Grundgesetze part ii

Gottlob Frege

The Basic Laws of Arithmetic Part II Proofs of the Basic Laws of Number

Very Roughly Translated by Richard Heck and Jason Stanley ©2004 Richard Heck and Jason Stanley

II. Proofs of the Basic Laws of Number Introductory Remarks §53. I wish to emphasize, with respect to the proofs which follow, that the explanations which I regularly lay out under the heading “Analysis” ought only to serve the convenience of the reader; they could be omitted without taking anything away from the force of the proofs, which are to be found, in their entirety, under the heading “Construction”. The rules to which I refer in the Analyses are presented in §48 under the corresponding numbers. The laws ultimately to be proven are to be found at

the end of the book, in a special table, together with the Basic Laws summarized in §47. The definitions of Part I,2 and others are also summarized at the end of the book. First we prove the proposition: The number of a concept is the same as the number of a second concept, if a relation maps the first concept into the second and if the converse of this relation maps the second into the first.

A. Proof of the Proposition

a) Proof of the Proposition

§54. Analysis According to definition (Z), the proposition

( ,32) ( ,32 )

is a consequence of



and




( ,32 ) This proposition is to be derived with (Va) and by Rule (5) from the proposition 

( ,25) If we exchange ‘u’ and ‘v’ in ( ), and write ‘ q’ instead of ‘q’, then we obtain 



( ,32 ) which is to be proven with (IVa). We need for this purpose the propositions 





One such relation is evidently a relation composed from the p-relation and the q-relation, as the following picture makes clear: For the extension of a relation composed from the prelation and the q-relation, I introduce the abbreviated notation ‘p q’, by defining

( ) This proposition agrees closely with ( ). To derive ( ) from ( ) by Rule (7), we need the proposition 





(B)

( ,31) We thus first seek to prove the proposition ( ). It results, by contraposition (Rule 3), from the proposition

The proof now depends upon the proposition






( ,19) In words: “If the p-relation maps the w-concept into the uconcept and the q-relation maps the u-concept into the v-concept, then the p q-relation, which is composed from these two relations, maps the wconcept into the v-concept”. Furthermore, we need the proposition 



( ,25 ) 

which follows, by Rule (5), from




( ,25 ) which can be reduced to ( ) with the proposition 



( ,25 ) Better to recognize the sense of this proposition, we transform it by contraposition into 

( ,24) To begin with, we attempt to prove proposition ( ). We can conclude from definition ( ) that two things must be proven, namely, first,









( ) To simplify the exposition, instead of ‘concept whose extension is to be indicated by “u”’ I say ‘u-concept’; instead of ‘relation whose extension is to be indicated by “p”’, ‘p-relation’; instead of “the objects falling under the w-concept are functionally related via the p-relation to those falling under the uconcept”, “the p-relation maps the w-concept into the u-concept”. We can now render ( ) in words thus: “If the converse of the p-relation maps the uconcept into the w-concept, and the p-relation maps the w-concept into the u-concept, if moreover the qrelation maps the u-concept into the v-concept and the converse of the q-relation maps the v-concept into the u-concept, then there is a relation which maps the w-concept into the v-concept and whose converse maps the v-concept into the w-concept”. 

( ,9)

and, secondly,

( ,17) 

( ) follows from



( ,9 ) by Rule (5). In order to express ( ) in words, it is simpler first to transform it by contraposition into



2



for ‘f( )’ in (VIa) and replacing ‘a’ with ‘f(a)’. ( ) follows by Rule (5) from



( ,1 ) which is to be proven with (IVa). For this we need the propositions 

( ) If we say, instead of “object which is indicated by ‘d’”, just “d”, then our proposition, in words, is: “If d falls under the w-concept and if the wconcept is mapped into the u-concept by the prelation and if the u-concept is mapped into the vconcept by the q-relation, then there is an object, which falls under the v-concept and to which d stands in the p q-relation”. The proof will have to be founded upon the proposition



( ',1 ) and






( ',1 ) The first follows by contraposition, in accord with Rule (3), from 

( ,5) In words: “If d stands to e in the p-relation and if e stands to m in the q-relation, then d stands to m in the p qrelation”. This is to be derived from the proposition

We now write (IIb) in the form

( ,4) which follows from definition (B). To prove it, we need the proposition

and so we see that ( ') follows from it with (IIIe). Proposition ( ') follows by contraposition from




( ',1 ) 












( ,2) because the left-hand side of the definitional equality (B) is a double value-range. ( ) is reducible to the proposition 




( ',1 ) 

and this follows, by Rule (5), from



( ,1) which is to be derived from definition (A). According to this definition, 

( ',1 ) This proposition follows by Rule (7), and with (Vb), from 

( ,1 ) is to be proven. This must be done with (VIa) and the proposition1 



which, via interchange of subcomponents, is merely a special case of (IIIc). We now construct the proof accordingly. Concerning the derivation of (2), it remains to be remarked that, at the first citation of (1), ‘f( )’ is replaced by the function-mark ‘f( ,b)’, according to Rule (9). (IIIc) is thus to be thought of in the form

( ,1 ) 



by taking



1

[There is misprint at this point in the text: It should contain the assertion-sign, as it does here.] 3



At the second citation of (1), it is to be thought of in the form

( ) 

where ‘ .f( , )’ occurs instead of ‘f( )’; ‘b’, instead of ‘a’; ‘ ’, instead of ‘ ’, by Rules (9) and (11). 







( ) 

(VIa): ______________________________ §55. Construction ( ) 

Vb

(IIIa): _________________________

(IIIc): - - - - - - - - ( ) 

( )::______________________________

( )















(1)

1 (IIIc): _______________




( )

( ) (1):: _______________________ (2)

( ) 

(IVa): ______________

(IIIc): ____________________





























(3)

( )

( ) 

(3): _________________________

IIIc (IIb): __________

(4) (IIIc): ______________________________ ( ) 

( )

( ) ( ): _________________




( )



(IIa): - - - - - - - - - - - - - - - ( )


(IIIa): ______________________________ ( ) 

( ) 

(IIIi):: ____________________________ 4

This proposition is easily proven with (IIa) and (5). What is essential is thus to derive proposition ( ) from ( ). This can be done with


(5)



( ,6)

§56. Analysis To prove the proposition




which follows from (3).

§57. Construction 3

( ,9 ) (§54, ) we must go back to definition ( ). From this, we derive the proposition 





(IIIa): - - - - - - - - - - - - - - - -











( ,8) To reach ( ) from this, we must have the proposition








(6)



(6): ____________________________ ( ,9 ) 



which follows by Rule (5) from

(Ib): - - - - - - - - - - - - - - - - -

(7)

( ,9 ) 






We can also write proposition ( ) as follows: ( ) (IIa): - - - - - - - - - - - - - -


( ,8) To reach ( ) from this, we need the proposition 















(8)

5


( ,9 ) 

which follows by Rule (5) from (IIa): - - - - - - - - - - - - - -

( ,9 ) ( ) 5

(Ie): - - - - - - - - - - - - - - - - - -

(11)




( )








(8): - - - - - - - - - - - - - - - - - -










11

( ) 

(9):: - - - - - - - - - - - - - - - - - - - - -

(12) ( ) 

(8): - - - - - - - - - - - - - - - - - -

§58. Analysis We must now prove the proposition

( ,17) i.e., “the relation composed of the p-relation and the q-relation is functional if the p-relation, as well as the q-relation, is functional”. According to definition ( ),

( ) 















(9)

3




( ,17 ) is to be proven, which results, by Rule (5), from 

(IIIc): - - - - - - - - - - - - - - - -















(10) ( ,17 ) 



From definition (B), it is easy to conclude

( ) 

(10): ___________________________

or

( ,15) 

With these propositions, one easily gets from

( ) 6

follows with (IIIc). If one applies proposition ( ) in the form 

( ,13) then one proves proposition ( ), from which we can reach our proposition ( ), as we have seen.

( ,17 )






to the proposition



§59. Construction ( ,17 ) from which ( ) follows by contraposition. Now, proposition ( ) follows by Rule (5) from






(IIIa): ________________________

( ,17 )

( )



This proposition is derived with

(IIa): - - - - - - - - - - - - - - - - -

( ,15) 

from




( ) (IIa): - - - - - - - - - - - - - - -

( ) 

(IIa): - - - - - - - - - - - - - - -

( ,17 ) in a manner similar to that in which ( ) is derived from ( ). In the course of this, the ‘c’ will be replaced by the ‘r’. Hence, the difference between the letters ‘b’ and ‘c’ is necessary; otherwise, by Rule (5), the ‘r’ would not be introduced just in those places where ‘c’ currently stands, but also in those where ‘b’ stands. Now 






















(13)















(14)

6

( ,13) 

follows from our definition ( ) and from this,



(14): _________________________

( ,17 ) 

7















(15)

( ) 

(IIIc): _________________________















(16)

( )

13

(IIIc): _______________ ( )


(15): - - - - - - - - - - - - - - - - - - ( ) (13):: - - - - - - - - - - ( ) 




( ) ( ) 

( )

( )





(16): - - - - - - - - - - - - - - - - - - - - - -








( ) 

(15): - - - - - - - - - - - - - - - -

8










(17)

)2

7




( ) (12): - - - - - - - - - - -

(Id): - - - - - - - - - - - - - - - - (18)

(19)

(17): - - - - - - - - - -

( ) (18): - - - - - - - - - -

2

[There is a slight misprint at this point in the text: In the main component, there is a right parenthesis to the right of the ”u”.] 9

b) Proof of the Proposition

and End of Section A

§60. Analysis We now have to prove the proposition


( ,25 )

( ,24) (§54, ). This depends, according to definitions (E) and (B), upon the derivation of the proposition



(§54, ). 



§61. Construction


( ,24 ) For this purpose, we can make use of the proposition

Va



(IIa):: - - - - - - - - - - - - - - - -

( ,20) which we will require, in any case, and which can be proven by means of two applications of proposition (Va). To apply proposition (Va), we must have the proposition 

( )




( ) ( ,24 ) 

(Va): - - - - - - - - - - - - - - - - - -



which follows with (4) from






( ,24 ) ( ) is to be proven with (IVa). For this we need the propositions 








(20)



(3): __________________



(21) (IIIc): ___________________ ( ,24 )










and








(22)

21 (IIIa): ____________________ ( ,24 )











We derive these from the proposition 23

( ,23) which follows easily from (E). The thus proven proposition ( ) we use, as indicated in §54, in the proof of the proposition 

(IIa): - - - - - - - - - -

10










(23)

( )

( )



( )




( )



(IVa): ________________

(23):: - - - - - - - - - - - - - -

( ) 

( ) 

( ):: ____________________________________ 

( )

( ) (IIIc): ___________________________________

( )

(4):: __________________________________





( ) 

















22

( )

(IIa): - - - - - - - - - -

( ) 

(20): ___________________________________

( )

( )

(IIIc): __________________________________ ( )


(22): - - - - - - - - - - - - - ( ):: _______________________________

( ) ( )


( )

(IIIc): __________________________



( ) 

( ): _____________________________

11

§62. Analysis To prove the proposition (§54, ) from (25), we need the proposition (§54, )

(24)



(IIIa): __________________




( )

( ,31)

(19):: - - - - - - - - - - - - - - - -

By (11), we have to prove the propositions




( )


(IIa): - - - - - - - - - - - - - - - - -

( ,27 ) 

and ( ,30) 




( ) follows by Rule (5) from















( ) 

19 ( ,27 ) 



By (8), we now have

( )

( )





It remains to prove

( ):: - - - - - - - - - - - - - - - 




( ,27 ) which follows, by Rule (5), from ( ) 

( ,27 )


We write (IIa) as follows

( ) ( ) 

and so see that the proposition ( ,26) must be derived, which can easily be accomplished with the use of (22). 

(25)

12

§63. Construction 22


( ,29 ) 

Now we have, by (13),

(22):: - - - - - - - - - - (26) (IIa): - - - - - - - - - - - -

( ) 

From this, ( ) follows with the proposition 

( ,28) which follows from (23) in a manner similar to that in which (26) follows from (22). After we have proven proposition (§62, ) in this way, we will use it to eliminate the subcomponent ‘I q’ by the amalgamation of subcomponents. We will then have achieved the goal of our section A, as presented in §54. §65. Construction

( )










( ) (8): - - - - - - - - - - - - -



23 ( ) 

(23): - - - - - - - - - - - (28) (13): - - - - - - - - - - - -

( ) 

(11): - - - - - - - - - - - - - - - - - ( ) (28):: - - - - - - - - - - - - - -

(27)

§64. Analysis We are still missing proposition (§62, ). We next prove




( )



( ,29) from which our missing proposition follows with (18). By (16), we must prove

( ) 

(16): - - - - - - - - - - - - - - - - - - - (29)


(18):: - - - - - - -

( ,29 ) 

or (30) (27): - - - - - - - - - 13

(31) (25): - - - - - - - - - - - - -

( ) (IVa): - - - - - - - - - - - - - - -




( ) (25):: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ( )3 

( ) 

(Va): - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( ) (IIIc): __________________________________________ (IIIc): __________________________ 

( ) 

( ):: ____________________________

( ) ( ):: ____________________________

(32) ( )


3

[Due to an apparent omission, there is no transition indicator between this proposition and the next: The inference is plainly a universal generalization, however.] 14

B. Proof of the Proposition

a) Proof of the Proposition

§66. Analysis To prove the proposition, that the relation of a number to an immediate successor is functional, or, as one can also say, that, for every number, there is no more than one which follows immediately after it in the number-series,1 we have to use proposition (16) and so must derive ( ) 

is to be proven, a proposition which follows, by several applications of contraposition and the introduction of German letters from

( ,71 ) which follows from




( ,70) From definition (H), it is easy to conclude

( )

( )



Accordingly,2



This proposition can be derived from

1

Cf. §43.

2

( ,66) By the just proven proposition (32), we need for this purpose only to demonstate the existence of a relation which maps the u-concept into the v-concept and whose converse maps the v-concept into the uconcept. Though it admittedly remains to be proven, that there is a relation which maps the concept into the concept

[The explanation which follows does not match the formal proof very well. The previous proposition, ( ), is a contrapositive of Theorem 68. The next, ( ), is closely related to Theorem 67, but the formal proof does not pass through ( ). It is difficult to know whether the formal proof itself was altered after this explanation was written (or conversely), or whether Frege explains the proof as he does in order to aid comprehension.]









15

follows from the identity of the numbers belonging to these concepts. Now the v-concept differs in extension from the concept only in that

able to associate infinitely many objects with infinitely many, although of these infinitely many associations only a few could actually be carried out were association a creative act of the mind. Thus, the postulate would be better understood as something like: ‘every object is associated with every other object’, or ‘there is an association between any two objects’. What would such an association be, were it not subjective, something first created by our act? But a single association of one object to another is also not what concerns us here, and what is appropriate to the [example of the] auxiliary line in geometry; for we need a family of associations, as one might say, what we have been calling, and will continue to call, a relation. The desired association is thus achieved, if we have found3 a relation, in which the object b stands to the object c, which maps the

the object c, which does not fall under the latter concept, falls under it; and the u-concept differs in extension from the concept only in that the object b, which does not fall under the latter concept, falls under it. From this, we must be able to conclude that there is also a relation which maps the u-concept into the v-concept. If one were to follow the usual practice of mathematicians, one might say something like: we associate the objects, other than b, which fall under the u-concept, with the objects, other than c, which fall under the v-concept, by means of the already known relation, and we associate b with c. We have thus mapped the uconcept into the v-concept and vice versa. Hence, according to the just proven proposition, the corresponding numbers are the same. This is admittedly much shorter than the proof to follow, about which many, who misunderstand my intent, will complain, on account of its length. What then are we really doing if we associate for the purpose of proof? Evidently, something similar to when we draw an auxiliarly line in geometry. Euclid, whose method can in many cases serve as an example of rigor, has for this purpose his postulates, which allow one to draw certain lines. But the drawing of a line may no more be construed as an act of creation than may the determination of a point of intersection. On the contrary, in both cases, we only bring into consciousness, only grasp, what was already there. For the proof, it is essential only that there is something there. Euclid’s postulates thus have, for the proofs, the force of axioms, which assert the existence of certain lines, of certain points. Because we here want everywhere to penetrate to the deepest foundations, we must thus ask upon what the possibility of such an association rests. If, following the precedent of Euclid, one wanted to lay down a postulate, then it could run something like: ‘it is postulated: every object is associated with every other’, or ‘it is possible to associate every object with every other’. However, this may be conceived no more as a psychological proposition, as if it affirmed a capacity of our minds, than may a postulate of Euclid; for, as such, it would be false, since not all objects, and not all things, are known to us equally well. Thus, something subjective would enter, which is wholly foreign to the subject. Also, one must be

concept into the

concept, and whose

converse maps the latter into the former. Moreover, a q-relation, which maps the concept into the

concept and whose converse

maps the latter into the former, can be presumed known. Of this relation, however, it is known neither whether b stands in it to some object, nor whether some object stands in it to c. We can indicate a relation in which all those pairs of objects stand, which stand in the q-relation, and in which b stands to c. This is the relation.4 It

indeed has the other desired properties, but whether it or its converse is functional remains unknown, so long as we know nothing more about the q-relation.

3

Following the example of geometry, one could say “constructed”; however, one must always remain aware that this is not a creation. 4

Instead of

substantial change, write

we could, without

Compare

with this what is said about ‘or’ and ‘and’ in §12. 16

It would be possible, e.g., for b to stand in the qrelation to an object d different from c. Then b would stand in the given relation to two objects, namely to c and to d, and this relation would not be functional, even though, by assumption, the q-relation is. To avoid this, we must find a relation which shares the properties important for us with the qrelation, yet in which b stands to no object and no object stands to c. The ( ,64 ) From the two propositions ( ) and ( ) and the above mentioned proposition







relation is such a relation. Writing, for purposes of brevity, ‘w’ for and ‘z’ for we have the following proposition to

( ) 

or prove: “If there is a q-relation which maps the wconcept into the z-concept and whose converse maps the latter into the former, then there is also a relation, which does the same, but in which b stands to no object and in which no object stands to c, as long as b does not fall under the w-concept and c does not fall under the z-concept”. For the derivation in Begriffsschrift, it is simpler to prove the proposition which results, from this one, by contraposition:

( ,49) our proposition ( ) follows. Proposition ( ) follows easily from (Z) and 





( ,49 ) 

and this from (IIIc) in the form

and the proposition ( ,48) This proposition is easy to prove, by showing that Identity is a relation which maps every concept into itself and whose converse does the same. The propositions 

( ,55) Further, we will then derive the proposition: “If there is a relation which maps the relation

( ,39) 

( ,42) are thus to be derived. Instead of ( ) we next prove the somewhat more inclusive proposition 

into the

relation and whose converse



maps the latter into the former and which is so constituted that b stands in it to no object and no object stands in it to c, then there is a relation, which maps the u-concept into the v-concept and whose converse maps the latter into the former, so long as b falls under the u-concept and c falls under the vconcept”. For the consequent, one can also say: ‘then the number of the u-concept is the same as the number of the v-concept’. After an application of contraposition, the proposition appears thus:

( ,38) which we also will need later. For this, we need the propositions 

( ,38 )



and ( ,35)

17

These follow from (IIIc) and (16) with (2). To prove ( ), we write (IIa) thus:

( ) 

It still remains to show ( ,37) which is to be inferred from (IIIe) with (2). The main component can, by means of the subcomponent easily be transformed into

.

( ) 

(16): - - - - - - - - - - - - - - - - - - - - - -

§67. Construction 2 (IIIa): _____________________































(34)

IIIc

(33)

IIa

(33): - - - - - - - - -

( ) (34): ______________








( )








(35)

2 (IIIc): _____________________

(IIa):: - - - - - - - - - - - - - - - - -

















(36)

IIIe (36): ______




( ) (33):: - - - - - - - - - - - - - - - - - - - -

(37) (IIa): _______________

( ) (IIIa): ___________________ ( ) 

(IIa):: - - - - - - - - - - - - - - - - - - - -

18

§69. Construction 2


( ) (IIa): - - - - - - - - - - - - - - - - - - - - -

( ) (20): ________________________


( ) (IIIa): _______________________

( ) 

( ) 

( ):: ___________________________










( )








(40)



(11): - - - - - - - - - - - - - - - - - - - - - -

IVa

( )

( )

(20): ___________________



(35):: _____________________




( )

















(IIIa): _________________

(38)

IIIe

( ) 

(40):: _____________________ (41)

( )

(IIIc): ___________________

(38): ____________________ (39)

( ) (39):: ___________________

§68. Analysis To prove the proposition








(42)








(42)

15

we use the proposition

(IIIc): ___________ (41) which we derive from the propositions

(43) (IIIc): - - - - - - - - -


(40, 41 ) The former will be inferred from (2) and definition (E), the latter, from (IVe), with the use of (20), in both cases.








5










(44)

[There is a misprint at this point in the text: The right parenthesis in the antecedent is missing.] 19




( ) (44): __________________________

(47):: - - - - - - -

( )

(49)

§70. Analysis We now prove the proposition




( ) (IIa): - - - - - - - - - - - - - -

( ) 

















(55)

(45)

(Cf. §66, .) We saw in §66 that

43 indicates a relation which agrees with the q-relation in respect of the properties important to us, but in which b stands to no object and in which no object stands to c. We therefore write (IIa) in the form

(IIIa): - - - - - - - - - -

















(46)

(46): _________________________

( )

















(47)

45 (42,39):



















and now have, among others, the proposition (48)

(IIIc): _______ ( )

20

Now, we have, by (IIa)

(52) to prove, which follows by contraposition from

and it remains to prove, with (36)

(51) For this, we need the proposition


(50 ) whereby the subcomponents ‘ d=b’ and ‘ a=c’ appear. From the last two propositions, we conclude

(50 ) Now, we have by (8)

(50 ) by rule (7). If we were now to introduce the German letter ‘a’ by rule (5), then we would not arrive at the desired destination, because of the subcompoent ‘ a=c’, which must be picked up in the scope of the ‘a’. Now we have, by (IIIa) 

Accordingly, something like

(cf. 50 ) is to be proven, where I have yet not put the assertion-sign, since more conditions (subcomponents) are to be added. In true proofs, of course, expressions with Latin letters must not appear without the assertion-sign; here, where it is a question of preliminary exploration, it may be permitted. The last proposition results, by rule (5), from an expression like 

and the subcompoent ‘ a=c’ can be replaced by ‘ c z’, by rule (8). Later, the sucompoent ‘ d=b’ is to be replaced by ‘ b w’.





§71. Construction Ie

(If):: - - - - - - - - - - - - - -

(cf. 50 ) 

21

(36): - - - - - - - - - - - - - ( ) 

(IIIa): _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _




( ) (IIa): - - - - - - - - - - - - - - - - - - - - - - - -

( )

( ) 

(IIIa): _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _

( )


(11): - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( ) 

(50)

§72. Analysis We have to remove the subcomponent

( ) 

(8): - - - - - - - - - - - - - - - - - - - - - - - - - - - -

from proposition (50). This is done with the proposition

22

( )

(51 ) for whose proof we use proposition (34). For this we need the proposition





(50): - - - - - - - - - - - - - - - - - -

(51)

(51 ) which follows easily from (13).

§73. Construction

(52)

13 §74. Analysis To prove the proposition (Ib,Ib):: = = = = = = =

(54) we next write (51) thus ( )

by exchanging ‘q’ with ‘ q’, ‘c’ with ‘b’, ‘z’ with ‘w’. We now have to prove




( )

(53 ) 

(34): - - - - - - - - - - - - - - - - - - - -

which follows with (40) from

( ) 

(53 ) This proposition is to be proven with (20). For this we need the proposition

(18):: - - - - - - - - - - - - - - - - - -




23

If

(53 ) 

(Id,Ic):: = = = = =

which follows from (21) and the proposition (53 ) 

This is to be proven with (IVa).

( ) 

(IVa): ________ §75. Construction 21 (IIIh): ___________________ ( ) 

( ):: ___________________________ 

( ) (IIIa): _________________________________

( ) 




( ): ________________________

( )




















( )



( ) (20): _____________________________________

( )


(IIIc): ______________________________________

( ) 

(40):: ___________________________________________

( ) 

(IIIa): _________________________________________

24

( )

















(Ic): - - - - - - - - - - - - - - - - - - - - - - - - -

(53)

51


( )

(53): _______________________________ ( ) 

(IIIe):: __________________________

( )

(54)



§76. Analysis If we write propostion (IIa) as in §70, then we see that the propositions

















( ) 



(55 ) 

and

(53): ______________________________

(55 ) are missing; the latter can be reduced to the former with (53), which is to be proven with (33) in due course.

( )





(IIa): _____________________________

§77. Construction 33

(Id): - - - - - - - - - - - - - - - - - - - - - - - - - - -

( ) ( ,54): - - - - - - - - - - - - - - - - - - - - - - - - - - 

25

(55) (49): - - - - - - - - - - - - - - - - - - -

( )


(52): - - - - - - - - - - - - - - - - - - - - - - - - - - - -

(56) ( ) 

26

b) Proof of the Proposition

and End of Section B

§78. Analysis We now prove proposition ( ) of §66, which results, by rule (5), from


( ,63) 

and then the proposition

( ,64 ) For purposes of understanding, it is convenient to consider the proposition 

(64 ) To prove the former, with (11), we first need the proposition 




( ) which results by contraposition of ( ). By (32), it will be sufficient, in order to prove that the number of the u-concept is the same as the number of the vconcept, to indicate some relation which maps the uconcept into the v-concept and whose converse maps the latter into the former. We have already become acquainted with such a relation in §66. Therefore, we will first derive the proposition

( ,61 ) The cases d=b and d=b are to be distinguished. We write (8) in the form 



( ) 

from which

27

This proposition may easily be proven from (1) with (IVb).

§79. Construction IVb (IIIc): ___________________

( ) easily follows. From these we easily reach our proposition for the case
d=b by contraposition, once we have proven

( ) (1):: _______________________ (57) (IIIc): ____________________








( ,60 )









(58)

57 (IIIa): _____________________



At this point, we write (IIa) in the form

















(59)

I

( )

(36): - - - - - - - - - - -



so we must prove

( ) (IIa): - - - - - - - - - - - - - - - - - - - - - -

( ,60 ) which can easily be done with (I) and (36). Accordingly, then 

( ,60 ) 













must be inferred from (Ia) in the form










( )

Ia ( ) 

and the proposition

(58): _______ ( ,58) 

( )

which follows from






( ):: - - - - - - - - - - - - - ( ,57) 

28

( ) 




( ) Thus

( ,61 ) is still to be derived; it follows with (36) from 



( ) 

(8): - - - - - - - - - - - - - - - - - - - - - - - - -


( ,61 ) 

This proposition results from (Ia) in the form

(I) and (IIIe).

( ) 

(59): - - - - - - - - - - - - - - - - - - - - - - - - - - - §81. Construction IIIe (I): _______

( )

(60)

(Ia): - - - - - - - -

§80. Analysis We now prove the proposition


( ) (36): - - - - - - - - - - -

( ) 

(IIa): - - - - - - - - - - - - - - - - - - - - - -

( ,61 ) which, united with (60), leads to proposition ( ) of §78. We write (IIa) in the form 



29

§82. Analysis We are missing the proof of the proposition

( ) 

(60):: - - - - - - - - - - - - - - - - - - - - - - - - - -

( ,63 ) (cf. §78, ). To show this with (34), we need the proposition 



( ) 




( ,64 ) which we prove, by rule (8), from the propositions

( ) 

( ,62) 

and

( ) (11): - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( ,63 ) 

( ) follows from (13) with (I) in the forms and 

(61) Yet the subcomponents arise first. With (Ia) in the form

30

and



and

these can be replaced by ‘ e=b’. ( ,63 ) 

§83. Construction

which follows in the first instance from

13




( ,63 )

(I,I): = = = = = = = =



We now write (I) in the form

( ) 

so we can afterwards make use of (Ia) in the form

( )




and then easily reach our proposition ( ) by contraposition and with (IIa). We then replace ‘a’ with ‘d’ in proposition ( ) and, with (IIIa) in the form

(I,I): = = = = = = = = = =

reach our goal. (62)

§85. Construction Ia

§84. Analysis To prove proposition ( ) of §82, we observe that 

(I): - - - - - - - - indicates the truth-value of ’s standing in the qrelation to , or ’s coinciding with b and ’s coinciding with c. If we now take b for , then only the latter of these two cases can arise, if no object stands to b in the q-relation; i.e., must then coincide with c. Accordingly, one will be able to prove the proposition





( )




31




( ) (IIa):: - - - - - - - - - - -

( )


(34): - - - - - - - - - - - - - - - - - - - ( ) 

(IIIa): _____________

( ) 

(61): - - - - - - - - - - - - - - - - - - - ( ) 

(IIIa): - - - - - - - - - - - - -

( ) 

(18):: - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( ) 

( ):: - - - - - - - - - - - - - 

(63)

§86. Analysis At this point, we have proposition ( ) of §78 to prove. To derive ( ), we exchange ‘q’ with ‘ q’, ‘b’ with ‘c’, and ‘u’ with ‘v’ in (63). We thus obtain (63) in the form

( ) 

(62): _ . _ . _ . _ . _ . _ . _ . _





( ) and we need only produce the proof of the proposition

32

( )

(64 )





(20): ___________________________________

which is similar to that of §75, . 

§87. Construction

( ) (IIIc): __________________________________

21 (IIIh): ___________________

( ) (IIIa): _______________________________

( )


(40):: ___________________________________










( )










(IIIa): __________________________________

I

( ) 

(63):: - - - - - - - - - - - - - - - - - - - - - - - - ( ) 

(IVa): _______

( )

( )





(32): - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( ):: __________________________ 

( ) 




( ): _______________________

( ’)6 

( ) 

(63):: - - - - - - - - - - - - - - - - - - - - - - - - - - - 6

[The index for this proposition appears accidentally to have been omitted.] 33

( ) (65):: ______________________________ ( ) 




( )

( ) 

(66) (IIIa): - - - - - - - - - - - - - - - - - - - - - - - - -

( ) (IIIc): _____________________________ ( ) 

(56): - - - - - - - - - - - - - - - - - - - - - - - - - - - - -




( )

















(64)

IIIe (I): _______ ( )








(58): _______ (65) (64): ______________

34










(67)

(14): _______________________________ ( ) 

(68): - - - - - - - - - - - - - - - - - - - - - -

(68)

( ) 

(67):: - - - - - - - - - - - - - - - - - -

(70) (69) (IIIc): __________________ ( ) (16): ___________________ (71) ( )




( )

35

. Proof of the Proposition

a) Proof of the Proposition

This proposition can be derived from1

§88. Analysis We want now to prove the proposition that, for each number, there is no more than one which immediately precedes it in the number-series. This reduces to the proposition

( ) By proposition (32), we need only exhibit a relation which maps the relation into the

(
,89
) We here introduce the expressions following from definition (H), so we have

relation and whose converse maps the latter into the former. The subcomponent  u=  v' now tells us, that there is a relation which maps the uconcept into the v-concept and whose converse maps the latter into the former. Let the q-relation be such a relation. Now, we know of the relation that no object stands

( ) in it to c and that b stands in it to no object.2 Furthermore, no object stands in this relation to n, and m stands in this relation to no object, if m stands to c, and b stands to n, in the q-relation, since the q-relation, as well as its converse, are functional. This relation maps the -concept into the

a proposition which results, by contraposition and the introduction of German letters, from

( )

1

[As in §66, the sketch of the proof has not closely followed the formal proof itself. The following proposition, (  ), is however closely related to Theorem 87  .] 2

See §66.

36

-concept, and its converse

maps the latter into the former. By (32), then, the number of the latter concept is the same as the number of the former. We can then hope to reach our goal with (66). We first devote ourselves to the proof of the proposition

( ,84) We write (51) in the form

and so we see that the proposition




( ,81  ) still remains to be proven, since the proposition

(83  ) ( ,80) then we could apply it twice and thereby reach the proposition

poses no difficulty. Now, if we had proven the proposition



37

(  ,81
) We still need the propositions ( ,73) and ( )

(
,72 )

(IVa): ______



to be able to replace

with

( )

These propositions are derived first.

(  ):: __________________________

§89. Construction If

( )

(I): - - - - - - - -

( )




(Va): ________________________

( ) 

(IIIa): _______________________ (
)

 

( )

1 (IIIh): ____________

38



(72)

(
)

stands in the q-relation; but n, which does not fall under the -concept, falls under the v-

( )

concept.3 If an object which stood to n in the qrelation were to fall under the -concept,

( )

that would require that there be no object which fell under the -concept to which it stood in

(74)

the q-relation. But this case is excluded by the subcomponents   b (n q)' and  I q'.

(Va): ______________________ (IIIa): ____________________

 






73 (IIIc): ____________________

 

§91. Construction



1

(75)

(IIIc): ____________

74

 



(77)

13 (72): - - - - - - - - - - - - - - - - - - - - - - (22):: - - - - - - - - - - - -

(78) (
)

(22):: - - - - - - - - - - - -

(75): - - - - - - - - - - - - - - - - - - - - - - -

(79) (IIIa): ____________ (76)

§90. Analysis We now prove proposition ( ) of §88. If the qrelation maps the u-concept into the v-concept, then there is, for each object falling under the u-concept, an object falling under the v-concept to which it stands in the q-relation. Now every object falling under the -concept falls under the u

3

concept, and so there is, by our supposition, for every object falling under the -relation,

[As a careful consideration of the proof, and the following remarks, will show, this supposition is not in fact required for the proof. It is, however, helpful, for purposes of understanding, to make this supposition: For, if n does not fall under the v-concept, then the theorem follows immediately, since the -concept then

an object falling under the v-concept to which it

just is the v-concept.] 39

(
) (I):: - - - - - - - - - - - - ( ) 

(8): - - - - - - - - - - - - - - - - - - - - -

( )

( )

( ) (77): - - - - - - - - - - - - - -

( ) (58): - - - - - - - - - - - - - - - - - - ( )

(
) ( ) (IIa): - - - - - - - - - - - - - - - - - -

( ) (11): - - - - - - - - - - - - - - - - - - - -


( )

40

(80) (80): - - - - - - - - - - - - - - - - - - - - - - - - - -

(µ) (18):: - - - - - - - - - - - - - - - - - - - - - - - - -

(
) (76): - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( ) (51): - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

  1 (IIIa): ____________

 



(81)

Id



(82) (
) (82):: - - - - - - - - - - - - - - -

41

( )

( ) (65):: ________________________

( ) (81): ________________________

( ) (  ):: ___________________________________________________________________

(83) (23): - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

(84)

42

b) Proof of the Proposition

§92. Analysis From (83) we can derive the proposition

(85  ) with (53) and (22). From this proposition and (84) we then reach, with (66), a proposition with main component

[Theorem 85 ].

§93. Construction 83

(53): - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

(
) (22): - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

43

( ) (32): - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( ) (84): - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( ) (66): - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -




( ) ( )

44

(86 ) which follows from (77). In the same manner as we removed  n', we also remove  m' from our proposition and then prove, as has been indicated, with (80) and (32), the proposition (85)

§94. Analysis The last two transitions already hint at the method which will henceforth be employed. In ( ), we had n and m as auxiliary objects similar to the auxiliary lines of geometry. They should not appear in our proposition, and so must be removed. This occurs, as always, by this means: one shows that there is something with the required property; or as is more convenient in the Begriffsschrift, that, if there is not something of the kind, then one of the assumptions we made does not hold. We now undertake to prove the proposition

(87  ) We then remove the subcomponents   b (c q)' and b (c q)' with rule (8). After the subcomponents  I q'  and  I q' are removed with (30) and (18), we reach the goal of our section (b) with (49); and after that, as has been indicated in §88, we reach, with (68), the proposition “ I f”.














§95. Construction 77




(
,86 ) By this means, we do indeed introduce the subcomponent  b (c q)'; but we can also prove the proposition, with the contrary subcomponent   b (c q)', which is otherwise unchanged.4 To derive (
) from (8), we need the proposition

(
) (IIIa): - - - - - - - - - - - - -

4

[Frege’s discussion here is difficult to follow, because he is here referring not to an analogue of proposition (
), with the “contrary subcomponent”, but to an analogue--proposition (87  ), which he mentions below--of proposition (86), the proposition with which we will be left once we have removed  n' and  m' from the proposition (85). Indeed, the relevant analogue of proposition (
) is not provable, as can be verified. [I am, indeed, inclined to think that this passage may have been misprinted.]

( )

( ) (IIa):: - - - - - - - - - - - - - -

45

( )

( )


( ): - - - - - - - - - - - - - - - - - - - - - - - - - - - - ( ) (8): - - - - - - - - - - - - - - - - - -




( ) (85):: - - - - - - - - - - - - - - - - -

(
) (22):: - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( )

( ) 

 



32

( )

(80):: - - - - - - - - - - - - - - - - - - - - - - - - - 46

(86)

( ) 

(49): - - - - - - - - - - - - - - - - - - - - - - - - - -

(
) (22):: - - - - - - - - - - - - - - - - - - - - - - - - - -

( ) (IIIc):: ___________________________ ( ) (80):: - - - - - - - - - - - - - - - - - - - - - - - - - -

( )

( ) (86): _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _

(
) (68): - - - - - - - - - - - - - - - - - - - - - - - - - -

( ) (30,18):: = = = = = = = = = = = = = = = = = =

(87) (IIIc): __________________

( )

(
)

( )




( )

(IIIc): __________________

47

(88) (23,23): = = = = = = ( ) (
)

( ) (16): ____________________ ( )

(89)

(68): - - - - - - - - - - - - - - - - - - - - - - (Ie): _____ ( )

(
) (71):: _______ (90)

48

. Proofs of Some Propositions about the Number Zero a) Proof of the Proposition

§96. Analysis We now prove the proposition that no object falls under a concept the number belonging to which is 0. The proposition mentioned in the heading is somewhat more general, since the function-letter ‘f’ does not indicate only concepts. Our proposition follows easily from the proposition

By definition (
), we have to prove

which can be done with (49), by deriving

For this, we use (8) and need the proposition

which is easily obtained from (58).

(I): ___________________

( )

(94)

(93  )

(49): - - - - - - - - - - - - - - - - - - -

(93  )

(91): _________________

94


(IIIc): _____________

( )

  

(94)

(82): - - - - - - - - - - (91)

(95)

IIIe

(58): _____ (92) (I): _____________ ( )

(8): ________________

( )

(93)

(93  )

§97. Construction

  

( )

( ) ( )

49

b) Proof of the Proposition

and Some Consequences §98. Analysis The proposition mentioned in the heading is somewhat more general than that which we express in words: ‘If no object falls under a concept, then zero is the number which belong to this concept’. We first prove, with (32) and (38), the proposition

92 (Ia): ____________

(IVa): ___________

(96) and then prove IIa

  

( )

( )

(97 ) (Ia): - - - - - - - §99. Construction IIIf

(  ): - - - - - - - - - - - - - - -

(IIa): - - - - - - - - - - - - - - - - -

( )

( )

(38): - - - - - - - - - - - - - - - - - - -

(IIIc): ____________________

( )

( )

(96): - - - - - - - - - - - - - - - - - - - - - - - - -

( )

(91): ___________________

( ) ( )

(97)

(41):: ______________________

(32): - - - - - - - - - - - - - - - - - - -

(38): - - - - - - - - - - - - - - - - - - -

  

( )

§100. Analysis We first extract some simple consequences from (97) and then devote ourselves to the proposition: ‘If a number is not zero, then there is a number immediately preceding it in the number series’; in signs:

( )

( )

(107) We first derive the simple proposition (104)

(96)

For this we need the proposition

50

(103) which follows from definition (H). 10

  

(99)

  

(100)

§101. Construction 94 (I): __________

(58): - - - - - - - - -

( )

( )

(97): - - - - - - - - - - - - - - -

(100): _________________________

( ) (IIa): - - - - - - - - - - - - - - - - - - - - -

93

  

( )

(98)

(IIa): - - - - - - - (IIa): - - - - - - - - - - - - - - - - - - - - -

( )

( ) ( )

( )

  

(97): - - - - - - - - - - -

(101)

IIIe

( )

(101): ________

   51

(102)

IIIe (102): _________________________

(105)

(103)

(106) (IIId): _____________

(IIa): - - - - - - - - - - - - - - - - - - -

( )

( )

(104) ( )

(97): - - - - - - - - - - - - -

( ) (107)

52

E. Proofs of Some Propositions about the Number One §102. Analysis We prove the proposition

( )

(97): _____________________

(  ,113) which we express in words thus: ‘There is an object which falls under a concept if one is the number of this concept’. Were this not correct, then it would follow from proposition (97) that the number one coincided with the number zero. It is to be shown that this can not be. We prove for this purpose the propositions

( )

(101): ____________________

( )

(109,I)::











(  ,110)

(110)

(  , cf. 108) Of these, (  ) follows from (101) with definition (I); (  ), from (68) with proposition (93).

(IIIb): _______ ( )

(108):: _________ §103. Construction 93 97

  

(111)

(I): __________

(112) ( )

(IIId): - - - - - - - - - -

( )

(111): ___________ (68): _______________________

  

( ) (108)

§104. Analysis With (110) and (71), we can easily prove the proposition that a number is one if it follows immediately after zero in the number-series. To prove the proposition

IIIe

(77): _____

  

(109)

82 (58): ____________

we make use of (49) in the form ( ) and now need the proposition

53

(  ,117)

18 By (79) and (18) we have the proposition

and we make use of the proposition

(  ,117  )

  

(116)

(79): - - - - - - - - - - - - - - -

(  ,117  )

(115,115):: = = = = = = = = =

( )

(115) which is easily derived with (77) and (8).

( )

§105. Construction 13

( )

(110,71)::







  

(114)

IIId

(49): - - - - - - - - - - - - - - - - - - - -

( )

(77): ___________

( ) (116): _________________

(8): - - - - - - - - - - - - - -

( )

( )

( )

( ) (117)

I

  

(115)

(IIIc): ___________

54

§106. Analysis We now prove the proposition Ie (  ,122) That is: ‘One is the number of a concept under which an object falls, if it follows, generally, from the fact that an object a and that an object d fall under it, that a is the same as d’. This proposition is a consequence of the proposition

  

(Ib,Id):: = =

(IVa): ____

which we reduce to the following:

( )

( )

(  ,121)

or (  ,118)

( )::____________________

( )

As the mapping relation, the relation presents itself. This relation and its converse must be proven to be functional.



  

( )

§107. Construction IIIa ( ) (Va): ________________________ (Ib,Ib):: = = = = ( )


(33,33): = = = = =

( )

( )



(Va): __________________________ ( ) 

(IIIc): __________________________ ( ) ( ) 

(IIIc): ________________

(16): _____________________________

( )

55

( ) 

( ): - - - - - - - - - - - - - - - - - - - - - - -

( )



(40):: _____________________________

( ) 




36

  

( )

( )

(11): ___________________________

(Ic):: - - - - - - - - - - - - - - - - - - - - - -

( ):: _________________________________

( ) 

(IIIe):: _____________________

( )

( ) 

(32): ________________________________

( ) 

(82):: - - - - - - - - - - - - - - - - - -


( ):: - - - - - - - - - - - - - - - - - - - - - - - - - - -

( )

(  ')



( ):: ________________________________

(  ')



IIa

  

( ) 

  

(118)

118 (116): ___________________ (119) (IIIa): ___________

  

IIIa

(IVa): - - - - - ( ) 

(109):: _______________________

56

(120)

(IIa):: - - - - - - - - - - - - - - - - - - - -

(77): _________________________

( ) (122)

( )

( )

(96): - - - - - - - - - - - - - - - - - - - - - - - - -

(120): ___________________

( )

( )

(121) (IIa):: - - - - - - - - - - -

( )

( )

( )

57

Z. Proof of the Proposition

a) Proof of the Proposition

§108. Analysis The proposition quoted in the main heading says that no object which belongs to the number-series beginning with zero follows after itself in the number-series. We could also say: “No finite number follows after itself in the number-series”. The importance of this proposition may be made more evident by means of the following considerations. If we determine the number belonging to a concept ( ), or, as one normally says, if we count the objects falling under the concept ( ), then we successively associate those objects with the number-words from one up to a number-word N', which will be determined by the associating relation's mapping the concept ( ) into the concept “member of the series of number-words from one' up to N'” and by the converse relation's mapping the latter concept into the former. “N” then denotes the desired number; i.e., N is this number. This procedure [of counting] may be carried out in various ways, since the associating relation is not completely determined. The question thus arises whether, by another choice of this relation, one could reach another number-word M'. Then, by our stipulations, M would be the same number as N,1 but, at the same time, one of these two numberwords would follow after the other, e.g., N' would follow after M'.2 Then N would follow in the number-series after M, which means that it would

follow after itself. That is excluded by our proposition concerning finite numbers. We prove it with the propositions

( ,144) 







(145 )





and






This last is a special case of



(126) which says that the number 0 does not follow in the number-series after any object. This we shall prove first. For this purpose, we need the propositions (108) and (125) This proposition says that an object follows in the q-series after no object if no object stands to it in the q-relation. To prove it, we need the proposition







1

[The reason is not just that both M and N would be numbers of the s. Rather, the objects falling under ( ) can be correlated one-one with both the number-words from ’1’ up to ’M’ and with the number-words from ’1’ up to ’N’. Since, generally, n is the number of number-words from ’1’ up to n, M=N.]


(123) which follows from (K) with (6). We then replace the function-mark F( )' with and have to prove the propositions 




2

[This in turn follows from proposition 243 of Grundgesetze or Theorem 133 of Begriffsschrift. Here, Frege appears just to relying upon the intuitively evident claim that the series of numberwords is linearly ordered.]

(124 ) 

and

58

(124 ) both of which follow from ( ) 

(124 ) 

( ) 

§109. Construction

(123): _________________

K















( )



(6): ______________________________ ( )

(  ): _________________

(124) ( ) 








(IIb): - - - - - - - - - - - - - - - - - -








(125)

108 ( ) 

(125): __________ (126)















(123)

IIa

( ) 

(I): ________________

59

b) Proof of the Proposition

and End of Section Z

§110. Analysis The proposition

§111. Construction K ( ,145 ) 



results by contraposition from

(10): __________________________

( ,145 ) This proposition can be derived from the propositions 



( , cf. 143) 

and








(  ,132)








(127)

IIa

by placing d' for c' and f' for q' in them. We prove (  ) from the propositions 







( ,130) (123): - - - - - - - - - - - - ( ,132 ) 

and

(128)

( ,129)


(IIa): - - - - - - - - - - - - - - - -

which follow easily from ( ), (K), and (123). 

( ) 

60

( ) 

(129): _ . _ . _. _ . _ . _

( )

(132)



(127): - - - - - - - - - - - - - - - - - -















§112. Analysis We now have to prove proposition ( ) of §110. It is a special case of

(129)





(143) In words: “If a number (b) follows after a second number (a) in the number-series and follows immediately after a third number (d) in the number-series, then the third number (d) belongs to the number-series beginning with the second number (a)”. Evidently, the corresponding proposition would not hold, in general, in an arbitrary series. It is here essential that predecession in the number-series takes place functionally (88). We rely upon the proposition that, if in some (q-)series an object (b) follows after a second object (a), there there is an object which belongs to the (q-)series beginning with the second object (a) and stands to the first object (b) in the series-forming (q-)relation. In signs:

(6): ________________________















(130)

IIa

(I): ______________

( ) 

( ,141) If one now knows that there is no more than one object which stands to the first object (b) in the (q-)relation, then this object must also belong to the (q-)series beginning with the second object (a). To prove this proposition, we make use of (123), by replacing the function-mark F( )' with We also need the two 






( ) 

(127): - - - - - - - - - - - - - - - - -

propositions (131)

(IIIa): __________

( ,141 ) 

and ( ) 

(130): - - - - - - - - - - -

61



( ,138  ) The former follows with (IIa) from the proposition 

( ) 

(123): - - - - - - - - - - - - - -

(140) which is a consequence of definition ( ). The latter results, by introduction of the German e' and contraposition, from 



(138 )

( )





By (IIa), we now have

It still remains to show

(138 )

( )





or

(127): - - - - - - - - - - - - - - - - - - -

( ,137)








( ) is a consequence of








(133)

131 ( ,136) and

(IIIa): ___________

( )

( ,134) This proposition is to be proven in a manner similar to that in which (132) was proven.






(130): - - - - - - - - - - -

§113. Construction

( ) 

(133): _ . _ . _ . _ . _ . _

IIa

















(IIa): - - - - - - - - - - - - - -

(10): _______________________

62

(134)















(135)

135















(138)















(139)















(140)

135

(Ia): - - - - - - - - - - - - - (136)

(I): - - - - - - - - - - - - -

(134): - - - - - - - - - -

(137)

IIIe (139): ______

( ) 

IIa

(IIa): - - - - - - - - - - - -

( ) 

( ) 

(140): ________________ ( ) 

( ) 

( ) 

( )

(138): __________________

(141)

( ) (123): _____________________






 IIIc

(88): - - - - - - - - - - - 63








(142)

( ) 

(144): ________________

( ) 

( ) (126): ___________ ( ) 

(142): - - - - - - - - - - - - -

(145)

( ) 















(143)

130

(130): - - - - - - - - - - -

( ) 

(128): _ . _ . _ . _ . _ . _















(144)

132

(143): - - - - - - - - - -

( ) 

( ) 

64

H. Proof of the Proposition

§114. Analysis We want to prove the proposition that the Number which belongs to the concept member of the number-series ending with b follows after b in the number-series, if b is a finite number. With this, the conclusion that the numberseries is infinite immediately follows; that is, it immediately follows that there is, for each finite number, one following immediately after it. We first attempt the proof with proposition (144),  by replacing the function-mark F( )' with
( (
∪’ f) f)'. For this we need the proposition3 






as a conseqence of d a f',5 for which (IVa) is to be used. It is thus to be shown that the same numbers which belong to the number-series ending with a first number (a), without being this number, belong to the number-series ending with a second-number (d), if the first number (a) follows immediately after the second (d) in the number-series. For this it is necessary to establish and 







as consequences of d (a f)'.6 But it turns out that another condition must be added, since  b=a' is to be proven as a consequence of b (d ∪ ’ f)' and d (a f)'. By (134) we have 

( ) Placing, in (102), (a ∪ ’ f)' for u', and a' for m', and for c', we thus obtain 

















If b coincided with a, then the main component would transform into
a (a ’-f)'. By (145), this is impossible if a is a finite number. Thus the subcomponent
0 (a ∪ ’ f)' is also added. Admittedly, the application of (144), as we wanted, thereby becomes impossible; but, with (137), we can replace this subcomponent with
0 (d ∪’ f)' and derive the proposition

from which we can remove the subcompoent



with (140). The question arises whether the subcomponent



can be  established as a consequence of d (a f)' and d ( (d ∪ ’ f) f)'. By the functionality of progression in the number-series (70), we have 





( ) 

We thus attempt to determine whether ( ,152) 

from (144), which then takes us to the goal. In order to have the proposition

admits of establishment as a consequence of d (a f)'.4 This must be done with (96). For this it is necessary to establish 

3

This proposition is, as it seems, unprovable, but it is not here being asserted as true, since it stands in quotation marks.

5

[See here Theorem 148.]

4

6

[See here Theorem 149.]

[See here Theorems 148 and 148 , respectively.] 

65

(  ,148 ) completely, we must next appeal to proposition (137) in the form















( ) 

134

The proposition

(IIId): - - - - - - - - -

( ,148 ) then remains to be proven. By (143), we have

( )





(145): - - - - - - - - - - - -

For this purpose, we need the proposition

( ) (If): - - - - - - - - - - - - ( ,147)

as well, which follows easily from (130).

§115. Construction

( ) (137): - - - - - - - - - - - - -

130

(IIIf): - - - - - - - - - - ( ) ( ): _ . _ . _ . _ . _ . _ . _ . _ 

(146)

( )
















(147)

(77): - - - - - - - - - - - - - - - - - - - - - - - - -

147

(143): - - - - - - - - - - -

( ) 

(IVa): - - - - - - - - - - - -

66

(148)

( ) (96): - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 

§116. Analysis To prove proposition ( ) of §114, we replace the function-mark F( )' with 






(149) (102): - - - - - - - - - - - - - - - - - - - - - - - - - - -

We then have to prove

( ) 

(140): ________________________

( ) 

(IIIc): ________________________ (151 ) which can easily be done with (137). For the transition to ( ) compare p. 68 [of §51]. 

( ) 

(70): - - - - - - - - - - - - - - - - -

§117. Construction IIa ( )

(137): - - - - - - - - - - - - - - -

( ) (IIa): - - - - - - - - - - - - - - -

(150)

( ) 

67

140 (Ie): __________

( ) 

(144): - - - - - - - - - - ( ) 

(I): - - - - - - - - - - - - - - - - - -

( ) 

(151): - - - - - - - - - - - - - - - - - ( ) 

(137): - - - - - - - - - - - - - - - -

( ) 

(Ib): - - - - - - - - - - - - - - - - - ( )















(152)

150

( )

(152): _____________________

(153)















(151)

68

§118. Analysis The proposition

§119. Construction 126

(154)

(130): _________

remains to be proven. By (102), we have

( ) 

(IIIf): - - - - - - - - - Here we can apply (140). We then also have to prove the proposition

( ) 

(58): ____________

(154 ) We use proposition (97), in that we show that no object falls under the -concept.

( ) 

( )

This follows easily from

(97): ______________________

(154 ) That is, only 0 belongs to the number-series ending with 0. This proposition follows from (126) and (130).

( )



(102): ____________________ ( ) (140): ________________ (154) (153): ________________ (155)

69

. Some Consequences

§120. Analysis First of all, we can easily conclude from (155) that, for each finite number, there is another which immediately follows it. That is to say that the number-series beginning with 0 continues without end. Furthermore, we prove a proposition, which grounds our counting, in that it says that, if n is a finite number, then n is the number which belongs to a concept, if a relation maps this concept into the number-series up to n, inclusive, excluding 0, and if the converse of this relation maps that numberseries into this concept. This proposition follows easily from the proposition















(159)

155 (158): - - - - - - - - - - - - - -

(160) (IIIa): - - - - - - - - - - - - - - - - - - - -

( )

(160)



(32):: - - - - - - - - - - - - - - - - - - - - - -

which we prove with (87) and (155).

§121. Construction 155 (161)

( ) 

(IIa): - - - - - - - - - - - - - - (156)















(157)

87

(158) (IIIa): __________________

70

I. Proofs of Some Propositions about the Number Endlos a) Proof of the Proposition

§122. Analysis There are numbers which do not belong to the number-series beginning with 0, or, as we also say, which are not finite, which are infinite. One such number is the number of the concept finite number; I intend to call it Endlos and denote it with “∞”. I define it thus:

§123. Construction 126 (IIId): _________ ( )




(M) ’ f is, of course, the extension of the concept finite 0 ∪ number. The proposition mentioned in the heading now says that the number Endlos is not a finite number. We prove it, as is indicated in §84 of my Grundlagen, by showing that the number Endlos follows after itself in the number-series, which, by (145), no finite number does. First, it is to be shown that Endlos stands to itself in the f-relation:

(134): - - - - - - - - - -

(If): - - - - - - - - - - - -

(  ,165)

( )

We reduce this proposition to

which follows from the propositions

(22):: - - - - - - - - - - - - - -

(  , cf. 165  )

(  ,162) and

( )

( ) (137):: - - - - - - - - - - - - -

(  ,165  )

To derive (  ), we have to show, by (11)

( ) (  ,162  )

which is easily reduced to the proposition

( ) (59):: - - - - - - - - - - - - -

( ,162 ) which may be decomposed into the proposition

( ) (IIa):: - - - - - - - - - - - - - - - - - - -

( , cf. 162  ) and (137). 71

§124. Analysis Instead of proposition (  ) of §122, we first prove the following proposition:1 ( ) (163) For this we need the proposition

(
)

(163 ) which is reducible to the proposition

(163  )

( )

This follows easily from (142).

(156): - - - - - - - - - - - - - - - - - - - - - -

§125. Construction 130 ( ) (Ig): - - - - - - - - - - - - - - - - - - - - - - (142):: - - - - - - - - - - -

(µ) (77): __________________________

 



( )

22 ( )

(IIa): - - - - - - - - - -

( )

(82): ___________________

( )

(11): ____________________________

( ) (71):: __________________________________

1

[There is a misprint at this point in the text, which has here:

(162) Frege is here talking about Theorem 163; as can easily be seen, the above is unprovable.]

72

( )

( ) (162):: __________________________________

(164): __________________________________

( )

(  ): - - - - - - - - - - - - - - - - - - -

(IIIc): _______________________________ ( )

( ) ( )

( )

(23):: - - - - - - - - - - - - - - - - -

(101): - - - - - - - - - - - - - - - - - - - - - -

( )

( ) (22, ):: - - - - - - - - - - - - -


(58): ____________________

( ) (140):: ____________ ( )

(165) (131): __________

 



(166)

145 ( ) (11): ___________________________

 



(163)



(164)

(166):: ____________

(167)

(44): ________________

 

( )

89 (163): ____ ( ) (32): ____________________________________

73

b) Proof of the Proposition

§126. Analysis We now prove the proposition: “If Endlos is the number of a concept, and if the number of another concept is finite, then Endlos is the number of the concept falling under the first or under the second concept” with (144), taking, in place of the function-mark F(  )',

For this purpose, we distinguish the case in which c falls under the u-concept from the opposite case. We thus have the propositions2

We have first of all to derive the proposition

(  ,172 ) In the second case, we also need the proposition

(  ,172 )


(171) which follows easily from (165) and (69). (  ,172  ') We have, by (IIa)

§127. Construction IIId

(If): - - - - - - - -

To this, we can apply (159). To obtain the desired main component, we must prove the proposition (IVa): - - - - - - -

( )

[Theorems 172 and 172 do not correspond precisely with these. Nor do any others, however, since there is a misprint, or mistake, at this point in the text. See below.] 2

(  ,172  ')

74




 

( ) (Id):: _________________________



(170)

165 (170): _________ ( )

(77): _______________________ (168): _ . _ . _ . _ . _ . _ . _ . _ . _

( )

 

( )

If

( )

(Ic,Id): = = = = =

(96): - - - - - - - - - - - - - - - - - - - - - - - - - - -

(IIIa): - - - - - - - - - - - - - - - - -

 





(168)

( )

(171)

( )

69

(IVa): ________

 

(169)

Ic

(IIIf):: - - - - - - - - - - - - - - - -

(I):: - - - - - - - ( ) (IIIf): - - - - - - - - - - - - - - - -

75



( )

(
)

( )

(IIId):: _ . _ . _ . _

( ) (96): - - - - - - - - - - - - - - - - - - - - - - - - - - ( )

(If): - - - - - - - - -

(  )3 (IIIc): - - - - - - - - - - - - - - - - - - - - - - - - - -

( )

 

(µ)



(171): - - - - - - - - - - - - - - - - - - - - - -

Id

(I):: - - - - - - - -

 



( )

Ia (  ): - - - - - - - -

( ) (I): - - - - - -

( ) (  ): - - - - - - - - - -

(IVa): - - - - -

( )

3

[There is a misprint, or slight error, at this point in the text. (172,  ) is said, in the text, to be derived from ( ) and axiom (Va), but these plainly do not suffice. Axiom (Va) would suffice, together with (IIIh), but it is rather easier to use (96) at this point. Frege may well once have had the more complicated inference at this point, but then changed it, without altering the citation.]

( ) (77): _________________________

76

( ) 

(IIIc): - - - - - - - - - - - - - - - - - - - - - - -

(  ):: - - - - - - - - - - - - - - - - -

 



( )

( )

(  ): _ . _ . _ . _ . _ . _ . _ . _ . _ . _

( ) 

Id

(IVa): _______ (75): ___________________

(  ’)

( )


(If):: - - - - - - - - - - - - - - - - - - - (IIa):: - - - - - - - - - - - - - - - - - - - - - -

( )

(  ’)



(IIIb):: - - - - - - - - - - - - - - - - - -

( ) 

(IIIh): - - - - - - - - - - - - - - - - - (159):: - - - - - - - - - - - - - - - - - -

(  ’)

( ) 

( ): _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _

(  ’)

( ) 

( )


(Va): - - - - - - - - - - - - - - - - - - - - -

77

(µ’)

(  ’)

 



(144): _________________________

 



 



(  ’)

Ia

( ’)

108

(IVa): ______

(IIIa): ________ ( ’) (IIIa): __________

( ’)

(  ’)

I

(97):: - - - - - - - - - -

( ’):: - - - - - - - - - - - -

(
’)

(  ’): - - - - - - - -

(  ’)

( ’)

( ’)

(94):: - - - - - - - - - - - - - - - - - - -

( ’)


(77): ______________________

( ’)

(  ’)



78

( ’) 

(96): - - - - - - - - - - - - - - - - - - - - - - - - - - ( ’) 

(IIa): - - - - - - - - - - - - - - - - - - -

( ’) 

(IIIf):: - - - - - - - - - - - - - - -

( ’) 

(IIIa): - - - - - - - - - - - - - - (IIIe):: _________________

(  ’’)

( ’)


(172)

(  ’): - - - - - - - - - - - - - - - - - - - -

( ’) 

c) Proof of the Proposition

§128. Analysis We can render the proposition to be proven in words thus: “If Endlos is the number of a concept, then the objects falling under this concept can be ordered in an unbranching series which begins with a certain object and, without coming back on itself, continues without end”. If ∞ is the number of the u-concept, then there must be a relation which maps the concept finite number into the u-concept and whose converse maps the latter into the former. Let the p-relation be
such a relation; we now ask whether the ( p f p)relation, taken as the series-forming relation, would meet our requirements, taking, as the beginning member of the series, the object to which 0 stands

in the p-relation. With (17), (18), and (71) we easily prove the functionality of our series-forming relation. That the series continues without end, we will be able to derive from (156) and (8).

§129. Construction 71 (17): ___

(17): - - - - - - - - -

79

( )

137 ( ) (18,18): = = = = = = = =

 



(22): ___________

( )

(173)

5

(  ):: - - - - - - - - - - - - -

(5): - - - - - - - - - - - - - - - -

( )

(174) ( )

( ) (IIa):: - - - - - - - - - - - - - - - - - (
) ( ) (Ia): - - - - - - - - - - - - - - - - - - - - - ( ) ( ) ( ) (156): _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _

(8): - - - - - - - - - - - - - - - - - - - - - - -

( ) (µ) (22): ________________________

 



( )

( )

80

For the proof of (  ), we go back to the objects, perhaps m, b, and c, which stand to x, d, and a in the p-relation. That there are such objects is to be shown with (15). Moreover, b appears twice: first, in that m stands to it in the t-relation, and, secondly, as standing in the q-relation to c. The following picture may make matters more perspicuous:

( ) (8): - - - - - - - - - - - - - - - - - - - - - -

(175) §130. Analysis
That no object follows after itself in the ( p f p)series can not be proven, but only that no object falling under the u-concept follows after itself in this series, if the u-concept is
mapped into the concept finite number by the p-relation. We are content, for the present,
with such a series, and later will define, with our ( p f p)-relation, another relation, which shares the remaining properties in question with it, but which has, in addition, the property that no object follows itself in its series. We prove the proposition

It must be concluded, from the functionality of the converse of the p-relation, that there is only one single object of this kind which can come into question for us.

§131. Construction 174

(IIa): - - - - - - - - - - - - - - - - - (178) from the propositions

(  ,177)

( )

and

(IIIc): _____________________

(  ,178  ) These we prove with (123), and we need, for this purpose, the proposition (78):: - - - - - - - - - - - - - - - - - - - - - -

( )

(  , cf. 177  ) which we derive from the more general proposition

( )

(  ,176 )

81

(
) (IIa):: - - - - - - - - - - - - - - - - - - - - - ( )

( ) ( ) (15): - - - - - - - - - - - - - - - - - - - - - - -

( )

( )

(15): - - - - - - - - - - - - - - - - - - - - - - - - -

( )

(IIa): - - - - - - - - - - - - - - - - - - - - - - - - -

(µ) (15): - - - - - - - - - - - - - - - - - - - - - -

( )

( )

82

 



( )

131

( ) (15): - - - - - - - - - - - - - - - - - - - - - - - - - - -

(174): - - - - - - - - -

( ) ( ) ( )

( )

(15): - - - - - - - - - - - - - - - - - - - - - -

( )

( )

 



(176)

( ) (15): - - - - - - - - - - - - - - - - - - - - - -

133 (
)

( ) ( ) (176): __________________ (  ): ________________________ ( )

( )

(177)

(123): - - - - - - - - - - - - - - - - - - - - - -

83

§132. Analysis We now prove proposition (  ) of §130, by con
cluding, from the functionality of the p-relation, that there would be only one object which stood to x in
this relation, if x stood to itself in the ( p f p)relation; while, by (15), there would have to be at least one such object, which would follow after itself in the number-series and which, by (145), could not then be a finite number. From this, it would then follow, by (8), that x could not fall under the
u-concept, if the u-concept were mapped by the p-relation into the concept finite number.

(IIIa): ___________________

(13):: - - - - - - - - - - - - - - - - - -

( )

( )

§133. Construction 145 ( ) (22):: - - - - - - - - - - - - - - - - - -

(IIIa): __________

( ) (13):: - - - - - - - - - - - -

(
)

( ) ( ) (15): - - - - - - - - - - - - - - - - - - - -

( )

(15): - - - - - - - - - - - - - - -

( )

( )

(µ)

( )

(8): - - - - - - - - - - - - - - - - - - - - - - -

(23):: - - - - - - - - - - - - - - - - -

84

( )

from which (  ) will easily be derived in the manner of (177). To prove
(  ), we conclude from the functionality of the p-relation, that
there is only one object which stands
to x in the p-relation; and, since x stands to y in the ( p ∪ ’ q p)-relation, we conclude that there is such an object, which belongs to a qseries which ends with an object n which stands to y in the
p-relation. Thus, if the object m stands to x in the p-relation, then it will also belong to the qseries ending with n. Furthermore, we prove the proposition

( ) (177):: - - - - - - - - - - - - - - - - - - -

( ) (18):: - - - - - - - - - - - - - - - - - - -

( )

(179) and arrive at our goal, by taking as the v-concept, in 
’ q)-concept. this proposition, the (m ∪

(178)

§135. Construction IIIc

§134. Analysis It still remains to be shown that all members of our series fall under the u-concept and, conversely, that all objects falling under the u-concept are members of our series. These are the two propositions

and

(13):: - - - - - - -

( )

(  ,181)

( )

(  ,186) where, more generally, the q-series beginning with m is taken for the number-series. We prove (  ) from the propositions

(8): - - - - - - - - - - - - - -

( ) (18):: - - - - - - - - - - -

(  ,180) and

 

(  ,180 )

85



(179)

22 (IIIa): _____________

(13):: - - - - - - - - - - - - -

( )

( )

(
)

( )

(22):: - - - - - - - - - - - - - - - - - - - - -

 

( )



( )

137

( )

( )

(15): - - - - - - - - - - - - - - - (176): __________________

(179):: - - - - - - - - - - - - - - - -

( ) (µ) (144): - - - - - - - - - - - - - - - - - - - - - -

( )

( )

(174):: - - - - - - - - - - - - - - - - - -

( )

(22,140): - - - - - - - - - - - - - - - - - - -

( ) (15): - - - - - - - - - - - - - - - - - - - - -

86

(180 ): - - - - - - - - - - - - - - - - - -

(180)4

(  ,185  )

(  ) easily follows. §137. Construction ( )

139

(18):: - - - - - - - - - - - - - - - - - - - -

(13):: - - - - - - - - - - - - - - - - - -

( )

(181)

§136. Analysis We have now to prove proposition
(  ) of §134. Since the is mapped by the p-relation into 
u-concept the (m ∪ ’ q)-concept and y falls under the u-concept, we can conclude that there is an object (n), to
which 
y stands in the p-relation, which falls under the (m ∪ ’ q)-concept, i.e., which belongs to the qseries beginning with m. We then prove the proposition

(18):: - - - - - - - - - - - - - - - - - - - - - -

 



( )

(182)

IIa (Ia): - - - - - - - - - - - -

(  ,186 ) with (152). We need for this the proposition

( )

( )

(  , cf. 186  ) ∪ ’ q)-concept is mapped into the uconcept by the p-relation, we conclude that there is an object (e) to which d stands in the p-relation, if d belongs to the q-series beginning with m. From this and from the proposition


Since the (m

(8): - - - - - - - - - - - - -

 



174 4

[The next inference is marked, in the text, just by “ ”, but I have altered the marking to indicate more clearly which such proposition is intended.]

(22):: - - - - - - - - - - - - - - - - - 87

(183)

(184) (137): - - - - - - - - - - - - - - - - - ( )

(IIa):: - - - - - - - - - - - - - - - - - - - -

( )

(152): - - - - - - - - - - - - - - - - - - - - - - - - - -

( )

(182):: - - - - - - - - - - - - - - - - - - - - -

( )

( ) ( )

( ) (IIa): - - - - - - - - - - - - - - - - - - - - - (185) (183): - - - - - - - - - - - - - - - - - - - - - ( ) (23,23): = = = = = = = = = = = = = =

( )

( )

(
)

( ) (22):: - - - - - - - - - - - - - - - - - - - - - - 88

(187) (Ib): - - - - - - - - - - - - - -

( ) (8): - - - - - - - - - - - - - - - - - - - - - - - -

(188) (13): - - - - - - - - - - - -

( )

( )

(188):: - - - - - - - - - - - - -

( )

(186)

§138. Analysis We now define, as was indicated in §130, a relation such that no object follows after itself in its series and which otherwise
shares the properties important for us with the ( p f p)-relation.

( ) (16): - - - - - - - - - - - - - - - - - - - -

 

(N) We show that the (u∩ _ q)-relation has those properties, if the q-relation has them, and that no object follows after itself in the (u∩ _ q)-series, if no object falling under the u-concept follows after itself in the q-series. We first prove the propositions



(189)



(190)

189 (173):: - - - - - - - - - - - - - -

 

(  ,189) 187 (  ,195) The first poses no difficulty; (  ) can be decomposed into the propositions

(Id): - - - - - - - - - - - - - (191)

(  , cf. 193) (  , 194)

(192)

§139. Construction

( ) (125): - - - - - - - - - - - - - -

(6): _______________________ 89

 



(193)

( )

188 (133): - - - - - - - - - - -

 

( )

 

(195)

178

(195): - - - - - - - - - - - - - - - - - - - -

( )

(123): __________________



(196)



§140. Analysis We now have
to prove that, under our assumptions, the (u∩ _ ( p f p))-series continues without end. This amounts to the proposition

( )

188 (131): - - - - - - - - - - - ( )

(  ,199  ) It is to be proven with the proposition

( )

(  ): _________________

(197)

 



which follows from (N).

(194)

194

§141. Construction

(IIa): - - - - - - - - - - - - (10): ______________________ ( )

( )

(Ie):: - - - - - - - - - - - - - -

( )

 

(193): _ . _ . _ . _ . _ . _ . _ . _ . _

90



(197)

137

(181): - - - - - - - - - - - - - - - - - ( ) (175): _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ ( ) (197): - - - - - - - - - - - - - - - - - - - ( )

(198) (186):: - - - - - - - - - - - - - - - - - - - - - -

( )

( )

(199)

§142. Analysis It still remains to prove the proposition

(IIa):: - - - - - - - - - - - - - - - - - - - - - - -

( ) (  ,206 ) This proposition is reducible, with propositions (181) and (186), to the propositions (  ,201)

( )

and

(  ,206  ) Of these, (  ) can be derived from (194), while (  ) is to be proven with (152). For this, we need the proposition

( )

91

 



(203)



(204)



(205)

140 (22): __________ (  ,206  ) which follows from (198) and (137). We then easily reach the end of our section (c).

 


(IIIc): _______________ §143. Construction

 

130 137

(198):: - - - - - - - - - - - - - - - - - - - - -

 



(200)

136 ( )

(194):: - - - - - - - ( )

(200):: - - - - - - - - - - - -

( ) (139): _ . _ . _ . _ . _ . _ . _ . _

 



(152): - - - - - - - - - - - - - - - - - - - - - - - - - - - -

(201)

( )

201 (181): - - - - - - - - - - - - - - - - - - - - ( ) (140):: __________________________

 



(202) ( )

130 (186):: - - - - - - - - - - - - - - - - - - - - - (135): - - - - - - - - - - - - - - -

92

( ) (IVa): - - - - - - - - - - - - - - - - - - - - - - -

( ) (202):: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( ) (203): __________________________________________

( )

(
) (Va): - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( ) (164): __________________________________________

( ) (IIa):: - - - - - - - - - - - - - - - - - - - - - - - - - - -

(µ) (IIa):: - - - - - - - - - - - - - - - - - - - - - - - - - - -

93

( )

( ) 

( ) (196,190): = = = = = = = = = = = = = = = = = =

( ) 

(204):: ____________________________

( ) (199): _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ ( ) 

( )5 ( ) 

(49): - - - - - - - - - - - - - - - - - - - - - - - - - - -

( )


( )

(205): ______________________________




(183): - - - - - - - - - - - - - - - - - - - - - - - - - -

5

[The following transition sign was omitted from the text.]

94

(206) (207)

d) Proof of the Proposition

§144. Analysis We now prove the converse of proposition (207), namely, that Endlos is the number which belongs to a concept, if the objects falling under this concept may be ordered in a series, which begins with a certain object and continues without end, without coming back on itself and without brancing. What it is essential to show is that Endlos is the number which belongs to the concept member of such a series; in signs:

pair (n;y) belongs to our series beginning with the pair (0;x), then n stands to y in the mapping relation to be exhibited. We now define the pair thus: ( ) The semi-colon is thus a two-place function-sign. The expression ( ; )' is accordingly co-referential [gleichbedeutend] with ' if ', ', and ' denote objects. For the extension of the relation, which, in the way explained above, is, as I say, coupled from the p-relation and the qrelation, I introduce a simple sign, by defining:













(  ,262) For this, we use proposition (32) and have to establish that there is a relation which maps the numberseries into the q-series beginning with x and whose converse maps the latter into the former. It is most natural to associate 0 with x, 1 with the next member of the q-series after x, and thus always associate the next number with the next member of the qseries. Each time, we combine a member of the number-series and a member of the q-series into a pair, and we build a series from these pairs. The series-forming relation is thereby determined: a pair stands in it to a second pair, if the first member of the first pair stands in the f-relation to the first member of the second pair and the second member of the first pair stands in the q-relation to the second member of the second pair. And so, if the








(O) Accordingly, indicates our mapping relation and its extension. We now define ( ) 

The propositions

95

and

(  ,262  ) are then to be proven. Instead of (  ) we first prove the somewhat more general proposition

(  ,256)

(  ,254) which we can then also apply in the proof of (  ). We use (11) and accordingly must derive the proposition

The proposition

is easily derived from the proposition

and with it we can reduce (  ) to

p-series q-series m x . . . . . . c d o a It is to be shown that: “If the q-series beginning with x continues without end and if there is an object (d) which, with c, forms a pair which belongs to the (p∪ _ q)-series beginning with the pair m;x, then there is also an object (a) which, with o, forms such a pair, so long as c stands to o in the p-relation”. We first prove the proposition

(  ,241  )

( ,241  ) ( ,235)

( ,211)6 which we can, in light of definition ( ), write as 

( ,240) We prove this proposition with (144), by substituting, for F(  )', the function-mark

(  ,209) This can easily be proven with the proposition

We then need the proposition

(µ,208) which follows from definition (O).

(
,237  ) To aid understanding, I present a pictorial representation of the p-series and the q-series, next to one another:

6

96

Here A' is written for m;x'.

§145. Construction (208) (137): - - - - - - - - - - - - - - - - -

(100): _________________________________

 



(209)



(10): ______________________

( ) (IIa): - - - - - - - - - - - - - - - - - - - - - - - - - - -

 



(210)

210

(IIa): - - - - - - - - - - - - - - - - - - - - - - - - - - -

(209): - - - - - - - - - - - - - - - -

( )

(210): ____________________

(IIIe):: ________________________

( ) ( ) (211)

( ) (IIa): - - - - - - - - - - - - - - - - - - - - (IIa): - - - - - - - - - - - - - - - - - - -

( )

( ) (IIa): - - - - - - - - - - - - - - - - - - - - ( )

(IIIe):: ___________________

( )

(IIa): - - - - - - - - - - - - - - - - - - - - -

( )

97

( )

( ) (23): __________________________ The proposition

( )

(  ,226) is to be proven first, from which, furthermore, the proposition

(212) (  ,230 ) which we also need, follows. From the propositions

§146. Analysis The subcomponent

( , cf. 217)

in (212) is now to be removed. For this we use proposition ( ) of §144, which follows from

We first prove the proposition

( ,224)

(  ,234)

( ,225) we can easily extract a proposition, which differs from (  ) only in that, in place of A', o;a' stands. We can then replace the main component, consisting of the first two lines, with

(  ,232) which we will also use again. For this we need the proposition

To remove the subcomponent we use the proposition (cf. 214) which follows easily from (O). (  ,231) which is similar to proposition (123) and can be proven with it. We write (123) in the form

98

§147. Construction §148. Analysis To prove propositions ( ) and ( ) of §146, we use (213) and need for this purpose the proposition

(14): __________________________________ (  ,223  ) which we can prove with the proposition

 



(213)

IIa

(218) This follows from ( ).

(IIa): - - - - - - - - - - - - §149. Construction

( ) (Ia): - - - - - - - - - - - - -

(IIIc): _________________

( )

(IIIa): _________________

( )

( ) ( ):: ________________________ ( )

( )

(Vb): - - - - - - - - - - - - - - - - - - - - -

(213): - - - - - - - - - - - - - - - - - - - - -

 



(214)

( ) (33): ______________________________________

(IIIh): _______________ (82): ______________________

 



( )

( ) (33): __________________________________

(215)

215

( )

(33): _______________________________

(IIIa): - - - - - - - - - - - - - - - - - - - - (216)

(IIIc): ______________________ (217)

99

( )

( )

(Ie):: - - - - - - - - - - - - -

( )

(IIIe,IIIe)::










( ) (213): - - - - - - - - - - - - - - - - - - - (218)

(Id): - - - - - - - - - - - - -

 

( )



(219)

218 (223) (Id): - - - - - - - - - - - - - - - - - - (Ib): - - - - - - - - - - - - -

 

(220) (IIIh): - - - - - - - - - -



(224)

223 ( )

(IIIc): ______________

(Ib): - - - - - - - - - - - - - - - - - - ( )

(219):: - - - - - - - - - - - - -

(225) (IIa): - - - - - - - - - - - - - - - - - - -

(221) (IIIc): - - - - - - - - - - - - -

 



(222) (IIa): - - - - - - - - - - - - - - - - - - - - - -

( )

222

( )

(222): ______________ (224):: - - - - - - - - - - - - - - - - - - - - -

100

IIa (IIa): - - - - - - - - - - - - -

(217): _________________________

( )

( )

(Ia): - - - - - - - - - - - - - -

( )

( ) (IIIb): - - - - - - - - - - - - - - - - - - - - - -

( )

(213): - - - - - - - - - - - - - - - - - - - -

 



( )

(228)

216 (IIIa): ______________________

(214): - - - - - - - - - - - - - - - - - - - -

( )

 



(229)

IIa

( ) (IIa): - - - - - - - - - - - - - - - - - - -

(226)

( ) (226): - - - - - - - - - - - - - - - - - -

 



(227)

101

( )

( ) (217): ____________________________

( )

(230)

(IIIb): - - - - - - - - - - - - - - - - - - - - - - - - -

(123): - - - - - - - - - - - - - - - - - - - - - - -

( )

( ) (229): - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( ) (228): - - - - - - - - - - - - - - - - - - - - - - -

(227):: - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( )

( ) (231)

102

§150. Analysis To derive proposition (  ) of §146, we replace the function-mark F(  , )', in (231), with

The propositions needed for this purpose we easily prove from (133) and (131).

 



( )

Ie

§151. Construction 133

(131,131):: = = = = = = (Ie): - - - - - - - - - - ( )

(133): - - - - - - - - - - - - -

( ) ( )

(  ): ____________________

(Ib,Id):: = = = = = = = = =

( )

(232) (Id): - - - - - - - - - - - - - - - - - - - (233) (136): - - - - - - - - - - - - - - - - ( )

( )

(200):: - - - - - - - - - - - - - - - -

 



( )

139 ( )

(219):: - - - - - - - - -

(231): _______________________

(  ): _ . _ . _ . _ . _ . _ (234) (210): ___________________

103

 

(235)



(239)

239 (237): ______________________ (236) (212): _ . _ . _ . _ . _ . _ . _ . _ . _ . _ ( ) (22): ____________________________ ( ) ( )

( )

 



(240)

236 (22): _____________________ ( )

( ) (IIa):: - - - - - - - - - - - - - - - - - -

(144): - - - - - - - - - - - - - - - - - - - - - - - - - -

( )

( )

(Ig): - - - - - - - - - - - - - - - - - - - - - ( )

 



(237) ( )

140 (240): - - - - - - - - - - - - - - - - - - - - -

(210): _______________

 



(238)

IIa ( )

(238):: ___________________

( )

104

( )

(  ,243 )

(11): - - - - - - - - - - - - - - - - - - - - - - - - For this, we use the proposition

(  ,242)

(241) which we derive from the propositions §152. Analysis To obtain our proposition (  ) of §144, we have to replace the subcomponent

(  ,242  )

and

in (241) with another. The train of thought is the following. If the pairs (b;d) and (b;a) belong to the (p∪ _ q)-series beginning with (m;x), then either (b;d) must belong to the (p∪ _ q)-series beginning with (b;a) or (b;a) must follow after (b;d) in this series, so long as the (p∪ _ q)-relation is functional. If either (b;a) followed after (b;d) or (b;d) followed after (b;a), then b would have to follow after itself in the p-series, which would violate our subcomponent

(  ,242  ) §153. Construction 137

Thus, only the possibility that (b;d) coincides with (b;a) remains. Then also d coincides with a. Accordingly, we need the proposition ( ) (123): _______________ (  ,243) In words: “If a first object and a second object belong to the p-series beginning with a third object, then the first object precedes the second or is a member of the series beginning with the second, if the seriesdetermining relation is functional”. We prove this proposition with (144), by replacing the function-mark F(  )' with

 



( )

13

(139): - - - - - - - - We then have to prove the proposition ( )

105

 

(  ): - - - - - - - - - - -

( )

(  ): - - - - - - - - - - - - -



(242)

( )

131 (IIIc): __________

( ) (130):: - - - - - - - - -

( ) (144): - - - - - - - - - - - - - - -

( ) (Ia): - - - - - - - - - - (
) (I): - - - - - - - - - - - - ( ) (242): _ . _ . _ . _ . _ . _

 



(243)

232 ( )

(I):: - - - - - - - - - - - -

(Ib): - - - - - - - - - - - - - - - - - (244)

 



( )

 



(245)

136

133

(244):: - - - - - - - ( ) (200):: - - - - - - - - - - - - - - -

106

 



( )

139 ( )

(220):: - - - - - - - - (246):: - - - - - - - - - - - - - - - - - - - ( )

(  ): _ . _ . _ . _ . _ . _

 



(246) (247,247):: = = = = = = = = = = = = =

( )

(6): ________________________

 



(247)

243

( )

(130): - - - - - - - - - - - - - - - - - ( ) (16): - - - - - - - - - - - - - - - - - - - - - - - ( ) (245,245):: = = = = = = = = = = = = (248)

§154. Analysis We now prove the proposition

( ) (219): - - - - - - - - - - - - - - - - - -

(  ,252) For this we need the proposition

(IIa):: - - - - - - - - - - - - - - - - - -

( )

(  ,251) which is derived from ( ). With (13), we extract from it the proposition

107

( )

(  ,252  ) We introduce, D' for o;a', and A' for e;i', and then, after the introduction of German letters, apply (213).

(IIIa): ____________

§155. Construction

( )

(IIIc): _______________

 



(249)



(250)

221 (IIIa): - - - - - - - - - - - - -

 

( ) (250): ________________

IIIh (IIIa): ___________________

( ) ( )

( )

(IIIa): ________________

(Va): - - - - - - - - - - - - - - - - - - - - -

( ) (249): _______________________

(249): _________________

( ) ( ) (251)

(13,13): = = = = = = 108

( ) (213): - - - - - - - - - - - - - - - - - - - - - - - - - -

( ) (213): - - - - - - - - - - - - - - - - - - - - - - - -

( )

( ) (µ)

( ) (IIIa): ____________________

(16): - - - - - - - - - - - - - - - - - - - - -

( )

(252) (248): - - - - - - -

(
) (253) (241): - - - - - - - - - - - - - - - -

 

109



(254)

145 (255) (254): _____________

( ) (71):: __________________________________________

(256)

§156. Analysis In (256), we have proposition (  ) of §144. (  ) is now also to be proven. For this, we use (254), by writing q' for p', 0’ for x', x' for m', and f' for q'. The subcomponent

§157. Construction 144

which then appears, we remove with (156). It still remains to derive the proposition

(230):: - - - - - - - - - - - - - - - - - - - - - - -

(  ,259) where, more generally, x' is written instead of 0’, and p' is written instead of f'. This proposition is reducible to

We derive (  ) from

(  ,259  ) (229): _______________________________

( )

(  ,258) which we prove with the proposition

(229): _______________________________ (  ) follows from (230) and (144).

(  ,257)

110

( )

( ) (257): ______________________________

 



(257) ( )

209

(140):: ____________________ (258) (IVa): ____________________

( ) (258): ________________________________________________ (203): ________________________________________________ (203): ___________________________________________ (210): _______________________________________

( ) ( ) ( ) ( ) ( )

(Va): __________________________________________

( )

( ) (Va): ___________________________________________ (
) (IIIc): _____________________________________________ ( ) ( ):: ________________________________________

( )

(IIIc): _____________________________________ (µ) ( ):: _____________________________________ 

(259) (IIIc): ___________________________

111

 



(260)

I

(IIa):: - - - (205): _____________________________

( )

( )

 



(262)

(261)

(IIIa): ________________________

156 (22): ______________ ( )

( )

( ) (254): __________________ ( )

(261,71): - - - - - - - - - - - - - - - - - - - - - - - - -

(260): _________________________________

( )

( )

( )

( ) (32): - - - - - - - - - - - - - - - - - - - - - - - - - - - -

(263)

( ) (256):: - - - - - - - - - - - - - - - - - - - - - - - - - - -

112

  
  
 

!#"   
  




$#%&
' () !
*,+ - .0/1 24365
798
: /<;=8#7>=?,3@?,AB/C3 D
EFG/H79IJ;2C5 7K8L5@8 I?,AB?8L3 I?,E?M247AB8NAOG/M2 IPAQ8L3@/#;3@24E/ MRG;AOG2HA AO@/C3 D
EFG/H7S8 =2 18#3@1/45
AQ?I)=?,3@?,AB/?=TAO@/8LFVU-/ 14ABI)=2MM?W3GXND
3GY
/47S?,AE2R F@/Z8#79Y
/47K/Y6?,3[2C\V]^`_a,bcdD
3 F
792H3G1HOG?,3@X@;3G8#AT1 8LE?,3@X F@21e68#3[?,ABI/M=fI-/H79?/ I)g>OG?14O6F@/X
?,3GI)g)?WAO[21/47AB2?W3 8#FHU-/ 14AQ243@Y6/43@Y I)g)?WAO[21/47AB2?W3[8LFVU-/1HA9h.0/C3@// Y6243 24F
F
79/:
?2HAB?8#3[=8#7>8#D
7i5
D
75@8 I/IJ;g>OG?14O[g)/?,3
A798
YLD@1/ 2I)=8 MM8 g)IHj

g>O@?14O[g)/C5
798
: /g)?,AO6AOG/C5
798#5G8
I-?WAB?8#3GI

c4Ÿ; `  Ÿ`f

cP¡; G¢ š f c£`; `-ž f cP¡fi?I>79/Y#D@1?,F@M/CAB8NAO@/CEN8#79/X
/43@/47K2M`5 7K8L5@8 I?,AB?8L3

k=l;Tmn;`op247K/8LFVU-/1HABI)243@YNqr?I>AOG//4s
AB/H3GI?8#3[8 =2 79/ M24Ad?8L3t;AO@/43 u lSvc-mwWoyx qrf u I2R
I>AOG2HATlzFG/ M8#3GX
I>AB8NAO@/Cq){BI-/H79?/ I>FG/ X ?W3 3@?,3@X6g)?,AO my2H3GY6/H3GY
?,3@X6g)?,AO|o);g>O@/43@1/CAO@/Cq){79/ M24AB?8L3[?I =}D
3@14AB?8#3@2M243@Y[opY
8 / I>3G8#AT=8 MM8 g~?,ABI/M=?,36AOG/Cq{ I/479?/IHh.0//4s
5 7K/II>AO@?I)?,3[g)8#79Y
I;F
79?/ =MR@;AO
DGIHj€-‚ ƒ b#a,„
…†G\n‡,„[‡,ˆ
bŠ‰‹d\JbŒJ],b4\ŒJŽL…
…@],…†) Œ-„@^‘n‡,„Š’T€4h.0?WAO AO@/C3G8#AB24Ad?8L3[I86/4s
5@M2?,3@/Y@;8#D 75
798#5G8
I-?WAB?8#36EN2 RNFG/ =8#7ED@M24Ad/Y62I)?W36AOG/CO@/2 Y ?,3@XhG“24E/MR@; mZw”oyx q ?I>AO@//4s
AB/43@I?8L3[8
=TAO@/Z1 8L3@1/H5 AQ„ ƒK• b –#‡ ƒ b
=8#7Sg>OG8
I/˜AOG/C5
798
8 =g)/C3G/ /YNAO@/C5 7K8L5@8
I-?,Ad?8L3

c-™; `Hš ™Tf }= 7K8LE›g>OG?1HO6AO@/ZI9D
FG1 8LE5@8L3@/43
ABI)g)?MM`AO@/436F@/ 79/HE8 :
/Yh.0/Y
/479?: /c-™Tf=B798#Eœcd Lž f?,36AO@/=8#7E

c¤G;  šž f

c¥G;  š# f ¦SR[c
fK;AO@?I>E2RNF@/ZY
/1 8LE5@8 I/Y6?W3 AB8N5
798#5@8 I?,AB?8#3 cd  
 f42 3@Y

243@YNAO@/C5 7K8L5@8 I?,AB?8L3




$#%& >    


.0/Y
/47K?: /c
fi=B798#E

F@RN79/4E8
: ?W3GXNAO@/IPD
F@18#EZ5@8#3G/H3 A

c
S;  š# f c ž f-j

c  ; 
f

c}kY fj`{{{{{i{{{{{i{{{{{i{{{{{

.0/C5
798
: /c   f g)?,AO6AO@/C5 7K8L5@8
I-?,Ad?8L3

c  ž f

c #   f  < ž

!

c J;    f g>O@?14O[?I>7K/Y#DG1 ?,F@M/CAB8N5 7K8L5@8
I-?,Ad?8L3|c #ž f9;?,36AO@/=8#7E c}k1fj`{{{{i{{{{{i{{{{{i{{{{{i{ 243@YNAO@/C5 7K8L5@8 I?,AB?8L3

c™Tf ¦SR[cd žG f9;g)/CO@2:
/

c —;  f

c 
 

.0?WAO|cd ž ž f9;g)/C5
798
: /

cPxQ; —™Tf g)?,AO[g>O@?1HO[g)/Z1 24367K/4E8 :
/CAO@/ZI9D
FG1 8LE5@8L3@/43
A —?,79I9A;g)/Y
/479?: /CAO@/?,EEN/ Y ?24AB/18#3@I-/LD@/43@1/ I)8 = 8#D 7SY
/=?W3G?,Ad?8L3|c—f-h


-ž

!



f



c ž
ž f-j cd



fj`{{{{i{{{{{i{{{{{i{{{

cHŸ`f

c}k1fj`{{{{i{{{{{i{{{{{

c}kBFGfj`{{{{{i{{{{{i{{{

c

c™Tf

 
f

cHŸ`f c
 š f

c  š 

c}k=fjj`{{{{{i{{{{{i{{{{{i{{{{



¢
f

!

c}kk2 f-j{i{{{{{i{{{{{i{{{{{i{{

cP¡f



c}kY fj`{{{{{i{{{{{i{{{

 

!

c}kk2 f-j{i{{{{{i{{{{{i{{{{{i{

cJ£f

c}kY fj`{{{{{i{{{{{i{{{{ c

f

c}kkk/f-jj 

c¤
f

c#@f !

c  f


 

c ž f  

!

c  š fj 

cd

fjj`{{{{{i{{{{{

c}kkk2f-j h  h  h  h  h 

cd #   fjj`{{{{{i{{{{{i{{{{{i{{{{

c™Tf

c   f cd  ž fj`{{{{i{{{{{i{{{{{i{{{{{

cd 

f 

cd #ž fjj`{{{{{i{{{{{i{{{{{i{{{{{

c
f

c f

!



fj`{{{{i{{{{{i{{{{{i{{{ cf

c™Tf

cd 

cJ¥ f

c™Tf

c  ž ž

cd #ž fj`{{{{i{{{{{i{{{{{i{{{{{

c¤
f

; 

¢
f-jj

cd ž@ fj`{{{{i{{{{{i{{{{{i{{{{

cd   Gfjj`{{{{{i{{{{{i{{{{{i{{{{

cHŸ`f

c}kk2 f-jj{{i{{{{{i{{{{{i{{{{{i{{{{

cP¡f

cJ£f




c  f

c-™;  š f g>O@?14O[g)/C5
798
: /g)?,AO|c   
f9;F@RN79/45@M2 1?,3@XNAO@/ =}D
3@14AB?8#3t{dE247Ke u  c G;¥ f u g)?,AO

cd

c9xf


fjj`{{{{{i{{{{{i{{{{{i{{{{{i{{

c

—8L7AO@?IJ;g)/C3G/ /YNAO@/C5 7K8L5@8 I?,AB?8L3

f


`OG?I)?I>79/Y#D@1?,F@M/CAB8NAO@/C5 7K8L5@8 I?,AB?8L3

g>O@?14O[=8 MM8 g)I)=B798#E

c—f

8N5 7K8 :
/c£f9;g)/CD@I-/C5
798#5G8
I-?WAB?8#3|cd

c f

c4Ÿ;  š— f

cP¡;  š— f

c£`;  š— ¡f žG f?,36AO@/Z=8L7E

243@Y6I9OG8
gpAO@24A};8L3[8#D 7S2 I-I9D E5
AB?8#3GIJ;Œ)12H363G8#AQ=8
MM8
g 2=BAB/47S–?,36AO@ / G{BI-/H79?/ I;I?,3@1/CAO@/43t;FGR[c ž @f9;Œ)g)8#DGMY F@/M8#3GXNAd8NAOG / @{}I/47K?/I>F@/X
?,3
3G?,3@X6g)?,AO[„@;243@Y 18#3@I-/ LD@/43
ABMR6„6g)8#DGMY6=8
MM8
g 2=BAB/47S?,AdI-/ M=?,36AO@ / @{ I/479?/IHh

c
f $#%  () !
*,+ -

`OG/I9D F@18#E5G8#3G/H3 A

$#%% >    
Lž

EDGI9A3G8
gpF@/C79/4E8
: / Y[c}1 =-h  V @š f-h `OG?I)?I)Y 8#3G/g)?,AO AO@/C5 7K8L5@8 I?,AB?8L3

cd ž Gfjj`{{{{{i{{{


c š f

cd 
¢G; 

f-jj

c
¢
f    

cJ£f

!

cd

fjj`{{{{{i{{{{

c}kkk1f-j h  h  h  h  h  h 

cd

c™Tf



¢G; 
f-j

c¤
f

c š
f  š


!

cd 

;  #  -f jj

cJ¥ f

cd ž Gfjj`{{{{{i{{{{

c
f c™Tf

c f cd ž@ fj`{{{{i{{{{{i{{{

cHŸ`f

kk2

!

c}kkk2f-j

c}k2fj`{{{{i{{{{{i{{{{{i{{{{{i{{{ cd  ž fj`{{{{i{{{{{i{{{

cP¡f


Hš

cf

c  f

c  -f j h  h  h  h  h  h  h  h  h  h  h 

c

c9xf

c}kk2 f-jj{{i{{{{{i{{{{{i{{{{{i{{{{{i{

cd    fj  c

cf

c f

c -ž
fjj 

c}kYG; k}F@f-jj 

c
`f

c—f

c}k/fjj`{{{{{i{{{{{i{{{{{i{{{{{i{{{{{i{{

cd
¢G;  ž f-jj

c f

f


c}kBFGfj`{{{{{i{{{{{i{{{{{i{{{{{i{{{{{i{{ c}k/fj`{{{{i{{{{{i{{{{{i{{{{{i{{{{{i{{

f 

c
`f

c—f

c š— f !


¢

 
cd

fj`{{{{i{{{{{i{

cd  

fj`{{{{i{{{{{i{{{

c
f

cd  fj  h  h  h  h  h  h  h  h  h  h  c™Tf

c f cd 

¢ fjj`{{{{{i{{{{{i{{{

cHŸ`f

cP¡f cf

c
f-j{i{{{{{i{{{{{i{{{{{i{{{{{i{{{{ cd 

fjj`{{{{{i{{{{{i{{

c}kkk2f-j{{{i{{{{{i{{{

c -ž
fjj 

cJ£f

c š @f

c  f

cd  
 fjj`{{{{{i{{{{{i{{{{{i{{{{{i{{{{{i{

c™Tf

cHŸ`f

cPxf !

c}kk2 f-jj{{i{{{{{i{{{{{i{{{{{i{

cP¡f

c}kk2 f-j{i{{{{{i{{{{{i{{{{{i{{

cJ£f

cd š— fj 

c}kk2 f-j{i{{{{{i{{{{{i{{{{{i{{{{{i{{

cPxfj`{{{{i{{{{{i{{{{{i{{{{{i{{{{

c¤
f

cJ¥ f




c


f

c™Tf

c

fjj 

c š f



c š# f 




!

c š
 f

cd šž fj 

c  @f-j{i{{{{{i{{{{{i{{{{{i{{{{{i{{{{{i{{

c™Tf cd š# ;  š# f-jj

c 

c™Tf

f-jj

cHŸ`f

 fjW{{i{{{{{i{{{{{i{{{{{i{{{{{ cd —

cd  @š fjj`{{{{{i{{{{{i{{{{{i{{{{{i{{{{{

cHŸ`f c š
š f $#%
() !
*,+ -

8N79/45@M2 1/CAO@/ZI9D
FG1 8LE5@8L3@/43
A g)?,AO Qc    u 
f u ;g)/IPD
FGI9AB?,AD AB/    u     cd š
š f h O@/C5 7K8L5@8 I?,AB?8L3 

u =8L7 u  u ?,3

c šž f 

!

c-™; G¢ ¡f ? I)/ 2I?MRN5
798
: /H3hkB3[8L7KY /H7iAOG2HA u  u IPO@8#DGMY6/H3 Ad/47AO@/ 5
798#5G8
I-?WAB?8#3[?,365GM21/8
=  u   u ;AO@/C5 7K8L5@8
I-?,Ad?8L3@I  c Hš ¢ fK;

c  Gfjj`{{{{{i{{{{

c š#  f -žG 

!

243@Y

c4Ÿ; G¢   f

c9¡;1=hGG¢
f 247K/CAB8NFG/CD@I-/ Yh.Š?,AO|c    fK;g)/C5 7K8 :
/CAOG/CE8#79/ X
/43@/4792 M 5
798#5@8 I?,AB?8#3t;g>OG?1HO[247K?I-/ I)=}7K8LEœc9¡`fTFGRNAOG/ 79/H5GM21/HE/43
AT8
=  u g)?,AO u ^ u 243@Y68 = -u g)?,AO u _ u h.0/ AO@/4363@// YG;=8L7AO@?IJ;AOG/C5
798#5G8
I?,AB?8#3

cd š#  fjj`{{{{{i{{{{



g>O@?14O[=8 MM8 g)I)=B798#Eœcd
¢
f243@Y[c ¢
f-h

c£`; G¢
¡f

cHŸ`f

cd š#  fjj`{{{{{i{{{{{i{{{{{i{{{{{i{{{{{

$#%
>    
¢ cd
¢ fj`{{{{i{{{{{i{{{{{

c}kf-jj{{{i{{{{{i{{{{{i{{{{{i{{{{{i{{

cP¡f

c™Tf

c    fj 

cJ£f

$#%
 () !
*,+ -

8N5 7K8 :
/C5 7K8L5@8 I?,AB?8L3|cHŸ`fi8
= 5
798#5G8
I-?WAB?8#3GI

c}kf-jj

c -ž
fjj 





;g)/CD@I-/CAO@/

c¤
f c ¢ ž f243@Y[c Hš ¢ fh.0/C5 7K8 :
/c-™Tfg)?,AO|c Jž
ž f-h

c-™; G¢#@f

$#%
& >    


cJ¥ f

¢ ž c š ¢
f

-ž


!

cG¢

cd š ¢ fj 

 

!

cG¢
f c ¢ fjj`{{{{{i{{{{{i{{{{{i{{ 


f

c}kf-jj{{{i{{{{{i{{{{{i{{{{{i{

cd š#  fjj`{{{{{i{{{{{i{{{{{i{{{

c™Tf cd@¢

; 
f-jj

cd 

c Hš ¢ fjj`{{{{{i{{{{{i{{{{{i{{{

;  #   f-jj

cd 

cJ£f

cP¡f

cJ£f

cG¢ ž f   ž

c}kf-jj{{{i{{{{{i{{{{{i{{{{{i{

c™Tf

cHŸ`f

cHŸ`f

cP¡f

c -ž
ž fj 

; G¢#@f-jj

!

c¤
f

cd
 ; 
 f-jj c -ž
fjj 

-ž


cJ¥ f

cG¢#@f

cd

cG¢  f

c}k
)2fj`{{{{i{{{{{i{{{{{i{{{{



¢G; 
f-jj

c™Tf

!

cd@¢#Gfj 

cd  ž fj`{{{{i{{{{{i{{{{{i{{{

 

cHŸ`f

cd@¢ ž f-jj

cP¡f

cJ£f

c}¢ 

fj`{{{{i{{{{{i{{{{{i{{{{{i{{{{{i{{{{{i{{{{{i{{{{

c¤ f

cG¢   f !

  š

cd š
š fj`{{{{i{{{{{i{{{{{i{{{{{

c -ž
ž fj`{{{{i{{{{{i{{{{{

c ¢ ž fjj`{{{{{i{{{{{

c™Tf cd@¢
; 4š ¢
f-jj

cG¢

f 

c}kkk2f-j

c™Tf

cHŸ`f

c™Tf

cd@¢   fjj`{{{{{i{{{{{i{{{{{i{{{{{i{{{{

c ¢  fj  h  h  h  h  h  h  h  h 

cP¡f

cHŸ`f cd@¢ š f cP¡f  ž

"  
 
  
-


! 

 
 
 $#%  () !
*,+ - .0/C5
798
: /C5
798#5G8
I?,AB?8#3|c£f8 =   Gš g)?,AO|c —8L7AO@?IJ;g)/C3G/ /YNAO@/C5 7K8L5@8 I?,AB?8L3

g O@?14O[=8 MM8 g)I)g)?,AO|c}k )2 f243@Y[c}¢ > 5
798#5G8
I-?WAB?8#3GI 

f=}7K8LE

AOG/




f-h

c ; 
@f 243@YNAO@/M?e
/˜7K/4E2?W36AB8NFG/C5
798
: /H3;g>O@?1HO[12436F@/ Y
8L3@/g)?,AO|c   fh

c-™; ` f

$#% >    
  c @¢ fj`{{{{i{{{{{

c4Ÿ; `
™Tf

c4Ÿ`f?I>7K/Y#DG1 ?,F@M/CAB8NAO@/˜5
798#5@8 I?,AB?8#3

c™Tf

cP¡; ` Ÿ`f

c   fj 

c£`; 
@š f g>O@?14O[?I>Ad8NFG/Y
/479?: / Y6=}798#Eœc ž Gfh.0/=?W79I9AT8#F Ad2?,3

k}AQ?I>3@8 gr3G/ 1/ I-I247KRNAB8N5 7K8 :
/CAO@/˜5
798#5@8 I?,AB?8#3

c¤G; 
 f

cP¡f `

`OG?I)=8 MM8 g)I)=B798#E

AO@/C5 7K8L5@8 I?,AB?8L3

cHŸ`f

c¥G; 

 f

!

cd  f-jj{{i{{{{{i{{ cJ£f

c S; 

  f g>O@?14O[?I)I?,E?M2H7AB8[cd žG f243@Y612H36FG/Y
/479?: / Y6=}798#E ?,A9h.z79?,Ad?,3GX[c ž@ f AO D@I

cP¡f-j

c¤
f

cG¢
¢
f

c 

f

AO@/C5 7K8L5@8 I?,AB?8L3@I !

c  ; G¢
¢
f   

kkk=

@¢ ¢

c  ¢ fj`{{{{i{{

c   fj`{{{{i{{{{{i{{ c 
 f




cd

fjj`{{{{{i{{{{{

!

c @¢ fj`{{{{i{{{{{i{ c 
 fj  h  h  h  h  h  h  c™Tf cd ž@ fjj`{{{{{i{{{{{i{

c   fj 

c™Tf

cHŸ`f

c 
#ž f

cHŸ`f c 

 ; 

 f-jj

c™Tf

cP¡f `

!

c 

fjj`{{{{{i{{{{{i{{

cd
@f-jj{{i{{{{{i{{{{

cHŸ`f

cJ£f

cP¡f-j

c   fj`{{{{i{{{{{i{{

c 

  f

c¤
f

 ž 

!

c 
@f c

;

cd

fjj`{{{{{i{{{{{i{

c™Tf

c 
 f 
 

c 
 fj  h  h  h  h  h  h 

f-jj


!

cHŸ`f

c 

 f

c 

!

 



; š ¢
f-jj


cd

fj`{{{{i{{{{{i{

c 

 f c `
f  


c 
 fj`{{{{i{{{{{i{{{

!

c  Gfj`{{{{i{{{{{

c™Tf

c™Tf

c

fjj  c 
@š f   ž

!

c -ž@  ; 

c `
f

c}kkkY fj`{{{{i{{{{{

c 

f-jj`{ { { { { { { { {



fjj 

c}k=fj`{{{{i{{{{{i{{{ c 
@š fjj`{{{{{i{{{{{i{

c ` @f

c 
¢
f

cd c}k1#; kY
f-jj

c™Tf



¢ fjj`{{{{{i{{{{{i{

c™Tf

c™Tf c}k
)2fj 

c 
fjj`{{{{{i{{{{{i{{{{{i{{{{{i{{{{{i{{{{{i{{

c 
f-j

cP¡`f

cJ£f

c¤ f

 

cHŸ`f

cB¢ 

f-j

c

c}kkk2f-j{{{i{{{{{i{{{{{i{{{{{i{{{{{i{




fjj`{{{{{i{{{{{i{{{{{i{{{{{

c}kkk2f-j{{{i{{{{{i{{{{

cd@¢ š fjj`{{{{{i{{{{{i{{{{{i{{{

c}kkk2f-j{{{i{{{{{i{{{{{i{{{{{i{

cJ¥ f

c ` f cHŸ`f

c™Tf c `Hš f c `-ž f

$#%
' () !
*,+ -

`OG/M2 IPAABg)8NA7K243@I-?,Ad?8L3@I)I/479:
/=8#7AOG/C7K/4E8 :
2M8
= … h.Š/˜3@8
gp5
798
: /CAO@/C5 7K8L5@8 I?,AB?8L3

c™Tf

c-™; ` ¢
f g)?,AO|c Jž
ž fi2H3GYN3@// Y6=8#7AOG?I>5 D
75@8 I/CAO@/˜5
798#5@8 I?,AB?8#3

g>O@?14O[g)/Y
/479?: /=}7K8LE

cHŸ`f

cd
¢
f-h

c  ¢ f

$#% >    

¢ c `  f





!

cd š#  fj 

cd #ž fjj`{{{{{i{{{

c   fj`{{{{i{{{{{i{{{{{i{{{

c `  f c}kk2 f-jj{{i{{{{{i{{{{{i{{{{{i{{

c™Tf

c `
 f cHŸ`f c™Tf

 š

c ` ¢
f cP¡f

c    fj`{{{{i{{{{{i{{{{{i{{{{{i{{

c 
f

c Hš fjj`{{{{{i{{{{{i{{{{{i{{{

cJ£f c™Tf

c¤
f

c   fjj`{{{{{i{{{{{i{{{{{i{{{{{

c  f

$#%   () !
*,+ -

`OG/I9D F@18#E5G8#3G/H3 AQ?,3|c   fTED@IPA3G8
gpF@/ 79/HE8 :
/Yh.0/C5
798
: /=8#7AO@?I>5 D
75@8
I-/CAO@/C5 7K8L5@8 I?,AB?8L3

cJ¥ f

c
f g)?,AO|cB¢
f9;`c

f9;`cd #   fK;243@YNAOG/C5
798#5G8
I?,AB?8#3

c-™;     Ÿ`f

c f g O@?14O[=8 MM8 g)I)=B798#Eœcd > 5
798#5G8
I-?WAB?8#3 

¢
f243@Y[cd 
fg)?,AO6AO@/

c4Ÿ;    f

cf g>O@?14O[g)/C5
798
: /g)?,AO|c Jž
ž f-h $#% % >    


c  f

c -ž
ž fj 

cP¡;  
@f

 

c9xf kk2

!

c -ž
ž fj  c -ž
fjj  cPxfj 

c


c™Tf

f

c—f

cd @¢



¢G; 
f-jj

c  
@f

c}k/fj`{{{{i{{{{{i{{

c™Tf

c   f

c}k2fj`{{{{i{{{{{i{{{{{i{{{ cd 

fjj`{{{{{i{{{{{i{{

c}k=fj`{{{{i{{{{{i{{{{

cd #   fjj`{{{{{i{{{{{i{{

cHŸ`f

c™Tf

cP¡f

cHŸ`f

cP¡f

cd



cJ£f

fj`{{{{i{{{{{i{{{{{i{{{{{i{{{

c¤
f cJ£f cJ¥ f

c}¢
f-j{i{{{{{i{{{{{i{{{{{i{{

c
f

c  ž f

c f

c  ¢ fj  h  h  h  h  h  h  h  c}¢
f-j{i{{{{{i{{{{{i{{{

c™Tf

c}kkkBFGfj 

c 

f 

c™Tf c  ¢ fj`{{{{i{{{{{i{{{{{i{

c 

fj  h  h  h  h  h  h  h  h 

c}kkk2f-j

cHŸ`f cHŸ`f

c   
f

cP¡f c    f




. Proof of the Proposition

§172. Analysis We may attempt to render the proposition to be proven in words thus: “If the number of a concept is finite, then the objects falling under it may be ordered in a simple series, which runs from a certain object up to a certain object”. This expression of the proposition is imperfect in so far as, so rendered, the proposition seems not to hold for the number zero. We could, however, accept as a series-forming relation one which is such that no object belongs to its series running from to , since what definition (P) requires, in order that an object should belong to this series running from to , never occurs. We have, by (314)


To prove ( ), we need, among others, the following propositions: 

( ,329)








and



Accordingly, there is a relation which maps the (1; u f)-concept into the u-concept and whose converse maps the latter into the former. Let the prelation be this relation. We now show that we can take the ( p f p)-relation as the series-forming relation. It is first to be shown that every object falling under the u-concept belongs to the ( p f p)-series running from x up to y, where 1 stands to x, and u stands to y, in the p-relation. More generally, we write, instead of ’1’, ’m’, instead of ’ u’, ’n’, and instead of ’f’, ’q’, and prove the proposition 

( ,330 )









These we derive from













( ,328) by once letting r coincide with x, s with m, and b with a, and, the other time, letting c coincide with y, a with m, and b with s, and then writing ’a’ for ’s’ and ’c’ for ’r’. Compare this with the following picture: 

( ,334) 

which follows with (8) from

In order to prove ( ), we use (152), replacing the function-mark ’F( )’ with 

( ,334 ) The following picture may aid understanding: 





131

Moreover, as in the proof of (186), propositions (183) and (185) will be applied, by means of which the subcomponent will be introduced. ( ) 

§173. Construction

(285):: - - - - - - - - - - - - - - - - - - - - -

183

(273): - - - - - - - - - - - - - - - -

( ) ( )

(322):: - - - - - - - - - - - - - - - - - - - - -



(271,265):: = = = = = = = = = =

( ) 

(185):: - - - - - - - - - - - - - - - -

( )


( )

( ) 

(152): - - - - - - - - - - - - - - - - - - - - - - - - - - - ( )

(I):: - - - - - - - - - - - - - - - - - - - - - 132

 



(329)

140 (328): ________ ( ) (I):: - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( ) 

(269,270):: = = = = = = = = = = = = =

(
) (182):: - - - - - - - - - - - - - - - - - - - - - - -

( ) 

(274): - - - - - - - - - - - - - - - - - - -

( ) 

(IIa): - - - - - - - - - - - - - - - - - - - - - - -

( )

(329):: - - - - - - - - - - - - - - - - - - - - -

 



(328) (330)

140 (328): __________

( ) 

(269,270):: = = = = = = = = = = = = = 133

§174. Analysis To remove the subcomponent we prove the proposition ( )

(15): - - - - - - - - - - - - - - -

( ,332) 

which, with (177), we reduce to

( )

(IIIa): _________________

( ,331) We show that there are objects s and a such that s stands to x, and a stands to y, in the p-relation and a follows after s in the q-series. From the functionality of the converse of the p-relation, it then follows that s coincides with m, and a coincides with n, and also that n follows after m in the qseries. Also, the subcomponent 

( ) 

(79):: - - - - - - - - - - - - - - - -

( ) is to be removed. This is easily done with (17).

§175. Construction IIIa ( )


(15): - - - - - - - - - - - - - - - - - - -

(78):: - - - - - - - - - -

( )

( )





 

( ) 



177

(331): - - - - - - - - - - - - - - - - - -

134

(331)

 



332

( )

( )

( )





(271):: - - - - - - - - - - - - - - - - - - -

 



(333)

17 ( ) (8): - - - - - - - - - - - - - - - - - - - - - - - - (17): - - - - - - - - - -

( ) 

(265,18):: = = = = = = = ( )


(18):: - - - - - - - - - - - - - - - - - - - - - ( ) 

(330): - - - - - - - - - - - - - - -

( ) 

( )

(23,333):: = = = = = = = = = = = = = = (334)

135

§176. Analysis We now have to prove proposition ( ) of §172. The proposition

§177. Construction



136 (174): - - - - - - - - -

( ) 

( ,341 ) remains to be derived. This is derived with (179) from the propositions 



( ) 

( ,338) 

and ( )

(15): - - - - - - - - - - - - - - - - - - - - - -

( )

( ,337) by deriving, from the functionality of the p-relation, that the same object, which stands to c in the prelation, both belongs to the q-series beginning with m and belongs to the q-series ending with n. Instead of ( ) and ( ), we first prove propositions with the same subcomponents, which have as their main components1









( ) 



(15): - - - - - - - - - - - - - - - - - - - - ( )

and With proposition (180), we pass to ( ). To pass to ( ), we need the similar proposition

( )






(177):: - - - - - - - - - - - - - - - - -



 

( ,335)

which we derive from (177).



( ) 

140 (184): _________ ( ) (IIIc): ____________________

1

[The propositions to which Frege is here referring are, respectively, 338 and 337 .]




136

(
) (130):: - - - - - - - - - - - - - - - - - ( ) ( ) 

( ): _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ 

( )

(335)

 






(15): - - - - - - - - - - - - - - - - - - - -

(336)

( ) 

(336): - - - - - - - - - - - - - - - - - - - - - -

23 (IIa): - - - - - - - - - -

 

( )



(337)

IIIa



(IIa):: - - - - - - - - - - ( ) 

( ) 

(78):: - - - - - - - - - - - - - ( )

(15): - - - - - - - - - - - - - -

( )

(IIIa): ____________________ ( ) 

( ) 

(79):: - - - - - - - - - - - - - - - - - - -

137

 



(338)

IIa

( )

(46): - - - - - - - - - - - - - - - -

( ) 

( )

(15): - - - - - - - - - - - - - - -

( ) 

(Va): - - - - - - - - - - - - - - - -

( )

(IIIc): __________________ ( ) 

 



(339)

( ) (15): - - - - - - - - - - - - - - - - - - - (339): _____________________________

 



 

( )



(340)

179




180 (273): - - - - - - - - - - - - - - -

( ) 

( ):: - - - - - - - - - - - - - - - - - - - - -

( )






(IIIa): ___________________

138

( ) 

(79):: - - - - - - - - - - - - - - - - - -

( )


(338): - - - - - - - - - - - - - - - - - - - - - - - - -

( )

( )

( )

( )

(337): - - - - - - - - - - - - - - - - - - - - (269,270):: = = = = = = = = = = = = = =

(
)

( ) 

(18):: - - - - - - - - - - - - - - - - - - - - - -

( ) 

(IVa): - - - - - - - - - - - - - - - - - - - - - - -

( )

139

(µ) (334):: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( )

( )

(340): - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

( )

( )






(IIa):: - - - - - - - - - - - - - - - - - - - - - -

( )

( )





(IIa):: - - - - - - - - - - - - - - - - - - - - - - - - - - -

(Ia): - - - - - - - - - - - - - - - - - - - - - - -

140

 

( )



(343)

140 (274): _________

(344)

 



(341)

140 (273): _________

( ) 

(8):: - - - - - - - - - - - - - -

(342) (8):: - - - - - - - - - - - - - ( ) 

(343):: - - - - - - - - - - - - - - - - - -

( ) 

(341):: - - - - - - - - - - - - - - - - - -

( )

(Ig): - - - - - - - - - - - - - - - - - - - - - -

( ) 

(Ia): - - - - - - - - - - - - - - - - - - - - - - -

( )

(324): - - - - - - - - - - - - - - - - - - - - -

( )

( )





(I): __________________________ 141

§178. Analysis We remove the subcomponent by indicating a series-forming relation such that no object belongs to its series running from an object up to an object, as was said in §172. One such relation is identity, since every object follows after itself in the series of this relation.

( )

§179. Construction ( )

37




(8): - - - - - - - - - - - - - - - - - - - - - -

(131): _____________ ( ) 

(272): ______________ ( )

( )





(Ia): __________________ ( )

(IVa): _________________

( )

 



( )

IIa

(
) (49): - - - - - - - - - - - - - - - - - - - - -

(Ia): - - - - - - - -

( ) 

( ) 

( ): - - - - - - - - - - - - - - - - - -

( )

(µ)

( )


(340): - - - - - - - - - - - - - - - - - - - - - - - - - -

( )

( ) 

(345) ( ) (IIa):: - - - - - - - - - - - - - - - - - - - - - - (
) (IIa):: - - - - - - - - - - - - - - - - - - 142

347 ( ) 

(314):: - - - - - - - - - - - - - - - - - - (348)

(µ) (345): _ . _ . _ . _ . _ . _ . _ . _ . _ . _ (346) (IIIf):: - - - - - - - - - - - - - - - - - -

 



(347)

143

     ! "#$%!&& ')(*,+.-/-0)132546+87946:<;3=

->4<+8*,2=6:<13+?2*A@CBD;3E(5*3FWIXPRZYAX[W\^]MNR_Va`LS9Xabc\5bdMNR9efASURZJ^VCJgihkj[l m J^VKonPjPpalLR_VS9Xabcg?R_MqQrSseJ5t_]ucbvW<MNb,Y3b,R9eb,\LSZXPbct9bdS9SUbAMqQxw f Va\zy{f3QDR_Va\5R9Y3b,Q>W
hkj[l^pO} m

hkj[lOp5f m





')*32522+ 6;c')*,2 625G/79+879254

( i')*32522+ 6;c')*,2 625G/79+879254

 ;3*9;5:64  bcY3fAVMNbdVP\bAMS9Xab~K5MqJ^KaJQR_SkR9J^VCRTV5S9MqJ5\O]aY,b,\RTVS9XPb ucf3R_VXPb3f,\R_Va`R_VCIDJ^Mq\5QDf3Q   gfcY,JOVPY3bAKSh"! m R9QDQN]5KabAMqJ^Mq\5RTVPfdSUb~SUJfcQb3Y,JOVP\ Y,JOVaY,bAKS h€e m WfAVa\R9g$) # Va\5t9JQDR9QS9Xab~V5]u~}abAMiJg SZXPbcg?RTMNQNS8W S9XabAVS9Xab~V5]u~}abAMiJg SZXPbcQ/b,Y,JOVa\R9QDb,R_SZXPbdMf3t9Q/J%) # VP\t9JQ JOMRZQDg?R_VaR_SUb_& n ' |Chkjap( m WIDbO) VPJIWR_VS9Xabcg?R_MqQrSKat9f,Y3bOW<S9XafASS9Xab i JO}+ * b3YASUQDg8f,t9tZR_VP`L]Va\5bdM SZXPb,! €Y,JOVaY,bAKSucf,|L}abcJ^Mq\5bdMNb,\

Q/bAMNRZb,Q}ab,`R_V5VaR_Va`IDR_S9X?> IXPR9YdXCg?f3t9t9Q]VP\bAMS9Xab,@ Y,JOVaY,bAKSsfAVa\LK5Mqb,Y,b3\5b3 Q  _R VSZXPb €Q/bAMqR9b,Qdn b~VaJ5I \b,g?RTVP b  hBA m

C

Y,Y,JOMN\R_Va`5tZ|PW>$DshEFDshG@IH m?m R9QS9Xab~S9M?]S9X[€ef,t_]abcJ5g J= Q}ab,R_Va`LS9Xabcg?R_MqQrSsJ^-} *b3YASsR_VS9XabK UQbdMNR9b3Q}Pb3`5R_VVaR_VP` IDR_S9? X > IXPRZYAXCg?f3t9t9Q]VP\bAMS9Xa, b @UY,JOVPY3bAKSNn C Va\Q/JPWIDb b,Q/Q/bAVSUR9f,tZt9|LXPf3e5bcJO]5MKMNJOKPJQ/R_SUR9JOVCR_VS9XabcQb3Y,JOVP\ Xab,f,\R_Va`< n  b~K5MqJ5ebcR_S.IDRTS9XS9Xab KMNJOKaJ5Q/R_SUR9JOV

R_VCfcQ/R_u~Kat9bcQbdMNR9b3QDIXPR9YdX}Pb3`5RTVPQDIDR_SZXCf KafAM8SUR9Yd]Pt9fdM JO+ } *b3YASsfAVa\PWIDRTS9XPJO]S.Y3J^ucRTVP`L}af, Y )J^VCR_SkQb3t9gNWY3J^VSUR_V]Pb3Q IDR_S9XaJO]5SsbAVa\?€RZg SZXPbdMNbcfAMqb fdVP|Jg.S9XabAux8kucf3| Q/R_ucR9t9fdMNt9|L}abcJ^Mq\5bdMNb3\R_VCfcQbdMNR9b3QWIXaR9YAXCb3R_S9XabAM Y,JOVSUR_V]Pb3QDIDR_S9XaJ^]S.bdVP\JOMibAVa\5QDIDRTS9XCf~KPfdM?SUR9Yd]PtZfAM JO+ } *b3YASN n  R_S9XKMNJOKPJQ/R_SUR9JOVPQh9 j /0 m fdVP\C1h 0^
j ( m WIDb~S9XabAV MNb3f,YdXCJ^]Mi`5Jf,t?n6HXaR9QDJOMN\bAMqR_VP`fdMNR9Q/b,Qu J5QNS.Q/R_u~Kat9|R9g IDb~}Pb3`5RTVCIDR_S9XS9Xabcg?R_MqQrSsJ^} *b3YASsR_VS9Xab JOMNRZ`5R_Vaf,tQ/bAMNRZb,Q IXaR9YAXCg8f,t9t9Q]Va\5bdM SZXP b .UY,JOVPY3bAKS.fdVP\LSZXPbdVCf,t9IDf3|5Q KMNJY,b,b3\LSUJLS9Xab~VP+ b 25SsJ^} *b3YAS8WR_VSZXPbcJOMNR9`R_Vaf,tQbdMNR9b3QW IXaR9YAXCg8f,t9t9Q]Va\5bdM SZXP b .UY,JOVPY3bAKSN n  b~Xaf,ebcg?R_MqQrSSUJ QNXPJI S9XafASS9XabAMqb RZQDf g8R_MNQNSsJ^} *b,YdS8WR_VS9Xa b 3PUQbdMNR9b3Q }ab,`R_V5VaR_Va`IDR_S9, X 4aWIXPR9YdXCg?f3t9t9Q]VP\bAMS9Xa b .€Y3J^VaY,bdK5S€W R9gqWR_VS9XaR9QDQbdMNR9b3QW<SZXPbdMNbcR9QDfAVCJO} *b,YdS.IXaR9YAXCg8f,t9tZQ]5Va\5bdM S9Xa b .€Y3J^VaY,bdK5SsfASsf,tZt? n  XafASs\5Jb,Q  S9XabcJ^} *b,Yd6 S 5R9QSZXPb g?RTMNQNS.R_VSZXP b 3a€Q/bAMNRZb,Q}ab,`R_V5VaR_Va`IDR_S9, X 4IXaR9YAXCg8f,t9t9Q ]VP\bAMS9Xa b .€Y3J^VaY,bdK57S &Pucb,fd V 895Lu~]aQrSsg?f3t9t ]Va\5bdM SZXP b . Y,JOVaY,bAKSsfAVa\L}Pb3t9J^Va`LSUJLS9Xa b 3PUQbdMNR9b3Q}Pb3`5RTV5VaR_Va`IDR_S9, X 4;: }]5SVPJJO} *b,YdS.IXaR9YAXKMNb3Y,b3\5b3QDR_S.RTVS9XPRZQDQbdMNR9b3QDR9Q f,tZt9J5IDb3\LSUJg?f3t9t)]5Va\5bdMS9XP b .^€Y,JOVaY,bAKSNn  VCJ^]M Q/|^u~}aJt9R9QNuxW

Eh LW 0 O
am IXaR9YAXCIDbcb2KMNb,Q/QDR_VCIDJOMN\QS9X]PQ+  HXabAMNbcR9QVaJJO}+*b3YAS IXaR9YAX}ab,t9JOVa`5QSUJLSZXP b 3a€Q/bAMNRZb,QM?]VVPR_Va`g€MqJ^uM4L]5KSkJ 5fAVa\IXPRZYAXCg?f3t9t9Q]VP\bAMS9Xa b .UY,JOVPY3bAKS8WR9g.S9XabAMNbcR9QVaJ g?RTMNQNS.JO+ } */b,YASsR_VS9Xa b 3PUQbdMNR9b3Q}Pb3`5RTV5VaR_Va`IDR_S9, X 4IXPRZYAX g?f3t9t9Q]VP\bAMS9Xa b .UY,JOVPY3bAKS.fdVP\R9N g 5L}ab,t9JOVP`QSUJLS9Xa b 3P Q/bAMNRZb,Q}ab,`R_V5VaR_Va`IDR_S9, X 4&_n  g.S9XabAMqbcR9QDf~ucbAu~}abAMiJg.S9XP b 3a€Q/bAMqR9b,Q}ab,`R_VVPR_Va` IDR_S9, X 4IXaR9YAXCg8f,t9tZQ]5Va\5bdMS9XP b .^€Y,JOVaY,bAKS8W<S9XabAVCIDbcY3fAV SU f )5b3QS9XPRZQDf,G Q 5R_VKMNJOKPJQRTSUR9JOV Eh L m fAVa\LXPbdVPY3b~MNb,f3YAX S9Xabc`5Jf,tJ5gJO]5MiQ/b,YdSUR9JOVCf5 n  bc\bAMqR9e5b~KMNJOKPJQ/R_SUR9JOV Eh L m IDR_S9X h - j m fdVP\LVab,b3\g?JOMS9XPRZQS9XPb~KMNJOKPJQ/R_SUR9JOV

h OW 0 0P m b \R9QNSUR_Va`^]aR9QNXS9XPbcY3f,Q/b RTVCIXPR9YdXRQCg?f3t9t9Q]VP\bAMS9Xab c .UY,JOVPY3bAKS.gUMNJOu SZXPbcJOK5KaJ5Q/R_SUbcY3f,Q/b5n  VSZXPbcQ/b,Y,JOVa\

9R QS9Xab~S9M?]S9X[€ef,t_]abcJ5g<= Q}ab,tZJ^Va`5RTVP`LSUJLSZXPb €Q/bAMqR9b,Q }ab,`R_V5VaR_Va`IDR_S9X?> fAVa\g?f,tZt9R_Va`L]5Va\5bdMS9XPb,@i€Y,JOVaY,bAKS8W fAVa\LS9XabAMqb }ab,RTVP`LVaJJ^} *b3YASsIXPR9YdX}Pb3t9JOVP`QSUJLS9Xa b D







Y,f3QbvWIDbcQrXaJ5IWIDRTS9XK5MqJ^KaJQR_SkR9J^V h9j6lOpO} m W ZS XPfdS S9XabAMqb R9QVaJLucbAu }PbdMJg.S9XaJ b 3a€Q/bAMqR9b,QM8]5VVaR_VP`gUMNJOu 4L]KSUJ%Q IXaR9YAXCg8f,t9t9Q]Va\5bdM SZXP b .UY,JOVPY3bAKSNn


246G+8*,:61,+87924 j[l^pO}

hELW



IXaR9YAXCg8J5t9tZJ5IDQDb3f,Q/R9t9|g€MqJ^u

"  h



f m

L h

m

0 06y m

b~Vab,b3\g?JOM SZXPR9QSZXPb~KMNJOKPJQRTSUR9JOV

hBA m n

hOW

0

m

 246G+8*,:61,+87924

  

A p m                               h

h

"

f m



h+O m   

W  h

L h

h  g m  )   

m

h+O m

( m  

h10
m

  D46(_@aG79G  bcY3J^ucbvWR_VS9Xab~K5MqJ5JgJ5g.KMNJOKPJQRTSUR9JOV hO m J5g

SUJLS9XabcY,f3QbcR_VCIXaR9YAXRC Q g8f,t9tZQ]5Va\5bdMS9XPb^. €Y,JOVaY,bAKSNn Q R_SUQ/b,tZg RZQS9XPbdVS9Xab g8R_MNQNSucbAu~}abAMiJg.S9XPba 3 €Q/bAMqR9b,Q }ab,`R_V5VaR_Va`IDR_S9X, 4 IXaR9YAXCg8f,t9t9Q]Va\5bdM SZXPb. UY,JOVPY3bAKS : S9X]PQSZXPbdMNbcba 2 R9QNSUQDQr]aYAXCf~ucbdu~}abAMiJ5g.S9XabcQbdMNR9b3QAn b Xaf,eb

W

0( m  ) h

h10

p m

h10

m

  

SkJLK5MqJ5eb~S9XPb~KMNJOKPJQ/R_SUR9JOV

j
( 

 gnPS9XabcR_V5SZMNJ\^]aYASUJOMN|LMqbAucfdM  ) QAn



hkj[lOp5f m




/



  

hkj

/

h

ll m  )  

Wj

(

m

  

h10

j m

L h

h

0

m )     

h

0 
5m  )

m

h m

hNy m   

h+O m h m

" h

f m



   

h
m

" h

h  j m  zn zn zn zn zn zn zn zn zn

f m



h  m    

h>w m h

P

h m

m

h




(

j m )      

h m

h m

h 
p m               h10


 D46(_@aG79G  b~Xaf,eb~SUJLMNbducJ5eb~S9Xab

0 m

h m

"  h

}a|QrXaJ5IDRTVP`LS9XafAS 4R_SUQ/b,t9gIDJO]PtZ\g?f,tZt ]VP\bAMS9Xab. Y,JOVaY,bAKSsR9g.S9XPbdMNbcIDbAMqbcf u bAu~}abAMiJg SZXPb3a€Q/bAMNRZb,Q M?]VVPR_Va`g€MqJ^M u 4L]5KSk J 4
IDR_S9X h9j6l,j m n \bAMNRZe5b



VCJOMN\bAMSUJfdK5Kat9|Ch10



j[lvj

QN]}PY3J^u~KaJOVPbdV5S



Y m

  

hNy m

h  m )    

h m

Eh LW 0 O
y m p m WIDb~S9XPbdVXPf3e5b~SUJ

hOW 0 O

m

" h

f m



h  m   


 246G+8*,:61,+87924 h

P

l5l

m

hkj[lOp m )     h

L h

0 0 m)  

h m  

m

h10 O
am hkj
(

m  )  

h+O m

h10

h

h
m

0

p m )     

l (

h>w m


l



m

h10

0


5


" h

/ m



f m



L h

h

0 / m)  

h

"

f m



h10

m

h10

l m

   

m

   h

h

(

 

0 5m  zn zn zn zn zn zn zn

h+O m

0

l^f m

h10 m


  

   

"!   # $&%' (*),+'-.0/1)243 56)879):-;3 /12<)=)247?>A@'BC219)DEC&+GF HI@J D )KCL+C&+'ME>,/1)B>A7NC-O+P.QC&79P.9C:9P.Q):>A+P-;/1D )A/ 79') 56)A5@')A/12Q- KR79')=S'TU2<)A/VC)2@')M C&+ +CL+ME.QC&796WX.9'C:9PK>BB 3 +D )A/ 79'),Y TU:-O+':)Z7U[">2Q.Q>279)/1))A\ ZB>C&+')D]

^Ao [ pq'r os` KU/V-O5t.9C:9PC&7hK- BB- .Q279>779) ^ YvuS ` T/1)B>A7NC-;+PC&+PKf>:A7 >::-;5ZBC249')279')-O/VD)/1C&+'MED )A56>A+'D)DE-K C&7U[">A+'DE279'>A7w+'D B-2QC279),+ 35@')A/e- Kx56)5=@')A/V2Q-KR79')ST 2<)A/1C)2y[".9C:9PK>BBz3 +'D)/79),Y TN:-;+:)AZ 7f["CKRw+D B- 2QC2 79'),+3 5@)/- KR56)A5@)/12Q- KR79) ^ juS ` TU2<)A/1C)2 @')M C&++'C&+'ME.QC&79GcQ["CKec0C279')=KfC&/12175i)5@)/- KR79')=S'T 2<)A/1C)2@')M C&++'C&+'ME.QC&796WE.9'C:A9PKf>BB23 +'D)/79),Y T :-;+':)AZ 7f["CKR79')ST/1)B>A7NC-O+PC2QKN3+':A7?C-O+'>B?[">A+'DECKR+-;@{):A7hK- BB- .Q2Q>KN7N)A/eC&7N2<)BKCL+E79)S'TU2<)A/VC)2| _ 9'C2QC279) Z /1-;Z- 2CL7NC-;+P}O3'-O7?)DEC&+P-;3/ 9')>DC&+'M|(*),Z/V-j ) ^kR` .QC&79 ^ Hyp p ` |

^U_` (*)C&+'D C:>7N).QC&79ba cda79')KC&/V24756)A5@)/e-KR79')2)/1C)2 .9'C:A9PKf>BB23 +'D)/79)6g TU:-;+':)AZ 7h>A+'Di9>j),7N-iZ /1- j) 79'),7N.Q-iZ/V-OZ'- 2C&7?C-O+'2

^k [mln n `





   f f  _ ^ Hp p ` ]       

^ Hp ` ]              

^ > ` ]]sT T T T TT T T T TT T T T TT T T

^ ln n `

^ ln;q ` _

^  r;q ` ]sT T T TT T T T TT T

^ H l'n ` ]sT T T TT T T T TT T T

^ H ` ]sT T T TT T T T TT T T

^ Hr r ` ]sT T T TT T T T TT T T T TT

^ ln H `

^ ln `

^ D ` ]sT T T T TT T T T TT T T T TT T T T T

^ l'n n ` ]sT T T TT T T T TT T T T

^ ln r `

^ ln;l `


"!   # $&%' _ -iZ/V-j ),Z/V-OZ'-2
^A7U[ 3+'D)/ Z>/7NC:3B>A/e>2<243 5Z7?C-O+'2y[ 79')A/V)=C2Q>856)5=@')A/e- KR79)S'TU2<)A/VC)2@')M C&+ +C&+'ME.QC&79 - K.9'C:9PC&79'- BD279>7R)j)/1Ji56)A5@')A/e-KR79')STN2)/1C)2 /3 + +C&+'MEKU/V-O5 c 3 ZE7N - ;[".9'C:A9PKf>BB23+'D)/ 79),YOT :-;+':)AZ 7f[ @')B-;+'M27N-i79) ^ YvuS ` TN2)/1C)2/3 ++'C&+'MEKU/1-;5 c 3 ZE7N-i79'C256)A5@)/ | /1-;5 79'C2Z /1-;Z- 2A! 7 ;["CKCL7RKf>BB23 +'D)/79),Y TN:-;+:)AZ 7R>+D @')B-;+M 27N-i79')STN2)/1C)2@)MCL++'C&+'ME.QC&79GcQ[">B2@')B-;+M 27N-i79') ^ YvuS ` TU2<)A/VC)2@')M C&+ +CL+ME.QC&79Gcd| _ 9') >2<2135Z 7NC-;+[ 79'>A! 7 ,@)B-;+M 27N-i79')STN2)/1C)2@)MC&+ +'C&+M .QC&79GcQ[":>A+E@'),/1)AZ'B>:)Di@'Ji79')>2<243 5Z7NC-O+E79'>A7hC&7 @')B-;+M 27N-i79')STN2)/1C)2@)MCL++'C&+'ME.QC&796W["CKc0C279') KCL/121756)A5@')A/e- Kx79)S'TU2<)A/1C)2@')M C&++'C&+'ME.QC&796WE.9'C:A9 K>BB23 +D )A/ 79'),Y TN:-;+:)AZ 71|(*),Z /1- j),Z /1-;Z- 2CL7NC-;+ ^kR` .QC&79 ^ "H  ` >+Di+'))DEK-;/ 79C279),Z /1-;Z- 2


^ H H ` ]sT T T TT T T T TT T T T

^ l'np ` ]sT T T TT T T T TT T

^ ln `

^ ln `

^
^ lnp ` l'n"H



^ n ` ]               

^
^ Hr r ` ]sT T T TT T T T TT T T

^os`

^ ln `

^
p

^os`

^Ao [ml  q  ` _ E - D)/1Cj ),79C2Z/V-OZ'-22<) C&+P.9'C:A9
PKf>BB23+'D)/ 79),YOTU:-;+':)AZ 7hKU/1-;5 79') -;ZZ'-22<)| +E79)B>77N)A/4[".Q),9>j),79'),Z/V-OZ'-2
^ [ml  q os` C&+P.9'C:A9ba Ra;C2Q./VC&77N)+PC&+217N)>DE- KIa ^ Y;uS ` a | _ 9C2 K- BB- .Q2Q)>2CBJEKU/V-O5 ^ r;q;@ ` | '   <   

^ > ` ]] T TT T T T TT T T T TT T T T T

@

^ `

^  r;qO@ ` ] T TT T T

^ > ` ] TT T T T TT T T T TT T

^
^ `

^  [mlr n  ` + -;/1D )A/ 7N-iZ /1- j ) ^os` [".Q)>M >CL+PDC217NC&+'M;3C219E79'):>2<) P C&+P.9'C:A9 :-C&+':CD)2Q.QC&79
PKU/V-O5 79)-;ZZ'- 2C&7?)|(*) 79')A/V)K-;/1),9'>j ),79),Z /1-;Z- 2
^ l  q `  v "!   #$L%< +P-;/1D )A/ 7N-iZ /1- j )8Z /1-;Z'-22<)C&+P.9C:A
9 PK>BB23 +D )A/ 79'),Y T :-;+':)AZ 7h>A+'DiZ/V-j ),79),Z /1-;Z- 2




^ s[ml ;p `

^k [m l  q kR` C 217NC&+'MO3'C219CL+Mi79'):>2)C&+P.9'C:A9 E217N>+D 27N- PC&+ 79') ^ Y;uS ` T?/V)B>7NC-;+PKU/1-;5 79')-OZ Z'-22)v[".Q),Z/V-j ) 79'),7N.Q-iZ/V-OZ'- 2C&7?C-O+'2






^ [m l 'n os` 7N-iZ /1- j) | _ 9')K-;/56)A/eC2/V)D;3:C&@'B),7N- ^ l;pp ` [">+DEKf-O/ 79'C2Z 3/fZ- 2).Q),53217h249'-.079'>A7f[ 3 +D )A/e-;3/ >2<2135Z 7NC-;+2y[ E@)B-;+M 27N-i79') TN2)/1C)2@)MC&+ +'C&+M .QC&79Gcd|  -;/ 79>7f[".Q)249'- . 79'>A7xc @')B-;+'M27N-i79) T 2<)A/1C)2/3 + +C&+'MEKU/V-O5 c 3 ZE7N- ["KN/1-;5 .9'C:A9PC&7 K- BB- .Q279'>A7 i@)B-;+M 27N-i79') TN2)/1C)2@)MC&+ +'C&+M .QC&79Gc ^  n  ` |





 '   <   



.9')A/1)Ea Ra7N> )279),Z'B>:)=- KQa ^ Y;uS ` a [">A+'D

l;p 

^Ao [ml 'n `

j



^   ` ]]sT T T T TT T T T TT T

^   HA[ n ` ]]


^ l  H `

^ l  `

^ : ` ]       

^ `

^ l  l `





n  ^ H l  ` ]sT T T TT T T T TT T

^ l;p p ` ]sT T T TT T T T TT T

^  n ` ]]sT T T T TT T T T TT

^ l ;p `

^
'"! #$L%< (*),+'-.0Z/V-j ),Z/V-OZ'- 2C&7?C-O+ ^ ` -KeF H Hv|  -;/79C2 Z 3/fZ- 2),Z /1-;Z- 22QD - )279') Z /1-;Z- 2CL7NC-;+ ^os`

.9'C:A9PC2Q)>2 ` ]] T TT T T T TT T T T T

^ `

^k [ml  `

'   <    H l  ^  n ` ]]sT T T T TT T T T

^ l l ` ]]sT T T T TT T T T TT T T T T

^ `

0

^  ;p ` ]sT T T TT T T T TT T

^
^   q[ n ` ]]


^ > ` ]] T TT T T T TT T T T T

^  r;qO@ ` ]] T T TT T T T TT T T T TT

^os`

^ `

^ l  `

^ ` ]] T TT T T T TT T T T TT T T T TT

^ l;p ` ] | | | | | | |

^ `

^os`

^ Hp'q ` ]]        

^
`

^ `

^ l 'n ` O



'"! #$L%<  /1-;Z- 22B2-E219- .

^ k [mlrp ` ^"` | 7/1)5i>C&+'27N-

l'nOl

79>779')A/V):>A+E@'),+-i56)5=@')A/ . 9'C:A9PKf-BB-.Q2Q>KU7N)/ ECL+E79)S'TU2<)A/VC)2Q>+DE.9'C:A9  Z /1):)D )2 PC&+E79)S'TU2<)A/1C)2Q>A+'DE.9'C:A9PKf>BB23 +'D)/79) Y TN:-;+:)AZ 71|  K Q.Q)/1)243':A9P>,56)A5@')A/4[ 79)+[ @'JE-;3/ >2<2135Z 7NC-;+2y["C&7h.Q-O3'BDi9'>j )87?-i@)B-O+'Mi7N-i79')STN2)/1C)2 /3 + +C&+'MEKU/V-O5 c 3 ZE7N -  ]





^ Hp p ` ]       

^ Hp'q ` ]]      ^  n ` ]]sT T T T TT T T T TT T ^  'n ` ]sT T T TT T T T TT T T T

^
^os` ^ l  ` ^ lr;q `

^Ao [ml'rOl `

3779')A+Q.Q-;3BDE>B2<-i@)B-;+Mi7N-i79) ^ Y S ` TU2<)A/VC)2



/3 + +C&+'MEKU/V-O5  3ZE7?- E>A+'D[":-O+'2)}O3')A+ 7NBJ'[".Q-;3BDE>B2<@')B-;+Mi7?-ESTN2)/1C)2Q)A+'DC&+'MEC&+ i@'Ji79')8Z /1-;Z'-2


^ l'r ` ]sT T T TT T T T TT T T T T

^  [mlr;q ` .9'C:A9PC27?-i@),Z /1- j)+"|C&+':)v[ @JE-;3/e>22135Z 7NC-;+[ .Q-;3BDiZ /1):)D ) QC&+E79')STN2)/1C)2 [ E.Q-;3'BDi79')A+E9'>j ) 7N-EKf-BB-. C&7N2<)BK C&+E79')STN2)/1C)2 [">:>2<)=.9'C:9P.Q) )A\':BL3D )| _ -iZ/V-j ) ^ s` [".Q)D)/1Cj );[".QC&79 ^ Hpp ` >A+'D ^ l'n;l ` [ 79),Z /1-;Z- 2


^


^ > ` ]] T TT T T T TT T T T TT T T

^os`

^ s[ml  ` 
'   <    H ln ^   ` ]]sT T T T TT T T T

^ lr H ` 

^  'n ` ]sT T T TT T T T TT

rOq



^ l  ` ^ H l'n ` ]sT T T TT T T T TT 

^ l'r;q ` ]]sT T T T TT T T T

^ > ` ]] T TT T T T TT T T T TT T T

^ lr `

^
^ Hr r[fH" H ` ]]


^ l ` ]]sT T T T TT T T T TT T T T TT T

^ `

^os`

^
`

^ l ` ]sT T T TT T T T TT T T T TT T T T

^ `

^ `

^ l'r H ` ]sT T T TT T T T TT T T T TT T T T T

^ `

^ lr;l ` _



/

^ H q ` ]             

^ K ` ]]sT T T T TT T T T TT T T T TT T T

^ H ` ]]sT T T T TT T T T TT T T T TT T

^ H ` ]] T TT T T T TT T T T TT T T T TT T

^ H l;p ` ]]sT T T T TT T T T TT T T T TT T

^ l'r;l ` ]]sT T T T TT T T T TT T T T TT T

^
^ lrp `

^
^ l'n ` ] | | | | | | | | | | |

^ `

^ lr `

^
^ l'rn ` .QC&79E79'),Z/V-OZ'-2
.9'C:A9PC2/V)D;3:C&@'B),7N-

^k [mlr `

^Ao [mlr r ` _  9 C2QKf-BB-.Q2Q.QCL79 ^ H l ` |

3 7R.Q)KfC&/1217Z /1- j )879) 2<-O56).9'>A756-O/V)C&+Kf-O/f56>A7NCj ),Z/V-OZ'- 2C&7?C-O+

^os`

^  [mlr  ` . 9'C:A9P.Q).QCBBB>7N)A/ +'))D| _ 9)D )A/1Cj>7NC-;+PC2Q2CL5iCB>A/  7N-i79'>A7h- K ^ ln n ` |

 9HXabAMqb

(

RZQDf u R_VS9Xab~SUb2SNn 

R9QrSkf)b

RTVS9XPb~V]5u }PbdMNR_Va`fASS9XaR9QKPJR_VS


'   <    l'n"H

^


^   ` ]sT T T TT T T T TT T

^ H ` ]sT T T TT T T T TT T T

^ l'n[fHAr r ` ]]


^os`

^ l q ` ] | | | | | | | | | | |

^
^ `

^ `

^ H 'l ` ]       

^kR` ]]         

^ Hr r ` ]sT T T TT T T T TT T T

^os`

^ `

^
`

^ lr `

^ lr r `

^ `



l'rr

^ l'r n ` ]sT T T TT T T T TT T T T

^ lr `

l

^ f` ]] T T TT T T T TT T T T TT T T T TT T T T TT

^   ` ]]sT T T T TT T T T TT T T T TT T T T TT T

^ ` ^ H ` ]sT T T TT T T T TT T T T TT T T T TT T T T TT T

^ `

^ `

^ l  q ` '"! #$L%< (*),+'-.0Z/V-j ),79')8Z /1-;Z'-2
.9'C:A9E/V)D;3:)27N-i79'),Z/V-OZ'- 2C&7?C-O+

^
`

^k [ pq q



^Ao [ml  r  ` . 9'C:A9P.Q)D )A/1Cj)KU/V-O5 ^ l;pp ` >A+'D ^  r  ` |  -;/79C2Q.Q)  +'))Di7?-iZ/V-j ),79),Z /1-;Z- 2
`

.9'C:A9PKf-BB-.Q2QKN/1-;5



^  pl ` >A+'Di79')8Z /1-;Z'-2
^ [ml  `

^ s[ml  ` ^ H l q ` ]     

'   <    Hpq ^ l;p p ` ]   

^ l'n ` ]]sT T T T TT T T T TT T T

^ l  H `

^os`

^ `



^ `

H ln ^ 'l n ` ]]sT T T T TT T T ^ l  ` 

^ Hn ` ] TT T T T TT T T T TT T T T TT T T T l'nOq

^ f` ]] T T TT T T T TT T T T TT T T T T ^ l H ` ]sT T T TT T T ^  ` ]]sT T T TT T T T TT T T T TT T T T

^ `

^ l  `

^ l  l ` ^ l'r ` ]]sT T T T TT T T T TT T T T T

^
^ l ;p ` ^ > ` ]         ^ l'n H ` ]]sT T T T TT T T T TT T T T

^  p'l ` ]sT T T TT T T T TT T T T

^ l  `

^  r ` ]]sT T T T TT T T T TT T T T T

^ l 'n `

^ f` ] T TT T T T TT T T T TT T T T T

^ l[ml  ` ]]


^os`

^
/Op

^ `

^ `

^ ` ^ l q ` ]sT T T TT T T T TT T T T TT T T T TT T

^ `

^ l 'r ` 

^ l ` ]sT T T TT T T T TT T T T TT T T T TT T

^ > ` ]] T TT T T T

^ `

^
^ pq q ` l'nr

^ Hr r ` ]sT T T TT T T T TT T T

^ l  ` Hpq





^ pq H ` 

^ > ` ]]     

l'nr ^ H H ` ]sT T T TT T T T TT T T

^ p'q q ` ]sT T T TT T T T TT T T

^ pq `

^
^ pq l `

^ l'r ` ]]sT T T T TT T T T TT T T T TT T T T T

^os`

/

'"! #$L%< _ -ED)/1Cj );[".QC&79 ^ p'ql ` [ Z /1-;Z- 2
^os` ^k [ p'q  `

_ -i/1)A56- j),79')243 @':-;5=Z'-;+)+7 .Q)>B2<-i+'))Di79),Z /1-;Z- 2CL7NC-;+

^ `

.9'C:A9PKf-BB-.Q2QKN/1-;5

^Ao [ pq os` ^  pl ` | _ 9')=213 @:-O5Z'-O+')A+ 7N2

>A/V),7N-i@),/V)AZ'B>:)Di@'J

^ p'q l[mlOp p ` ]]
 |

^ `



 '    <    l'n  ^ > ` ]] T TT T T T TT T T T TT T T

^ pq `

^ : ` ]sT T T TT T T T TT T T T TT T T T T ^ pq;p `     q ^ >

^ pq'n ` ` ]] T TT T T T TT T T T TT T 



^
/,j

^
^ p'q HA[ pq;p ` ]]


^os`

^ p'q'n ` ]sT T T TT T T T TT T T T T

^ `

^ H H ` ]]sT T T T TT T T T TT T T T TT T

^ Hr r ` ]]sT T T T TT T T T TT T T T

^ l'n  ` ]] TT T T T TT T T T TT T T T TT TT T T T TT T T T TT T T T TT

^  ` ]                

^  ` ]                

^ pq `

^
^s`

^ `

^ `

^
`

^ n ` ]] T T TT T T T TT T T T TT T T T TT T T T TT T T T TT T T T TT T T

^ > ` ] T T TT T T T TT T T T TT T T T TT T T T TT T T T

^
`

^ `

^ pq'r `

/0

1+i')*,222 +"6;


;  D46(T@PG/79G

')*32 62G7Z+?7924

H)JLKMNJe5bxhO m IDR_SZX

H JLMNb3f,YdX6WgUMNJOuh
pal m W<S9Xab KMNJOKaJ5Q/R_SUR9JOVS9XafAS8W ) `R9e5bdVCJ^]Mif,Q/Qr]u~KSUR9JOVPQDfd}PJO]S$3fdVP\ DW;#)Va\5tZJ5QDR9QSZXPb V]5u~}abAMiJg.ucbAu~}abAMqQDJ5g.S9Xa b 3PUQbdMNR9b3Q}Pb3`5RTV5VaR_Va`IDR_S9X,4 IXaR9YAXCg8f,t9t)]5Va\bAMS9XaK b .^€Y3J^VaY,bdK5S8WRZgNWg?JOMib,ebAMq|LucbAu~}abAM Jg SZXP b 3a€Q/bAMNRZb,Q}ab,`R_V5VaR_Va`IDR_S9, X 4aW<S9XPbdMNbcR9QDfdVCJ^} *b,YdS IXaR9YAXCg8J5t9tZJ5IDQDR_SsIXaR9YAXCg8f,t9t9Q]Va\5bdM SZXP b .UY,JOVPY3bAKS8WIDb u~]aQrSsfAKKat9|Chkj /vj m n 'ib,g8J^Mqb IDbc\JQJaWIDbcIDfAVS SkJb 2K5Mqb,Q/QS9XPRZQ KMNJOKPJQRTSUR9JOVucJ^MqbcQr]aR_SUfd}Pt9|g8J^MiJO]5MK]5M8KPJQ/b,Qd n  b Xaf,ebOW
h

 

f m IDb~VPb3b,\LS9Xab~K5MqJ^KaJQR_SkR9J^V




MqbAKat9f,Y3b,\L}a|

= 3;= n

hw)W
 pNO m



24
b

0
/ m )   h

g?JOMS9Xab

h O 0 j5j m g€MNJOu

hPW
p m



3 m =OfAVa\ 3;=vR9QSUJL}ab

QN]5}aY,JOu~KaJ^VabAVSUQ=   h . IXaR9YAXS9XabAVCfdK5Kab,fdM/W = . u~]aQrSS9XabAVCR_VS9MqJ5\O]PY3b

IXaR9YAXCg8J5t9tZJ5IDQDIDR_SZX

 

QN]}PY3J^u~KaJOVPbdV5S

L m

h

hkjapp m  )   HXab KMNJOKaJ5Q/R_SUR9JOV

hkj
0 m



h
p m    

h  L W

 m I XaR9YAXCR9QVab,b3\5b3\g?JOMS9XPRZQDIDb KMNJeb  KMNJOKPJQRTSUR9JOVPQ

h+O m

0 5 m  zn zn zn zn zn zn zn h

IDRTS9XS9Xab

h O 0 jj m )     

h

"

f m



L h

h+O m

m

   

hOW
 p m

h
 p m

h
W
>
w m

/



  D46(T@PG/79G  b~VaJ5I K5MqJ5eb~K5MqJ^KaJQR_SkR9J^V h
m J5g j n6HXaR9Q KMNJOKPJQRTSUR9JOVCQf3|5Q+  RZg SZXPbdMNbcR9QVaJJ^}-*b,YdS.IXaR9YdXCR9QS9Xab g?RTMNQNS.JO}+/* b,YASkIXPR9YdXCg?Jt9t9JIDQDR_VS9XabP 3 UQbdMNR9b3QDf,g€SkbAM
IXaR9YAXCg8f,t9t9Q]Va\5bdM SZXPb . UY,JOVPY3bAKS8W<S9XabAVS9XabAMqbcR9Q fA}aQ/J5t_]SUb3t9|LVPJJO}- * b,YdS.IXaR9YAXCg8f,t9tZQ]5Va\5bdMS9XPb^. €Y,JOVaY,bAKS fAVa\IXaR9YAXCg8J5t9tZJ5IDQDR_VSZXPba 3 €Q/bAMNRZb,QDf,gUSUbAM
[Wf,QDt9JOVa`f,Q S9XabP 3 9MNb,tZfASUR9JOVCQ/fASUR9Q/g?RZb,QDJO]5Mif,Q/QN]5u~KSUR9JOVaQAn  S.R9QDQ/R_ucRZt9fAM SUJLKMNJOKPJQ/R_SUR9JOV h 0 l m fAVa\Y,fdV}Pbc\bAMNRZe5b3\g€MNJOuR_SNn  g QNSUfAVa\5QDRTVS9XPba 3 kMqb,t9fdSUR9JOVSUJLVPJJO}-* b,YdS8W<S9XabAVS9XabAMqb RZQVPJ JO}+ * b3YAS8WIXaR9YdXCg?Jt9t9JIDQDf,g€SkbAM
CR_VS9XabP 3 UQbdMNR9b3QWfdVP\ S9X]PQDf3t9Q/JLVPJQN]PYdXCJ^}- * b3YASsIXPR9YdXCg?f3t9t9Q]VP\bAMS9Xab.  Y,JOVaY,bAKSNn  ' ]S.R9gCQNSUfAVa\QSUJfAVCJO}+/* b,YASNC Q R_VS9XabP 3  MNb3t9fASkR9J^V[W<S9XabAVCIDbcY,fdVCQrXaJI S9XafASsfAVCJO}+/* b,YAS€WPWR9QVaJOS S9Xabcg?R_MqQrSsJ^}- * b3YASsR_VS9XabJa3 €Q/bAMqR9b,Q}ab,`R_VVPRTVP`IDR_SZXR6 Q W IXaR9YAXCg8f,t9t9Q]Va\5bdM SZXPb . UY,JOVPY3bAKS8WR9g
R9QVaJ^SS9Xabcg?R_MqQrS JO}+ * b3YASsg?Jt9t9JIDR_Va`f,g€SkbAM
CR_VS9XabP 3 UQbdMNR9b3QDIXPR9YdXCg?f3t9t9Q ]VP\bAMS9Xab . €Y3J^VaY,bdK5S7

fAVa\ChO m n


 24
ll5W  m h

6]5M8S9XabAM?u

J^MqbOWIDb~XPf3e5b~SUJ\bAMqR9eb

Eh LW
 j SZXPb~KMNJOKPJQRTSUR9JOV

P

( m)  

h 

 

L m

h

 

m

h
m



h m

   

h OW
m IXaR9YAXCY3fAVCb,f3QRZt9|L}Pbc\J^VabcIDR_S9X h9j
j m n6H)JLK5MqJ5ebxhEL m W IDb~K5MqJ5Y3b,b3\g€MNJOu#K5MqJ^KaJ5Q/R_SURZJ^V h10Pl
Pm  L h fAVa\LXaf,eb SZXPb~KMNJOKPJQRTSUR9JOVPQ

m

h
W
 joy m

fAVa\

SUJLKMNJe5bvWJ5gIXaR9YAX h
m g?Jt9t9JIDQDb,f3QR9tZ|g€MNJOu hw m R9QSUJL}abc\5bdMNR9eb,\IDRTS9X h 5m 

HXaR9QDR9QD\J^VabcIDR_S9X

h m

h+O m

h>w W
j w m h 0^j m n m h



     

h
m



/

h a 0 l
Pm )     

h>w m

h m

h m n



0^j

( m)  

h 

  h
 m

h
 m  )   

ll5W  m h

 

h m

h m h m

hNy m



   

h m

0
/l5W
h

p
am

 

h m

h
 j m

h  m

  f



hkjj m              

" h

f m



h

P m

L h

   

j 5m )     h

h m

h+O m

m

h

h
m

0

l m )     

h
m

h  0 pOj m  )  

//

h>w m

ll m  )   h

0

h


h

L

m

m h  m

j m  )  

h


h  ` m )     

h+O m

h
m

"

f m



" h

h>w m

f m



   

   

L

hkj , / j m)    

h
 m

h m h

W 0 

h


p m  )   

m

h

h m

m



0al 

h

/

h


( m   

L h

hNy m

m

h
p

m  )    

h+O m

h m

h
pal m )       

/(

h
m

/

h


m

= i  ')*,222+ 6;c')*,2 <2G/79+?7924


46=z2;31,+?7Z254

(546=


 D46(T@PG/79G  SMNbducf,RTVPQSUJL}abcQrXaJIVS9XPfdS SZXPb~V]5u~}abAMiJg ucbdu~}PbdMNQDJg.S9XPba 3 €Q/bAMqR9b,Q}ab,`R_VVPR_Va`IDR_S9X,4IXaR9YAXCg8f,t9t ]VP\bAMS9Xab . €Y3J^VaY,bdK5SsR9QDg8R_VPRTSUbOWR9gf3g€SUbdMQ/JOucb~ucbAu~}abAM h m J5g.S9XaR9QDQ/bAMNRZb,QVaJLucbAu~}abAMig8f,t9t9RTVP`L]VP\bAMS9Xab . 

 

 b~Xaf,eb~SUJ\5bducJOVPQNS9MNfdSUb~S9XafAS8W<}a|J^]Mif,Q/Qr]u~KSUR9JOVPQ>W S9Xabxh . 3 m €Q/bAMqR9b,QM8]5VVPRTVP`gUMNJOu {]5KSUJ Y3J^VSUf3R_VPQDf3t9t fAVa\JOVPt9|LSZXPJQb~ucbdu~}abAMNQDJg.S9XabJ3a€Q/bAMqR9b,Q}ab,`R_VVPRTVP` IDR_S9, X 4IXaR9YAXCg8f,t9t)]VP\bAMS9Xa b .€Y3J^VaY,bdK5S€WR9g R9QDf ucbdu~}PbdMiQr]aYAXS9XafASsf,tZt ucbdu~}abAMNQDJg.S9XaJ b 3a€Q/bAMqR9b,Q M?]VVPR_Va`g€MqJ^u ]KSUJ cIXaR9YAXCg8f,t9t)]5Va\bAMS9XaK b .^ Y,JOVaY,bAKS}Pb3t9J^Va`LSUJLS9Xabxh . 3 m €Q/bAMqR9b,QM8]5VVaR_VP`gUMNJOu ]KSUJ ;












Y,JOVaY,bAKSsg?Jt9t9JIDQWfdVP\R_VCY3f,Q/bcJ^]Mif,Q/Qr]u~KSUR9JOVPQDfd}PJO]5S 3fAVa\ fdMNbcQ/fASUR9Q/g?RZb,\6n6HXPR9QKMNJY,b3b,\QDIDR_S9X KMNJOKPJQRTSUR9JOV h 0O j / m R_VS9Xab g8J^M8u








h
jPp w m

6MNJOuh10

/ m

WIDbcb,f3QR9tZ|Y,JOVPY3t_]P\b~S9Xab KMNJOKaJ5Q/R_SUR9JOV

hOW
jap



L m

VCJOMN\bAMSUJJO}SUf,R_VCf~KMNJOKPJQ/R_SUR9JOVCIDR_S9X fAVa\fASS9XaR9QKaJ5R_VSsfAKKPt9|LSZXPb~KMNJOKPJQRTSUR9JOV f,Qu f,R_VCY3J^u~KaJOVPbdV5S8WIDbcIMqR_SUbxh

"

f m R_VSZXPbcg?JOM?u

/l

IXaR9YAXCIDbc\bAMNRZe5bcg€MqJ^u

h
p( m fAVa\



0

b~KMNJe5bxh>w m IDR_SZX

hkj


h
W
 l m

" h

(

hw)W


f m

h
 l m



    

m

m n


 24


h 

" h

f m



m

m

   

0 m)  

hkj


L h

L h

m  )   

h+O m    h
m j
(5p

0
/ 5m ) h

ll5W  m h

  



 

h
m

h
 f m )    

h
 j
(


( m



h
p(PW


( m   

/ 

h

L m

hO m

hkjj m                                        

h


m  )       

h
m

hw m

h
jap m



0Oj/ h  " f m                    

h 

/ m 

hEL m       

h
jap m  )       

(p

hO m



 D46(T@PG/79G  b~u~]aQNS



MqbAucJe5bcQN]5}aY,JOu~KaJ^VabAVS =  sh
 = 3 m = €g MqJ^u#S9Xabct9f,QNS KMNJOKaJ5Q/R_SUR9JOVon6HXPR9QDRZQDf,Y,Y3J^u P K t9RZQrXab,\IDR_SZX S9Xab~K5MqJ^KaJ5Q/R_SURZJ^V

h
m

Eh LW
jj  m I XaR9YAXCIDb~KMNJe5bcIDR_S9X h9j
0 m W<}a|QrXaJ5IDRTVP`LS9XafAS cY,fdV  VaJ^SK5Mqb,Y3b,\b R_VS9Xab3PUQbdMNR9b3QW<}ab,Y3fA]aQb~S9XabAVS9XabAMqb IDJO]Pt9\L}abcf~ucbAu~}abAMig8J5t9tZJ5IDR_Va`f,gUSUbAM vW







 24
h>w m

f

h 

( m  )

L h

m

 

h+O m h m

hkj


h
p0 m  )     

h m

h
j m

(

0 m)  

  

h
p

m  )   

h
j

m )     

h
m

h>w m

h m

0 5m ) h

h m

h
pOj m  )    

h m    

hNy m

h
jj m

h
j
0 m

  " h

f m



L h

m

 f

hkjj m                   

     

L h

h+O m

h 

( m

h

0 5m )

h+O m

    

  

h
j
0 m  zn zn zn zn zn zn zn zn zn zn zn h


/ m)  

m

h
m

h>w m

h
m

    

h

" 

f m                            

h
j
am

h>w m

h
j m

 h m

(Oj



" h

f m



h       

" h

L

( m)  

f m



h>w m   

h m     

m

h

hO m )      

h m

hNy m

h+O m hkjap
/ m )       

h 

/

h
j

)

h m       

m



hkjj m                               

h  m

h
j 5m )      

h
 f m              

L h

m

h+O m j
0

/ m

h

P



m

h

" 

h

(( m  

f m             

    

h
m

h m

(0

h
j
( m

h
j

/ m)  

h
m    

(

h
j[l m




N. Proof of the Proposition

For this we need the proposition §29. Analysis We now prove the proposition that a finite number belongs to a concept if a finite number belongs to a superordinate concept. Let the u-concept be the superordinate; the vconcept, the subordinate. We saw at the end of the first volume, that the objects falling under the uconcept may be ordered in a series which begins with a particular object and runs up to a particular object, if the number belonging to the u-concept is finite. We first replace the u-concept with the qseries running between x and c and consider the case in which, indeed, objects fall under the vconcept. Then, let m be the first member of our series which falls under the v-concept and n, the last member which falls under the v-concept. We use our proposition (422), from which we first remove ’m’ with (357). We replace the subcomponents and with the subcomponents and thus obtaining the proposition



For (x (v∩ _∪ ’ q))', the proposition



( ,433
) v' is then to be introduced with

(  ,434) which follows with (IVa), (191) and the proposition

( ,434  )

§31. Analysis The bottom subcomponent in (435) we replace with and

( ,436) which follows from (44). ’x’, ’c’, and ’q’ are then to be removed. Because (345) is not fully appropriate here, we derive from (345) the proposition

(  ,440) Finally, to remove ’a’, we use the proposition ( ,442) which follows from (97).

. Proof of the Proposition

a) Proof of the Proposition

§33. Analysis "The sum of two numbers is determined by them". The thought of the proposition in our main heading is most easily recognized in this expression, and it may be cited for this purpose, although the definite article in the subject actually anticipates the assertion of the determination, and although the word "sum" here is used differently than we will later use it in connection with numbers. Namely, we here call the sum of z and v, if no


converse maps the latter into the former. It is in order for this relation as well as its converse to be functional that we must make the assumption, about the q-relation, that an object which does not fall under the z-concept does not stand to any object in the q-relation, and the same must hold for the prelation. In order to prove the proposition in the sub_ q)’heading from (49), we show that the ’ (z∩ relation has the required properties, if the q-relation maps the z-concept into the u-concept and if its converse maps the latter into the former. Accordingly, we prove the propositions 




object falls both under the z- and under the vconcept. Also, infinite numbers hereby come into consideration. If we wanted to prove the proposition only for finite numbers, then another method would be more sensible. We first prove the proposition in the subheading, which says somewhat more than (49). By contraposition, a proposition follows from it which one may render as: "If the number of the z-concept coincides with that of the u-concept, then there is a relation which maps the z-concept into the uconcept and whose converse maps the latter into the former, and in which an object which does not fall under the z-concept stands to no object". Let the qrelation be such a relation, and let the p-relation be appropriately constituted with respect to the vconcept and the w-concept; then the -relation maps the concept into the



( ,448) 

( ,451) 

To derive ( ), we use, according to (11), the proposition 

( ,445 ) 



and for this the proposition

-concept and its

( ,445 ) 

which follows from (197). 176



§35. Analysis To prove proposition ( ) of §33, we need the proposition

which is again reducible to



( ,451 ) 

( ,451 ) 



b) Proof of the Proposition

§37. Zerlegung We now define:

for whose proof we need a proposition with main component ( )

and prove the proposition

We here distinguish the case in which e falls under the v-concept from the opposite.

(459) with (8) and (11).

§39. Zerlegung We remove the subcomponent with the proposition

( ,462) 

177



c) Proof of the Proposition

and End of Section

§41. Analysis To obtain a proposition with main component

§43. Analysis By making the changes in (463) indicated in §41, we obtain a subcomponent

we place, in (463), ’ q’, for ’q’, ’ p’, for ’p’, and exchange ’z’ with ’u’ and ’v’ with ’w’. We then use the proposition 

.

which we remove with the proposition



( ,464) 

( ,467) 

We need for this the proposition

( ,465) 

which we prove with (13).

178

O. Consequences a) Proof of the Proposition

§45. Analysis By leaving ’x’ and ’u’ in (469) unchanged, but by placing for ’v’ and

We can make use of this proposition in proving that every number stands in the f-relation to some number, which, in the first volume, was demonstrated only for finite numbers. Let us assume that the number n belongs to the w-concept. Then there is either an
object c which does not fall under it, in the fand then w stands to

for ’w’, and by introducing the subcomponents and , we easily reach the proposition in our main heading. If we then place

relation; or there is no such object, in which case, the (0 ∪’ f)-concept is subordinate to the w-concept,
and we find, according to proposition ( ), that w stands to itself in the f-relation. In order to prove the proposition in our main heading, we must show that
coincides with v, if the z-

for ’z’, and

for ’v’, and introduce the condition that c falls under the u-concept, we obtain the proposition

concept is subordinate to the v-concept.

( ,476)

b) Proof of the Proposition

§47. Analysis To reach proposition ( ) of §45, we must prove that coincides with

§49. Analysis In order to
obtain
the proposition with main component ’ u ( u -’f)’, we can not here make any use of (68); rather we need the proposition

if c falls under the u-concept. (475) 179

which we derive first. c) Proof of the Propositions

and

§51. Analysis As is stated in §45, in order to use (476), we need the proposition

§53. Analysis We further derive from the proposition (476), with (145) and (165), the consequence that no finite number belongs to a concept if the number Endlos belongs to a concept subordinate to it.

( ,479) which is derivable from (103). For this we need the proposition

( ,478)

180

I. Table of Basic Laws and propositions following in the first instance from them

(I, §18

(III, §20

(Ia, §49

(IIIa, §50

(Ib, §49

(IIIb, §50

(Ic, §49

(IIIc, §50

(Id, §49

(IIId, §50

(IIIe, §50 (Ie, §49 (IIIf, §50

(IIIg, §50 (If, §49 (IIIh, §50 (Ig, §49 (IIIi, §50

(IIa, §20 (IV, §18 (IIb, §25

180

(V, §20

(Va, §52 (IVa, §51

(IVb, §51

(Vb, §52 (VI, §18

(IVc, §51 (VIa, §52 (IVd, §51

(IVe, §51

181

II. Table of Definitions

(A

(B (Composition relation. §54, p. 72.)

(K (The following of an object after an object in the series of a relation, §45, p. 60.) ( ( (The relation, that an object belongs to the series of a relation, beginning with an object, §46, p. 60.)

(Functionality of a relation, §37, p. 55.)



(M (

(The number Endlos, §122, p. 150.)


(Mapping into by a relation, §38, p. 57.) (N

(E

(§138, p. 171.)

(Converse of a relation, §39, p. 56.)

( 

(The pair, §144, p. 179.)

(Z (The number of a concept, i.e., the number of objects falling under a concept, §40, p. 57.)

(O (Coupling of a relation with a relation, §144, p. 179.)

(H (Relation of a number to the next following, §43, p. 58.)

( 

(§144, p. 179.) ( 

(The number zero, §41, p. 58.) (I (The number one, §42, p. 58.) (P (The circumstance that an object belongs to a series running from an object up to an object, §158, p. 201.)

182

III. Table of Important Theorems













(1)













(77)







(82)



































(174)













(15)













(16)

(2)







(6)



















(13) If a relation is functional and if an object (b) stands in this relation both to a second (d) and to a third object (a), then the second object (d) coincides with the third (a).

(10)







(33)







(36)

(17)

















A relation composed from two relations is functional if each of them is.







(58)

(5)







If an object (d) stands to a second object (e) in a (p-)relation and if the second object (e) stands to a third (m) in a second (q-)relation, then the first object stands to the third in the relatons composed from the first and the second relations.1

























(18)

(8)

(11)

1

The translation which I append to the propositions of Begriffsschrift, although they reproduce their principal content, do not always exhaust the whole content.

(19) If a concept is mapped into a second by a first relation, and if a second concept is mapped into a 183

third by a second (q-)relation, then the relation composed from the first and the second relation maps the first concept into the third.













(22)













(23)







(94)

(108) In the number series, nothing immediately precedes zero. (97) If no object falls under a concept, then zero is the number of objects falling under this concept.

(49) The number of objects falling under a first (w-) conceptdoes not coincide with the number of objects falling under a second (z-)concept, if there is no relation which maps the first concept into the second and whose converse also maps the second concept into the first.

(107) For every number different from zero, there is a number immediately preceding it in the number series.

(32) The number of objects falling under a first (u-) concept is the same as the number of objects falling under a second (v-)concept, if a relation maps the first into the second whose converse maps the second into the first.



















(104)

If zero is the number of objects which fall under a concept, then no object falls under this concept.

(24) The converse of a relation, which is composed from a first and a second relation, is the composition of the converse of the second and the converse of the first.













(99) If zero is the number of objects falling under a first concept, then zero is also the number of objects which fall under a concept subordinate to the first.

(96)

(117) If one is the number of objects falling under a concept and if a first object falls under this concept and, similarly, a second object, then the two objects coincide.

(68) (71)

The relation of a number to the number immediately following it in the number-series is functional.

(121) If an object falls under a concept and if every object which falls under this concept coincides with that object, then one is the number of objects falling under the concept.

(89) The relation of a number to the number immediatley preceding it in the number-series is functional.

184

(133) If an object follows after a second in a series and stands to a third in the ordering relation, then the third also follows after the second in this series.

(122) One is the number of objects falling under a concept if there is an object which falls under this concept and if every object falling under the concept coincides with every object which falls under the concept.

(275) If an object follows after a second in a series and precedes a third in this series, then the third also follows after the second in this series.

(110) The number one follows in the number series immediately after the number zero. (111) The number zero is different from the number







one. (113) If one is the number of objects falling under a concept, then there is an object falling under this concept.













(123)







(129)













(302)







(126) The number zero is not preceded in the number series by any object.













(299) An object follows after a second in the series of a relation if the second follows after the first in the series of the converse relation.

(124) If an object follows after an object in a series, then there is an object which stands to the first in the ordering relation.



















(130)

(200) If an object belongs to a series beginning with a second, then either it coincides with it or it follows after it in this series.













(276)













(280)

(127)

(131) A first object precedes a second object in a series if it stands to it in the ordering relation. 185

























(335)













(180)

(144)

(134) If an object belongs to a series beginning with a second object and stands to a third in the ordering relation, then the third follows after the second in this series.

(132) If an object belongs to a series ending with a second and if a third object stands to it in the ordering relation, then the second follows after the third in this series.













(136)













(137)













(152)

(141) If an object (b) follows after a second (a) in a series, then there is an object which stands to the first (b) in the ordering relation and which belongs to the series of this relation beginning with the second object (a). (145) No finite number follows after itself in the number series. (pp. 137-144).

(322) If an object (d) belongs to a series ending with a second (y) and also belongs to the series, of the same relation, beginning with a third (x), the the second likewise belongs to the series beginning with third.

















(285)







(139)

(155) The number of members of the number-series ending with a finite number (b) follows in the number series immediately after this number (b). (157) For every finite number, there is an immediately following member of the number series.







(140) Every object belongs to the series, beginning with itself, of any relation.







(303)

(304) An object (n) belongs to the series of a (p-) relation beginning with a second object (m), if the second object (m) belongs to the series of the converse relation beginning with the first object (n). 186

(175) If Endlos is the number of a concept and if the number of another concept is finite, then Endlos is the number of the concept falling under either the first or under the second concept (p. 154).

(242) If an object (d) precedes a second (n) in a series whose ordering relation is functional, and if it stands to a third object (a) in this relation, then the second object (n) belongs to the series of this relation beginning with the third (a).

(167) Endlos is not a finite number.

(243) If an object (r) belongs to series, whose ordering relation is functional, beginning with a second object (m) and belongs to the same series beginning with a third object (n), then the latter (n) belongs to the series beginning with the former (r) or precedes it in the series. (306) If an object follows after zero in the number series, then it belongs to the number series beginning with one.

(307) If an object belongs to the number series ending with a finite number, then it is itself a finite number.

(296) If an object (y) belongs to a series, whose ordering relation is functional, beginning with a second object (i), and if the second object (i) follws after itself in the series of this relation, then the first (y) also follows after itself.

























(188)













(189)













(194)













(201)













(191)













(197)

(207) If Endlos is the number of object falling under a concept, then these objects can be ordered in an unbranching series which begins with a certain object and which, without turning back on itself, continues without end (p. 160).

(205)







Endlos follows immediately after itself in the number series.







(249)

(251)

187

If an object coincides with a second, and a third with a fourth, then the pair consisting of the first and third coincides with that consisting of the second and fourth.







(215)







(219) If a pair coincides with a second, then the second member of the first pair coincides with the second member of the second.

















(234)













(244)













(246)

(220)







(221)































(224)













(225)













(257)

(252) If a relation is functional as is a second, then the relation coupled from the first and the second is likewise functional.

(213)



















(208) If an object (c) stands to a second (o) in a (p-) relation and if a third object (d) stands to a fourth (a) in a (q-)relation, then the pair consisting of the first and the third objects (c;d) stands to the pair consisting of the second and fourth (o;a) in the relation coupled from the first and the second relation.

(231)

(233) If a pair follows after a second in the series of a coupled relation, then the second member of the first pair (d) follows after the second member of the second pair (x) in a series whose ordering relation is the second member of the coupled relation.

188













(209)













(258)













(247)



























(210)







(238)













(235)













(211)































(253)



















(269)

(270) If an object belongs to a series running from a second up to a third, then it belongs to the series of the same relation beginning with the second. (312) If an object belongs to the number series running from one up to a second object, then it is different from zero. (314) Every finite number is the number of members of the number series running from one up to it itself.

(259)

(263) Endlos is the number of objects falling under a concept, if these objects may be ordered in a series which begins with a particular object and continues without end, without branching and without turning back on itself (p. 179).

(274) An object (c) belongs to the series of a relation running from a second (m) up to a third (n), if this relation is functional, if the third object (n) does not follow after itself in the series of this relation, and if, finally, the first object (c) belongs to the series of this relation beginning with the second (m) as well as to that ending with the third (n).

(344) 189













(323)













(271)













(282)

(298) (p. 202) (325) The number of members of a series running from an object up to an object is finite. (327) If the objects falling under a concept can be ordered in a series which runs from a particular object up to a particular object, then its number is finite.













(340)

(348) If the number of objects falling under a concept is finite, then these objects can be ordered in a series which runs from a particular object up to a particular object.

Notes to Appendix III

190