AC-NF,

showing that

WE-IDB w + A C - N F

as described in [TS], 1.9.25, then

has a Dialectica interpretation into itself ;

the details are practically the same as for the interpretation of V*

described in

[H], w 2.

Then, after the manner of

obtain closure under Markov's rule for Next, we construct in fortiori a model for

IDB I

IDB I

a model

U

into

IT5], 3.8.5 we can

W~m-IDB m + AC-NF . K-ECF

WE-IDBm+AC-NF)

for

similar to

~-IDBW-AC-NF ECF

(hence a

itself, but simply

taking W~(a) E Ka,

IK(a,b ) m Vx(ax= bx)

otherwise as for

ECF,

of the constants

I, 91, 92, 93

of

E-HEO

and constructing in the obvious way representations of

E-IDB ~

(cf. the analogous construction

in 2.9.5 of IT5] ! a representation of

~

is found by applica-

tion of the recursion theorem analogue 1.9.16 of IT5] ; cf. again the construction of

[$ ]

in 2.9.5.

From this construction it follows that IDBI,

hence the closure of

closure of 4.2.

I DB 1

E-IDB w + A C - N F

~-ID~BW+AC-NF

is conservative over

under Markov's rule implies

under Markov's rule.

Now assume

(1) Since in

FI~Varx[A(~,x) V ~ A ( ~ , x ) ] , F I ~ ~ V ~ x A ( ~ , x ) C~

Va~(a=

.

~) ,

C~S ~ Va,x(A(a,x) V ~A(a,x)) , and if we write

A*(a,x)

for

T(A(a,x))

by the first elimination theorem of C~ ~Va,x[A*(a,x) V ~A*(a,x)

(T

[K,T], 7.2

],

as in w 7.1 of [K,T]) alsO,

378

A.S.

hence by the second elimination IDB I ~Va,x[A*(a,x) Also,

since

Troelstra

theorem of [K,T], 7.3

V

~A*(a,x)].

C SS ~ V~ ~ x A ( ~ , x )

by (I)

~D2B~ ~ V a ~ A * ( a , x ) and applying Markov's rule for

Since

A(~,x)

quantifiers

IDB I

was assumed to be a formula of Sa, ~e , hence the proof of

CS ~ A * ( a , x ) ~

A(a,x)

does not make use of the axiom assume the lawlike variables the

K - variables

V ~ x ( ~ ( ~ x ) / 0) ,

over

(cf. IT3]).

BC-F

V n m ( ~ n / O ~ om= ~(n*m)) ,

~,~

Therefore,

**

Now we note that (I ~ ) if we

~

if we let

K*

of

CSS,

A**(a,x)

T(B) <--~ B

and the

I DB I

hold except

be obtained from

to choice variables,

and

such that

then we obtain a model of

all axioms needed to prove

ranging over

CS ~

CSS.

being the class of

by changing all lawlike variables by variables

of

to range over the choice sequences

K* , K*

(cf. [K,T], 6.3) and (2 ~ BC-F

FI}~, it does not contain

A*(~,x)

K-variables

then

(~,x)

(from (21 and (I~ just mentioned),

hence

CA 5 v=~A(=,x) (by (2o)).

Thisestablishes

Markov's rule for

FI~.

Thus we have established Theorem.

FIM

is closed under

MR O

4.3.

For a direct proof of

MR

for

FIM,

O

theory of finite types containing the schema

we have to embed BR ~

FIM

for bar recursion

in

a

of type

0 , and constants for moduli-of-continuity ; this theory does not have an

A.S.

extensional model analogous to 4.4.

Troelstra

ECF

(cf.

379

[T5~, 2.6.7).

If we could establish, for all closed terms of type 2 in

(I)

WE-IDB 1 ~ZeVx1(e(x

I) =

t2X 1)

then we could establish closure of the whole of Markov's rule as follows.

WE-IDBI:

C~S (not only

The argument runs as before for

FIM

FIN)

under

till we

arrive at

~DB I ~ Va~xA*(a,x). By

m q-realizability,

one then establishes closure of

under the rule of choice (of. [T5], 3.7.2), W~E-IDB W + A C . N F

~-IDB ~+AC-NF

so for suitable

t2

~A*(a,t2a) ,

and then, by (I)

~-~DB ~ § AC-NF b ~eV~*(a,e(a)), IDB I ] - ~ e V a A * ( a , e ( a ) ) ,

and thus

hence

cs 5 ~eV~(a,e(a)). In

CSS V~(A(~,e(~))V ~A(~,e(~)))-~

(VaA(a,e(a)) ~-~ V~A(~,e(~)))

(cf. [K,T],

6.2.2), therefore CSS ~ SeV~A(~,e(~))

and so

CS ~V~xA(~,x).

A possible way to establish (I) might be the extension of the technique developed in IS], to establish closure of recursion.

~w

under the rule of bar

380

A.8.

w 5.

Troelstra

Markov's principle for functions.

Because of the presence of continuity axioms in

M1

CSS,

Va(A(~,6) v ~A(a,B)) & -7~S~A(~,~)-~ S~A(a,B)

C

is derivable from 5 1 9

Theorem.

MC :

CS ~ M c ~ M I

9

~

Proof.

C

Abbreviate

9

A(~,elB )

as

Ae(~,B) , and assume

VaS(Ae(a, S) V~Ae(a,6)). Then there is an

fEK

such that

e

(here

e

f(c~,6) abbreviates

f(J(~,S))).

Therefore

e

and thus

~Z~Ae(~,~

) .~ ~Ae(~,~)

is equivalent to (I)

~Z~(f(~,B)

Now let

=0) -- Z~(f(~,~) =0)

rn= kx.g(n,x) , (g

(in the notation of IT4]).

the notation of ~K,T]) Therefore

since

or

r n = ~x.(n) x

~ ( f ( ~ , ~ ) = O *-4

(lth(n) = 1) *-* Sn(fn,~ ) = O) , (I) is equivalent to

~n(f(~x.j((n)x,2X)) ~n(f(r

.

n,B) =0) -* Wn(f(rn, B) = O)

which is a consequence of V~B[Ae(~,B ) V ~ A

e

M C . Therefore, for all

(~,~)] & V~ ~ A

e

e

(~,6) -~ V B ~ A e ( ~ , ~ )

which by the schema of analytic data implies V2(V~[A(~,~) V-~A(~,~)] & - ~ A ( ~ , 6 )

-* Z~A(~,6))

.

A.S.

w 6.

Troelstra

881

Completeness of intuitionistic first order predicate logic

In [Krl] Kreisel defines validity

VaI(A)

for a formula

I~PC.

~(PI' .... Pk )

of pure first order intuitionistic predicate logic, constructed from the ri-placed

relation symbols

_ ~def VD~.. Val(A) where of

D

D ri

9

Pi ' I < i < k

. ~k* AD(P~, =

by

,P~) ,

""

ranges over arbitrary species, and (the

Pi

and

D

*

Pi

over arbitrary subspecies

may contain choice parameters) ; ~D

by restriction of the individual variables of

A=

to

is obtained

D.

Let "Proof" denote one of the usual arithmetized proof_predicates for intuitionistic predicate logic, then completeness is defined as VA(VaI(A) ~ Sp Proof(p,

A ))

and weak completeness as VA(VaI(A) - - ~ p Let us use

"Valc"

Proof(p,

for validity relative to species and relations not con-

taining choice parameters,

"Val"

also contain choice parameters. completeness corresponding to Comp', Comp~

~ )) .

for validity where Let

ComPc , Comp

Val, Val c

P' P~' "''' Pk*

may

denote the notions of

respectively, and, similarly,

the respective notions of weak completeness.

From theorem I of [Krl] we see that ComPc ~ C o m p ~

Comp' ~ ~

and by Proposition B in 2.~ (cf. also [Kr4], page IV- 4) CT ~ ~ C o m p ' ~ ~ Ccmp ~ ~ C o m P c . This also follows from the result in [Kr2] (for exposition see [D]) ~ but there a much stronger result is established, namely that not recursively enumerable, in other words

IPC

IA : Valc(A)}

does not even have an r.e.

axiomatizable extension which is complete. In footnote 16 of [Kr2] (p. 147) it is noted that permits to extend the result obtained there from principle is of course a consequence of

V~Sa(~

is

ComPc = a) .

Va ~ A a ~ V ~ A ~ to

Comp ; this

A.S.

382

Troelstra

w 7.

Miscellaneous additional remarks.

(i)

As is well known, Markov's schema

Mc

is false for lawless sequences.

It is worth noting, however, that it is also false for two kinds of projections of lawless sequences: (a)

it is false for the collection of all e E K, u E N

lawless, independent, contains all

n*(~)n

elVu(~ I ..... ~u ) ' ~I ..... ~u

considered in [TI], since this collection

which form a model for the theory of lawless sequences

(cf. [~2]), (b)

it iS alsO false for the universes

~

:

{el~: e6K},

~

lawless, for

the same reason (one is led to consider validity in all such universes in relating validity in topological models to intuitive validity (cf. [Kr2],

p. 137). (ii)

~n [Kr3], p. 126- 127 it is pointed out that the Dialectica inter-

pretation of bar recursion conflicts with that

CTo + ~ R

+ BRo + constructive

CT ; in ILl, 9.3 it is shown

~- rule

are inconsistent relative

to elementary analysis.

References

[D]

D. van Dalen, Lectures on intultionism, in : A.R.D. Mathias, H. Rogers (editors), Cambridge Summer School in Mathematical Logic. Berlin (Springer-Verlag) 1973, pp. I ~ 94.

[HI

W.A. Howard, A system of abstract constructive ordinals. J. S. L. 37 (1972), pp. 355- 374.

[J]

D. H. J. de Jongh, Formulas of one propositional variable in intuitionistic arithmetic. Report 7 5 - 0 5 of the Dept of Mathematics, University of Amsterdam.

[K]

S. C. Kleene, Formalized recursive functlonals and formalized realizability. Memoirs of the American Math. Soc. nr 89. Providence, Rh. I. 1969.

[Krl]

G. Kreisel, On weak completeness of intuitlonistic predicate logic. J. S. L. 27 (1962), pp. ~39- 158.

[Kr2]

- , Church's thesis : A kind of reducibility axiom for constructive mathematics, in : A. Kino, J. Myhill, R.E. Vesley (editors), Intuitionism and Proof Theory, Amsterdam (North-Holland) 1970, pp. 121 - 150.

[~r3]

G. Kreisel, A survey of proof theory II, in : J. E. Fenstad (editor), Proceedings of the Second Scandinavian Logic Symposium, Amsterdam (North-Holland) 1971, pp. 109- 170.

A.S. [Kr4]

[~,~]

Troelstra

383

- , Theory of free choice sequences of natural numbers, in : Stanford report on the foundations of analysis, Stanford 1963 (mimeographed, privately circulated only), section IV.

G. Kreisel and A. S. Troelstra, Formal systems for some branches of intuitionistic analysis. Annals of Mathematical Logic I (1970), pp. 229- 387.

[K,vl

S. C. Kleene and R. E. Vesley, Foundations of intuitionistic mathematics, especially in relation to recursive functions, Amsterdam (North-Holland) 1965.

ILl

H. Luckhardt, Extensional Godel functional interpretation. A consistency proof of classical analysis. Springer Lecture Notes Vol. 306, Berlin (Springer) 1973.

[s]

H. Schwichtenberg, Einige Anwendungen yon unendlichen Termen und Wertfunktionalen. Habilitationsschrift, ~unster (Westf.) 1973.

[TI]

A. S. Troelstra, Notes on the intuitionistic theory of sequences I, Indag. Math. 31 (1969), pp. 4 3 0 - 4 4 0 .

[m2]

- , Notes on the intuitionistlc theory of sequences III, Indag. Math. 32 (1970), pp. 245 - 252.

[T3] [T4] [~5]

- ,

An addendum.

- ,

Note On the fan theorem.

Annals of Mathematical Logic 4 (1971), PP. 437-439. J. S. L.

To appear.

(editor), Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer Lecture Notes Vol. 344, Berlin (Springer Verlag) 1973.