STUDIES I N LOGIC AND THE FOUNDATIONS OF MATHEMATICS V O L U M E 93
Ediiors
J. BARWISE, Madison D. KAPLAN, Los Angeles H. J. KEISLER, Madison P. SUPPES, Stanford A. S. TROELSTRA, Amsterdam
Advisory Editorial Board
K. L. DE BOUVERE, Sunta Clara H. HERMES, f'reiburg i. Br. J. HINTIKKA, Helsinki J. C. SHEPHERDSON, Bristol E. P. SPECKER, Zurich
NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK 0 OXFORD
ANDRZEJ MOSTOWSKI
FOUNDATIONAL STUDIES SELECTED WORKS V O L U M E I1
Editorial Committee
Kazimierz Kuratowski (Chairman), Wiktor Marek, Leszek Pacholski, Helena Rasiowa, Czeslaw Ryll-Nardzewski, Pawel Zbierski (Secretary)
1979
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Library of Congress Cataloging in Publication Data Mostowski, Andrzej, Foundational studies. (Studies in logic and the foundations of mathematics; v. 93) Some of the papers translated from Polish, French or German. “A bibliography of works of Andrzei Mostowski (compiled by W.Marek)”: p. 1. Logic, Symbolic and mathematical - Collected works. 2. Mostowski, Andrzej -Bibliography. I. Kuratowski, If. Title. 111. Series. Kazimierz. QA9.M7554 51 1’.3 77- 18025 ISBN 0444-85102-X (v. 1) ISBN 0-444-85103-8 (v. 2) go 59%
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Printed in Poland
Editorial Note The present selection of A. Mostowski’s work comprises within one edition the most important of his papers, written during nearly forty years of scholarly activity and scattered so far over various publications. We hope that it will gain the approval of all readers who concern themselves with the foundations of mathematics. In selecting the papers, the Editorial Committee aimed at including, on the one hand, the most important of A. Mostowski’s contributions to mathematics and, on the other hand. those papers which have preserved their topical interest and are the most frequently quoted ones. Most of A. Mostowski’s papers, which were originally published in English, have been reproduced photographically without any changes. In those papers, a double pagination occurs: at the outer corners the running pagination of the volume, and at the inner corner, in square brackets, the reference to the complete bibliography and the pagination of the original version. It is intended to facilitate the use of the international reference marks within each paper which refer to the original page number. Only a few of the papers included in this edition (namely those marked in the Bibliography with the numbers [I], [21, [31, [51, [61, [7l, [511, I7911 have been translated from Polish, French or German, so at the inner corners in those papers the references to the complete bibliography are given only. We are grateful to all publishers of the original papers here included for their consent to the reproduction of those papers, which has enabled us to prepare this edition of A. Mostowski’s work and present it to the readers. In particular the following permissions to reprint Mostowski’s papers were given : On absolute properties of relations, reprinted with permission of the publisher American Mathenptical Society from Journal of Symbolic Logic, copyright @ 1947, Volume 12, No. 2, pp. 33-42; fro0j.i of non-deducibility in intuitionistic functional calculus, reprinted with permission of the publisher American Mathematical Society from Journal of Symbolic Logic, copyright @ 1948, Volume 13, No. 4, pp. 204-207; Arithmetical classes and types of wellordered systems, reprinted with permission of the publisher American Mathematical Society from the Bulletin.
VIII
EDITORIAL NOTE
copyright @ 1949, Volume 55, p. 65; On the rules of proof in the pure functional calculus of the jirst order, reprinted with permission of the
publisher American Mathematical Society from Journal of Symbolic Logic,. copyright @ 1951, Volume 16, No. 2, pp. 107-1 I I ; On direct product of theories, reprinted with permission of the publisher American Mathematical Society from Journal of Symbolic Logic, copyright @ 1952, Volume 17, No. I , pp. 1-31; A proof of ffwbrand’s theorem, Journal de MathCmatiques Pures et Appliquks 34 ( 1959, pp. 19-24; Eine Verallgemeinerung eines Satzes von M. Deuring (A generalization of a theorem of M. Deuring), Acta Scientiarum Mathernaikawm Szeged 16 (1959, pp. 197-201 ; Concerning a problem of H. Scholz, Zeitschrift fur Mathematische Logik and Grundlagen der Mathematik 2 (1956), pp. 210-214; An example of a nonaxioniatizable many valued logic, Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik 7 (IY61), PF. 72-76; Concerning the problem of axiomatizability of the jield of real numbers in the weak second order logic, Essays on the Foundations of Mathematics, Magnes Press, Jerusalem 1961, pp. 269-286; Addition au travail “ A proof of Herbrand’s theorem” (An addition to the paper “A proof of Herbrand’s theorem”), Journal de Mathtmatiques Pures et AppliquCes 40 (1961), pp. 129-134; Representability of sets in formal systems, reprinted with permission of the publisher American Mathematical Society from Proceedings of Symposium on Recursive Functions, copyright @ 1962, Volume 5, pp. 29-48; The Hilbert epsiion function in many-valued logics, Acta Philosophica Fennica I6 ( 1963), pp. 169-1 88 ; On models of Zermelo-Fraenkcl set-theory satisJying the axiom of constructibiliy, Acta Philosophica Fennica 18 (1965), pp. 135-164; A transjnite sequence of w-models, reprinted with permission of the publisher American Mathematical Society from Journal of Symbolic Logic, copyright @ 1972, Volume 37, No. I , pp. 96-102; A contribution to teratology, H36paHHbIe BonpocbI ame6pb1 n n o r m n , HayKa, H o ~ o c n GEIPCK 1973, pp. 184-196.
Countable Boolean fields and their application to general metamathematics by
Andrz ej M o s t o w s k i (Wasszawa) (Translated from German orginal Abzahlbare Boolesche Korpzr und ihre Anwendung auf die allgemeine Metamathematik by M. Mqczynski)
In the present paper solutions of several problems of Tarski will be given. These problems were formulated by Tarski in connection with his general metamathematics,(') and i have been able to solve them mainly owing to his hints. General metamathematics is a special case of Boolean algebra. Hence there arises a further possibility of formulating theorems of general metamathematics as theorems of Boolean algebra.(2) This generalization seems to be very appropriate, since Tarski has shown, in connection with a very interesting work of Stone,(3) that the notions of metamathematics have simple and important counterparts in general Boolean algebra (see Theorem 2). Therefore it is useful to conduct the following considerations in Boolean algebra, and not in its interpretations (one of which is metamat hemat ics). The results of Stone have had yet another influence on theorems dealt with in the sequel. My complicated and difficult proofs have turned out to be superfluous, because long known tGpologica1 methods lead to the same result easier and faster. The whole ballast of my earlier proofs is put aside in the present formulation, and for the reader acquainted with the elements of topology the considerations given here will appear as direct consequences of Stone's results. Hence my considerations can hardly be regarded as discoveries. But they can serve as a striking illustration of the ( I ) See A. T a r s k i, Grundzuge des Systemenkalkiils, Erster Ted. Fund. Math. XXV, pp. 503-526, and Zweiter Teil, Fund. Math. XXVI, pp. 283-301. These papers wilt be referred to in the sequel as T I and T,. The majority of the theorems to be found in the present paper were cited in Tl, pp. 289-290. (*) See T t , Satz 4 and remarks on p. 511. Boolean algebrnr and their application to topology, Proc. (3) See M . H . S t o n e Nat. Acad. Sc. 20, No 3., pp. 197-202.
2
f11
FOUNDATIONAL STUDIES
interesting and quite unexpected confiections between such distant doinains as metamathematics and topology.
8 1. Next we give a precise definition of the notion of Boolean algebra, which will be the object ’of our investigations. DEFINITION a, An ordered quadruple K = [A, N , v , ’1 consisting of set A, a binary relation N , a two-argument operation v and a one-argument operation is called a generalized Boolean field provided that for a , 6 , c E A the following conditions hold: a
1. a
-- -- a,
2. if a b, then b - a , 3. if a b, b c, then a c , 4. a’, a v b E A , 5 . if a b, then a‘ b’ and ~ 6. a v b b v a , 7. (aVb)Vc a v ( b v c ) , 8. (a’vb’)’v (a’vb)’ a.(4) N
V C b Nv c ,
If the relation denotes the usual identity, then we say simply tbat K is a Booleanfield. If A < No,then we call K a countable BooIean$eld. We recall the definition of a deductive theory given by Tarski. N
DEFINITION b. An ordered quadruple T = [S, L , -+,-1 consisting of sets S, L, a two-argument operation .+ and a one-argument operation is called a deductive theory provided that the following conditions hold: -
I. 0 c 3 =g No, 11. L c s, 111. if x, y E S, then X, x IV. if x , y , r ~ S then , (x - y ) + [(y -+ 2) V. if x, y E S and x, x
--f
-+
-+
y ES,
(x
-+
z)], (X + x) -+ x, x E L.(5)
y E L, then y
-+ (X
-+y) e L ,
There is a close connection between the theory of generalized Boolean fields and general metamathematics(6) (i.e. the theory which investigates (“) This is a system of axioms of the algebra of logic given by Huntington. Comp.
E. V. H u n t i n g I o n, New sets of independent postulates for the algebra of logic, with
special references to Whitehead and Russell’s Principia Mathernatica, Trans. Amer . Math. Soc., 35., No. I . , pp. 274-304 and his Boole’an Algebra- a correction, ibidem. (’) See T I , p. 504, footnote. ( 6 ) See T I , Satz 4.fheorem I given here is a slight modification of the theorem of Tarski referred to.
111
COUNTABLE BOOLEAN FIELDS AND THEIR APPLICATION
3
the properties of deductive theories defined above). To state this connection
jn a precise manner, we assume the following two definitions, where T = [S, L, +,- J may denote an arbitrary deductive theory. c. For any u , b E S we put DEFINITION (a) a
Df -
c-)
b = (a + b) -+ (b 3 a), Df -
(p) UVTb = 4 4 6 ,
(y) a NTb if and only if a ++ b E L .
DEFINITION d. Iu, is the ordered quadruple [S, w T ,v T ,-1. THEOREM 1. If T is a deductive theory, then K T is a generalized countable Booleun jield. We begin the proof by remarking that when we imitate the well-known considerations of the propositional calculus, we can prove the validity of the following rules : If T = [S, L, +! -1 is a deductive theory and a , b , c E S , then we have
A. B. C. D. E. F.
attael, (a++b)+(bcra)eL, (a c-) b) + [(b c) + (a ++ c)] E L : ( a o b ) + (_ Z +_ + ~_ ) E_ L, (a-b)+(a+?-b+i2)EEL, a + b o b z a_____L , ++
G. a
-
+b +C
_ _ -
-a
-__ -
H. Z --+ g + if
-
-
-+
-2 - a
b+ C g L ,
E 15.(~)
Making use of the above rules, we can easily prove further that KT is a countable generalized Boolean field. To begin with, from condition I of Definition b we infer that S is an at most countable set. From Theorem A and Definition c it follows that if a E S, then 1. a NTa. The conditions: 2. if a NTb, then b NTa and 3. if u =Tb and b N ~ C , then a w T C (for u , b , c E S ) follow from Theorems B and C in view of condition V of Definition b. Condition 4. : if u NTb, then a, a v T bE S, follows directly from (’) As can easily be seen from the conditions 11-V of Definition b it follows that every theorem which arises by substitution into a valid formula of the propositional calculus is an element of the set L . To derive such a theorem it suffices to base the derivation upon statements which arise by substitution into the axioms of the propositional calculus (by IV they belong to 15)and upon the rule of separation provided by V.
4
FOUNDATIONAL STUDIES
111
Dehitions b and c. From Theorems D and E we infer that 5. if a wTb, then Ti WTb and a v T c w T b v T c . Finally, the conditions: 6. ( u v , ~ ) wT(bv,a), 7. [(uv,b)v,c] -T[av.r(bv,c) and 8 . (?iv,b)v,(avb) mTa are direct consequences of Theorems F, G, H . Hence we obtain Theorem 1 on the basis of Definition a. Theorem 1 shows that with every deductive theory it is possible to associate in a definite way a generalized Boolean field. Hence every theorem about this field is simultaneously a metamathematical theorem. On the basis of Theorem 1 we obtain the connection mentioned above between the theory of Boolean fields and general metamathematics. It is clear that between these domains we can also easily establish a converse relation, which allows us to interpret metamathematical theqrems as theorems about Boolean fields. But we will not use this in the sequel. Before we go into the proper subject of this paper, we would like t n recall some algebraic definitions. Let K = [ A , -, v , ’1 be a generalized Boolean field. Df
DEFINITION e. a . b = (a’vb’>‘.(*) DEFINITION f. We call a set I an ideal in K if and only if I is a non-empty subset of A which satisfies the following conditions: if a , b E Z, then a v b E I, if a E Z and a b, then b E I , if ~ E Z b, E A , then a - b e / . DEFINITION g. We call Z a prime ided in K if and only if Z is an ideal in K,Z is not identical with A and, moreover, every ideal J which contains Z is either identical with A or identical with I. DEFINITION h. We call a a generating elemePtt of the ideal Z if and only if Z is an ideal in K and for every element b of Z there exists an element c of the field K such that b a . c. DEFINITION i . We call 1 a principal ideal of the field K if and only if I is an ideal in K and there exists a generating element of I. We would like also to define the notions of isoniorphism, hornomorphism and subfield. Let K ’ = [ A , -, v , ‘1 and L = [ B , =, + , - I denote two generalized Boolean fields.
-
-
(*) The Definition e is not correct, becausc the dependence of the operation . upon the field K is not marked in it. However, this little deficiency will not lead further onto any confusion.
PI
COUNTABLE BOOLEAN FIELDS A N D THEIR APPLICATION
5
DEFINITIONj. We say that the field L is
hornomorphic with the.field K if and only if there exists a relation 0, with A as its left domain and 8 as its right domain, which satisfies the following conditions: ( a ) i f a , b E A , c , d E B , a - b , c ~ d a n d a a cthen , bcd, apb and ugc, then b = c, (y) if apb and ccd, then (avb)o(c+d), ( 8 ) if apb, then u'pb -.
(p) if
DEFINITION k. We say that the field K is isomorphic with theJield L if and only if there exists a relation Q, with A as its left domain and B as its right domain B, which has the properties (a),(p), (y), ( 6 ) of Definition j and, moreover, satisfies the condition (E)
if bea and cpa, then b
-
c.
-
DEFINITION 1. We say that L is a subfield of K if and only if 8 c A and theoperations V , ' and also the relation restricted to the domain B are identical with the operations +, and the relation I, respectively.
.
With respect to Boolean fields these definitions coincide with the usually accepted ones. There is a close connection between the notions of general metamathematics and the algebraic notions given above which-as we mentioned before-was established by Tarski. This connection shoys up in the following theorem.
THEOREM 2 (of Tarski). Let [T = [S,L,
< , *] be two deductive theories. Then
-1 and TO=
4,
[SO,40.
a) The set X is a deductive system of the theory T( 9 )i;f and only if the set X is an ideal in the field K T , b) The set X is a complete deductive system of the theory T ( ' O ) if and only if X is a prime ideal in the field KT , c) The set X is an axiomatizable deductive system of the theory T(") if and only i f X is aprincipal.idea1 of theJield KT. d) The theories T and To are of the same structural type(12)if and only if thejelds KT and KToare mutually isomorphic. e) The homomorphism of the field KTo with the field KT is equivalent to the following condition: there exists a deductive system X of the theory T (lo)
('I)
(I2)
Sze Ti,Definition 5. See T,, Definition 13. See TI.Definition 9. See
Tz.p.
288.
6
FOUNDATIONAL STUDIES
such that the theory T, = [ S , X, as the theory To.
4,
r11
is of the same structural type
-](I3)
P r o o f. a) We assume that Xis a deductive system of the theory T, i.e. (4 LCXCS, ( P)
from x, x
-+
y E X it follows that y EX.(")
We want t o show that X is an ideal in the field KT. For any elements E S we have x -+ [ y -+ ( x -+ y ) ] E L ; hence, on the basis of (a), we [y + (x -+ 3)]E X . From this, applying (p) and Definition c, we have x obtain if x, y E X , then x v , y EX. (Y) x, y
--f
If x, y E S and x C I y E L , then x -+ y Definition c we obtain the conclusion
E
L. Hence in view of ((3) and
if X E X and x w r y , then y E X .
(a')
For any elements x, y E S we finally have x -+ X 7 E L. On the basis of Definition c, taking into account the notation accepted in Definition e, we can write this in the form x -+ x - y E L . On the basis of (u) and ((3) we conclude that (E) if x e X , y ~ S then , X-YEX.
The conditions (y), (6), (E) show that the set X i s an ideal in the field K T . Since the set of elements of this field is identical with the set S, we infer that
(0
x c s.
-
We notice that if x, y E S, then x -+ [ y -+ (x ~ y )E ]L . From this we conclude that if x , y E L, then x y E L holds. Since for x E S we always have x . X E L, in view of Definition c we obtain .
(r)
if ~ E S ,L. ~then E x . 2 wry.
By assumption and Definition f , X is a non-empty set. Let x be any element of X. According to the assumption we have x . x E X , from which by ('0) and Definition f it follows that if 1' E L then y EX. Hence we ha\e
(8) (I3)
L cx. We omit here the easy proof of the theorem stating that the quadruple [S, X,
+,-I is in fact a deductive theory. This proof is implicitly contained in Tarski's paper.
Comp. Theorem 21 and the remarks on p. 522. ("1 See TI,Satz 5.
[I1
COUNTABLE BOOLEAN FIELDSA N D THEIR APPLICATION
7
-,y E S. According - -
Finally, let us assume that x , y E X and x , x
to
Definition c and e we then have X * y = (%vTy) = + -T x + y, which by Definition f yields X * y E X.From x E X we obtain x y E X; further we X . since x * y v T X - y w r y , if follows on the basis have x * y v ~ x - y ~But of Definition f that y EX. From this, in connection with (C) and (8),it turns out that X is a deductive, system. We obtain b) simply from the definition of the complete system and Definition g. As regards c) we first observe that for x, y E S we have Y ) [(X Y ) c*Yl E L . We assume that X is an axiomatizable system of the theory T,i.e. X is a system and there exists an element x E X such that for every y E X we have x y E L . In view of ( 1 ) we can put this condition in the following form : (x
(1)
-+
++
+
-+
-
for y E X we have (Z
(4 Since
- y - y = (X vTF) = X -+
mTX
+ y ) -T
y.
y , we infer from (x) that p
w T x * y ; hence x is a generating element of the ideal X.
Now let us assume that X is a principal ideal of the field K T . Then on the basis ofa) we conclude that X i s a system of the theory T.Let x be a generating element of the ideal X. Hence for every y E X there exists an element z E S such that y w T x *z. From this we conclude by well-known rules of Boolean algebra that x . y w T x . ( x . z) -rX . z y y , i.e. y - T x * y , which implies by ( 1 ) that x .-+ y E L. Therefore the system Xis axiomatizable. d) We assume that the theories T and To are of the same structural type. Then there exists a (multivalued) relation p with the set S as its left domain and the set So as its right domain, which has the following property: (A) if xoxo,yeyo, then the conditions x equivalent .(’7
-+
y
E
L and xo < yo E Lo are
We. make the assumption: xe.xo, y@yo and x vTyezo. Since ( x v T y ) L, in view of ( A ) we obtain zo < xo E L and analogously z0 < yo E Lo, which shows that zo < ( X ~ V , €~L ~ o . ~Let ) z denote an arbitrary o. ( X , V ~ ~ ~ ~ ) pcounterimage of the element x o v T o y o ,i.e. ~ @ x o v T o ~Since < xo E Lo, we obtain z .-+ x E L and z --* y E L, which gives z (XVTY) E L On the basis of (A) we infer that ( x o v T o y o )< zo E L,,’which shows that +x E
-+
(Is)
See Tz, p. 289.
8
[11
FOUNDATIONAL STUDIES
if xexo,yeyo, x v ~ y e z o ,then zo “ T x o v T o ~ O . Quite similarly we prove that
(PI
(P’)
if xexo, yeyo, z@xOvToyO, then z
NTXVTY.
Further, we assume xexo, Xpyo and denote by y any element satisfying the condition y p x t . For any t E S we have ( x v T Z ) + t E L ,which implies on the basis of (A) that for every to E So we have ( x ~ v ~ < ~ to Y E~Lo. ) We obtain hence the condition yo < X $ E L o .
(v)
For every to E Sowe have (xovT0x$) < to E t owhich , implieson the basis of (A) and (p’)that for arbitrary t E S we have ( x v , ~ )+ t E L, i.e. y 4 X EL.Hence on account of (A) we obtain x$ < yo E Lo; further, according to (v), we get xX -To y o . Hence we have proved :
(El
if xpxo, @ y o , then yo -Toxg.
Now we define a relation in the following way: x@xo if and only if there are elements y , yo such that yeyo, y mTx and yo -T0xO. The relation @ defined above clearly satisfies the following conditions: (0)
(4
the left domain of
is S, the right domain is So,
if x ’ l x o , ~N T X , Y O - T ~ x O , then
YFYO.
Let us assume x i x , and
[email protected] there are four elements u , v , uo, vo for which we have x - T u , x - T v , u0 -ToxO, 00 -TO yo and Ueuo, 0 ~ 0 Further, we have u + w E L and w + u E L, which implies on the basis of (A) that uo < wo E Lo, v0 < uo E Lo, i.e. xo Hence we have shown that if x@xo,xGy0, then xo Similarly one proves that
(P)
(6)
if x i x o , yijx,, then x
NT~YO.
NTY.
From (p) and ( x ) we infer that
(9
if ~ 5 x 0yeyo, , then From ( x ) and (E) we infer that (0)
K . ,
(XVT
~ ) ~ ( X OYO), VT~
if x$xo, then S2ext.
According to Definition k these conditions show that the fields KT and are mutually isomorphic.
.
[11
COUNTABLE BOOLEAN FIELDS A N D THEIR APPLICATION
9
Now we assume that there exists a relation
satisfying the conditions --* y E L is equivalent to the condition x ~ T ( x v T ~ ) and , the latter is equivalent to the relation xo N T , ( x O V T , Y ~ ) . Indeed, from (z) it follows that if x w T ( x v T y ) , then (xvTy)$x0, from which we obtain xo ~ ~ , ( X ~ AnalV ~ ~ ~ infer , ~ ~that ) x N ~ ( XVry). ogously, from the assumption xo ~ T , ( x ~ vwe This implies
(o), ( x ) , (p), (a), (4, (u) and such that xpxo, yeyo. The condition x
(9) if xsx,, yeyo, then the conditions x --+ y E L and xo < yo E Lo are equivalent. We have thus shown that the isomorphism of the fields KT and KT, impliCs the identity of the structural types of the theories T and To. e) We denote by X a deductive system of the theory T and by Tx the theory [S, X, -+,-1. As we proved in a), X is an ideal in the field KT. We define a rglation e in the following way: a@ holds if and only if u is an element of the field KT/X,(16)b an element of the field KTx, and we have b E a. Then we can easily prove that this relation satisfies the conditions of Definition k. Hence it follows that the fields KT/X and KTx are isomorphic. Now we assume that for a certain system X of the theory T the theory Tx is of the same structural type as the theory To. On the basis of d) the fields KTx and KT, are isomorphic. Hence, in view of the isomorphism between the field KTx and K,/X proved above, the fields K T o and KTlX are also isomorphic. But since the field K r / X is homomorphic with the field KT(''), the field KT, is also homomorphic with the field KT. Conversely, let us assume that the field KT, is homomorphic with the field KT. Then !here exists an ideal X in the field KT such that the residue class field KT/X is isomorphic with KTo. From the isomorphism of the fields KT/X and KT, it follows that the fields KT, and Kr, are isomorphic, which on the basis of d) proves the identity of the structural types of the theories To and T,. As can be seen from Theorems 1 and 2, every theorem about countable Boolean fields has its counterpart in general metamathematics. Therefore our further considerations can be restricted to Boolean algebra. ("'1 We denote by &/X,as usual, the residue class field modulo X.
(") This theorem is derived from the well-known algebraic theorem about the homomorphism of rings by taking into account the connection between Bookan algebra and the theory of rings discovered by Stone. Comp. M. H. S t o n e, Subsumptioft of ihe theory of Boole'an algebras under the theory of rings, Proc. Nat. Acad. of Sci. No. 2, pp. 103-105.
~
~
)
-
10
ill
FOUNDATIONAL STUDIES
0 2. First of all we introduce the following definition given by Stone: DEFINTION m. For each Boolean field K = [A, , v ,'1 let 6(K)be
-
the topological space formed of all prime ideals of the field K, where the notion of neighbourhood is defined as follows: if I is a prime ideal in K, then a set U of prime ideals in K is a neighbourhood of I if and only if 1 E U and there exists an element x E A such that U coincides with the set of prime ideals in K which do not contain x. Several theorems,concerning the space G ( K ) can be found in the work of Stone. As far as one can see from his concise treatise, they are only formulated for Boolean fields in the narrow sense, and not for generalized Boolean fields. However, Stone's results can easily be extended so as to hold for those fields as well. Therefore in the sequel we shall base our considerations upon the generalized theorems of Stone.(') First we state the following theorem of Stone: THEOREM 3. I f K is a Booleanjeld, then: a) the space G ( K )is bicompact and totally disconnected,('s) b) every neighbourhood of any point is open and closed. From b) we obtain COROLLARY 3c. The space 6(K)is ~ero-dimensional.('~) As can be seen from Definition m, the power of the neighbourhood system of the space G(K)is not greater than the power of the set A. This implies that if the field K is countable then the space 6 ( K ) is separable. Therefore the space G ( K ) (for a countable K) is homeomorphic with a subset of the Cantor set C.('") The class of all those subsets of the set C which are homeomorphic with G(K)will be denoted by S ( K ) . (I8) A topological space E is said to be bicompart if for every class R of open sets in E satisfying the condition X = E there exists a finite part of it K* such that
X
X€R
=
E. One proves that for every part X of a bicompact infinite space Ethereexist.
XER'
a-point x -
u-x
=
E
E such that every open set 0 in E containing
x7
x
satisfies the condition
A topological space E is said to be fofally disconnected if for any two different elements E there exist two disjoint sets X, Y closed in E which satisfy the conditions .uEX,yEY,X+Y=E. ( I " ) One can easily prove that the condition dimG(K) = 0 follows from Condition a). (*O) See, e.g. K . K u r a t o w s k i, Tupo/ogie I , Monografie Matematyczne, War-
x, Y E
szawa-Lwow 1933, p. 124, Th. VI.
(11
11
COUNTABLE BOOLEAN FIELDS A N D THEIR APPLICATION
We prove the following lemma: LEMMA 4. If K is a generalized countable Boolean field and X then the set X is closed in the interval [0, 11 = I.
E
S(K),
P r o o f . W e a s s u m e t h a t x , E X ( n = 1 , 2 , ...) and l i m x , = x . Since n+m
the set X is bicompact, there is an element y E X such that every open set Vin X containing y contains infinitely many elements x,,.(~*)If the identity x = y were false, then there would exist two disjoint open sets W,,W,in Z such that x E W,,y E W,.Since x = limx,, almost all elements xn would belong to W,, i.e. at most a finite number would belong to W, .X, although this set is open and contains y . Hence the supposition x # y leads to a contradition. Accordingly, x = y and consequently x EX. COROLLARY 5 . Every countable field K is isomorphic with the field of all closed and open subsets of a certain closed zero-dimensional linear set.
P r o o f. Let X E S(K); according to Stone(3) the field K is isomorphic with the field of closed and open subsets of the space G(K).Since the spaces G ( K ) and X are homeomorphic, it follows that the field of all closed and open sets in G(K) is isomorphic with the field of all closed and open sets in X , which was to be proved. It is known that a closed set in I has the power < KOor 2xo.(21)In view __ of Lemma 4 we infer hence that if Kis countable, then 6 ( K ) < H,,or G(K) = 2”o. Since the set of the elements of the space G ( K ) is identical with the set of all prime ideals of the field K, we infer that also this set either is countable or has the power of the continuum. Taking into account Theorem 2b, we obtain the following theorem.
6. The power of the set of the complete systems of an arbitrary THEOREM deductive theory is either < KO or 2No. Now let us assume that the field K is countable and let us denote by G 6 ) ( K ) ,where 6 < Q, the derivative of order of the space G(K),and by ct the smallest ordinal for which we have @“)(K)= G(’)(K). Then, according to Cantor’s theorem, the following equality holds :
(”) See F. H a u s d o r f f, Crundziigeder Mengenkhre, Veit u.Comp. 1914, p. 370, Satz IV. (’’)
See K . K u r a t o w s k i , I . c . , ~ . 115.
12
FOIJNDATIONAL STUDIES
Dl
This partition corresponds to the partition of the set of the prime ideals of the field K . LEMMA 7. The set G(O)(K)- G(')(K)is identical with the set of allprincipal ideals of the field K . P r o o f. Let us consider a generating element a of the principal ideal I and the set U of all those prime ideals Z of the field K for which a ' Z J. According to Definition m the set U belongs to the neighbourhood system of the space G(K),where we clearly have I E U . If I E U, then a' E J, which, by a known property of prime ideals,(") implies a E I, and hence I c J. The ideal I-as a prime ideal-has no proper divisors; we infer hence that I = J , which implies U = ( I } . Accordingly, the point I of the space G(K) has a neighbourhood consisting of only one point, and hence it is an isolated point of this space, i.e. I E G(O)(K)- G(')(K). Now let us assume, conversely, that I is an isolated point of 6 ( K ) . Hence there exists a one-point neighbourhood of I.On the basis of the definition of neighbourhood in the space G(K)it follows that there is an a such that a E I and a E J for every prime ideal J different from I. The ideal (a')(24) is equal to the intersection of all those prime ideals of the field K which contain a'. (") Since I is the unique ideal with this property, we have I = (a'), and consequently I is a prime ideal. In the sequel we give a certain application of the Cantor partition. Let us assume that Tis a deductive theory. From Lemma 7 and Theorem 2 we infer that the pair of cardinal numbers
is identical with the characteristic pair (a, U) of the theory T.(26) From this it is easy to determine the values which can be taken by the numbers a, U. Namely if a < KO (i.e. if the space G'(KT)contains only a finite number of isolated points), then we have G(')(KT)= G(*)(KT)and consequently U = G(R)(KT).Hence u = 0 or u = 2xo, because G(*)(KT), being a perfect set in the separable space, has the power 0 or 2w0.If a = KO, then the space G(Kr) contains at least one non-isolated point, and consequently U 2 1. Here we have u < K O if G'RJ(Kr)= 0, and U = 2 w o if G(*)(K,) z 0. (")See TZ,Satz 30 (3). We denote by ( x ) the principal ideal for which x is a generating element. (") See TZ,Satz 36. (z6) See Tz, p. 289.
(24)
[I1
COUNTABLE BOOLEAN'FIELDS AND THEIR APPLICATION
13
We have thus arrived at the following theorem.
THEOREM 8. The characteristic pair of an arbitrary deductive theory can assume the following values (where n denotes a finite cardinal): ( n , O), ( n , 2 X o ) ( ~ o ,n ) , (No,No), (No, 2"0).(~') We d o not intend to deal here with further investigations of the partition (A) (the most important task of which would be the explanation of the metamathematical meaning of systems belonging to the set G(s)(K)-G'5+1) (K) ( E 2 1) or to the set @ " ) ( K ) ) , because the notions obtained in this way are rather complicated and d o not seem to have much significance for general metamathematics.
0 3. Now we would like to investigate more closely the question ofisomorphism between countable Boolean fields. DEFINITION n. By the characteristic of a countable field K with only a countable set of prime ideals we understand the pair of numbers ( a ( K ) , n(K)), where a ( K ) is the order and n(K) the power of the last non-empty derivative of the space 6(K). If is clear that 0 < n ( K ) < No. Let K and L be two fields with only countable sets of prime ideals. Hence the sets X, and X 2 are linear countable closed sets. A necessary and sufficient condition for the homeomorphism of these sets is, according to Mazurkiewicz and Sierpinski,(") that the following equalities hold: (*)
a ( K ) = a(L),
n(K) = n ( L ) .
The homeomorphism of the sets X , and X, (or-which is equivalent-of the spaces G(K)and G ( L ) )is, according to Stone,(29) a necessary and sufficient condition for the isomorphism of the fields K and L. Hence the identities (*) express a necessary and sufficient condition for the isomorphism of the fields K and L. In this way we have obtained the following theorem.
THEOREM 9. Two countable (generalized) Boolean fields with a countable number of prime ideals are isontorpliic (f and only if their characteristics are equal. We give a corollary to this theorem. (") See T,, p. 289. (") See S. M a z u r k i e w i c z et W. S i e r p i n s k i, Contribution d la ropologie des ensembles dhnombrables; Fund. Math. I., pp. 17-27. ("9) M.H. S t o n e , Bwle'un algebras etc. (See('), Theorem IV,).
14
FOUNDATIONAL STUDIES
111
COROLLARY 10. Every countable Boolean .field with an at most countable set of prime ideals is isomorphic with the field of sets which are closed and open in a certain closed well-ordered linear set. P r o o f. Let.K denote a countable Boolean field with an at most countable set of prime ideals. Clearly there exists a linear closed and well-ordered
set X for which we have X(@’) = n ( K ) . Hence, on the basis of the theorem of Mazurkiewicz and Sierpinski cited in(28), the set X is homeomorphic with every set of the class S ( K ) , and hence also with the space G(K). The homeomorphism of the sets G(K)and X implies isomorphism between the field of all closed and open sets in X and the field of all closed and open sets in 6(K). But by Theorem IV, of Stone the latter field is isomorphic with K,which establishes the validity of the corollary. COROLLARY I 1. a) There are K , distinct types of isomorphism of countable fields which have at most K Oprime ideals. b) There are X, distinct structural types of deductive theories which have H, complete systems.(30)
P r o o f. a) follows from Theorem 9 in view of the fact that the set of the pairs ( a , n) where 0 < c( < 52 and 0 < n < KO is of power K , . b) follows from a) and Theorems 1 and 2d. COROLLARY 12. /f the characteristic pairs of the theories T and To are equal to ( K O ,n) (where 0 < n < KO),then the theories T and To are of the same structural type.(3i)
P r o o f. From Theorem 2 it follows that the fields KT and KTocontain KO principal prime ideals, and n prime ideals which are not principal ideals. By Lemma 7 each of the spaces G(KT) and G(KTo) contains a countable
set of isolated points and n condensation points. Therefore both fields have the characteristic ( 1 , n). According to Theorem 9 the fields KT and KT, are isomorphic, which by Theorem 2d ensures the equality of the structural types of the theories T and To Now let us cpnsider the isomorphism of countable fields which have 2N0prime ideals. Let K and L be two such fields and let G(K),G(L) denote the topological spaces corresponding to them. According to Formula A we have
(’3 See Tz, p. 289. t39 See Tz.p. 290.
[I1
COUNTABLE BOOLEAN FIELDS A N D THEIR APPLICATION
15
The kernels G(')(K)and Gts)(L) are homeomorphic, because they are two perfect zero-dimensional sets. If the spaces G(K)and G(L) are homeomorphic, then we have (1) a=j3 and also (when GI is not a limit number)
-
6'"- 1)( K ) -@a(K)= Gtfl- 1)(L) G-tn(L), (2) because the homeomorphism of two bicompact spaces implies the homeomorphism of all its derivatives. By means of a simple example one can see that (1) and (2) are not sufficient conditions for the isomorphism of the fields K and L, since they do not imply the homeomorphism of the spaces 6(K)and G(L). But we can obtain a sufficient condition of a relatively general nature, for that isomorphism if we assume besides the equalities (1) and (2), the closedness of the scattered parts of the spaces G(K)and G(L).In order to prove that this strengthened condition is in fact sufficient, we need only to recall the theorem of Mazurkiewicz and Sierpiliski mentioned in the footnote(28) and to make use of the well-known topological theorem stating that the homeomorphism between closed sets X and Y and also between X* and Y* (where X is disjoint with X* and Y with Y*) implies the homeomorphism of the sums X + X * and Y + Y * . As a direct consequence we obtain the following theorem: if u = fi G 1 and G("-')(K)- G(")(K)= G@-l)(L) -G(fl)(L)< H,,,then the fields K and L are isomorphic. On the basis of Lemma 7 this theorem admits the following algebraic interpretation : THEOREM 13. If countablefields K and L have 2No prime ideals and among them only a j n i t e number of principal ideals, then theseJields are isomorphic if and on1.v if they contain the same number of principal ideals. On the basis of Theorems 2 and 1 this theorem has also the following metamathematical meaning: THEOREM 14. All deductive theories with the characteristic pair (n, Po), where 0 < n < KO,are of the same structural type. As a counterpart to Corollary 11 we also give the following theorem: THEOREM 15. a) There are 2*0 distinct types of the isomorphism of countable fields having 2No prime ideals. b) There are 2Ho distinct structural types of deductive theories,
16
FOUNDATIONAL STUDIES
[I1
To prove a) it clearly suffices to show that there are at least 2wo distinct homeomorphic types among subsets of the Cantor set C. The construction given below is an almost exact repetition of the construction of Mazurkiewicz and Sierpifi~ki.(~') Let A be a slosed bounded linear set, and Z and P the scattered and the perfect parts of A, respectively. For 6 < 0 we denote by Z, the 5-th cohere ~ c e ( of ~ ~the) set Z . A point x E A is called a point of order a if and only if x E P , x E ZL and x E Z; for 5 < a. We want to prove that if a set A* is a homeomorphic image of A and the point x is a point of order a in the set A, then the image x* of x is a point of order a in the set A*. Let Z* and P* denote the scattered and the perfect parts of the set A*, respectively. It is known that the homeomorphism between A and A* implies that P* is a homeomorphic image of P and Z* is a homeomorphic image of Z. The coherences of the set Z (resp. their derivatives) are carried by a homeomorphic mapping onto the coherences (resp. their derivatives) of the image set. From this we infer that if x E P Z i - Z ; , then we have x*
- kfla
EP*.n
€car
Z $ ' - Z z , which was to be proved.
Now let [il, ilr ..., i n , ...I be an arbitrary sequence of the numbers 0 or 1 and k a natural number. We consider the countable closed sets Tk 1 contained in the interval __2k+ , &hose (2k+ik)th derivative is the
[
-2k]
{ 2L+ }, and we denote by C, the Cantor sets in the inter-
one-element set ~-
'-],
where the end-points of c k may coincide with the [ 2 k : 2 ' 2k+l end-points of the corresponding intervals. As can casily be seen, the points vals
-
1
2k+ 1
(k
-
1 , 2 , ...) are the only points of finite order of the closed set
where the point
1
has the order 2 k + i k . 2k+ 1. From the lemma proved above it follows that the sets A i l , i,. _...in. ._. and A j , j , . ,...j , , ... are homeomorphic if and only if the sequences (32> (33)
See S . M a z u r k i e w i c z et W. S i e r p i n s k i , l.c.,pp. 23-27. See F. H a u s d o r f f , I.c.,p.227.
I11
COUNTABLE BOOLEAN FIELDS AND THEIR APPLICATION
17
[i,, ..., in, ...I and [ j , , ...,j , , ...I are identical. Since here the set*Ai,.i 2 . ....i n . ... is zero-dimensional for every sequence [ i t , i 2 , ...I, i.e. it can be topologically embedded in the Cantor set, there are at least 2"o topological types among the closed subsets of the Cantor set C. From Theorem 15 a) we infer that there are 2"o distinct structural types among the theories that have 2". complete systems, which directly implies Theorem 15 b) directly.
0 4. Finally we would like to consider the universality properties of the fields which have 2"o prime ideals. THEOREM16. If K is a countablefield with 2". prime ideals and L is an arbitrary countablejield, then L is homomorphic with K. P r o o f. We investigate the topological spaces G(K)and G(L)associated with both fields. It is known that the space G(L) is homeomorphic with a closed subset of the set C. The kernel G(')(K)of the space G ( K ) is homeomorphic with C, and the difference G(K)-@")(K)is clearly open in G(K).This implies that the space G(K)is homeomorphic with the complement with respect to 6(K)of an open set in G(K).On the basis of Theorem IV3 of Stone, this implies the homeomorphism of the fields K and L. As v. Neumann and Stone(34) have proved, a countable field K with which the field L is homomorphic contains a subfield isomorphic with L. Therefore from Theorem 16 we obtain the following theorem. 17. If K is a countablejield with 2"o prime ideals, and L is an THEOREM arbirary countablefield, then there is a subfield M of K isomorphic with L. As is shown by Theorem 2d, the following theorem is a metamathematical counterpart of Theorem 16.
THEOREM 18. If T is a theory which has 2"o complete systems and To an arbitrary deductive theory, then there exists a system X of the theory T such that the structural types of the theories To and Tx are equal.
N e u m a n n and M. H. S t o n e, The determination of representative (j4) J. v. elements in the residual classes of a Boole'an algebra, Fund. Math. XXV, pp. 353-378, Theorem 17.
On the independence of definitions of finiteness in a system of logic by
Andrzej M o s t o w s k i (Warszawa) (Translated from Polish original 0 niezaleinoici definicji skoriczonoici w systemie logiki by A. H. Lachlan)
The present paper constitutes an extract from a wider study concerning notions of finiteness of a set, in which we present solutions of a series of problems connected with the notion of finiteness and posed in the main part by A. Tarski. In the present paper we turn our attention to just one of these problems, that of the equivalence of the notion of finiteness defined by Frege and Russell with another notion of filiiteness first defined by Dedekind. On the basis of a specified system of logic we show that the equivalence of these two definitions cannot be deduced from the logical axioms alone, which is not more than definitive confirmation of a longstanding conjecture. Investigations of this kind are closely connected with the question of the independence of the axiom of choice, because as is well known the equivalence of the definitions referred to above is a consequence of the axiom of choice; thus as somewhat incidental corollaries of our theorems we obtain theorems about the independence of the axiom of choice and certain consequenccs of it which seem logically rather weak. These results are given in 3. For definiteness we carried out our investigations for the system of logic formulated by A. Tarski; we have given a short description of this system in 4 1, in 4 3 on pp. 65-67 we discuss the possibility of extending our consideratior s to other formal systems. In 9 2 we describe the method by which the results presented in 53 were obtained. Our results are strictly related to a number of other papers on logic (some published and some unpublished). Thus the theorems of Q 3 are connected with the papers of A. Fraenkel concerning the independence of the axiom of choice, being in part a strengthening and transfer of Fraenkel’s results to another system of the foundations of mathematics. The main idea of Fraenkel’s method is also present in our proofs, however the mode of application, which aroused a number of serious doubts in the case of
PI
ON THE INDEPENDENCE OF DEFINITIONS OF FINITENESS
19
Fraenkel’s papers, has here been completely changed. In this regard the author owes the basic idea to A. Lindenbaum, who in the course of our joint work on making Fraenkel’s proofs precise indicated the method, which as it turned out, constitutes the tool by which the desired results are obtained (cf. also the remarks on p. 67). As far as we know A. Tarski first suggested applying this method here. The method referred to is the method of “relativization of quantifiers” about which we write at length in 0 2. It was conceived by A. Tarski who together with A. Lindenbaum applied it to various investigations on axiom systems. Independently of Polish logicians Herbrand discovered a particular case of this method (cf. here the introductory remarks to 8 2). However despite this apparently rather rich history -of the method of “relativization of quantifiers” there is almost no mention of it in the literature, and consequently we are compelled to develop it ab initio giving all the details. From above it is clear that in this paper the author in large part had to describe ideas and methods due to others. Both in this work and equally in the formulation of the results which seem to be new in this context the author profited greatly from the advice of A. Tarski for whose willing and valuable help the author expresses his deep gratitude.
5
1. A system of logic
In the current section we intend to describe the system of logic with which the metalogical theorems to be established in the following sections are concerned. This language is a precise version of the well known system of Russell and Whitehead [I] and was first formulated in a completely precise form by Tarski [2] (cf. also Quine [l]). Because this language has been described and discussed many times we shall not go into the details of its construction and will be content to repeat a few of the most basic definitions. 1. Expressions of the language are finite words built from.the following symbols “C”, “N”, ‘‘p (1) , (2) “Xi”,“x;:”, “XI,,”,...,“x;’”, “x:’”, “x;;,”, ..., “X,:l”,“XI, ,“p;”, ... I1 ill97
The symbols (1) represent the names of the implication and negation signs and the universalquuntijier ; the symbols (2)--so-called vuriables-represent respectively names of individuals, sets of individuals, classes of
20
FOUNDATIONAL STUDIES
sets of individuals and so on, such that in the language we have at our disposal an unlimited number of symbols serving to denote individuals (these “X;,”,...), a like number serving to denote sets (i.e. are the symbols “X/”, “X;;”,‘‘Xi,\ ”,...) properties) of individuals (these are the symbols “X(”’, and so on.
2. In metalogic are considered the properties of expressions belonging
to the system of logic, and hence properties of certain Anite sequences built from the symbols (1) and (2) given above. It is convenient to introduce
(in metalogic) symbols serving to denote those expressions and the relations holding between them.
2.1. The symbol of the language consisting of the letter X with kstrokes superscript and 1 strokes subscript we denote by the symbol X,” (for k, 1 = 1, 2 , 3, ...). The symbols “ C , “ N , “W we will denote simply by the symbols “C”, “ N ,‘‘IT“. As a rule we will denote arbitrary expressions by the letters a, @, y , 6, A, p, Y ; the letters a , @, ... are thus variables (of the system of metalogic) whose values are arbitrary expressions. It is worth pointing out that the symbols X:(k, 1 = 1 ,2 , ...) which denote variables of the system of logic, are constants of the system of metalogic-in contrast to the symbols a, @, ... The expression, consisting of the two expressions a, @ written one after the other ( a in the first place, B in the second), we denote by
[email protected] for example “CX:X;NX:” is the name of theexpression
“cx,”x/;”X,”’.
2.2. In 2.1 we explained the intuitive meaning of a number of symbols of which we shall be making continual use, profiting from the most varied properties of the notions denoted by the symbols, and thus for example from laws such as (a@)y = a(8y) and many others. In a systematic construction of metalogic one treats the symbols introduced in 2.1 as primitive expressions of a special theory (namely metalogic) and defines their meaning axiomatically. A system of axioms, sufficient for the construction of the metalogic which interests us here, has been given by Tarski (cf. [2], p. 100 and [3], p. 289) who is also responsible for the general idea of constructing metasystems in axiomatic form. All the laws which we mentioned above and on which we will continually rely can be derived from the axioms cited. However, having regard to the great technical difficulties which would be associated with a detailed derivation of metalogical laws from axioms, we will not justify our theorems formally and will rely on purely intuitive arguments. Let us add that besides the axiomatic approach one may also use another method for the precise treatment of metalogic: this is the method
PI
ON THE INDEPENDENCE OF DEFINITIONS OF FINITENESS
21
of arithmetization which W e 1 introduced (cf. his paper [l], and furthermore Carnap [I], pp. 47-51 and 69, Tarski [2], p. 100 mentions the possibility of interpreting the axioms of metalogk in arithmetic which is basically the same as the method of arithmetization). 2.3. To make our notations perspicuous we will introduce a series of symbols which will allow us to match the symbolism with that of Hilbert (in the definitions below k, I, nz run through the natural numbers): 2.31.
a --* PE Cap;
2.32.
EENa;
2.33.
a v b z a --t /I;
2.34.
a&pEZv$;
2.35.
at,@E(a+fi)&v+
2.36.
(X;)a Z Z 7 X f a ;
2.37.
(EX/)a Z f ;
2.38.
X:cX&+l52!XX:X;+';
2.39.
X f l d X ~ ~ ( X F + ' ) [ X f & ' + '+ X : E X ~ + ' ] .
~
a);
In connection with these definitions cf. Tarski [2], Def. 1-4. Let us remark that the expressions X:sX;+' and X / I d X k are certain functions of the three number variables I, in, n ; it would thus be more in keeping with the traditional manner of denoting functions were we to use symbols of the kind i?k,l,m, Idk,l,m instead of X/d'k+', X f l d X k respectively. However, having regard to the great lack of clarity of such a method of denotation in logic, in the definitions 2.38 and 2.39 we have used a different notation which in our opinion makes it easier to decipher logical schemas. Analogous remarks apply to a number of later definitions,
3. One defines the idea of a propositional function inductively: the expressions XFEXF (k,I , m = 1 , 2 , ...) we call propositional functions of rank 1 ; if a, B are propositional functions of rank at most n then the expressions a --* B, 5,(X:>a ( k , 1 = 1 , 2 , ..,) we call propositional functions of rank at most n+ 1. The expression a is a propositional function when there exists a number n such that a is a propositional function of rank n; the class of these expressions we denote by S. A variable Xf,occurring at a certain place in the propositional function a can be eitherfiee ot bound at this place; Tarski [2], Def. 6, gives a defi-
22
PI
FOUNDATIONAL STUDIES
nition of this notion. The set of variables which are free (resp. bound) at least one place in a we denote by Fr(a) (resp. Gb(a));further we set Vr(a) = Fr(a) Gb(a).
+
4. In conclusion we wfilldescribe the idea of consequence for the system of logic under discussion. 4.1. By L we denote the set containing the following expressions: + (a+ j3), (2 -,a) + a, ( a + 8) + [(j3 arbitrary a,j3, y E S ) ;
4.11. Z
4.12. (EX:+') ( X f ) [X:&X;+' vided Xi+ E Fr (a) ;
c-)
+
y)
a] for m , k , I = 1 , 2 ,
+
(a + y)] (for
... and a E S pro-
4.13. ( X f )(XfeXi+I c1 X / E X ~ + + ~ ) ZdX;+l ( k , I , m , n = 1 , 2 , ...).
We call the propositional functions of the set L the axioms of logic. 4.2. We say that the expression a results from #? through substitution of the variable X i for the variable X: if a differs from j3 only in that, at places in j3 where Xf occurs as a free variable, X i occurs as a free variable in a. We write this a = Sbk,, (j3); iterations of the function Sb we denote briefly (j3). (Tarski [2], Def. 8, gives a more in the form a = Sbrnj;,,,&... exact description of the operation of substitution.) 4.3. We say that the expression a results from the expressions B, y by when y = j3 + a.
cut,
4.4. We say that the expression a results from the expression j3 by adjunc-
tion of the quantijier when there exist expressions y, 8 and a variable X: E Fr(y) such that j3 .= y -+ 8 and a = y + (X38.
4.5. We say that the expression a results from the expression j3 by dropping a quantifier when there exist expressions y, 8 and a variable X: such that j3 = y -+ ( X f ) 8 and a = y + 8.
4.6. Let M denote an arbitrary set of propositional functions. We say that a is a consequence of rank 1 of the set M,when a E M + L . If a, j3 are consequences of rank at most n of the set M, then every expression y resulting from a and j3 by one of the operations 4.2, 4.3, 4.4, 4.5 we call a consequence of rank at most n + 1 of the set M. An expression a, for which there exists a natural number n such that a is a consequence of rank n of the set M, we call a consequence of M; the set of consequences of M we denote by FI(M); the set M(L) we denote briefly by T and its elements we call theses.
PI
ON T H E INDEPENDENCE OF DEFINITIONS OF FINITENESS
23
In this way we have defined for the system of logic the two essential notions characterizing every formalized language: the' idea of meaningful expression (3) and the idea of consequence (4.6). (This idea lies at the root of some of the papers of Tarski about the methodology of the deductive sciences; cf. particularly Tarski [5], and further Carnap [l], pp. 120-123.) 5. The next part of this section we denote to the introduction of a series of symbolic abbreviatiocs of the type of Definitions 2.31-2.39 and the demonstration of theses constructed with the help of these abbreviations. The proofs that these propositional functions really are theses (ke. belong to the set T ) may be obtained either by transferring to our system of logic derivations given in Principia Marhematica or by formalizing simple mathematical arguments; thus we will not overburden the paper by presenting these rather tedipus formal deductions. 5.1. First of all we give some lemmas concerning the propositional function X:ldXL introduced in 2.39.
5.1 I. LEMMA.a) X['IdXf&X[ZdXi b) X$IdX,P E T.
-,X P I d X l E T ;
5.12. LEMMA.Let a E S,X,! e Fr(a); let
denote an expression resulting from a by replacing the variable Xf at one place where it occursfree in a by another variable X i which is free at this place in /?. From these assumptions follows X:ldXL + (a f* ,!?) E T. The proof of this lemma was given by Tarski and presented in seminar tutorials in 1934; we will not reproduce the proof here which is obtained by induction on the rank of the propositional function a. 5.2. At this point we introduce the two following abbreviations (p, k, I = 1, 2 , ...) : 5.21.
Xi+'
E X/+'~(X[)[X[EX~+~ --t X [ E X / ' + ' ] ;
5.22. X f + l c Xi'+'
X[+I
c XP+'& X m X p ' .
The propositional function X[+I c Xt+' expresses the idea that the set of the ( p + I)-th type denoted by the symbol X [ + ' is included in the set of the (p + 1)-th type denoted by the symbol Xf+ ; the propositional function X [ + I c Xf+Iexpresses the idea that strict inclusion obtains between the sets mentioned. It is worth pointing out that the use in Definition 5.22 of the same symbol "c"which we use to denote the relation of inclusion ought not to lead to misunderstanding, since we will be using different
24
PI
FOUNDATIONAL STUDIES
symbols to denote sets (and in particular the symbols M, N, P, A, B, ...) from those used to denote variables of the system of logic. Thus for example " M c N" is an expression stating that the relationship of inclusion (not strict) holds between the sets M and N, yet ''X? c Xi" is a name of the expression N c f v N r z x ;CXIXI'xi x,',"Nl7x: ' c x : , x' /X ; ' x ; y .
5.3. We will specify the next abbreviation by the scheme 5.31. P(X,P+'; X p , X $ g ( X , P + l ) (X,P+*eX,P+' -((X~)(X,PEX,P+~ -Xnp/dXf') v ( X i ) [X,P&Xf+I CI (x:ldxp v X,P1dX.q)]}) where n = max(1, m)+ 1. It is easy to see that the propositional function on the right-hand side of Definition 5.31 says that the set denoted by the symbol is the ordered pair of objects denoted by the symbols Xf', Xk. (As usual we define the ordered pair bl,p 2 ] of the objects p I ,pz by the scheme bl,p z J = ( { p , } , { p l , p Z } } ;cf. Wiener [I], Kuratowski [I], p. 171.) 5.4. Now we introduce abbreviations for several more propositional functions. 5.41. R
(xi+ z ( x i +' ) [ X I +zEx[+ .+ (EX:) ( E X [ + )P(x,P+ ;xi,xi+ 3)
3
I
2
5.42. Z ( X [ + ' ; Xf', X i ) z (EX:+') [P(X[+2; X / , X i ) &X[+2~X[+3];
5.43. D ( X [ + 3 ;Xp+1)DL(X[)[x[&X~+Ic, (EX[+I)z(Xl+3; x,p,X,p+l)];
5.44. a(xif3; xp+l)~(x,q)[x,pexp+1 ++ (~x[+,)z(x,p+~; x[+,,xi)]; 5.45. J ( x ~ +
(x,P) (x[+I ) (x[+z) { [ Z (x,P+J; x!, x[+
& Z ( X j + 3 ;X [ , X i + * ) v
&
Z(x[+3; xi+,,X i ) & Z(X[+3;Xl+2,Xal +%+tZW+22).
The propositional function expresses the idea that the set denoted by the symbol Xf+3 contains as elements exclusively ordered pairs, i.e. is a binary relation. The propositional function Z(Xf+ ;Xf', XE) expresses the idea that the ordered pair of objects denoted by the respective symbols Xf, xz is an element of the set denoted by the symbol thus in particular if this set is a binary relation R, then the propositional function Z ( X { + 3,!';A ):'A expresses the idea that the relation R holds between the objects denoted by the respective symbols X!, X i . The propositional funcX/+'),a(X!+3; X/'+l) respectively-in the case where the tions D(X[+3; set denoted by the symbol is a binary relation R-express the idea
[21
25
ON THE INDEPENDENCE OF DEFINITIONS OF FINITENESS
that the set denoted by the symbol XI+' is the domain, respectively range, of R. With the same assumption the propositional function J(X[+3) expresses the idea that the relation R is one-to-one. 5.5. In turn we will define a propositional function expressing inductiveness of a set (cf. Russell and Whitehead [l], *120.01 and 02). 5.51. O(X,'; X:, XA)
5.52. S(X,')
(X,") ((A';+
(XA+ L)[X,!,+ EX,'
-
[(X,') (XleX;+ 1)
@'A+ -+
EX: v
IdX,,?,)];
X2+ EX,"]&
Bt (Xt+I ) (X,'+Z) (Xk9 [O(Z+ I ;X2+2 XL) &Xi+ 2EXk3 X,'+ L M I .+ X,'&XCJ) The propositional function O(X2;X?, X,,?,)says that the set denoted by the symbol X,' results from the set denoted by the symbol X: by the addition of a single element, namely the element denoted by the symbol A',?,. The propositional function S(X,') states that the set denoted by the symbol X,' belongs to each class which contains the empty set and is closed with respect to the operation of adding one element, i.e. the set is finite (inductive) in the sense of Principia Mathematica. 9
+
5.53. We present without proof a few lemmas concerning these two
propositional functions :
5.531. LEMMA.[(X,')X,'&X,']
-+
S(X2)E T .
5.532. LEMMA.O(X,';X?, XA) & S(X?)
-+
S(X,') E T.
5.533. LEMMA.a) (EX,') S(X2) E T ; b) (EX;) [S(X,') & X,,?,&X,']E T. 5.534. LEMMA.(EX,') S ? ) [(X,') (A'; EX?) S(X:)J E T. 5.535. LEMMA.(X:)[X,'&X,' t* X , ' & X ~ V X , ' E X ~...VVX; EX$)]& & S(X$ & ... & S ( X t ) 4 S(X2) E T for p = 1 , 2 , ... 5.536. LEMMA.S(X,') ++ ( X , " ) { ( X , ' + , ) [ ( X ~ ) ( X , ' & X , ' .++~X,'+,sX,"] ) & 8c (X,'+ I ) (Xi+2) W,') [Xi+1 && S(X,'+ 1) 2) & & O(X;+2; X,'+l, Xi)-+ X;+2~X;] .+ X,'eX,") E T . The intuitive content of these lemmas is quite clear; Lemma 5.536 presents a certain necessary and sufficient condition for the inductiveness of -+
-+
w;+
a set.
5.6. At last we give the final group of abbreviations which will play an important part later on: 5.61. G(X2)ZJ(X:)& R(X:) & (EX:)[(X:)(X,'&X,2)& D(X:; X x ) & 84 a v : ; X31;
26
[21
FOUNDATIONAL STUDIES
5.62. B1(Xi;Xi,XA) E G ( X t ) & Z(Xi'; X : , X i ) ;
xi+,)
B i + 1 ( H ; X +',X~+')~.f(XL4)&(X:+,)(X:+*)[Bi(X2; Xi+, , -+
where X: # Xi+1and X,'# X;+';
(x,+'&X:+'*X:+2&X;+1)]
5.63. Ni(X2;X / ) Z B i ( X : ; X i , Xf) (for X,* # X i ) ; 5.64. N*(X&
x:) E (x,')[x,lex:+N' (X2;X,l)]; -
5.65. C(XL+';X ~ + l ) ~ ( X ~ ) [ X & k ' ; 't +,' X&Xi+'].
The content of these propositional functions is extremely simple: the function G(X,O)says that the object denoted by the symbol X,* is a function mapping the class of all individuals one-one onto itself. Each such function f maps an arbitrary individual p to Ythef-image of p" denoted by the symbol f(,u),the set p = {p',psi,...} into "thef-image of p" {f@'),f(p"),...}, and generally each set p of type i is mapped into its 'Timage" which is also a set of type i and contains as elements allf-images of elements of p. Cf. Tarski [6], p. 431, also Carnap and Bachmann [l], 92. The propositional function Bi(X$;Xf, X a says that the set denoted by the symbol X,* is a function f mapping the set of all individuals one-one onto itself and that thef-image of the set denoted by the symbol Xi is the set denoted by the symbol XA; the propositional function Ni(Xt; Xi) says that the set denoted by the symbol X 2 is a one-one function f mapping the set of all individuals onto itself and that thef-image of the set denoted by the symbol Xi is Itself.The meaning of the functions N*(X$; X:) and C(XL+', Xi+') should not be in any doubt. 5.66. We give without proof a few simple theses concerning these functions: 5.661. LEMMA. Let a be a propositional function whose free variables are X Z , ...,X z , and let the variables X:;, ..., X e satisfv: the variable X:, when substitutedfor X l in a at all places where Xp9' isfree in a, does not become bound at any of these places (i = I ,2 , ... ,n). Finally let Xf # X:;, Xr",', ..., X4qyX e . Then
Bpl(X~;X~~,x~)&sp,(X~;x~,xp,.)& ...&B,,(Xf; Xg?,X$) +
[a * Sbrpliglftiq2::: ~:i&ll E T .
This lemma was proved by Tarski and Lindenbaum and is given in their paper [I] (Th. 1). It is not difficult to reconstruct the proof if one proceeds by induction on the rank of the propositional function a.
PI
O N T H E INDEPENDENCE O F DEFINITIONS OF FINITENESS
5.662. LEMMA. G(X:) &N*(X,4; X?) & X: E X:
+ N*(X>; X :) E
27 T.
This lemma states that from the axioms one can deduce a proposition with the following intuitive meaning: if the function f maps each element of the set M into itself and N is a subset of M, thenfmaps every element of N into itself.
-
5.663. LEMMA. G ( x $ ) & ( X j ) ( x j E X f ) + Nz(X,"; Xi')E T . 5.664. LEMMA. G(Xt)& (Xf)[X,%Xi
S ( X f ) ] + N,(X$; X j ) E T .
These lemmas state the deducibility from the axioms of logic of propositions with the following meanings: the set of all individuals and the set of all inductive sets are mapped onto themselves by each one-one function which maps the set of all individuals onto itself.
5.665. LEMMA.VX: # Xi, then ( E X L + & f i ( X ixi, ; Xi+,> E T; b) G(X2) + (EXi+,,,)Bi(X$;Xi+,,,, Xl)E T . a) GV:)
-+
5.666. LEMMA. If X t # X;,Xi,Xi then a) Bi(Xt; X i , X i ) & B~(x:;X:, xi) + x j l d ~E; T; b) Bi(X2; X i , Xi)& Bi(X2; Xi, Xi) + XildX;' E T . These four lemmas which one proves by induction on i together express the deducibility from the axioms of logic of the following proposition: for each function f mapping the set of all individuals one-one onto itself and for each set p (of arbitrary type) there is a uniquely determined f-image of ,u and a uniquely determined finverse image of p.
5.667. LEMMA. C ( X f ;X f ) & (EX:)[C(X:; X?)& S(X:)]
-, S(Xkz) E
T.
The correctness of this lemma follows from Lemma 5.12 and the remark that the one-oneness of the operation of complementation can be demonstrated in our system of logic.
5.668. LEMMA. C(Xi;X:)
c-,
C(X:; Xi) E T .
This lemma says that the proposition: the complement of the complement of a set is the same set-is deducible from the axioms of logic.
6. In the preceding paragraphs we have given a concise description of the language which is the object of consideration in the later part of this Paper. We could not give all the theorems and definitions which have been introduced for the study of this language, the reader can find them in Tarski's Papers [2] and [3]. We mention here one definition:
28
FOUNDATIONAL STUDIES
PI
6.1. The i-th increase in type of a propositional function u is the propositional function which arises from the expression a by the substitution of the variable X."+ifor each variable X," E Vr(a) ( i = 1 , 2, ...) (cf. Giidel [I], Def. 33, p. 184). The i-th increase in type of a propositional function a we will denote ai. As was shown by Tarski, the following holds: 6.2. LEMMA. If a E T and i is a natural number then aiE T .
The proof proceeds by induction on the rank of consequences of the set L. Profiting from the notion of increase of type we can state the following lemma : 6.3.
~EMMA.
[(EX:)S(X:)] -+ [(EXf)Sm)]' E T.
This lemma says that the first increase in type of the axiom of infinity is a consequence of this axiom, or in other words that from the axiom of infinity follows the existence of infinitely many sets of the first type. It would be superfluous to give a formal proof of this thesis here; see Russell and Whitehead [I I, *I25.24.
8
2. The method of relativization of quantifiers
The purpose of the theorems presented in this section is to sketch the method which we will apply in 0 3 to investigate what implications obtain between certain definitions of finiteness. This method-below we call it the method of relativization of quantiJiers-has been known for some time and was used to obtain the results mentioned in Tarski [a], footnote 20, and in Lindenbaum-Tarski [I], p.22. Herbrand ([I], Chap. 3, $ 3 ) also applied this method in certain investigations concerning the so-called problem of solvability. All these papers either treat relativization of quantifiers very cursorily or mention only results obtainable through its application. Because working with the ejtensively developed apparatus of this method is essential for our further considerations as well as for a number of other investigations in the field of metamathematics we give a rather exhaustive description of the method of relativization of quantifiers below. 1. To begin with we have to introduce a series of auxiliary operations on expressions which are essential to a precise description of the method under discussion. 1.1. Let n denote an arbitrary natural number. We introduce a function
121
ON THE INDEPENDENCE OF DEFINITIONS OF FINITENESS
29
p, which with every propositional function a associates a new function p,(a), namely, the one obtained from a by substituting for each variable X ~ VrCa) E the variable Xf+.( k , I = I , 2, ...). The function p,, which we need not describe more precisely, has the following properties.
1.2. LEMMA. Let n be a natural number, N be a finite set of variables, and a , B E S. Then a) for all sufficiently large n Vr(p,(a))* N = 0; b) p,(a 4 8) = PA^) pn(B) and = p.(aj; c) p,((Xlk)a) = (Xlk+,>p.(a);d) is a is an instance of B then p,(a) is an instance ofp“(j3); e) XF e V r ( a ) ifand only i f x f + n E Vr(p.(a)). +
The proof of this lemma i s obtained by an easy induction which we may omit. 2. It will be convenient to have the following definition:
2.1. a) t is the class of sequences 0 = [e,, 8,, . . . , O n , ...I whose n-th term for n = 1,2, ... is a propositional function with one free variable X ; . Df b) e(x;)=sb;,,(e,) for p . n = I , 2, ... and e = [e&, ..., en, ...I E t. We will denote sequences in t by the symbol 0 possibly with various superscripts; in keeping with this 8. will denote the n-th term of the sequence 8. In connection with definition b) notice that the notation “8(X;)” introduced there is somewhat unconventional: in essence e(X:) is a function of two natural number variables and a variable running through t, thus we should really use a notation such as @,,p(t9) (cf. here 0 1, 2.3). In practice we will only use Definition b) in cases where X:Z Vr(f3,,)or p = 1. In the first case the operation of substitution Sb,”,,@,)involves replacing the variable X; by Xl at those places in 8. where Xl is free in 8. in the second case we have simply e(X;) = 8., Df 2.2. Let 8 denote a sequence of the class f. We set o t ( a )= a for each propositional function a of rank 1. Suppose that we have already defined @a) for all propositional functions a of rank G k and let a be a propositional function of rank k + I . From the definition in $ I , 3 one of
(1) (2)
a=B-,y,
-
8, a = (xk)B a
=
(3) holds, where @, y are certain propositional functions of rank < k and I, m natural numbers. In case (1) we set o ~ ( a ) ~ o o -+ , *ot(y), ( ~ ) in case (2) Df o:(a) -oo (a), and in case (3) we set
30
FOUNDATIONAL STUDIES
Df O;(E) = a
if
X,!, E Vv(O(X:)).
In this way with each propositional function a is associated an expression o m .
2.3. Let a E Sand XF;, ...,X k denote all free variables of the function a. Further, suppose that k l d k2 < ... Q k, and that Ir < l i + ,when ki = k i + l Df (i = 1,2, ..., m- 1). If X$ E Vr(O(Xp)) for some i then we set oo(a) -- a, and otherwise oo(cr)2 ! e(x::)& ... & e(x:;) -+
0; (a).
We say that the function oe(a) arises from a by relativization of quantijiers to the sequence 0. 2.4. To explain this operation suppose that a is a proposition, i.e. Fr(a) = 0. The function p,,(a) is now also a proposition with the same meaning as a; these two propositions differ only with respect to the variables used.
However the structure of the proposition oo(p,,(a)) is different: if we choose n sufficiently large then (X:)[O(X/) + o;(o)] occurs in the proposition oO(p,,(a)). Hence one can say that the proposition oo(p,,(a)) says of the individuals of the k-th type which satisfy 0(X:) exactly what p,,(a) (and hence a) says about all individuals of the k-th type (k = I , 2 , 3 , ...). Thus the proposition oo(p,(a)) expresses exactly the same idea as a but with respect to another “world”-namely that in which the notion of “set (or individual) of the k-th type” is replaced by the more restrictive notion of “individual of the k-th type satisfying 0(Xf)”. From this one sees clearly that there is a connection between the method of relativization of quantifiers and so-called proofs by interpretation which one uses all the time in studying systems of axioms (cf. Fraenkel [2b pp. 340-343). However, while with the method of proof of interpretation we are restricted to studying systems of axioms in a system of logic which itself is unchanged by the interpretation, we can apply the method or relativization of quantifiers to the logic itself. Because of this the method of relativization of quantifiers may be applied for example to questions of mutual independence of logical constants, rules of deduction, or logical axioms to which the usual method of interpretation cannot be applied. 2.5. To facilitate our further work we introduce one more auxiliary definition
[21
ON THE INDEPENDENCE OF DEFINITIONS OF FINITENESS
2.51. a E So if a E S, 8 E t, and Vr(a)
-
q
i=I
31
Vr(e(Xi)) = 0, where q de-
notes the greatest number such that at least one of the variables of type q belongs to Vr(a). Next we prove a series of lemmas
2.52. LEMMA. If 8 E t, a E S, then for aN suficiently large n, pn(a) E So.
For the proof we recall Lemma 1.2 a) and Definition 2.51.
If 8 E t, a E S, then Fr(oe(a)) = Fr(o$(a)) = Fr(a). 2.53. LEMMA. The proof is obtained by an easy induction from Definitions 2.2 and 2.3.
2.54. LEMMA.If 8 E t, a, B E So and B = Sb::lml ::: :;lm,(a) then o;(B) ::::;lmp(o,*(a)) and ~ o VE Fl( ) {oe(a>}). Here we only sketch roughly the main idea of the proof. If GL is a function of rank 1, then a = X,"EX;+' for some k, I, m,and X:, XL+' Z Vr(O(X,"))+ +Vr(e(X:+')) since a E So. Thus having regard to 2.2 and 2.3 = Sb::,,,
(1)
o,*(a) = X:EX?~,
oe(a) =
e(x:)&e(xi+l)+ x:EX?'.
From the assumption that /I is an instance of a, 9, = XjeX;+' and from /3 E Soit follows that X i , Xikf1Z Vr(O(X:))+ Vr(e(X:+')). Thus bearing in mind 2.2 and 2.3 o,*(B) = x ~ E x ; + ' ,oe(/I) =
e(x;)&e(x;+l) + X$EX:+',
whence by (I) the correctness of the lemma for functions of rank 1 follows immediately. Suppose that k 2 1 and that we have demonstrated the truth of the lemma for functions a of rank < k. Let a E So be a function of rank k + 1 and B E So be an instance of a. Bearing in mind Definition 8 1, 3 we have
(2) or (3) or (4)
a =
7,
a=y+6, a = (XqP)Y
where y, 6 are functions of rank < k which belong of course to So while p, q are natural numbers. The substitution which takes a into B, takes y into y', 6 into 6' and each variable occurring in y1 or 6' occurs in either a or B; hence it follows that y', d1 E SO. The functions y l , 6' are thus instances of the functions y, 6 of rank < k,belonging to So and themselves
32
PI
FOUNDATIONAL STUDIES
belong to So, which from the induction hypothesis proves that the substitution mapping a into /Imaps o t ( y ) into o:(y') and $(6) into oQ(6,). Now we must examine in turn the cases (2), (3), (4). If case (2) holds, then #l = y' and ___
.m = oe*(r'>,
o t ( a ) = .,*cy>,
whence we immediately conclude that the substitution which takes a into
p takes o,*(a) into o:(/?). Denoting the free variables of the function a by X:;, ..., Xf; (where k , < k2 < ... < k , and ti < ti+, if ki = ki,, for i = I , 2, ..., m-1), from the assumption that a E S B and 2.3 we have: o,(a) = e(X,";)&
... &O(X;:)
-,o$(a).
The substitution taking a into /? thus takes oe(a) into an expression of the form
e(x:;)& ... &e(x;:).+ o,*(p). and so belongs to N( {oe(a)}). The variables X i ; , ..., Xi: are all the free variables of & Thus in view of 2.3 and the assumption p E q0 odp)
-
{e<X,":> ... &e(xi:)
-+
438)) E T ,
whence we deduce that oO(p) E FI( { O g ( t l ) } ) . The reasoning in cases (3) and (4) is entirely analogous to that used for case (2) and so we shall not treat them here.
LEMMA.If 0 E t, a,?! , E SO, rhen s ( B ) E ~ t @@(a ( -,8), oo(a), ( m : ) e ( X : ) , ( E X : ) ~..N .,(EX;)B(X?), ), ...I). 2.55.
P r o o f. We denote
W B ) = {X,4' ..* ,'yp"-")
(1)
9
9
(2)
Fr(p)-Fr(a)= {$;,
(3) Then we have
Fr(a)-Fr(p) =
..., X k } , {XL;, ..., X;;}.
(4)
(X,";,...)
xq c
(X$ ..., x;:}
and
{ X i ; , ..., Xi:, X i ; , ..., Xg:}. From the assumption that a , B E S,, we deduce by 2.2, 2.3 and the identity
(5)
Fr(a
Fr(a) =
+
-
8) = Fr(a)+Fr(P) that
(6) oe(a)
[O(Xi;) & ._.&O(X:,)&f3(X;;) & ... &e(X::)
-, o:(a)] E
T,
PI
ON THE INDEPENDENCE OF DEFINITIONS OF FINITENESS
33
34
FOUNDATIONAL STUDIES
If case (3) holds, then-as
is clear from (1)-
PI
(4) Fr(a -+ (.Yap) = {X:;, ... ,A';)(X,"}. Thus if Xf;, ..., X? denote all the variables amongst A';, differ from Xi,then-as is clear from 2.2 and 2.3(5)
o,(a
-, (x,p@)
++
+
(O(X:,I)& ... &O(x,:)
( X X W , P ) 4691)) +
-+
E
..., : X which
{o;(E)
T.
The expression (2) gives
e(x:;)&... &O(X;qn.)&o:(a)
-, o,*(p) E F I ( { o ~ ( ~p)}), -+
from which results (6) e(x;;)&...&8(Xk)&o:(a) -,[e(x;)+ oQ(@)] E FI( (oo(a p)}). The variable X,P-being different from X;;, .. , Xf'-is not free in the exOn the strength of the hypotheses and Lemma pression O(X;;)& ... &€J(g;). 2.53 X,P 2 Fr(o:(a)) so thatwe may apply the rule of adjunction of a quantifier to the thesis (6), obtaining -+
.
(wf;) & ... &e(x&Wb)
-+
or equivalently
e(x;;)& ... &e(x$)-, {o:(a)
-+
(x;)[W,p) oe*(8)1 E FI( (oda @)I), 4
+
(x;)[e(x:) o3@)1>E Ff({ o d a @>I). -+
+
Bearing in mind (5) we deduce that Lemma 2.56 is good for the case under consideration. Suppose next that case (4) holds. In this event we obviously have Fr(a 4 (Xi)@) = Fr(a -+ p) = {X;;,..., >;:'A and in the light of 2.2 and 2.3 holds
(7)
oo(a -+ (X;)@) ++ (O(X;;) &
CX,",[W,.) -+ From the expression (2) follows -.*
ecx;;)& ... & e(x2)& O Q ( N )
... &O(X:$
-+
(@(a)
ot(P)l))E T . -+
oQ(p) E Ff({ o d a -+
@I})
and a fortiori
e(x;;)& ... & e(x,":)& o,*(a) -.+ [o(x;)-+ o;(p)] E N( (odor PI}). +
The variable X,P, being different from all of X:;, ..., Xi:is not free in the antecedent of the above implication of the strength of the hypotheses and Lemma 2.53. Thus adjoining a quantifier we get
w:;) & ... &e(X::)
&o m
-+
(x:)recX:)
-+
o,*(p)iE ~q { o o ( ~-,@)I)
35
ON THE INDEPENDENCE OF DEFINITIONS O F FINITENESS
[21
or B (xi;)& ... & e(x::) tot (a) -+ (XXW,P) 0 : which from (7) proves the lemma in case (4). +
+
2.57. LEMMA. 0 E t, a oo(a
-+
-+
(~111E F[( {oo(a-+ B) 11,
(Xi))8 E So, then
B) E FI( (00
(a
+
<X,'>B), (-W')Wf)>).
P r o o f . Let Fr(a
-+
(X,P)B) = {X:;,
..., Xi,}.
In keeping with the hypothesis and Definitions 2.2 and 2.3 oo(a --+ -+
(x,p)B)
C,
(e(X;;) & ... & e
@,*(a)-+ S) E Se
and
6)) E W M ) . Again the existence of such a number follows from ( I ) and Lemma 2.52. Applying Lemma 2.56 we conclude easily that pn(y
-+
o,(pn(y
+
oe(Pn(Y
-+
(x:)a)) E N((o,(pn(y
+
a))}),
whence follows oo(pn(y (X,")d))E F / ( M ) or oO(pn(B))E F / ( M ) . Finally if case (5) holds then again we denote by m a number sufficiently large that for all n 2 m -+
P.(Y
+
and
( X ~ P VE) SO
oo(pn(y
+
( x : ) ~ E) )W M ) .
The existence of this number results from (1) and Lemma 2.52. On the basis of Lemma 1.2 b), c) Pn(Y
-+
(x:)6)
= pnty)
-+
(XqP+n)Pn(d),
Y ~ ( B= ) pn(y)
--*
pn(6).
Applying Lemma 2.57 we deduce that o,(pn(y)
-+
pn(6)) E M ( ( o , ( p n ( y ) -+ (X;+n)pn(d)),
(~xf)e(x~)}) 9
which from the hypotheses of the lemma immediately gives oO(pn(y .-+ 6)) E
WW,
i.e.
o,(pn(B)) E
N(W.
From the above discussion we infer that from the correctness of the expression (1) for a number k 2 1 follows its correctness for k 1. Hence by induction we deduce the correctness of the expression (1) for arbitrary k 2 1. Because for each function B E F / ( M ) there exists k 3 I such that B is a consequence of rank k of the set M, it follows from (1) that for each B E F / ( M )there exists m such that o,(pn(B)) E F / ( M ) for all n 2 m, q.e.d.
+
38
[21
FOUNDATIONAL STUDIES
From Lemma 3.2 it follows that to check condition (1) of Lemma 3.1 (of course only in case (EXf)O(Xf)E FI(M) for p = 1 , 2 , ...) we may confine ourselves to proving the corresponding condition for functions of the set M+L. About the functions in M we can say nothing until M is more narrowly prescribed. However, we can give conditions which imposed on the sequence 8 ensure that the property under consideration holds for functions in L. 3.3. LEMMA. If 8 E t and there exist functions a, @, y
-
E S such that b
=(a+jl)+ [(B-y)+(a+y)]orS= (%+a)-aorb=a+(Z-tB), then there exists a number m such that oo(pn(6))E T for all m 2 n.
P r o o f . We will treat only the case 6 = ( a B) + [(B -+ y ) + ( a remaining cases can be treated in a completely analogous manner. From Lemma 2.52 we deduce that for n sufficiently large p.(a), p,(/?), p.(y) E So. Denoting the free variables of p.(6) by ,;:'A ..., X$, on the strength of 2.2 and 2.3 we have -, y)], because the
nO(pn(6)) +
* (e(x;:>& --.&e(xz) { [ o t ( ~ * ( a ) ) o t ( ~ n ( ~ ) ) ] +
[(O:(PJB)
+
o:(PAY)))
+
+
(o,*(Pn(a))
-+
O:(PAY)))])) E T ,
whence we immediately conclude that oo(pn(6))E T.
3.4. LEMMA. If M c
p r p = I ,2,3,
(1)
s,
..., then for
a =
8 E t and d(xf+l) &
x~Ex:+~ -,O(xf) E F(M)
every propositional function
(Xf)(Xf&,+'-X~EX;+I) -,X?'IdX:+'
there exists a number i such thal oe(pj(a))E Fl(M)for all j
2 i.
P r o o f. Let i be so large that pj(a) E So for all j 2 i and let j denote a number 2 i. Setting q = I+j, r = m+j, s = n + j for brevity, from (l), Definition 1.1 and Definition 1, 2.39 we obtain pi(.) = (X,")(X,"EX:+' Lx,"&x~+') + (~:+2)(~:+1&+2 + x:+1 Ex;+Z)*
Noticing that pj(a) E So, on the strength of 2.2 and 2.3 from this expression we obtain oe(pj(a)) (e(X;+') & e(x:+l)-+ {(x,") [e(x,") -, (x;&+l -x,"&x:+71-+ (x:+*)[e(x;+z) -+ (x:+lex:+2-,x:+iex:+2)j)) E T, C,
or after an easy rearrangement
[21
(2)
-
ON T H E INDEPENDENCE OF DEFINITIONS OF FINITENESS
oe(pj(a))
39
{O(X!+l)&e(x:+l) & (x,")[e(x,") -,(x,"Ex!+~
+-. x;Ex:+1)1+
(x!+z)[e(x;+2,&x;+i&x;+z-,x:+1&x;+21} T.
for short let us denote (3) (4)
An%(x:+l)&x;&x,k+l -,qx;)
( n = I , 2,
lcm,n~(x;)[e(x;) -, (x;&x;+1 .-.x;&x."+1)1 (111,
...I, = I , 2, ...).
Then we have (5)
I,,, & e(x:+1) & x;&x: 1 -,e(x:)& x;&x:+ 1 E T (n = I , 2, ...), +
T
(m,n = 1,2, ...).
p,, &O(x,") &xpL~xi+~ + X~EX;+I E T (7) From ( 5 ) and (6) follows
(m,n = 1,2, ...).
(6)
~ , , , , & ~ ( X ; ) & X , ~ E X-;,+X$Y;+l ~
E
A , & ~ , . , & ~ ( x , ~ + ~ ) &-x,x;Ex:+i ;~x~+~ E T, while from ( 5 ) and (7) follows
~,&~~,~&e(x,k+l)~cx,kExf+l-, xiEx;+lE T . Linking these two expressions we obtain (8)
~ ~ a ~ ~ & p , , , s c e ( ~ ~ +-+ l (x;Ex;+~ ) t e ( x ~-X.+X:+~)E +1) T.
From (3) and the hypotheses of the lemma it follows that A,, 1, E FI(M), thus the expression (8) qives pr.s&6J(X,k+1) &
O(x,k+l) + (x~Ex:+I
c-,
X,".EX;+') E FI(M),
whence by adjunction of a quantifier pr,5&o(x:+1)
&e(x:+i) -, ( X ; ) ( X ~ ~ X ; + ~ ~ X ,E"FI(M). &X,~+~)
Bearing in mind Q I, 4.13 from this it follows that
e(x:+l)ste(xf+1)aPr,,-+ x:+1id~,+1 EF4W , and thus a fortiori (cf.
0 1,
2.39) that
e(x;+l)&e(x:+*)&p,,, -, (x:+z)(e(x;+z)&x:+i&x~+z +
x,"+1&x:+2)E FI(M).
Using (4) and (2) this expression immediately gives o,(pj(a)) E FI(M), q.e.d. 4. As is clear from the last two lemmas to complete the discussion of propositional functions in L we still have to consider functions of the form
40
PI
FOUNDATIONAL STUDIES
9 1, 4.12, i.e. so-called pseudodejnitions. This is the most difficult point of the whole proof and the conditions we know, which when imposed on the sequence 8 ensure that (I) of Lemma 3.1 holds for these propositional func-
tions are of an extremely special kind. Instead of giving these conditions we will define a particular sequence 8 which we use in the next section and we will show that for it the desired conditions hold. In 6.2 we will point out how our reasoning can be generalized so that it applies not only to this special sequence 0 but to a certain (yet very narrow) class of sequences from t.
4.1. DEFINITION. a) K(Xj';Xd)~ S ( X ~ ) & ( X ; : , ) [ G ( X j 4 , , ) & N * ( X i 4 ,X, ;i ) for arbitrary natural numbers i, j, m such that X:# Xf;
+ N,(X,?+, ;A';)]
b) O*, E((EXi)K(X: ;X i ) ,
e*i+lZ ( X ~ ) [ X , %.-, Xe,;i]&+ ~ (EX;)K(X~+~; x;); Df
c) o*(a) Df ot,(cc), o(a) - o,*(a)
for a E S .
Definition 4.1 b) defines inductively the sequence O,, which-as we notice at once-belongs t o the class t. We will show that this sequence possesses certain properties we need and to this end we first present a few auxiliary lemmas: 4.2.
LEMMA.i f 0
= 8, then
(x:)e(x:)E T .
P r o o f. From Definition 4 1, 5.64 X ; & X : & N * ( X t ; X,')
-t
N , ( X , 4 ; X,')
E
T,
whence immediately follows X,'EX~
--+
(X,")[G(X,")&N*(X,4; X i ) + NL(X,4;X:)]E T .
From this thesis and Lemma Q I , 5. 533 b) it follows immediately that
(EX;) { . S ( X i ) & [ G ( X : ) & N * ( X , 4 ; X,')
-+
N , ( X , 4 ;X i ) ] ) E T ,
i t . O(X,') E T; from this expression we deduce (X,')e(X:) E T, q.e.d. 4.3. LEMMA. / f e = e,, then
(Exi)e(.k'i) E T for
i =
1 , 2 , ...
P r o o f. For i = I the correctness of the lemma follows from 4.2. suppose that i = j + I where j > 1 and for brevity let us write - .-
-
ct(X{+') D' (X:)X{&x{+l;
(1)
we obviously have (2)
a(~:+l)
-,(x:)[x:~x:+*ecx:)]E T , --+
PI
41
ON THE INDEPENDENCE OF DEFINITIONS OF FINITENESS
and
-
EX{+'
a(X{+l)--+
which as we easily see gives a(X:'+I) & G(X:)
(3)
+
)ET,
1
K + (X,"; Xi+') E T
(cf. 0 1, 5.62, 5.63). From (3) it follows that a(X{+')
4
(X:)[G(X,*) & N*(X,*;Xi) + 4+, (A',";X{+')] E T ,
which bearing in mind Def 4.1 a) gives a(X{+l)& S(X,') --t K(X{+';X i ) E T . Recalling Lemma
9 1,
5.533 a) we deduce from this that
a(X{+I) 3 (EXf)K(X{+';X i ) E T , which together with (2) and Definition 4.1 b) gives a(X{+')+ + 8(X{+')E T. Because as we see from (1) and 0 1, 4.12, (EX{+')a(X{+') E T, the last expression gives (EX:'+l)O(:i+l) E T, q.e.d. 4.4. LEMMA. If 8 = O* p = 1 , 2 , ...
then
O(xf+l) & , ~ f e X f ' +-~, O(Xf)E T
for
The proof is immediate from Definition 4.1 b). 4.5. LEMMA.If the variables Xf:, ...,Xf:, Xil, ..., Xi",X: are all direrent from each other, then
[(EX,JK(Xj:;X&31 [(J%:)K(X.; 4318c f.. [ ( H ; y ( X k ; X;")I 3 ( E X i ) [ K ( X j ; ;X i ) & K(X';; X i ) & ... & K ( x ; ; Xi)]E T for n = 1 , 2 , 3 , ...
P r,o o f. On the strength of Lemma 9 1, 5.662 G ( ~ ~ + l ) & N * ( ~ ~ + l ; XG ~X)z &+N*(Xi+,;X;&)E X~~ T
for k = 1 , 2 , ..., n. Hence we deduce that for k
q.e*d*
6. Summing up the results of our discussion we arrive at the following theorem: __
6.1. THEOREM. If 6 E S, the set {(EX:)S(X:)} is consisten/, and there exists a number m such that o(p,,(S)) E F / ( { ( E X : ) m ) for all n 2 m,
then S i i Fi( {(EX:)S(X:)}).
The proof is immediate from Lemmas 3.1, 3.2, 4.3, 4.6 and 5.4.
52
PI
FOUNDATIONAL STUDIES
6.2. Analysing the proof of Theorem 6.1 it is possible to generalize it somewhat. Let us introduce two arbitrary propositional functions a(X,), /3(A't)-one having a single free variable of type 2 and the other having a single free variable of type 4. By analogy with 4.1 let u's define
L&;XI
a < X >& CX;+1) U V ~ + ~ ) & WX!+ -+
eu,,(x:>
(EXiWa. g ( ~ :;~
;
N~(x;L,, ;xj)]
t),
(for Xj
z xi),
ea,,(X1i + lI-( E XiI)[ X'1&X'+ 1l eu,,(Xf)I& (EXZ)Ku,,(Xf+';XI). (For simplicity we suppose here that the substitutions required for writing these expressions are permissible, and thus for example that X;+l does not become bound in /?(X,*) when we substitute it at those places in the function /?(X:) where X,4 is a free variable, and so on.) Now one can prove without difficulty the following generalization of Theorem 6.1 : if 8 E S, M is a consistent set of propositions, M' a subset of M, a(X.Z), /3(Xf) are propositional functions such that Fr(a(X,Z)) = {X.'}, Fr(p(X:)) = {X,*}, 8 =,,O,. and if the following conditions are satisfied +
(1)
(2)
(EX,l)a(x,Z)E F & W , (X:) [BW.') + G(X,*)I E F W W , a(X,21)& ... &a(Xi)&(X:)[Xi&? c,( X : E X , ~ V... v v
(3) (4)
X:eX;JJ+ a(X,) E FI(M) for
s = 1 , 2 , 3,
...,
for each a E M' there exists m such that Oe(p,(a)) E FI(M) for n 2 m, there exists m such that 0e(pn(8))E FI(M) for n 2 m
then 6 Z FI(M'). Here, if (EX:)S(Xi) E M , then for there to exist a number n 2 rn it sufices thut
m suck that oe(pn((EX:)S(x:))) E F ~ ( Mfor )
(X:)[S(X:)
-,€J(X:)JE Fl(M)
and
(X:)O(X:) E FI(M).
The proof of this theorem is essentially the same as that of Theorem 6.1. However, to give a strict proof of the independence of certain propositions it is necessary to call on a still more general form of Theorem 6.1. This next generalization turns on enriching the language of logic with certain new constants denoting individuals or sets of individuals of various types, which causes modification of the notions of propositional function, proof, and so on.
7. The treatment of the method of relativization of quantifiers which we presented in the preceding paragraphs will completely suffice for the purposes of 0 3. However for other investigations it may prove too narrow.
PI
ON T H E INDEPENDENCE OF DEFINITIONS OF FINITENESS
53
We will present here an example of a notion for whose definition we must rely on a somewhat more general treatment of the method of relativization of quantifiers. This time we will consider a class t* of sequences 8 = [el ,02, ...,On, ...I, where Fr(8,) = (X;,X i > for n = I , 2, ... and by analogy with Definition for 8 E t*. Next we keep Definitions 2.2 2.1 b) we take 8(X;, Xi)%b"p#,) and 2.3 unchanged except that 0 denotes a sequence not in the class t but in t*. In particular let 8" denote the sequence in t* defined inductively as follows:
e:%:
-.
&XI,
e:, ,E! (z)EX',&x;+ e : ~ . We say that f is an internal property of a set if there exists a sentence a such that the set M has the property f if and only if it satisfies the propositional function oe,(p2(a)) (for the notion of "a set M satiflying a pi'oposirional function" which occurs in this definition cf. Tarski [3], pp. 308-3!2, 348-352, 354-358, and 398). The idea defined above comes from Tarski and Lindenbaum (cf. [I], p. 22). It is helpful in expressing certain theorems, e.g. from the area of the methodology of topology. The idea of a term which is nexus with respect to a given set of formulas also mentioned by Tarski ([6], footnote 20) can be precisely defined with the help of a construction similar to the above.
0
3. Proofs of independence
Here we apply the theory described in the last section to establish the mutual independence of several sentences. The key to our reasoning will be a theorem stating the independence from the axioms of logic of a proposition asserting the existence of an infinite set of individuals (of type 1) with infinite complement. From this theorem we will next deduce the independence of the axiom of choice from the axioms of logic as well as a theorem saying that the proof of the equivalence of Dedekind finiteness with the usual notion cannot be carried out without calling on the axiom of choice. 1. To start with we establish several lemmas.
1.1. LEMMA.I f 8 = 8,, then
e(x:) {s(x:)v (EX;) [c(x;; x:)& s(x:)l)E T . t)
54
PI
FOUNDATIONAL STUDIES
The proof of this lemma rests on two theses which we give without deducing them formally from the axioms: (1)
(2)
C(X,’; X:) & G(X:) & N*(X:;X i ) + N2(X:;X:) E T;
so&(X3) [C(X,2;X : ) -+
+
S(X31
(Xt)[S(X:) -+ (EX;)(G(X;) & N*(Xi;X:) & N(X:; Xi))]E T .
Thesis (1) states the deducibility from the axioms of logic of a proposition with the following content: if a function, which maps the set of all individuals onto itself, maps each element of a certain set into itself then it maps the complement of this set into itself. Thus the proof of this theorem presents no difficulties. Thesis (2) asserts the deducibility of the following sentence: if the set M and its complement N (with respect to the set of all individuals) are both infinite, then for each finite set P there exists a one-one correspondence f from the set of all individuals onto itself such that f(p) = p for p E P and,f(M) # M. (We are employing here the usual ambiguous notation:f(p) denotes the value of the function f for argument p when p is an individual, while f(M)denotes the set of values which the function f takes on the set M.) For the proof it is enough to observe that in viewofthe hypotheses thesets M - PandN- Parenonempty. Let m e M- P, n E N - P,and f denote a function such that f(m)= n,f(n) = m, and f(9) = 9 for q # m, n. Then f maps the class of all individuals into itself and i s one-one and onto. For p E P f ( p ) = p since m,n iE P. The equalityf(M) = M fails, because n E ~ ( Mand ) n E M . Formalizing the reasoning indicated above we obtain a proof of the thesis (2). Using the theses (I) and (2) it is now not hard to prove the lemma. First of all we have
s(x:) .+ e(x:)E T
(3)
on the strength of Lemma 8 2, 5.2. The expression (1) gives C ( X f ;X : ) & S ( X i )
-+
S(Xt)&(X:)[G(Xi)& N*(X;;Xi) -P
whence on the’basis of Defnition (4) Lemma
Nz(Xd; X:)I E T ,
0 2, 4.1 a) we deduce that
(EX;) rc(xt ; x:) & S(X:)] 2, 4.2 yields
~ E X ; ) K ( X;:xi)E T .
-+
(x:) (x:&x:-+ e(x:))E T .
On the basis of Definition Q 2,4.1 b) from this thesis and from (4) we obtain (5)
(EX;) [c(x;;x:)& s ( x ; )-, ~ e(x:) E T .
55
ON THE INDEPENDENCE OF DEFINITIONS OF FINITENESS
I21
O n the strength of Definition Q 2, 4.1 a) the expression (2) is equivalent to
S(X:) & (Xi)[C(X:; X:)
+ S(xfl]+
(EX;)K(X:; X,) E T.
Hence by contraposition and changing the variable Xg to
Xi
we obtain
(EX,)K(X:; X i ) + (S(X:) v (EX:) [CtX,’;X:) & S(X:)]} E T. Since e(X:) + (EX:)K(X:; Xt)E T in view of Definition thesis gives
8 2,
4.1 b) this
e(x:)-+ {s(x:\v (EX;) [c(x:;x:) & s(xi)l)E T . (6) Combining the expressions (3), (3, and (6) we obtain the desired result.
-
1.2. LEMMA.If 0 = 8, and C(X& A’:) E So fhen
c(x;;x;)
o*(C(Xi;X i ) ) E T .
-
P r o o f. In view of Definitions 8 1, 5.65, 0 2, 2.2, Q 2, 4.1 c)
o*(c(~ x:)) : ;= (x;)[e(x;)-,(x;&x: x:&x,3]. However, because O(X;) from this that
E
-
T o n the strength of Lemma Q 2, 4.2 it follows
o*(C(X;; X:))
which proves the lemma.
1.3. LEMMA.~ f8 ’= 8, then
( X ; ) (XAEX; HX,!,EX:)
e(x:)& CCx,’;x:)
--*
E
T
e(X,’)
E
T.
P r o o f. From Lemmas Q I , 5.667 and Q 2, 5.2 it follows that
c(x:;x:) & (Ex:)[c(x:; x:) & S(X:)I-, e(x;)E T .
(1)
On the strength of Lemma 1 . 1 by substitution we obtain
( ~ x : ) [ c ( xx::); & s(x:)].+ e(x:)f T , whence
C(X:;X,’) & S(X:)
+ @(Xi)E
T,
+ O(X:) E
T.
or on the basis of Lemma 9: 1, 5.668 (2)
C(X,’;A’:) & S ( X : )
From ( I ) and (2) we deduce at once (3)
c(xf;X : ) & {S(X:)v
(EX,Z)[C(X,I;X : ) &S(X,’)]) --* O ( X 3 ) E T .
I n view of Lemma 1 . 1
o(x:)+ (s(;y:)v (Ex,I)[c(x:; X : ) & s(x:)I} E T .
56
FOUNDATIONAL STUDIES
The expression (3) thus gives
e(x:)& c(x;;x:) -+ e(x,l)E T,
q.e.d.
1.4. LEMMA.There exists a natural number rn such that for n 2 m
o(p.(Ex:) {s(x:) (X,2)[C(X,2;X:) P r o o f . Let us take
+
s(X2)l)))E
-
62 ! (EX?) {S(x:) & (X,’)[C(X,2; x:)
(1)
T.
__
S(X,’)l}.
-+
Then it is easy to check from the Definitions Q 1, 5.52, 5.65 that
v:+
w.+
pn(6) = (EX,‘, 1) {S ( X +3 2 ) [c(x:+ 2 ;xn+ 1 ) -b 2111 (2) for n = I , 2, ... On the strength of Lemma 2.52 there exists a number m such that (3) p,(6) E So. for n 2 m .
From (2) and (3) on the basis of Definitions Q 2, 2.2 and 2.3 we get (4)
o(pn(6))
= (EX,+ ,) +
(em?+I ) & o*(W;+
[o*(C(X.+2;e + d )
+
where for brevity we have set 8
= 8.,
1) & W ~ + {W,’+J J
o*(s(x.2+2))1})
From (3) it follows that
S(X.2+,), S(X,2+2), cw:+z;
- c+L ) E
So.
Thus calling on Lemmas 1.2 and Q 2, 5.3, and applying the rules of propositional calculus, from (4) we obtain o(pn(6))
++
(E~,+,){~(X,?,,)& s(x,2,,-)&(~,?+~)[e(~n2+~) & &C(X,’+z; X.”+i)
-+
S(X,Z+2)1}E T .
Transforming this thesis with the help of Lemma 1.3 we obtain o(pn(6))
(Ex,?+1) {e(X:+
I
) & S ( X + I ) & (Xi’+ 2 ) [C(x,2+ 2 ;X i + 1)
S(xn2+2)1) E T. Lemma 1.1 states that the right-hand side of this equivalence is the negation of a thesis. Thus the left-hand side must also be the negation of a thesis which means that o(pn(6)) E T which by (1) proves the lemma. From Lemma 1.4 we easily obtain the following theorem: +
-
1.5. THEOREM. Ifthe ser {(EX:) S ( X ? ) } is consistm, then ( E X ? ) ( S ( x : ) & (Xi)[C(X,; X:)
-+
S(xt)]}
-
z F l ( { ( E x ; ) i 3 T ) )).
For the proof we take S = (EX:){S(X:) & (X,’)[C(X,’;X:)
+
S(X,’)l)
[21
57
ON THE INDEPENDENCE OF DEFINITIONS OF FINITENESS
in Theorem $ 2, 6.1. The hypotheses of this theorem are satisfied on the strength of Lemma 1.4 and thus so is the conclusion S Z Fl({(EX:)S(X:)}), q.e.d. Theorem 1.5 is the first of the results mentioned in the introduction of the present section. It says that if adjoining the axiom of infinity to the axioms of logic produces no inconsistency, then in the system so extended it is impossible to prove the existence of an infinite set whose complement is infinite. Chwistek [I], p. 138, mentions this theorem but without proof. L-
2. We now apply Theorem 1.5 to investigate the relationship between Dedekind's definition of finiteness and the definition of inductiveness. To this end we first introduce three definitions 2.1.
X,+
-x;+ 52 I
x,*
3) J(X,+ 3) & D(X,p+3 ;
(EX,+
1)
&
qx,+
3;
X,"
1)
;
2.21. Sl(X;+')~(X,+~)(X,~:){(X;~l')[x~~:&x,~: ux;;.: c x,+y & &X,+2
2.22.
E
x,;.:
-
s,(x,+')E(x~::)[xg:
& q + 2 ,;+I
- x,::
&X,$:
-+
G
x,: c
X,p+2};
,;+I
+ ,,+I
c X P, ++I'] .
The propositional function 2.21 says that the class of subsets of the set denoted by is not of the same power as any proper subclass, while the function 2.22 says that the set X:+' itself is not of the same power as any proper subset. Thus the propositional function S2(X;+')is a formalization of the well-known definition of finiteness proposed by Dedekind ([l],p. 17). The propositional function S,(X;+l)is a formalization of a p r o p erty, which from an intuitive point of view, also takes in just the finite sets, and thus may also be taken as a definition of finiteness. Amongst many others Tarski ([I], p. 93, Def. 111) put forward this definition. The following implications hold : 2.3. LEMMA.a) S,(X,P+') + S,(X,P+') E T, b) S(X:)
-,S,(X:)E T.
The proof presents no difficulty (cf. Tarski [l], p. 94). We will now show that the converse of the implication in Lemma 2.3 a) is not a thesis. To this end we first note the following obvious lemma: +
2.4. LEMMA.Thefirst increase of type of the propositional function S2(X:) Sl(X:) i s S2(X:)--* S,(X:).(Cf. $1, 6.1.)
-
We next give two lemmas stating that two particular functions are consequences' of the set {(EX:) S(X:)}. Giving a formal proof of this would be exceptionally tedious, since the propositional functions involved are
58
PI
FOUNDATIONAL STUDIES
somewhat complicated, thus we content ourselves with an intuitive sketch of the proof which would then have to be formalized. This lemma says that a class containing all finite (inductive) sets being infinite in the sense of Definition 2.22 is a consequence of the axiom of infinity. In other words: the class A of all subsets of the set of inductive sets is Dedekind infinite (when we assume the axiom of infinity). Formulated this way the theorem is obvious because the class A contains all natural numbers (i.e. equivalence classes of finite sets under the relation of having the same power), and the class of all natural numbers is Dedekind infinite in the presence of the axiom of infinity (cf. Russell and Whitehead [l], *124.12; from the axiom of infinity it follows that the set of natural numbers has power No).
2.6. LEMMA. (X:) [ X : E X ; c-, S(X:)] & S,(X?) This lemma says that if the class of all finite sets is Dedekind infinite, then there exists an infinite set whose complement is also infinite. The intuitive idea of the proof is the following: if the class of all finite sets is Dedekind infinite, then there exists an infinite sequence [K,, K 2 , ..., Kn,...I of distinct finite sets. Let L be an arbitrary finite set; because its class of finite subsets is finite, there exists a number a such that for m 3 a K,,, is not a subset of L. The least such number we denote by a(L) and by induction we define a sequence of natural numbers rn as follow:
From the definition of the function a(L) it follows that (1)
rn
since Krmis a subset of
rm
k= I
Kk.Further we have
1.
(2)
Kr*+,-
< rn+l
xo k= I
( n = 1 , 2 , 3,...).
If m < n, then by (1) r m + , < r n , and thus Kr,+, c
k= I
Kk which yields
[21
59
ON THE fNDEPENDENCE OF DEFINITIONS OF FfNITENESS
and thus a fortiori (3)
m
rl
- k = I Kk) = 0 (K,,,, - k= 1 Kk)* (Krm+, Denoting W,,
= Kr,+,-
rn
k= I
for m < n, m yn = 1, 2,
._.
K,, (n = I , 2, 3, ...) we have W, # 0 and
W,,, . W,, = 0 for m , n = 1 , 2, 3,
..., m
c WZk,
# n. Then taking W' =..
5 . c
k= 1
c W,,-, we easily conclude that both the sets W' and W" are OD
W" =
k= I
infinite (even in the sense of Definition 2.21) and W' . W' = 0. Thus the set W' is infinite and its complement is infinite which proves the lemma. (The existence of sets W,, satisfying the conditions W,, # 0 and W,,, W, = 0 for m, n = I , 2 , ... and rn # n is a consequence of a theorem of Tarski; cf. Lindenbaum and Tarski [2], Theorem 68.) Relying on the three lemmas above we can now easily prove the following theorem :
--
2.7. THEOREM. If ((EA':)S(X:)} is a consistent set, then &(X:> E q ( ( E A ' 3S<X:>>). P r o o f. Let us suppose that (1 1
sz (A':) s, (A':) 4
+ S,(A':)
I).
E Fq { (EA':)S ( X 3
Then in view of Theorem 2 a) of Tarski 121
(EA'?)s(x:)
-+
[&(A'?)
+
S, (X:)] E T .
In accordance with Lemma 9 I, 6.2 the first increase in type of this propositional function is also a thesis:
-
[(EX:)S(X:)]'
-+
[S z(x ? ) 4 sL(x?)]' E T.
In view of Lemmas 2.4 and $ I , 6.3 it follows from this that
szcx:> s, (A':)E FI({ (EX?) S(x:)))
which is
s,(x:)
(2)
+
s,(x:)E FI({ (EX?)S O } ) .
From (2) and Lemmas 2.5, 2.6 we easily deduce that
(x:)[X:EX:
-
(-*
-
S(X31 4 (EX?)(S(A'38c (A',")[C(X,Z ; x:) -+ S(X22)I)
q{cJ=3sm))Y
60
which since (EX:)(X?)[X,’eX? * S(X:)J E 7’gives
(3)
[2J
FOUNDATIONAL STUDIES
(EX:)
{rn)( X 3 fC(Z X:) lk
;
--+
-
s(xt)l} E q { ( E x ; ) s ( X : ) } ) .
-
Relying on Theorem 1.5 we deduce from (3) that the set {(EX,)S(X:)} is inconsistent. Thus assuming consistency of this set entails the failure of (l), q.e.d.
2.8. We can express Theorem 2.7 as follows: in the system of logic it is impossible to prove a theorem which says that if an arbitrary set A is Dedekind finite, then the class of its subsets is also Dedekind finite (provided that the axiom of infinity is consistent). Analagous questions related to the axiomatic system of set theory were posed by Tarski ([l], p. 94) and Fraenkel ([2], p. 321 and [3], p. 36). The implication S2(X:) + S,(X:) is also mentioned by Russell and Whitehead ([l], *124.58) who pointed out that if it were a thesis then one could prove that every Dedekind finite (irreflexive) set is finite in their sense (inductive). In connection with this problem see also Theorem 3.1 below. Theorem 2.7 seems to demonstrate the superiority of the notion of finiteness, as defined by Russell and Whitehead [l], *120.02, to the notion of “irreflexiveness”, i.e. the notion of finiteness in the sense of Dedekind at least when we rely on a system of logic such as that used in the present paper. Indeed, regarding finiteness as inductiveness we can easily develop in our system of logic the whole theory of finite sets (cf. for example Tarski [I]). However if we wished to define finiteness in the manner proposed by Dcdekind then we could not even prove such a simple theorem as that which says that the class of subsets of a finite set is finite (cf. Tarski [I], P. 92). In connection with Theorem 2.7 it is maybe worth calling attention to the assumption of consistency of the set {(EX:)S(Xf)} which plays such an important role in the proof of this theorem. At first glance it may seem rather unnatural that a theorem about the mutual independence of two definitions of finiteness should be proved under an assumption about the consistency of the axiom of infinity. It is clear however that, for the correctness of Theorem 2.7 as we formulated it above, assuming the consistency of the axiom of infinityis essential. This simply follows from the fact that if { (EX:)S(X:)} were inconsistent, then every propositional function, and __ thus in particular S2(X:) + S , (X:) would belong to the set FI( { ( E X ? ) S ( X ? ) } ) . Somewhat less obvious is the fact that the theorem (1)
S2(X?)-+ S, (X:)Z T
121
ON THE INDEPENDENCE OF DEFINITIONS OF FINITENESS
61
turns out to be equivalent to the consistency of the axiom of infinity. In one direction assuming the consistency of the axiom of infinity implies (1) as follows immediately from Theorem 2.7. In the other direction (1) yields the consistency of the set
m)
(2) Fl( { ( m > S , ( m c?c (cf. Tarski [2], Theorem 3b), whence we deduce that the set {(EXf)S(X:)}, which is contained in the set (2) by Lemma 2.3 b), is also consistent. Thus the question as to whether the expression (1) is correct or not turnsout to be equivalent to the axiom of infinity, although in (1) there is apparently no reference to the admissibility, inadmissibility respectively, of the notior; of an infinite set. From this it is clear that in trying to solve the question (1) we meet exactly the same difficulties that one meets in trying to carry through u strict proof of the consistency of infinitistic mathematics. On the basis of the theorems of Godel ([l], Th. XI) we can even state that the question, whether (1) is correct or not, cannot be answered in the usual system of metamathematics in which we work only with variables of finite types (if we accept that this system is consistent). We could carry through the proof of (1) only by equipping the system of metamathematics with variables of transfinite types, because in such a system one can prove the consistency of the set {(EX:)S(X:)} (cf. Tarski [3], pp. 401 and 402). By the same token in this enriched system of metamathematics one can prove the theorem which arises from Theorem 2.7 by omitting the assumption that the axiom of infinity is consistent. Hence we see that proving the expression (I), or strengthening Theorem 2.7 by omitting the hypothesis, requires extremely powerful logical tools. Yet the proofs of Theorem 2.7 and of the theorem i f the set {(EX:)S(X:)} is Consistent, then S,(X:) -+ S , ( X : ) E T can be carried out completely elementarily, because the reasoning carried out in the present paper can be formalized without difficulty using only variables of the lowest logical types, and even in the arithmetic of natural numbers. Finally let us observe that Theorem 2.7 can be strengthened as follows:
if { ( E x ~ ) iss a~consistent } set, then S2(Xf)+ S,(Xf)EFl({(EX:)S(X:)}) for p = 2 , 3 , 4 , ...
For the proof it suffices to point out that
(X,P)[S2(X~)-'S1(Xf)]-+(X:)[Sz(X:) S,(X:)]E T forp = 2 , 3 , 4 , ... -+
62
FOUNDATIONAL STUDIES
and apply Theorem 2.7. 3. A simple corollary of Theorem 2.7 is the following
-
3.1. THEOREM. Ifthe set {(EX:)S(X:)} is consisteni then S,(X:) + S(X:)
E FI({(N) SV:) 1). P r o o f . If
s,
(X:> S ( X 3 E "{(EX:)S(X:)}) -+
then on the strenflh of Lemma 2.3 b) would hold Sz(x:>
-+
s,(X3E N ( { ( W )Sex:>>),
which in light of Theorem 3.7 is inconsistent with the assumption that the axiom of infinity is consistent. Theorem 3.1 says that in the system of logic considered here Dedekind's definition of finiteness is not equivalent to the usual one. From this theorem it is easy to conclude that the axiom of choice is independent of the axioms of logic. For brevity we take 3.2. apg(Xf+z)((Xf+l)[Xf+1&Xf+2 -+ (EXf) (Xf&Xf+')j & & (xf+l)(X~+')[Xy+'&X~+z &x,p+l&xf+z & (Exf)(X{&x:+'&
& Xf&X,p+ 1) + xy+'IdX;+ '1 -+
-.+
(EX{+1) (X;+ 1) {X;+ 1,xy+2
(EXf)(Xf)[X;&Xf+l & X!&X5f1 ++ XfZdX3)).
ap
is a formalization of the axiom of choice for sets of type p Definition 3.2 we immediately deduce
+I.
From
3.3. LEMMA. 8' is thefirst increase in type of the proposhion 8'.
As is well known, from the axiom of choice it follows that every Dedekind finite set is also finite in the usual sense; more precisely the following lemma holds : 3.4. LEMMA. 8'
-+
(X:) [Sz(X:)-,S ( X : ) ] E T .
The proof of this lemma follows from the formalization of Whitehead's and Russell's argument [I], *124.56. From Lemma 3.4 we immediately deduce 3.5. LEMMA. r f the set {(EX:)s(x:)} is consistent then 8'
S(Xf)})and '8 E T.
E FI('((EX:)
Otherwise we could conclude from Lemma 3.4 that S z ( X ? ) which contradicts Theorem 3.1.
E F/({ (EXf)s(x,2)))
-+
S(X:.)
I21
ON THE INDEPENDENCE OF DEFINITIONS OF FINITENESS
63
From Lemmas 3.5, 3.3, and Q 1, 6.2 we can further deduce: 3.6. THEOREM. If the set {(EX:)S(X:)} is consistent then 3' Z T.
This theorem states the independence of the axiom of choice from the axioms of logic. Arguing as in the proof of Theorem 2.7 we could even prove that 8' Z FI(((EXi)s(X:))) (assuming that the axioms of infinity is consistent). The same remarks we made in connection with Theorem 2.7 can also be made about the role of the consistency of the axiom of infinity in Theorems 2.1 and 3.6. Since ap + 8' E T for p = 1 ,2, ... as one may easily check, as a further corollary of Theorem 3.6 we obtain: dp Z T for p = 1,2, ..., and 8'Z FI({(EX:)S(X:)})for p = 1,2, 3 , ... (assuming the consistency of the set {(EX:)S(Xf)}). 4. The method of relativization of quantifiers can be applied to a number of other problems similar to those considered in Sections 2 and 3. In particular this method turns out to be strong enough to handle the question of the mutual independence of a sequence of definitions of finiteness put forward by Tarski ([I], p. 94). Applying methods close to those applied in the present paper one can show that all these definitions are inequivalent and all are weaker than the usual definition. This state of affairs might suggest that the usual definition of finiteness is the strongest property characterizing the notion. However deeper investigation shows that this is not SO. More precisely the following theorem holds: for each propositional function u such that Fr(a) = { X I } , whichfurfils the conditions
(Xp'+l)[X,!+l~X:- ( X d + l I d X : v X,!+,ldX: v ... v Xpl+lldX,!)l +aeT ( p = 1 , 2 , ...) it is possible to construct a propositionalfunction /3 such that Fr@) = { X : } (I)
and
(2) (3)
( X : + , ) [X;+,&X:
-
(X;+, IdX: v X j + , IdX: v
... v X j + lIdXj)]
- - + B E T ( p = 1,2,-..) / ~ - + u E T ,U - ~ B E T .
Thus in particular taking for u the function S(X:) we obtain a function /3 expressing a certain property f of sets. From (2) every set contJining p elements ( p being any of 1 , 2 , 3, ...) has the property f, but as cne Sees from (3) f is not equivalent (in the system of logic considered here) to the notion of inductiveness, being in fact stronger. These theorems will be published elsewhere, here we will discuss briefly the possibility of extending the methods and results of the present paper
64
FOUNDATIONAL STUDIES
PI
to systems of the foundations of mathematics other than the one considered above. Our main concern is with transferring these results to the realm of axiomatic set theory. It turns out that amongst the systems of axioms for set theory known at present are some to which our methods apply after suitable modification, and also some for which these methods yield no results. To the first group belong systems of axioms which-roughry speaking-do not exclude the existence of infinite sets whose elements are not sets. In particular this includes the original axioms of Zermelo ([I], pp. 262-267) and also various more precise versions of this set of axioms such as the system given by Skolem [I], 0 1 and Quine [I]. Although these systems contain no assump tion about the existence of infinitely many non-sets, it is not hard to show that adjoining such an assumption to the system does not lead to inconsistency (cf. Quine [I], remarks on Def. r6, p. 48). Further we can enrich these systems with the so-called axiom of replacement (cf. for example Fraenkel [2], p. 309). The proofs of the inequivalence of various definitions of finiteness and also the proofs of the independence of the axiom of choice in these systems rest on the same idea as the proofs given in this section. Logical relationships expressible in the language of set theory are invariant under a one-one map of the set M of all objects which are not sets into itself. Speakingsomewhat more precisely, if we introduce a propositional function O(x) which says that there exists a subset N of M containing almost‘all elements of M such that the set denoted by x is invariant under every permutation of M which fixes each element of M-N, and if we then relativize quantifiers to O(x), then all theses of set theory remain true propositions (theses) of set theory after relativization, while a proposition such as the axiom of choice becomes the negation of a thesis. The main difficulty one meeks in carrying out this plan lies in formulating precisely the propositional function e(x) in terms of the axiomatic theory of sets. This turns on faithfully expressing the condition “ x is invariant under a permutation of the set M . Here we cannot go into the details of the construction of this propositional function. It rests on the theory of ordinal numbers which can be developed within axiomatic set theory (cf. von Neumann [I] and [2], pp. 710-7211 and further on the possibility of numbering (within set theory itself) all sets by ordinal numbers. In this numbering objects which are not sets receive the number 0, and a set composed of objects whose associated ordinals are < E receives at most E. This unique theory of logical types within the theory of sets is due to Mirimanoff [I] and as yet has not been fully
121
O N T H E INDEPENDENCE OF DEFINITIONS OF FINITENESS
65
expounded in the literature. Zermelo ([2], $ 3 ) and Tarski ([3], p. 397, footnote 106) make some remarks on this theme. From the above sketch the reader will easily notice that the method referred to is closely related to the papers of Fraenkel on the independence of the axiom of choice (cf. especially his paper [I] and also [3] and [4]). Essentially the idea of proving the independence of the axiom of choice by investigating mappings of the set of individuals into itself is due to Fraenkel. However, Fraenkel’s realization of this idea is insufficient to say the least. Amongst various methodological obscurities the formulation, of the notion of invariance of a set with respect to a mapping of the set of all non-sets into itself, as given by Fraenkel and later modified b y his student Merzbach ([l], pp. 33-38), contains only the intuition behind this notion and is completely inadequate for formalization. As a result Fraenkel’s whole argument has only the value of intuitive hints. The argument sketched above arose as a result of attempts made by A. Lindenbaum and the author of the present paper to make Fraenkel’s proof precise. This led not only to the elimination of the obscurities from Fraenkel’s argument but also to its application to other axiom systems logically more correct than Fraenkel’s rather incorrect system and having the axiom of replacement besides. Besides this it was possible to obtain results going beyond the theorems of Fraenkel such as establishing what implications hold between various definitions of finiteness. Applying a similar idea to a system of logic we also arrived at the results to which the present subsection is devoted. It is also worth pointing out the following fact: the axioms of set theory together with the axiom of replacement allow us to prove the existence of very large powers, incomparably greater than those it is possible to construct in a system of logic. One might feel that the method of proof of the independence of the axiom of choice sketched above is independent of what powers can be constructed in the system considered. However, after closer study this problem turns o u t to be a good deal more intricate, because, as Tarski showed, when the axioms are augmented by a suitably constructed proposition stating the existence of very large powers the axiom of choice ceases to be an independent proposition (see Tarski [4], pp. 85 and 86). The above remarks apply only to systems of axioms in which the existence of non-sets does not lead to inconsistency. To other systems, in which one can prove that the only object not containing any element is the empty set, o u r methods do not apply. This includes the systems of Fraenkel (121, 9 16), von Neumann [2], Robinson [ I ] and Bernays [I]. Zermelo ([2], pp. 38 and 45) says that such systems are less suitable as a basis for applying
66
FOUNDATIONAL STUDIES
PI
set theory to mathematics than systems which admit the existence of nonsets. However the metamathematical investigation of these systems is considerably more interesting and considerably more difficult than the investigation of systems in which non-sets occur. In the latter systems problems, which are so deep in another setting, such as the existence of undecidable propositions or the independence of the axiom of choice, reduce to facts which are essentially rather trivial. References Papers cited i n the text nre denoted by the author’s name and number according to the list below.
B e r n a y s, Paul [I], A system of axiomatic set theory, part I , Journal of Symb. Logic 2 (1937), pp. 65-17. C a r n a p. Rudolf [I], Logische Syntax der Sprache (1934). and B a c h m a n n, Friedrich [I], Uber Extremalaxiome, Erkenntnis 6 (1936), pp.
-
166-188.
C h w i s t e k, Leon [I], Granice nauki (1936). D e d e k i n d, Richard [I], Was sind und was sollen die Zahlen, fifth edition (1.923). F r a e n k e I, Adolf [I], Uber den Begr’i “definit” und die Unabhiingigkeit des Auswahlaxioms, Sitzungsberichte d . Preuss. Akad. d . Wiss., Phys. Math. KI. (1922), pp. 253-257. - [2], Einleitung in die Mengenlehre, fourth edition (1928). - 131, Sur I’axionte du choix, L‘Enseignement Math. (1935), pp. 32-51. - [4], Uber eine abgeschwachte Fassung des Ausnwhlaxioms, Journal o f Symb. Logic 2 (1937), pp. 1-25. G o d e I, Kurt [I], Uber formal unentscheidhare Satze der Principio Mathematicu und verwandrer Systeme I, Monatshefte fur Math. u. Phys. 38 (1931), pp. 173- 198. H e r b r a n d, Jacques [I]. Rerherrhes sur la thiorie de la dPmottstration,Travaux de la Soc. des Sci. et des Lettres de Varsovie 33 (1930). K u r a t o w s k i, Kazimierz [I], Sur la notion de I‘ordre dun., /a rhPnrir des ensembles, Fund. Math. 2 (1920), pp. 161-171. L i n d e n b a u rn, Adolf and T a r s k i, Alfred [I], ilher die Beschrunktheit tler A.usdrucksmittel deduktiver Theorien, Ergebnisse eines math. Koll., Book 7 (l934), pp. 15-22. - [2], Communication sur les recherches de la thiorie des ensembles, C. R. des Gances d e la SOC.des Sci. et des Lettres d e Varsovie 19 (1926). Classe I l l , pp. 209-330. M e r z b a c h, Julius [I], Bemerkungefc zur Axiumatik der Mengenlehre, Marburger 1nauguraldiss. ( 1925). M i r i rn a n o f f, Dimitry [I], Remarques sur /a thiorie des ensemh1e.s et les antinomies rantoriennes I , L’Enseignement Math. 19 (1919), pp. 209-217. N e u m a n n, Johann voi: [I]. Zur Einfuhrung der transfiniten Zahlen, A& Litt. at Went. Univ. Hung. I (1923), pp. 199-208. - [2], Die Axiomafisierung der Mengenlehre, Math. Zeitschrift 27 (1928), pp. 669-752. Q u i n e, W. V. [I], Set theoretic foundation fur Logic, Journal of Syrnb. Logic 1 (1936), pp. 45-57.
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ON THE INDEPENDENCE OF DEFINITIONS OF FINITENESS
67
R o b i n s o n, Raphael M. [I], The theory of classes; a modification of von Neumann’s System, Journal of Symb. Logic 2 (1937), pp. 26-36.
R u s s e 1 1, Bertrand and W h i t e h e a d, Alfred North [I], Principiu Mathematics,
Vol. 2, second edition (1927). S k o 1 e m, Thoralf [I], ober einige Grundlagenfragen der Mathematik, Skrifter utgitt av Det Norski-Videnskaps-Akademi i Oslo, I. Mat.-Nat. K1. 4 (1929). T a r s k i, Alfred [l], Sur les ensemblesfinis, Fund. Math. 6 (1924), pp. 45-95. - [2], Einige Betrachtungen iiber die Begriffe der w - Widerspruchsfreiheit und w- Vollstandigkeit, Monatshefte fur Math. und Phys. 40 (1933), pp. 97-112. - 131, Der Wahrheitsbegriffin den formalisierten Sprachen, Studia Philosophica 1 (1 936), pp. 261-405. - [4], Uber unerreichbare Kardinalzahlen, Fund.Math. 30 (1938), pp. 68-89. - [5], Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften I, Monatshefte fur Math. und Phys. 37 (1930), pp. 361-404. [6], Z badaii metodologicznych nad definiowalnoiciq termindw, Przeglqd Filozokzny 37 (1934), pp. 438-460. W i e n e r, Norbert [I], A simplification of the logic of relations, Proc. of the Cambridge Phil. SOC. 17 (1912-1914), pp. 387-390. Z e r m e 1 0, Ernst [I], Untersuchungen iiber die Grundlagen der Mengenlehre I, Math. Ann. 65 (1908), pp. 261-281. - [2] uber Grenzzahlen und Mengenbereiche, Fund. Math. 16 (1930). pp. 29-47.
-
On some universal relations by
A. M o s t o w s k i (Warszawa) (Translated from German original h e r geirisse tmherselle Relationetr by M.J. Maczynski)
By a relu!ion we understand every set I' of ordered pairs (x, y ) , The set Irl consisting of the left domain and the right domain of all pairs (x, y) E r is called t h e j e l d of r. TWOrelations r and s are said to be isonzorpliic, denoted by r s, if there is a functionfmapping 11'1 in a one-to-one manner onto Is1 in such a way that the conditions (x, y> E r and (f(.~),.f(y)) E s are equivalent for all x, y E 11'1. For every relation r a n d arbitrary sets m. n c 11'1 we put
-
r, =
E
(I-. Y >
[y ~ m ]r , .
r" =
E
[x E I I ] . r ,
r;, =
(I.",)".
<x.Y )
A relation I' is said to be universal for the system 6 of relations if I' E 6 and for every s E 6 there exists a set m c Irl such that s r.: Here we would like to announce the existence of universal relations for some systems of relations with a countable field, namely for the following systems: (53) the system of transitive and antireflexive relations I' with li.1 < Xo: (2)the system of transitive and reflexive relations r with < No: (2*)the system of relations r E 2 satisfying the condition
-
(*I
if (x, y) E r and (y, r ) E r then x = y; (%TI) the system of transitive relations r with 171 KO; (m*)the system of relations r E llJz satisfying the condition (*).
80
FOUNDATIONAL STUDIES
[61
basis of R, then clearly B’ = B + ( e ) is an ordered basis of R (in the trivial case of e = 0 this is omitted) and the condition p(B’) = 1 holds. Consequently, we have A = p(B’) = I , which was to be proved. To conclude this section we would like to present concrete examples of rings with a basis of a given’type.
THEOREM 1 .I. If < is a relation which orders a set M into type a, then the set system consisting of I the empty set, 2” all subsets of M of the form n
E [ai = x
i=l x
or ai i x < bi]
constitutes a Boolean ring with a basis of type a, under the operation X . Y as multiplication and the operation X @ Y = ( X - Y ) + ( Y - X ) as addition.
5
2. Isomorphism and homomorphism of rings with an ordered basis
THEOREM 2.1. If R is a Boolean ring with an ordered basis of type a, then a ring S is isomorphic to R if and only if S is a Boolean ring with an ordered basis C of the same type a.
P r o o f. Let S be a ring which is isomorphic to R and letfbe a function which maps R onto S isomorphically. On account of 1.1 a) S is a Boolean ring and from [B] = R it easily follows that S = [f(B)]. Further we have Onon E f ( B ) , since f(0) = 0 and Onon E B. Further it is clear that the inclusions a c b and f ( a ) c f(b) are equivalent, which proves that the set f ( B ) is ordered, clearly in the same manner as B. Hence C = f ( B ) is an ordered basis of S of type a. Accordingly, the condition given in the theorem is necessary. To prove that it is also sufficient, we show that every order-preserving map of the basis B of R onto the basis C of S can be extended to an isomorphism betweenR and S.Namely, we extendfby the requirement
+
where a , ... +a, is the canonical representation of the element a # 0. From Theorem 1.4 it easily follows that the function f thus defined maps the whole ring in a one-to-one manner onto the whole ring S. Further, from (1) and (2) we infer that 0 is the only element of R which is mapped onto the zero of S. It remains to show that both isomorphism properties,
(61
HOOI f \ \ 1 < 1 \ ( ! \
!\
81
ITH A N ORDERED BASIS
are satisfied for all a, b E R. By (1) these formulas hold when one of the elements a, b is equal to 0. If b E B and a, ... +a, is the canonical representation of a, then the canonical representation of a+b is
+
either b+a,+
... +an or
a,+
... + a i - ,
... f a ,
+ ~ i + ~ +
depending on whether n o aj (f= 1, 2 , ... , n) is equal to b or we have b = a i . In view of (2) it follows that (3) is valid for b E B . By simple induction we show that this also holds when b is of the form b, +b,+ ... +b, (b,, ... ..., b, E B ) . In virtue of Theorem 1.4, property (3) holds for all a , b E R . SinceB is ordered, for arbitrary x , y E B we have either x c y or y c x , and each of these inclusions implies the inclusionf(x) c f ( y ) orf(j) c f ( x ) . This implies the validity of (4) for a, b E B. Now if a, b are arbitrary elements of R and a, ... +a,, and b, ... +b,, respectively, are their canonical representations, then on account of (3) we have f(a. b)
+
~m
=
i=l j = l
+
f(ai.bj). Since (4) holds for the basis elements, we conclude
further that
which was to be proved
THEOREM 2.2. I f a ring S is a homomorphic image of a Boolean ring R with an ordered basis B, then there exists a set C c B such that the rings S and [C] are isomorphic.
P r o o f. Let f be a function that maps R homomorphically onto S. We put (1)
K(a) = B . E [ f ( a ) = f(b)] b
for a E B
and using the axiom of choice we pick up from each set K(a) an element a*
(2)
a* E K(a)
for a E B .
Since Onon E B, we infer from (1) and (2) that 0 does not belong to the ordered subset C of B :
(3)
C=
E [uEB]. U*
Consequently, C is an ordered basis of the ring [C].If x = f ( u ) , y = f ( b )
82
FOUNDATIOYAL STUDIES
161
are elements of f ( B ) ( a , b E B) such that x c y and x # .y, then we have also a* c b* and a* # b*. Otherwise we would have b* c a* since C is ordered, and we would obtain from ( I ) and (2) the inclusion y =f(b) = f(b*) c ,f(a*) = f ( a ) = x . Since S is a Boolean ring, this would imply x = y, a contradiction. Now if a* c b" and a" # b* hold, then by (1) and (2) we obtain x = f(a) = f(a*) c f(b*) = f(b) = y and x f y, which proves that the inclusions x = f ( a ) c f ( b ) = y and a* c b* are equivalent. As can easily be shown, this implies that the sets C andf(5) are similarly ordered; consequently, the rings [C] and Lf(B)] are isomorphic. The proof will be completed when we show that Lf(B)] = S. The inclusion Lf(B)] c S is obvious. If a is an element of S, then we clearly have a E [f(B)] whenever a # 0. If a # c, then we put a = f ( b ) , where b # 0 and b E R. By Theorem 1.4 we have b = b , + ... +b, for some b , , ..., b, E B and then a =f(b) =f(b,)+ ... +f(b,) E [f(B)].This proves that S c [ , f ( B ) ] . Hence Theorem 2.2 has been proved. R e m a r k 2.3. A more exact analysis of the proof of Theorem 2.2 shows that in Boolean rings with an ordered basis the so-called representation problem is solvable for every ideal.( 2 ,
5 3.
Hereditary atomic rings and rings with a scattered basis
We precede the discussion of rings having a scattered basis by some lemmas which concern general kinds of Boolean rings.
DEFINITION 3.1. a) An element a of a Boolean ring R is said to be an atom of R. denoted by a E At(R), when 0 is the only element of R which is included in a and is different from a. b) A Boolean ring R is called atomic if for every element a # 0 of R there exists a n element b E At(R) such that b c a . c) A Boolean ring is called hereditarily atomic if every homomorphic image of R is atomic. d) A Boolean ring is called atomless if R contains at least two elements and At(R) = 0.
NOW we shall give necessary and sufficient conditions in order that a Boolean ring be hereditarily atomic. For this aim we prove some lemmas.
LEMMA3.2. a ) Evrry lirreditarily atomic ring i s atomic; b) no atomic (*) For this problem see J . v. N e u m a n n and M. H. S t o n e , Fund. Math 25 (1935), p. 353 ff.
161
83
BOOLEAN RINGS WITH AN ORDERED BASIS
ring is atomless; c) e w y ring that is a homomorphic image of a hereditarily atomic ring is itself' liereciitarily atomic. LEMMA 3.3. Let R be a Boolean ring, 0 E A c R and let F be a,function associating with every pair .x, y of elements of A f o r whicli .r c y , x # y , an element F ( x , y ) SUCII that x c F ( x , y ) c y , x # F ( x , y ) # y . Under ihese assumptions there exists a set B c A of type q provided that A contains at least two elements. The proof is straightforward. 3.4. If'no subring of a Boolean ring R is atomless, then none of the LEMMA rings (a), where a E R, contains (I set of type q.
P r o o f. The existence of a subset of ( a ) of type q clearly implies the existence of an atomless subring of R . LEMMA3.5. If R is a Boolean ring and none of the sets ( a ) , d i e r e a contains a set of type q, then every subring of R i s atomic.
E
R,
P r o o f. We suppose that the theorem does not hold. Then in agreement with Definition 3.1 b) there exist a subring S of R and a E S such that the ring S. ( a ) does not contain any atom of S. With the aid of the axiom of choice we conclude hence that there exists a functionf associating with every non-zero element b E ( a ) . S an element f ( b ) such that f ( b ) c b, 0 # , f ( b ) # b. Hence if we put A = S . ( a ) and F ( x , y ) = x v f ( x + y ) for x, ,v E A , x c y , .x # y , then all the assumptions of Lemma 3.3 hold and consequently there exists a set B c ( a ) . S c ( a ) of type q, which was to be proved.
LEMMA3.6. If no subring of a Boolean ring i s atomless, then R is liere&tarily atomic.
P r o o f . We suppose that R is not hereditarily atomic. Then there exists an ideal I in R and an element a of the residue class ring R / I which does not contain any atom of R /I . Consequently, there exists a functionf associating with every non-zero element b c a of R / I an element f ( b ) E R / l such that f ( b ) c b , 0 # f ( b ) # b. Now from every residue class b mod I we pick up an element g(b), where we can make the choice in such a way that g ( 0 ) = 0. Further we put A =
E
[b E R / I , b c a]
g(b)
Then
(2)
A c R ,
QEA,
A 2 2
84
FOUNDATlONAL STUDIES
161
Now let (3) x, J'E A , .Y c J*, .v # y and let x*, y* be the residue classes mod I of the elements x, y. We put F ( x , y ) = x + g ( f ( x * + y Q ) )* ( X f ? ' ) . From (3) and (4) it follows that x c F(x,y) c y. (5) From (3) and ( I ) we infer that x* c y*, x* = .v*. Consequently, we have x * + y * # 0, i.e. 0 # f(X* + y * ) # X * +y*, f(X* + y * ) C .V*+ J'". (6) Now if we had F ( x , y) = .Y, then we would have g ( f ( x * + y * ) ) . ( x + y ) = 0, from which by passing to residue classes we would obtain f ( x * + +y*)(x* + y * ) = 0, i.e. by (6j it would follow f ( x * +y*) = 0, hence by (6) we have (4)
(7) m , Y ) # x. If we had F(x, y) = y , then we would have x f y = g ( f ( x * + y * ) ) . ( x + y ) , from which by passing to residue classes we would obtain the equality x * + y * = f ( x * + y * ) . (x" + y * ) , i.e. x* + y * c f(x*+.y*). Since this contradicts (6), we have
(8)
F(x,y) # Y . From (2), (9, (7), (8) it follows by Lemma 3.3 that there exists a set B c R of type 71. Hence every ring (b), where b E B, contains a subset of type q, which shows by Lemma 3.4 that at least one subset of R is not atomless, which was to be proved. To formulate the next lemma we need the following definition: DEFINITION 3.7. For every Boolean ring R we put a) At,(R) = At(R); b) At€(R) is the set of those elements of R which are carried by the homomorphism R -+ R / ( C A t t ( R ) ) onto atoms of the image ring; t<E
c) M R ) =
(C At,(R));
d) x ( R ) is the smallest ordinal number, I. for which At,(R) = 0 . ( 3 ) (') Definition 3.7 assumes n transparent meaning when one takes into account the results of Stone (Trans. of the Amer. Math. SOC.41 (1937), p. 375 ff.) and interpretsevery Boolean ring as a ring of sets which are biconipact and simultaneously open and closed in some topological space S = S(R). Then the number x ( R ) can be characterized as the order of the last derivative ot'the space S.
161
85
BOOLEAN RINGS WITH AN ORDERED BASIS
R e m a r k 3.8. The existence of the number x(R) can be ascertained as follows: if R as a Boolean ring, t < 5 two ordinal numbers and a E Ata(R), then the' homomorphism R + R/A,(R) carries the element a onto 0. On the other hand, the same homomorphism does not carry any element of At,@) onto 0, which shows that At,(R) Atc@) = 0 for t# c. With the aid of the axiom of choice it follows hence that at least one of the sets At,(R) must be empty.
-
LEMMA3.9. If R is a Boolean ring and t an ordinal number > 0, then for every a E A,(R), a # 0, there exist an ordinal number t < 6 and elements bl , ..., b, E At,(R),
cl,
..., c,
E
c At,(R) such that a
t . It follows that A - { b } = b , because otherwise we would have A - {b} e A- {a}. Thus a e b for every a E A - {b}. The end of the proof is the same as above. From this lemma we easily obtain the following ') It follows from this definition that every set whose elements are not sets is normal. 6) This axiom states that there is no sequence Z ~ , Z ~ , . of . . sets such that Zn+ieZn for n=1,2, See, e. g., E. Zermelo, Fund. Math. 16, 1930, p. 31.
...
[lo], 140
AXIOM OF CHOICE FOR FINITE SETS
95
Lemma 2. I f there is a choice function for a v e y class of normal sets with n elements, then [ n ] is true. Proof. Let K be an arbitrary clms of sets of the power n. Divide X in two parts El and Kay including to Xlthose A P X for which the alternative (i) of lemma 1 holds and to X a the remaining A's. Lemma 1 enables us to distinguish a particular element b in any set of the class Kl.The principle of choice is thus true for this class. For every A E Kzthe class A* of all differences A - { a } where a runs over A is a normal set with n elements. Accordingly to our supposition we may distinguish a particular element W(A*) of this class. Denoting by @ ( A )the unique element of A -W( A*), we have @(A)B A, what proves that @ is a choice-function for the class 4. Thus there are choice-functions for claslaes XIand I&and ., consequently there is such a fsnction for their Bum, i.e., for the class X,q. e. d. 2. For every set X denote by P(X) the class of all subsets of X and put m
O ( X ) = P , ( X ) + P , ( X ) + P , ( X ) + .*. Theelementsof O ( X )willbecalledobjects with t h e b a s e x . 9
If T e P + ( X ) - ~ P b ( X ) , we shall say that T is of degree q+1; b=o
if Z'eP,(X), we say that the degree of IT is 0. Len8ma 3. I f IT is an objed with the ba8e X of &?pee q+1, and U P I T , then U P O ( X )and the degree of 17 ie O, there are t < q elements V1,Va,..-,Vr such that V , P X and VlB V, e a V , e U. Hence, if U were an element of 2, we would have
...
V , E V * E . .E. V ~ E U E TV,, a X and T r X ,
which is imposeible, because X is a normalbt. If l3 were of degree 0, we would h a w U P IT and U P X, P ! P X, which is again imposeible.
96
[lo], 141
FOUNDATIONAL STUDIES
Two above lemmas enable us to characterize the objects with the normal base in the following way: (i) Objects of degree 0 are identical with the elements of X; (ii) Objects of degree q + l are identical with sets of the object8 of degree less than p + l , one at least of these objects having exactly the degree q 6 ; . Emy proofs of ihese assumptions may be omitted here. The characterisation given in (i) and (ii) is more preferable than the primitive,one, because it makes possible proofs and definitions by induction. Indeed, from (i) and (ii) follows that in order to prove that every object with a normal baw X has a given property P it is sufficient to show that loevery element of X has this property and 2O if all elements of a set A have the property P, then A has this property too. Analogous remmks apply to definitions by induction. It is well to note that (ii)is, in general, false for objects with a non normal base 1). 8. The set {1,2, n] of first n integers will, for brevity, be denoted by (%). A and B being any two sets of the same power, we denote by A 2B the clam of all one to one mappings of A on B. If A has rc elements (% finite), then A 2 A is the group of all permutations of A, and is isomorph with &=(%) : (n). (%) 2A is the clae~of all one to one functions defined on (n)and talting on values from A. I shall uae letters f,g,h, ... to denote functions of the class A :A and letters (p,p,x,... to denote functions of the o h s
...,
( n )2tm).
Let A and B be two normal sets of the same power and f a function of the class A Z B . For any object X with the base A I shall define its image f ( X )by induction on X: (i) if X E A, f ( X ) denotes the value of f for the argument X ; (ii) if f( P)is defined for any element P of X,then f(X)- [PeXb i.e. the set of all f( P) such that P e X. It follows eaey that f ( X ) is an object with the brtee B.
-A
s) Our ,objeota of degree q" form the Bame aa the qtb layer (.,Schicht") eonsidered by Zermelp, loo. cit. 5 ) , p. 36. The difference is only this, that we do not auppoae that the lowest layer is built up from elements which are not sets. ') Example: X=(o,b,(o.b)). The degree of @.b) is 0, though it is a set of two objecta of degree 0.
[lo], 142
97
AXIOM OF CHOICE FOR FINITE SETS
By induction we show that, if A, B and C are normal sets of the same power, and if f e A Z B, g e B 2 C and X E O(A),then
l ( X )= x 9. f - - ' ( f ( X )= ) x; It follow8 at once from these formulas that the set of all functions f A 2 A for which f ( X ) = X , is a group. This group will be cdled the s y m m e t r y g r o u p of X and denoted by G ( X ) . Lemnza 6,I f A is a normal set and X , Y two objects with the baee A, then the symmetry group of the ordered pair < X ,p> 0) is ctmt a W in G ( 1 ) and G ( P ) . Proof. If f B A 2 A, then f ( < X , Y > )= < f ( X ) ,f(P)>.Hence, if f ( < X , Y > )=<X,p>, we must have f (X) = X and f ( Y) = Y, i. e., f belongs t o G(X)and G ( P ) . Lemma 6. I f A and B are two normal sets of the same power, X e O ( A )and Q e A 2B,then the symmetry group of q ( X ) is cpcf(X)v-l. This follows from equivalences:
f ( 9 ( X ) )= f q ( X ) ;
-
-
(fa(x)=v(x)j = {Q-'fr(x)=X) = k ' f v a(x)}* (f evQ(X)v-')*
(fe@(o(X)))
4. Let A be any normal set and a an element of A. The symmetry group of the object has of coursea fixpoint a. Thus, if we are able to choose an element from a normal set A, we can &o construct an object X with the base A whose symmetry group has at least one fixpoint. We shall now show that, for finite A the converse theorem is true: Lemma 7. l'h<ereis a funclion B ( A , X )defined for finite normal 8& A and X r o ( A ) 8 w h that if the symmekvy group of X has fkpoints, then e ( A , X ) belong8 to A. Proof. Suppose that A has n elements. The set O ( ( n ) ) of objects with the baee ( I Z ) may, of come, be well ordered. Let (1) B1,B8,B97'.* be a sequence formed of all elements of O((n)). 8 ) fg denote8 the composed function fg(s)=f(g(o));f-Iis the function inverse to f, and 1 the identical function l(z)=z. 9) The ordered pair ie defined a8 { ( A ) .( A , B ) } .Generally, we denote by O a d {t e m > = {p(t) E 3)for e v e y t , then m=(p(o). It is further easy to show that K E C E , for &q and that from y E (E E K: follows y E 9 s . The least property of Kt is wiled transitivity. We may now define the new meaning of ,,setuc' and ,,idividolaEs". As individmls in the new sense we assume the elements of KOand 8% sets in the new sense the elements of any K: with 5>0. 14. Before going further we shall still introduce a convenient terminology.).1 Let I be any concept definable in terms of the primitive concepts ,,d', ,,indi~,id~aZcc and ,,Setc'. If we replace these primitive concepts by their new memiup, we obtain a new concept 3 which is, in general, different from E. This new concept will be called ,,pseudo Z''. We can thus speak about ,,pseudo-inclusion", ,,pseudoproduct'' etc. Pseudo-individuals reap. pseudo-sets mean the same as individuals reap. sets in the new sense.
-
)'l This terminology has been introduced by K. G o d e l in his lectures 8t the University of Vienna in 1937.
",
1%
AXIOM OF CHOICE FOR FINITE SETS
109
If the pseudo-concept E* related to pseudo-sets or paeudoindividuals a,y,z, coincides with the primitive concept E related to 5,y,4 we shall say that E is an absolute concept. Examples thereof me given in the following Leen9na 11. ErdEotdng eoncepte are absdocte: (i) inclusion; (ii) p~oductof two setq (iii) the rdation between two sets: their produs h d y k &m~~otS~(k=O,1,2,3). We have now to prove that we get true propoeitions if we r e p h in the axiom of set theory aJI concepts by the correspondent putlo-concepts. It will be sufficient to outline this theorem for two axioms, since its detailed proof h a been given elaewhere'o).
...,
...
,...
18. One of the axioms states that if x isa set, there is another set y=P(a) (clam of subsets of a) such that, for any 1, t e $ if and only if t is 8 subset of a. In order to show that this axiom remains valid for the new ~81~8 of8 the primitive concepts, we.must show that if x is a pseudoset, there is another pseudo-set y such that if 1 is any pseudo-individud or pseudo-set, then 1 E y if and only if t is pseudo-included in a. Let us assume as y the clam of those subsets of x which are pseudo-sets themselves. Jt is plain that if t is a pseudo-individual or (I pseudo-set, then t E y if and only if t is included or (what is by lemma 11 (i) the same) pseudo-included in o. It remains to show that y is a pseudo-set. Assume that a e Kt. Every subset of m being an element of Akt+1, we follow that y C El+, and consequently y E M ~ + z . By our hypothesis there is further a number p such that if 0" e a" and Q , = v ~ = . . . = ~ 1, ~ =then +)=x. Suppose that Q e Q" and ~ l = ~ z = . . . = I~ f q tCz, = lthen , (p(t)Cq(x)=xand vice versa. By lemma 10 (i) (p(t) is a pseudo set if and only if t is one. Thus {t L y} P (q(t) B y}, i. e., 'p(y)=y, which proves that y satisfie8 the invariance-condition (2) and is consequently a pseudo-set m). A. Mostowski, Fund. Math. 82, 1939. p. 221-262. The relation .,y ir tku cluw of alttub-& of a? is not ahsolute in the mum htrodueed in 14. It follows that if we only know about a domain D of nets that it contains P ( z ) with every of its element z,we cannot still be sure that the ariom ,,for m y set 2 t h e i s the elare of it, wb-8et.s" relativited to the domain D is valid. "his is one point which I do not understand in the works of Fraenkel about the independence of the axiom of choice. Sea footnote') and the litteratnre quoted in thin paper. '0)
'1)
110
FOUNDATIONAL STUDIES
[lo], 155
16. As the second axiom for which our theorem will be proved we choose the axiom of substitution. It states that if x is a set and E(u,t)is any relation between sets or individuals t , u, and if there is for any u E s exactly one t such that E(u,t ) holds, then there is a set y suah that 1 e y if and only if an u e x exists for which the relation f(u,t ) holds. In order t o eliminate the concept of an ,,arbitrary relation" pz) we admit only such Z which may be defined in terms of primitive relations: (31 ,$v4 w", ,,v is an individual", ,,v is a set" and of logical operations: Negation, colzjunction and quantifiers (bounding sets or individuals)
It is well to remark that E,which is involved in the formulation of our axiom, may depend upon another sets or individuals a,b, ...,h which play the r61e of parameters. y is then a function of x and of these parameters. We shall need the following Lemma 12 m). Let E(u,t,a,b, ...,h ) be a relation of the above type and let Z*(u,t,a, b, h) be the corresponding pseudo-relation. Let further u,t,a, b, ...,h be pseudo-sets or pseudo-individuals and (peGo. Then
...,
(4)
E*(U,t, a, b,
"',h) = E*((p(u),Ip(t),(p(a),(p(b), ...,W)).
Proof. The lemma is of course true for the primitive relations (3). If i t is true for relations t' and H, it is also true for relations ,,non-=-'< and ,,Z and H".
It remains to prove that if (4) holds for a relation E, it holds also for the relation
H(u,t,b,...,h) =2' E(u,t,a,b,...,h). a
**) If we wish to retain this concept, we must give to our system of axioms
a larger logical basis (variables of the second type). The proofs of independence are still possible. but must be modified a little, because it is neceesary to relativize to a model not only the primitive concepts of axiomatic eystem but also the logical concepts. *') This lemma is essentially due to A. Tarski and A. L'ndenbaum, Ergebn. eines math. Kolloq., 7, 1934, pp. 15-22.
[lo], 156
AXIOM OF CHOICE FOR FINITE SETS
111
...,
Suppose that pseudo-individuals or pseudo-sets u,t, b, h fulfin H*(u,t,b,...,h). There is then a pseudo-set or a pseudo-individud a such that Z*(u,t,a,b,...,h) and hence by ( 4 )
*: (%w, d t ) ,d a ) ,N ,".,d h ) ) . There is thus a pseudo-individual or a pseudo-set a'=p(a) such that I * ( v ( U ) , ~ ( t ) , a ' , ~..., ( b ) , i. e. fi*(o?(u),dt),d b ) , This proves the implication
m),
M*(U,t,
...,m).
b., h)*H * ( d u ) , v ( t ) , m ,...,m).
Replaring here q by q - - l and u,t,b,...,h reap. by p ( ~ ) , q . ~ ( t ) , r p ( b ) , .,.,q(h) we obtain the converse implication and lemma 12 is proved. We pass now to the axiom of substitution. We have to show that if x is a pseudo-set, a,b, ...,h are pseudo-sets or pseudo-individuals and E(u,t,u,b,...,h) is a relation of the type described above such that for any u E X there is exactly one pseudo-set or pseudoindividual t for which :*(@,@, h), then there is R pseudo-set y such that
...,
( 5 ) t r y if and only if there is an
2cez
for which i*(u,t,u,b,A).
For u E 5 denote by f( u ) the unique t for which E*(u,t, a, b, ...,h), and let y be the set of all j ( u ) where u runs over 2. Since it is obvious that this y satisfies ( 5 ) , we have only to show that y is a pseudo-set. Suppose that IC e Kt . If u E Z , then f(u)is a pseudo-set or pseudoindividual and there are ordinals 7 such that f ( u )eKV. Let C(u) be the least such q, and let 5 be the least ordinal exceeding all ( ( u ) with u E Z. Then f ( u )r Kg for every u E X , i. e., y CKg or y E 61~+,. We now show that y satisfies the invariance-condition. Let q(r),p(a), ...,a( h) be numbers such that (6)
gJ(x)=z, p ( a ) = a ,
. . ., pl(h)=h
for any 4p e G" for which the first q(s),p(e), ...,p( h) terms yk are equal to 1. Let p be the gr.eatest of the numbers q(m),q(a),...,p( h). If is such that q . ~ ~ = ~ ~ = . . . = q ~we ~ =have l , equalities ( 6 ) . I shall show that q(y)=y. In fact
,...,h.)]);
{ t E y ) = Z ( ( U E 5 ) *[Z*(u,t,a,b U
112
[lo], 157
FOUNDATIONAL STUDIES
this is by lemma 12 equivalent to J{(a f II
w)- CE*(~,cp(M4,dW? ...,QW)Jj
or in virtue of (6) to
z’{@ 4 f
- CI*(u,Pr(t),a;,b,...,~)J}= (At) Ylf
Hence (1 E y}aa{p(t) E y}, i. e. y=p(y), which proves that y is really a pseudo-set.
17. It is h o s t obvious that the proposition [a] will become false if we replace the primitive concepts by their new meanings. Indeed, since the axiom of choice for sets of power a is a consequence of [B] (comp. the footnote*)), we follow that if [a]were true in our model, the axiom of choice for sets of the power a would be true too. In virtue of lemma 11 this would mean that for every pseudo-set x whose elements are disjoint sets of the power B? there ie a pseudo-set y such that if z E x, then y z has exactly one element. This consequence is false. The pseudo-set s= {N,,N,,N,, ...) satifdies namely all hypothesies and there is no corresponding pseudo-set y, beoause if y has exactly one element ahin common with i V k (h= 1,2,...), then y doee not &My the invariance-condition. In fact, let Q be my integer. Since f3 has no fixpoints, there is a l t e G such that 4%+1)4=a;&1, and putting
.
Q
...,1,
= [I, 1,
X,
9tlks
we obtain an element
f
1,1,...]
Go such that
q(y)=t=y. Hence y cannot be a pseudo-set.
v,=q8=...=qq=
1 .and
18. In order to accomplish the proof of the theorem I11 we must still show that if z e 2,the proposition [ z ] remains valid in t h e model. In view of lemma 11 and footnote *) this means that for every pseudo-set X whose elements are disjoint sets of the power a, there is a pseudo-set P such that if P E X , the product P.P has exactly one element. Suppose that X is a pseudo-set (7)
X 8 RE,
that every element of X has B elements and that U - V =0 if U S;V and U , V e X .
[lo], 158
113
AXIOM OF CHOICE FOR FINITE SETS
It follows from (7) that there is a positive integer q such that q ( X ) = X for any q e G " for which ~ ) ~ = q * = . . . = q ~Let ~ = lf. be a subgroup of Q" consisting of all (p's for which p,=(p2= =(p,=l. For U , V E X we write U b V if there is a c p e r such that cp(U)=F. Since this relation is of course reflexive, symmetric and transitive, i t induces a decomposition
...
of X into t h e classes of abstraction R r A of -. Thus A is the family of all classes of abstraction and the relation holds between two elements U,V of X if and only if they belong to the same summand R of (8). Applying the axiom of choice (see footnote 17) I select from every R EA a particular element and call it ER. Hence
-
(9)
and consequently (10)
E R E R C X for R e d E R
h a s z elements.
Let H R be the subgroup of r containing all (p's suoh that Q(ER)=ERand let us write U w V if U,V z ER and if there is a cp B HR such that Q( U)=V. We may again decompose ER into a sum of classes of abstraction of w :
ER = &,-I-8 2 -I-
(11)
..+
The number r of these claeaes (which will, in general, be different for different R ) is finite in virtue of (10). Using again the axiom of choice, I select from every #J a particular element Tf and denote by Ej the group of those (peH~ for which q(!Tf) = !l', u). I shall show that Is; has exactly Ind(HRIKj) elements. Indeed, let 1
2
HR= Kj+SKj+6Kj+
(12)
be the decomposition of Yf,
(13) II)
HR
P ...+6Kj
into co-sets. Elements
1
twl),
2
W,)
9 a - 9
K f is, in general, not self-conjugate.
&Tf)
114
[lo], 159
FOUNDATIONAL STUDIES
are all contained in S,, because the relation w holds between them h I and TJ. They are all different, because from 6 ( T ~ ) = 9 ( I jwould ) 1
I
h
h
h
I
follow 8--+9(T,)=Tj or 6-'6 e K,, i. e., 6 E @K,. Hence 8, ha8 a t least p + l = I n d ( H R / K J )elements. On the other hand, if 17 e 81, there must be a q E HR such that Q(Ti)= U. Hence q belongs to 1
I
one of the summands 8Kf of (12). Hence q=&y, where y e XI,and I
I
consequently U=S(y(TJ))=S( TJ).We follow that 8,has no elements different from the elements (13),i. e., 8,has exactly p 1=Ind(HR/E,) elements. Formulas (10)and (11)yield now
+
z = Ind ( H R / & )
+ Ind ( H R / & ) + ... + Ind ( H R / K ~ ) ,
and since z E 2,we follow from the hypothesis made at the beginning of 12 that one a t least K, is equal to H R . This means that in every ER there is at least one U such that Q( U)= U for every Q e HR. Using still once more the axiom of choice,I select from every E R one such U and I call it U R . For every R EA we have therefore (14)
115)
URE E R , Q ( U R ) = U R for q E H R . Define now QR as the set of all
Q( U R ) where Q E
f and put
We shall show that this Y has desired properties. For the first, P is a pseudo-set. Indeed, from (7), (9) and (14) we follow that U R EK , (transitivity of KO. Hence, Q R C K , by lemma 10 (i) and consequently Y C Izg or Y E M ~ + II.f y e f , then ~ ( Q R= ) QR, because Y)(QR) is the set of all yq( OR)where cp E f , and the conditions yq E f and Q E f are equivalent. From this we follow that y( P)=P. Y satisfies thus the invariance-condition, i. e., it is a pseudo-set. It remains to show that if P E X, then P.Y has exactly one
element. Suppose that P e X. There must be a summand R of the decomposition (8) such that P E R , i. c., P - E R . Consequently there i3 a T e r suchthat Q ( E R ) = P Since . U R E E Rwehave , Q( U R ) E Q ] ( E R ) = P ; on the other side cp( UR)E P by definition. This proves that P.Y contains at least one element q( U R ) .
[lo], 160
115
AXIOM OF CHOICE FOR FINITE SETS
We shall now show that this element is unique, i. e., that
W E P.P, then W=(p( UR). Suppose that W E P . P. Since W EP, there is a S E A -such that W E QS and it follows that W has the form y( Us)where fy e f . . Since W E P=Q(ER),we have y( US)E ( ~ ( E Ror)
if
q-%'( US) ER-
(16)
Q
On the other side ~ - ' y (U S )E Q - ~ ~ ( E bys )(14). The sets ER and v - ' y ( E S )are therefore not disjoint. Being both elements of X they must be disjoint or equal. Consequently (p-llp(E~)=E~, which proves that ER Es. But E R e R, ES E 8 and the relation holds never between two elements of different classes of abstraction. I t follows thus R = 8 and (16) gives y( OR)e ~ ( E RBut ) . y( UR)EP(ER) by (14); ( ~ ( E Rand ) y(ER) have therefore an element in commpn and since they are both elements of X, they must be identical. This gives ~ ( E R ) = ~ ( E orRv)- l f y ( E ~ ) = Ei.~ e., , (p-*y e HR. By (15) we obtain now ~ - ' f y ( U R ) = UR,i. e., y( UR)=P)(OR) or W = q ( U R ) . Every We P.P is therefore identical with (p( U R ) , i. e., Pa P has exactly one element. The proof of theorem 111 is thus accomplished.
-
-
19. We shall now draw some consequences of theorem 111. Defthition 6. We shall say that a positive integer n and finite set 2 of such integers satisfy the condition ( M ) if for any decomposition of n into a sum of primes n = P,+P,f...+PS
(1)
there are 8 %on negative integers q,,q2, ...,q8 such that the sum plql+p,q,+ ...+p,q, i s contuined in 2.
Theorem IF', Condition ( M ) is necessary for the implication
[ZI -+ [nl. Proof. It suffices to prove that ( M ) is a consequence of (K). Let us suppose that 91. and 2 satisfy the condition (K), and let (1) be a decomposition of n into a sum of primes. Let q~ be the permutation
,...,P,)(P,+1,P,+2,...,pl+P,)
(1,2
- . (Pl+P,+...+Ps-l+1,pl+P,+...+P~-l+2 *
*
--
,...,n),
and let Q be the cyclic group composed of all powers of 9.The order h of G is equal t o the product of all different p's.
116
FOUNDATIONAL STUDIES
[lo], 161
Since G has, of course, no fixpoints, then there is accordingly t o (9) a subgroup H of G" and a finite number r of proper subgroups Kl,E2, Xrof H such that the sum
...,
Ind (HIIL,)
+ Ind (HIK s) + ...-tInd (H/ILr)
is contained in 2.It follows in particular that the indexes Ind (H/K,) are all finite. rn order to prove the theorem I V it is now sufficient; to show that if KC H C G" and if Ind ( H / K )is finite and greater than 1, then Ind (H/R) is divisible by one of the primes p,, p2,...,p,. Let (2) H = K + q ( 1 ) K + q ( 2 ) K + . . .+ q ( p ) I Z
be the decomposition of H in co-sets of IC ( p + l = Ind (H/K)). Every cpQ is a sequence [q!?,q,!?,~ $ 0...I , where almost all e?:') are equal to the unit 1 of G. Suppose that r# = 1 for j>qi and let q be the greatest of the containing dl numbers ql,qz, ...,p,. Let H* be t'he subgroup of such cp's that p&l=pq+z=...=l and let K* be the common part of H* and R. It follows that y(t),~(z))..,,@')are contained in H*. We shall show that the decomposition of H* into co-oets with respect to K* is
H* = E*
(3)
+cp("IZ* +v@)IC*+ ...+cp(p)K*.
Indeed, if q e H*,then cp E H, and there is an i < p such that ~ e y ( 0 . K or q~=p(oy, where ~ € 9It. follows that qh=cp$ly; for k=1,2, and since qh=cpio=l for k > q , we have also yb=l for k>q, i. e., 1y B H*.Hence y E 3*, and we follow from cp=cp'ntp
...
that
Q P
q.MK*. E* is therefore the sum of co-sets
K*, q m * , cp@)X*,. .
., #P)K*,
and these co-sets are disjoint, because they are contained in the corresponding co-sets E , y(')E, v ( ~ ) K , cp(p)E. Formula (3) is thus proved and we have
. . .,
(4)
Ind ( H * / K * )= p + l = Ind (H1.K).
€T* and K* -may be treated as subgroups of the direct product Q! x Q x ... x G=Gq of order hq. Ind (H*/lK*)is thus a, divisor of hq, i. e., it must be divisible by one at least pf. By (4)we follow
that Ind ( H / E )is also divisible by one at least pf, q. e. d.
[lo], 162
117
AXIOM OF CHOICE FOR FINITE SETS
80. Theorem V. If [Z]+[fl] and ifm i s the greatest of the numbars oeoacring ia 2, t b a < 8m8. This theorem state6 that, for given 2, there is only a finite number of n. such that [Z]-+[fi]. P r o o f . Suppose that [ Z ] - t [ n } and n&S,mx. By the 80 called Bsrtrand's postulateu) there are primes p , q such that nzl SEDS.(eke), e,(y)) f &I.
re&. s e a . S ( 4 d )
= EW,[ze&.ye&. re&.
E
Hence Jp(k, n) is Borelian and its class is finite by (iv), (v), and (vi). is the w. f. formula R(z, y), then J P ( h ,i) = Eryrs[z~e2. geeS.rt&. seDI. R,(h, i)] = E,,lze02. yet%. rfDZ.scDt. (c,(t.,a(r) = 211. 9
Use is made here of the axiom of choice.
If P
132
[141,41
FOUNDATIONAL STUDIES
CPN(R~ , RJ = *~u(q,.ro)(Rr , R3 3 (20 9 vo 3 r, 8) E K. Hence the set E,[%N(R,, R,) z = = YO] is an intersection of the fourdimensional Borel set K whose class is at most F, and of the plane z = % , y = yo . The set E,&(Rr, R,)] is therefore Borelian and its class is F ,
.
.
.
7. We shall apply the theorem to mme particular properties of relations.
(a). Let BO(R) = R is a well ordering relation. The set
0 = Er[rdh.BON(R~)] is known to be an analytic complement and not Borelian." It follows.that there is a model M such that A M and -(R)[BON(R) -= BOM(R)],i.e. (3R)[BOw(R). -BON(R) v -BOM(R) Box@)]. But it is easy to ~ e that e BON(R) 3BOU(R) and hence (3R)[-BON(R) BOM(R)].This means that there is a model M which fulfills all axioms and a relation R with field NOsuch that there are decreasing sequences" . - . n&nzRnl but no such sequence is in M. (b). Let R smor S be defined aa in Principia muthemutica, *151.01. The set
.
.
E&&&
(15)
.
SE&
.R, smor R.1
is analytic and not Borelian. In fact
R, smor R,
E
-
(*)[ZEDS R,
.
E
.
1 -+1 R, = &Z,IRrIRz].
The set E,,[zEDI. R. c 1 -+ 1 R. = &R,IR,] being Borelian, we conclude that the set (15) is anslytie. If this set were Borelian of class a, the set At = E,[r& R, is of type €1 would be Borelian of class a for any ordinal number [ (sincethere are s such that R. is of type [). Hence the set 0 would be expressible as a sum C r < o A c ,the At being Borel sets of class a. This is impossible, since 0 is not a Bore1 set." It results now from our theorem that there is a model M such that A M and two relations R, S such that -(R smorN S E R smoru s>. Since R BmOrM S 3 R morNS , it must be that R smorN S and -R smoru S, i.e., there are oneone relations T such that R = SIT but no such relation" is in M.
.
(c). Let F(R) mean that the relation R is well-founded, i.e., that there ia no decreasing sequence n&*Rn:. If the set E,[~&&FN(R)] were h d h , the set 0 would be also. Hence F(R) is not an absolute property. (d). It is easy to prove that the property of being inductive is an atwolute property of a set. Writing Fin(A) to indicate that A is inductive, we have n elements]. Hence the set E[reDl. E,[mD1. Fin(R,)] = xzJCr[r&D1.R, Fin(R,)] is Borelian of finite class. See K. Kuratowaki, Fundamento methematicor, vol. 29 (1937), p. 58. is defined 88 the set {#(a1, 11, +(na , 2 ) , +(n, 31,. sequence nl , n, , n~ , 1) K. Kuratowski in the paper cited in footnote 10, page 66.
10
11 A 11
-.
I.e., the set E+~,.,JmTnlis not in M.
,
*
1.
[141,42
133
ON ABSOLUTE PROPERTIES OF RELATIONS
Consider now the eimilrvity (equality of power) of two sets. The set E&&. x D 1 . R, sm R,] is Borelian of finite class, since it can be expressed aa a sum
.
Cmd{E,[r&Dl.R, haa n.ekmmts] X E.[s&D1 R, h a n elenenfa])
-
+
E,[r&Dl.Fin(R,)] X E*[S&l. Fin(R,)]. DI X This example shows that the theorem converse to that proved above is pmb&ly false, Bince the relation R sm S e m s not to be absolute. UIPTPEBSITX
or wAB8Aw
On the principle of dependent choices. BY
Andrzej
M 0 s t o w s ki (Warszawa).
Let us consider the following weakened form of the axiom of choice:
(X)
I
if 11'is a binary wlation and B a set +O aml if for every s e l l there is a y c I3 s w h that xBy,then. thfre is asequence x1,x2,...,xlI,... o j elptjitnts of H such that x,KxlI+~ for n = 1 , 2 , . . . I ) .
It will be proved here that the yencral asiow of choice (which we shall denote by (2))i s independmt of ( X ) , 1. e . , cannot be proved from (T) and the usual axioms of set-theory. An independence-proof has sense only with resppeet to a well defined formal system whose consistency is either proved or assumed as an hypothesis. Our proof applics only t o such systems of settheory as remain self-consistent after adjunction of the following axiom (AT)
there i s a %now,-dtmtnierable set of e l m f a t s which are .not sets.
It can be shown without difficulty that the system G described in one of my former papers 2, satisfies t'his condition. Hence we shall ta,ke 6 as a basis for our proof. I n order to prove that (2)is independent of (T)we have to coiistruct in a self-consistent theory GI a model in which all the axioms of 6 as well as the axiom (T) are fulfilled and in which the axiom of choice is false. *) This axiom ha5 been considered
by A . Tarski
iti
his recent paper
i l zionzatic and algebraic aspects of two theoren48 ow sums oj cardinals, this voliiiiie, p. 79-104. T:irski calls (2') the priiicipls of drpedent choices. L~ Fiiiit1;inwiita illntlieniatirae 32 (1939),pp. 201-253.
[15], 128
135
ON THE PRINCIPLE OF DEPENDENT CHOICES
We shall take as GIthe system 6 enriched by the axioms (N) and (2)but we shall make free use of many notions known from intuitive set-theory without defining them meticulously with the help of primitive notions of 6. Since it is known that the eonour result can be stated as follows: sistency of 6 implies that of GI3)), If G is self-consisttnt, then the iinplicatiola (!P)+(Z) i s not procable in 6. Let A be a non-denumtrable set of pairs {as,bs}
s c
S
whose elements are not sets. Lct A', be the set of all u,'s and a.11 b,'s:
..>
KO={. ..,as,b,, . and l e t Kt be defined by induction on
6 a s follows
(!$(X)=set of subsets of X). If f is a one-to-one mapping of h ' , onto itself and tji E h-,), then f ( m ) is defined as the value of f for the argument m . Suppose that f(n) is already defined for n ' E , and that rti K c - z K , . 4 q2> ...> P k > q k and they are the smallest numbers with this ~pruparty.According to lemma 4.31 the 1+1 - tuple ( E , n,, ..., n z ) belongs to J if and only if there is an a such that for every f~
:
Using equivalen'c-s (8) we obtain the !desired result.
Theorem 4.35. The set of numbers corresponding to true norr L a l sentences with 2k quantifiers in the prefix is equal to F
2 + 1 (h2J
186
FOUNDATIONAL STUDIES
Proof. Put 1'0
[29], 269
in theorem 4.34.
Theorem 4.36. The set of (numbers corresponding to) true sentences of the system S, belongs to class PE). Proof. First we evaluate bhe class of the set consisting of numbers of true ntonmal sentences. If V' is this set, then we infer from theorem 4.35 that V' is equal to the sum
which is of class P'? according to theorem 4.33. Since two equivalent e x p r e s s i m are evidently either both true o r both untrue, we infer that (the num'ber of) a sentence E belongs to the set V of (the numbers correspodding to) true sentences of the system S, if and only if f ( E ) is an element of V'; f is here the primitive recursive function referred t o in the formulation of theorem 4.21. Hence E € V = f(E)€V' and since V'€P(:, we obtain from theorem 1.62 the desired formula VEP". Theorem 4.36 gives an evaluation of the class of S,: it shows that this class in not higher than P,. IQ order to show that this class is exactly P,,, we must prove that V does not )beJong to any of the classes Pcf) or Q'i) with n<w. This will be proved in tRP next section. 4.4. Evaluation of the class of S, from below. We begin with the following definition Jl. A binary relation P is said to ,bedefinable tin S, if there is a sentence E with 4wo free variables x, and xe such that the following equivalence holds
m ~ =n (S,1 ,2,( E ) is true}. Lemma 4.41. I f E has two free variables x1 and x, and E' = S;, E, then
T a r s k i 1151, p. 312.
[29],270
A CLASSIFICATION OF LOGICAL SYSTEMS
187
Proof. Sirnce S i E as well as Sk E' have no free variables, we infer from lemma 3.52 that in order to establish the truth of
these sentences it is sufficient to show that they are satisfied by at least one sequence. Suppose first that Si E' is true and let CP be any sequence. Define Q!, by the equations CPl(j)=CP ( j ) for j # 1, @,(I) = p .
Since Q! satisfies S: E', we infer from lemma 3.55 t,hat CP' satisfies E ' . Denote by 'Pi the sequence which we obtain from identifying the second term with the first. Using lemma 3.54 we infer that C Pi satisfies E and since CPi (I) = CPi (2) = p we see using twice lemma 3.55 that CPi satisfies the sentence S:; E. Hence Sii E is true. The converse imlplication can be proved similarly.
(n)l is defiTheorem 4.42. I f the relation (&$) [rn non E nable in S , (or more generally in any system S containing S& then S, (cr S ) is not of class Pa. Proof. It follows from the hypothesis of the theorem that there exists a sentence E, with two free variables x, an'd x such that
Let T be th'e set of pairs (E, p ) such that E is the number corresponding to a sentence with one free variable x, and S; E is true. Put Eo= S.: E,. By lemma 4.41
{SiEbis true} = {Sd p" E, is true} = p non C F (11 a (p) and hence
188
FOUNDATIONAL STUDIES
(291, 271
Using nOw lemfma 1.61 we i,nfer that the set
is ‘not of class P‘:’ and thcerefore the sets6
does not belong to P t ) . If T were of class P t ’ , its intersection with the set
(4,b) [q = Elo]
were also of class P z ) 36. Since this intersection is identic with the set (25), we arrive at a contradidion which p m e s that T does not belong to the class PE). Observe now that (the number corresponding to) the sentence Sh E is a primitive recursive function of p and of (the number corresponding to) E :
Sb E=f(E, p ) fwmp. the introductory remarks to the proof of theorem 3.62). Denoting by V the set of (numbers corresponding to) true sentences and by A the set of (numbers corresponding to) sentences with on’e free variable xl,we obtain th? equivalence ( E , p ) E T ” [ ( E € A )and f(E,p)€V].
Tlhis equivalence shows that the class of the set T is the same as the olass of the set V. Indeed, the set of pairs (E, p ) such that f ( E , p ) E V has the same class as V according $0 Iemmla 1.62 and iintersecting it with the set of pairs (E, p ) such that E € A we do not alter its class since A is a primitive recursive sets7. Therefore V cannot be of class Pfx”, q. e. d. Taking now the results of theorems 4.36 and 4.42 together anld combining them with lemma 1.51 we obtain an exact evaluation of the class of the system S,: xi 56 57
M o s t o w s k i [9],p. 88. Ibid., p. 91. Ibid., p. 88-91.
[29], 272
A CLASSIFICATION OF LOGICAL SYSTEMS
189
Theorem 4.43. System S, is exactly of class P,. Theorem 4.43 evidently implies the Godel incompletness theorem f o r the system S,: no attempt to base arithmetk on a finite number of primitive recursive rules of proof an’d a set
A of axioms can result in a complete system unless A will be a t least of class P If A is of any of the classes P!’ or Q:’ with a finite n, there will be always santences tundecidable in the system.
f.
4.5. Arithmetic of real numbers S,. By arithmetic o f real numbers we understand a system which contains all the signs of the system S, and an infinite set of new variables (called classvariables). We choose the signs
as these new variables. Furthermone, we have in S, a new type of meaningful expressions: Xi (I‘) which can be read as: I‘ is an element o f Xi. These expressions as well as sentences of S, caa be combined by means of the strcke and the quantifiers (aj), @Xi) which bind the number variables and the class variables. We shall not set forth in detail the $definition of truth f o r this system. We only remark that we obtain this definition imitating the definition given in section 3.4 f o r the system S , and interpreting the class variables as names of arbitrary classes of integers. Theorem 4.42 remains valid in S, and its proof is unchanged. Therefore, if a is any ordinal such that the universal function F t) is definable in S,, then S, is not of class Pa.Now it ‘can !be shown that the functions Fg’ are definable in S, for all values o f tx lower than wl, the first non-constructible ordinal, and it follows that the class of S , cannot be determined unless we extend the definitions of classes p f ) wand QLk’ beyond wl. This negative result shows that there exists a profound difference between the structure of the set of true sentences of the arithmetic of integers anld that of the arithmetic of real numbers. It may be that this difference is cause,d by t,he same rea-
I
190
[29], 273
FOUNDATIONAL STUDIES
sons which have led the intuitionists and the followers of related trends to a distinction between the clear and constructive notions of arithmetic of integers artd the ccmplicated and rather mysterious notions of continuity N.
Bibliography [l] R. C a r n a p , Logiecha Syntax der Spraohe, Gchriften zur wimenechaftlichen Weltauffassung, Wien 1934. [a] A. C h u r c h, A note on the Entscheidungsproblem. Journal of Symbolio Logiq vol. 1 (1936). pp. v 1 (see also B correction thereto, ibid., PP. 101-102). 131 A. C h u r o h , The conetructive m o n d number cleee, Bulletin of the American Mathematical Society, vol. 44 (1938), PP. 224-239. [4] A. C h u r c h, Introduction to mathematical logic, Annals of Mathe. matics Studies, No 13, Princeton 194k [5] K. G b d e l , Die VollsUindigkeit der Axiome dee logischen Funktionenkalkttls, Monatshefte fir Mathematik und Phyeik, eol. 3’7 (1930), PR. %9-360. [6] K. G 6 d e 1, Vber formai unentscheidbare Stitze der Principia Mnthematica und verwandter Systeme I, ibid., vol. 38 (1931), PP. 173-198, [‘I S] . C. K I e e n e, Recursive predicatm and quantifiers, Traneactions of the American Mathematical Society, vol. 53 (1943). pp. 41-73. IS] K: K u r a t o w ski, Topologie I (2nd edition), Monografie Matematyczne, Warszawa Wroclaw, 1948, [9] A. M 0 s t o w s k i, On definable seta of positive interne, F u n d s menta M a k a t i c a e , vol. 34 (194’7). pp. 81-112. [lo] E. L. P o e t , Recursively enumerable sets of positive integers and their decision probkms, Bulletin of the American Mathematical Societp, Val. 50 (1944), pP. %-316. [ll] R. M. R o b i n 8 o n, Primitive recarsive functions, Bulletin of the American Mathematical society., vol. 53 (1947), pp. 925-942.
-
The difference between S, and the arithmetio of integers wonid probably remain the same even if we would decide to extend considerably the s y t e m S,. It has been namely pointed out by Dr G l i d e 1 that the identification of arithmetio of integers with the eystem s, is rather artificial. According to Dr Glide1 the ,,whole” arithmetic of integers Bhoula give a sufficient basis to ,,all” inducthe proofs and definitions and in particular to the inductive definitions of classee P‘:’ and for a w1 Evidently the notion of .,all” inductive proofs and definitions is rather vague and we must therefore conclude with Dr Glide1 that no satisfactory definition of arithmetic has been given as yet. Contemporary logio ie full of mch disappointing conclusions which indicate that we are still very far from a final solution of baeio problems of foundations of mathematios.
Qt’
, ,...} where i,j,k ,... are the f r e e v a r i a b l e s o f t or of m. Then Hf(u)=Hf(a,b,c,...) is what is usually called the value of t for the values a,b,c, ... given respectively to the f r e e v a r i a b l e s of t. The relation ti Stsf m holds if and only if the elements a,b,c, ... satisfy t h e matrix m in the domain 3t.1 of individuals. Lemma, 1. There eaists exactly one function H ( t , u ) = H f ( u ) siich that l o t r m s oaer t e r m s of s ; 20 H f ( u )considered as a function of 11 alone i s an %-function
with the set of ayguments B ( t ) ; 30 if t i s a v a v i a b l e , thew B f ( { < t , a > ) ) = s ; 1 0 if t=f,, them Hf(u)=F,, i===l, ...,a ; so if t=[gt(t,,...,tq,)], then H f ( u ) = G j ~ H f , ( i i,..., 1 ) Htqj(uq,)) where ? I , T 141 B(t,), n =1 ,..,,q j , j =1, ...,p. L e m m a 2 . There exists exactly one binary yelatiou Stsf such that Lo the counterdomain of k f consists of m a t r i c e s of s ; 20 for a fixed m a t r i a m the set E u [ uXtsf m ] is a n %-relation with the set of arguments B(m); 3O if n? i s the elenaentnry formzcln Lr,(fl,...,f,,,)), t!6e,t
[32], 142
201
ON MODELS OF AXIOMATIC SYSTEMS
So if in=[(33r)m1]and the v a r i a b l e 3i i s not f r e e in mlt then a Stsf m= u Stsf ml. If however 31 is free in mL, then zc
Rtsf m =there i s a n element a
+{
< 3i, a>) Stsf m,.
a such
that
Note that both lemmas are provable in S. I)efinit,ion14).A pseudo-model ( 2 ) is a Tea6 model of the secolzd kind of s in 8 if for every a x i o m [ai] of s and for every sequence u satisfying the conditions D ( u )= B ( [ a i ] )and D*(zc)C% the following formula holds u Stsf [a,], i - 1 , ...,6. Using this definition one can prove the following theorems: V 15). If ( 2 ) i s a real model of the second kind of 5 in S and if m i s a m a t r i x provable in s, then u Atsf m for an arbitrary sequence u sntisfginy the conditions D ( u ) = B ( m ) and D*(u)C%. VI16). lf at leant one yea1 model of the second kind of s in S uxists,. t h e n N ( s ) . Theorem V and VI as well as all the previous definitions and theorems belong to the system S. We now abandon S and pass to its syntax. We can then formulate the following statements concerning models of the second kind. These statements should he compared with statements 1-3 of section 2 , pp. 138-139: 1 . The general notion of models of the second kind is definable within S. 2 . Every individual model of the second kind is an element of the universe of discourse of 8. 3 . Models of the second kind can also be defined in cases where the number of axioms of s is infinite and statements 1 and 2 above also remain valid. From the circumstance that theorem VI has been proved in X x e infer that the following theorem holds: 171I . Zf the enisteiace of a real model of thv wcond Bind of s is pi*orahle i r i S , the)/ so i s the formwlit AT($) mpressitiy the cow.stvtr~t/cyof s. 14)
'6) '6)
Cf. T a r h h l 111'1, 1'. 8 . t7f. 'I'arslt I 1 I I], 1). 317, tlieorem 3, ant1 p. 35\. ('f. 'I'ariiki [ 1 I], p. 318, theorem 5 , and p. 3.5').
202
(321, 143
FOUNDATIONAL STUDIES
Hence models of the second kind enable us to obtain absolute consistency proofs whereas models of the first kind yield merely relative consist6ncy proofs. We note finally that just as in section 2 we can derive from the completness theorem of Godel the following theorem which is the converse of VII: VIII. If the sentence N ( s ) ia provable in. S , then 80 as the sentence stating the ezistence of at least one model of the second kind Of 8 i l 0 8. 4. Impossibility of a finite axiomatisation of set-’ theory. Let us amume ae in section 3 that 8 is an axiomatic sys-
tem of set-theory and let 8 be a system based on a finite number of axioms of i3. Since we It88ume the &-ruleboth in 8 and in 8 , we can aesume that the axioms of 8 contain no quantifiers and that k3 contains all functors occurring in the axioms of 8. From now on until the formulation of theorem IX we again aesume that our discussion takes place in system 8. Let 54f be an arbitrary non-void set such that if
ml ,...,mq,c%,
fi,...,fac
54f,
then g,(mL,...,m q , ) c N ,
P u t Fa=f,,...,Pa=f a and define the =-functions
ae sets of pair8
j=1, ...,B.
a, (j=1, ...,8)
,
where ml,...,mq, vary independently in 3f. Further let be X-relations defined by the equivalence {,***,)J
The a + / I + y + l (3)
Rh
( k = l , ...,y )
Rh Irh(ml, - - * 9 m P h ) .
tuple
nc, FI,...,FG,ax,...,@@, 4,...,R,
constitutes a peeudo-model of the eecond kind of 8 in 8. To show that this pseudo-model is a real model we remark that if [a,] is an amiom of 8 with the free variabzes 31,...,3k and if u ie any sequence {,..., < 3 k , u h > ) with u,, ..., u h . 3 i , then (4)
u 8tsf [a,] = a,(ul,
...,Wk)
WI, 144
ON MODELS OF AXIOMATIC SYSTEMS
203
(cf. "convention W" in Tarski [ll], p. 305). Since the right side of (4) is an axiom of X, we obtain u 8tsf [at]. This formula being provable in 8, we infer that (3) is a real model of 8 in 8. The construction carried out above is expressible in 8; therefore on using theorem VII we obtain: IX. If s i s a finitely axiomatizable szbb-system of 8, then the sentence N ( s ) is provable in S . Remarks. 1. The assumption made above that B and 8 both contain the &-ruleis not essential for the validity of theorem IX. Indeed, let s' and X' be systems without the mule which become equivalent to s and 8 after adjunction of that rule and explicit definitions. It is evident that N(s') is not stronger than N ( s ) and hence provable in X. Since N(s') is an arithmetical sentence in which the &-symboldoes not occur, it follows from the second 8-theorem of Hilbert and Bernays that N ( s ' ) is provable without the &-rule, i. e. in S'. 2. One might ask why our construction breaks down when 8 contains infinitely many axioms, e. 9. when s=8. To answer this question we recall that equivalences of the form (4) are provable in S for each ai separately. There are no means by which to express in S anything which could serve as a logical product of infinitely many such equivalences.
The following theorems are easy corollaries of IX: X. If 8 i s self-oonsistent, it is not finitely axiomatizable 17). Proof. Fir& of all we remark that there exists a finitely axiomatizable sub-system so of S which is at least as strong as the arithmetic of integers based on Peano'a axioms with the axiom of mathematical induction (conceived as an axiom-schema). To prove this we remark that this system of arithmetic is equivalent to the system (21) of Hilbert-Bernays [ 5 ] , p. 384. It has been shown by Novak and Wang [8], p. 90, that upon extending (Z) by the introduction of a new primitive notion and suitable axioms we obtain a system which is finitely axiomatizable. The new primitive notion is that of a predicative class of integers. The resulting system so is therefore certainly weaker than 8 since in 8 we have at our disposal the general notion of classes which satisfies all the
204
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[32], 145
axioms formulated by N o v a k and W a n g in their construction. Hence so is a sub-system of S. The existence of so being proved, we proceed as follows. According t o a theorem of Godel la) the sentence N ( s ) is provable in no self-consistent system s which contains ( Z ) . Hence if 8 is self-consistent and s is an axiomatizable sub-system of S which contains so, then N ( s ) is not provable in 8 . Since N ( s ) is provable in 8 according t o IX, we infer that systems s and X are not equivalent. Theorem X is thus proved. XI. If S i s self-consistent, it i s iu-ilzcomplete 19). Proof. Let X' be a system equivalent t o S 1)nt without the &-rule and with a fixed set of primitive functors and predicate8 fi. e . explicit definitions are not allowed in 8').Let al,a2,... be axioms of 8'. We can assume that no at contains free variables. P u t m(i)=[a,.n,. . . . . u , ~ ~ , . N ( ci =~l], ,2 , ..., andlet @(m) bethe matrix m ( s ) i s uizprocable i?z the first order functional calculus. It is easy t o see that this matrix can be written in purely arithmetical terms and is therefore a matrix of 8. According to IX, sentences @(l),O(Z),... are all provable in X whereas the general statement ($)@(a) is equivalent to N ( S ) and hence unprovable in S unless S is inconsistent. XI1 20). If 8 i s self-consistent, then there exist consistent but c~-inconsistentsets of arithmetical semtences. Indeed, sentences - N ( X ) , @(l), @ ( 2 ) ,O ( 3 )... form such a set.
5. Models of the second kind in the axiomatic theory of real numbers. llniost all we have said 111 sections 3 and 4 can be repeated wnen S is an axiomatic theory of real numbers. When speaking of the arithmetic of real numbers, we have in mind systems in which the class of integers as well as the developmentiof any real number into decimal (or other) fractions is definable and can be proved t o exist. G o d e l [3], theorem XI, p. 196. For the notion of o-completeness see T a r s k i [13]. The result obtained in theorem XI is of course not new. n o ) Of course this Iesult is not new either. See G o d e l [3], p. 190 and T a r s k i 1131, p. 108. 18)
19)
[32], 146
ON MODELS OF AXIOMATIC SYSTEMS
205
Note that the arithmetic of real numbers in its usual formulation is based on an infinite number of axioms because the axiom of continuity cannot be expressed otherwise as a schema. The notions of functions and relations do not occur explicitly among the primitive notions of arithmetic. Some particular cases of these notions, however, are definable in arithmetic and these particular cases are general enough to enable us to carry over the proofs given in sections 4 and 5 from set-theory to arithmetic. The procedure is as follows. First, we define one-to-one correspondences between integers and finite sequences of k integers, k=1,2, ... If an integer n is made ) p k is the k-th prime, to correspond with a k-tuple ( n l 1 . . . , ~ 2 band then we shall identify the integer p ; with the k-tuple ( ) t l , ...,n k ) . I n this way we obtain an arithmetical substitute for the notion of a finite sequence of integers. It is well known that we can effectively establish a one-to-one correspondence between reel numbers and sets of integers. I n other words we can find a matrix @(m,m) such that the following formulas are provable in X: O ( . r l t t ) 3n i s an integer, x ~ z / , - ( n ) [ @ ( dn,) =@(%",it)].
-
A real number % will be called a k-termed relation if (m)[@(m,~~)3(3m).n=p,m]. Integers 9z1, ...,nk are said to be in relati0.n s !i the integer n corresponding to the sequence ( i ? l l . . . , n b )SRtisfies the condition @(.;,p;). I n this case we write I I : ( ? L ..., ~ , nb). A real number II: is called a function with k argument8 if it satisfies the following conditions: z i s a binary relation,
4%, * 13(Em1 n 1 = PT ) , ( ' ~ ~ ) ( 3 ~ ~ z ) ~ ( ~ ~ i ~ ~ ~ ~ , .C(n1,
..I) .c(nl,n;)3n;=11;.
The value of the function m for the arguments ? i l , . . . , n k is defined as (tnz)x(p;,n,) where m is the integer corresponding to t h e sequence ul,...,9 2 1 . Having defined the notions of functions and relations, we can reconstruct without difficulty all the definitions and proofs which were given in sections 3 and 4. In this way we arrive at the following results:
206
FOUNDATIONAL STUDIES
1321, 147
XIII. If the system X of the arithmetic of real numbers is selfcowsistent and s i s a finitely axiomatixabb sub-system of X, then the sentence X ( s ) is propable i n S . XIV. The arithmetic of real numbers is ?lot finitely axiomatixable. 6. Models of the third kind. The method described in sections 3-5 does not apply in the case in which X is the system of the arithmetic of positive integers based on Peano's postulates. The failure of the method is caused by the fact that no model of the second kind is definable in the arithmetic of positive integers for a system s in which the existence of infinitely many individuals is provable. JIodels of the third kind which we shall discuss presently will enable us t o prove theorems similar to IX-XI1 for the ewe in which S is the system of the arithmetic of integers. These models were first, defined by H i l b e r t and Bernays who stated their definitions id a non-formal language*'). We shall present here an arithmetical counterpart of the Hilbert-Bernays definition in order to discuss the possibility of its use in a formal system A'. We shall make the same assumptions concerning the systems A' and s as in section 1. The system 8 will however be slightly enlarged by the adjunction of the symbols A and v, denoting the Boolean zero and the Boolean unit. Boolean addition and multiand . and, when many summands plication will be denoted by or factors are present, by the Z- and 27-symbols. Boolean complementation will be denoted by an upper index 1. For symmetry we put A Q = A and V o r ~ . A term r of s will be called a, constant term if no variable occurs in it. It is easy to construct in arithmetic (and hence in 8) a functor l' with one free variable such that the following formulas are provable in 8: if y is an integer, then T ( y ) is a constan-t term; if x is a constant term, then s = T ( y ) for some y. We shall now construct in X a function X(m,m) which enumerates all matrices t h a t can be obtained from a Q-matrim m by all possible substitutions of constant ternas for the free v a riables of m. We introduce first the auxiliary functors u, 8,and p.
+
31)
See H i l b e r t - R e r n a y s 151, pp. 33-36.
[32], 148
ON MODELS OF AXIOMATIC SYSTEMS
207
Let a be a functor of B with two free variables such that the formula = 2 ~ x 1.3ac2.~)-1. , ~ ~
....p2~,~)-i. ...
is provable in 8. In other words the definition of u(y,s) is obtained by expressing in 8 the following definition: a(y,x)=l+ (the exponent of the y-th prime in the development of a into the product of primes). For m y t e r m t we put X(t,", 1)= Sb t( 3 1 M 41,$1 11, f l ( t ,$7 y f1) = 8 b #(t, a,?//) (3g+l ' r ( d y +I,")))* This is clearly an inductive definition of the type which can be represented in 8 by a single functor. Hence S(t,m,y) is a functor of 8. Let p ( t ) be the functor of 8 defined as the largest integer y such that 3sr o c c u r s in t. We put
W , @=fw,",A t ) ) and call B(t,s) the x-th substitutiofi of t. 8(t,m) is clearly a functor of B. It can be proved in 8 that ( s , t ){ # ( t , $ ) i s a c o n s t a n t t e r m } . If e = [ r f ( t , , ...,f,)]
is a n e l e m e n t a r y f o r m u l a , then we put
& e , 4 = [rAflVl,s), -.,W pj14)l
and call 8 ( e , m ) the x-th substitution of e. The following statement is provable in 8: (z,e) {S(e,s) i s a p r i m e f o r m u l a } (cf. section 1, definition 8). We define now the x-tR substitution of an arbitrary Q - m a t r i x m. The definition proceeds by induction. If m=c, then 8(e,a) has been defined above. If m = [m,Im,l, then we put S(nz,a)= [S(m,,~)lfJ(m,,s)]. By a standard procedure we transform this inductive definition into an explicit one whin can be expressed in 8. We shall call a pseudo-model of the third kind or briefly a valuation, a functor di or 8 with one free variable such that the following formula is provable in 8: if p i s a p r i m e f o r m u l a , then D ( p ) = V or @ ( p ) = A .
208
FOUNDATIONAL STUDIES
[32], 149
Let @ be an arbitrary valuation. We consider a functor V n l a ( m , s ) of S with two free variables satisfying the following conditions: a) the first argument of Val@ runs over &-matrices and the second over arbitrary positive integers; b) the following statements (5) and (6) are provable in S:
T’u7.(e, a)= @(S( e , z)), Vazs([mllnL,],s)={VU?s(ml,s) .V a f + 1)Zz7 ( a))’ (the upper index 1 denotes here the Boolean complementation). It is easy to construct effectively a functor Va7a which satisfies the above conditions. All we have to do is to remark that conditions (5) and (6) can be considered as an inductive definition of Val@and that inductive definitions of this kind can. be transformed into ordinary definitions which are expressible in S. Definition. A pseudo-model @ is called a real model of the third kiizd of s itz 8 if the following formulas are provable in 8:
(5) (6)
Models of t8hethird kind are in some respects similar to models of the first kind. Indeed, it follows from the definition that 1. The general notion of models of the third kind is defined not in S but in its syntax (since @ was defined as a functor of S ) ; 2 . Every particular pseudo-model of the third kind can be defined within S (since each particular @ can be writteh down by means of symbols allowed in 8).The problem whether this pseudomodel is or is not a real model of s can be formulated and in particular cases solved in S. On the other hand 3. The notion of models of the third kind retains its meaning also for cases in which s contains an infinite number of axioms. bdeed, @ is a real model if the formula ( m ) [ ( m i s nsz asionz Of s ) 3 (a)(Va7*(m,s)=V } ]
is provable in 8. This definition is meaningful not only when the a s i o m s of s are finite in number but more generally when their set is definable in S. I n this respect there is an analogy between models of the third and second kinds. We shall now investigate the problem whether models of the third kind can be used to obtain absolute proofs of consistency.
[32], 150
ON MODELS OF AXIOMATIC SYSTEMS
209
Following H i l b e r t - E e r n a y sz 2 ) we shall call a Q-matrix rn uerifiable if (m)Vale(m,x)=v . The following theorem can then be proved: XV. The formula ( m ) { ( m is a &-matrix provable in s ) 3 ( m i s verifiable)] is provable i n 8. The proof of this theorem has been given by H i l b e r t - B e r n a y s [53, pp. 33-36. We note that t,his proof is straightforward for the case in which m can be obtained from the axioms by the elementar? calculus with free variables z 3 ) . Hence the essential step in the proof of XV consists in proving in S the implication: ( m i s provable in s ) 3 ( m can be obtained from the a x i o m s of s by means of the elementary calculus with free variables). This implication is established by Hi1 b e r t and B e r n a y s with the help of the first &-theorem. Another proof of XV has been given by LOB24). His proof is not finitary but can be translated in X along r i t h the H i l b e r t B e r n a y s proof. As a corollary from XV we obtain XVI * 5 ) . The following implication i s proaablc in S : if @ is a real model of s in 8, then N ( s ) . It follows from XVI that models of the third kind can be used when an absolnte proof of consistency is desired.
7. Impossibility of a finite axiomatization of arithmetic”). We assume in this section that X is the system of arith-
methic of positive integers based on Peano’s postulates and that s is a finitely axiomatizable sub-system of 8. We can assmie that axioms of s have been brought to the normal form
where
E v , < ( i) = rn(v,:,fi ( r u , c , i , l y ...t - I ’ v . : ~ , p ~ ( ~ , t , ~ ) ) .
H i 1 b e r t - B e r i i a y s [ 5 ] , p. 36. H i l b e r t - B c r n a y s [ 5 ] , p- 380. z4) Lo6 [ti], 1). 3i. theorem 34. 2*) H i l b e r t - U e r n a y s [5], p. 36, call S V I the “\Vf-tlieoreiii”. ne) The itiipossibility of a finite axioinatization of the ar.ithinetic of positivl. integers was first slioivi~by R y l l - S a r t l z e \ v s k i in [lo]. TlicwwIn S V I I I of tlit. preseiit paper is lionever sliglitly stronger t l i i i i i the rewlt of It! I l - S n r d z e w s k i . ”)
210
[32], 151
FOUNDATIONAL STUDIES
Further we can asume that constants, functors, and predicates of 8 occur also in 8. We shall denote by ev,s(i)the Cfodel number of Lemma 3. If Qi i s an arbitrary vodzcdiora, then formula is provable in 8:
(exponents I m well a the symbols 17 and 22 have here the Boolean meaning). Proof. Immediate from ( 5 ) and (6). L e m m a 4. Under the assumption8 of lemma 3 tht foUming equisalence i s provabb in 8:
-
v1
Val.(Ca,1,4=
(4{4 w ( i ) 3 ( 3 E E ) C ( Kt ( V , i ) )
*(Vd.(e,,s(i),a)= V f W ) ] } .
Proof. Immediate from lemma 3. Let t run over oonstant terms of 8. We consider a functor 8 of fl such that the following equations be provable in 8:
(7) (8)
s(f,,=f,,
i = i , ...,
Q(Csl(tl,...,h,)l) =gjt@(tl), ”’,W q , ) ) ? i=1,.*.,B.
It is easy to construct explicitly a functor satidying these conditions. Indeed, (7) and ( 8 ) contain an inductive definition which can be transformed into an explicit one and the definiens of the explicit definition thus obtained is the required 8. The intuitive meaning of the functor 8 can be explained a8 follows: Consider an arbitconstant term of 8; of course it denotes an integer. Let t be the Gi3del number of l”. Then e(t)is the integer denoted by I: We put e ( T ( o ) ) = $ ( x )Note . that 6 is a term of 8 with one free variable. @(a?) is of course the integer denoted by the m-th conetmt term (in the enumeration of constant term8 given by the function T).
r
r
Lemma 5. Let be a term of 8 , t it8 Cf6del number a d h the Zurgeut integer 8 w h that xh o m r s in The% the fouoCaing equation is provable in S : (9)
@( S( t ,5))=S t h t
r.
l‘( s#(
...,
a( 1,s)), %,JS(a(h, 2))).
[32], 152
ON MODELS OF AXIOMATIC SYSTEMS
211
Proof. If F = q , then t = 3i, LJ(t,a)=T(a(i,s)) and hence @(8(t,n))=8(a(i,m)).On the other hand the right side of (9) is &(u(i,a)). Hence the lemma is true in this case. If r=fr,then t=fi and the left and right sides of (9) can easily be shown to be equal to f i . Let us assume that the lemma is proved for terms TI, ...,r,, with the Godel numbers t,,...,f,,. Let r be the term q,(rl,...,I',,); the Godel number of T is t=[g/(h, ...,fp,)]. It follows from lemma 2 1 in section 1 and from (8) that equations
w ,2)= IIsAfV,, m),. m,,, $))I, s,(@(ml,$)), ...,@ (LJ(t,,,s))) * *,
(10)
@ ( L J ( t , 4= )
are provable in X. On the other hand if we put
ry)= B&t
rt(.C@(
...,z h /@(a(h, S ) ) ) ,
Q( 1,g ) ) ,
we obtain from lemma 2 1 in section 1 the equation (11)
...,$A /@(a(h, z)))=g,(l'v), ...,1;;))
b b 8 t r(sl/&( a(1,s)),
provable in 8. Since by the inductive assumption equations
=I'p)
@(S(ti,a))
are provable in LJ, we obtain the desired result by comparing form u l a (10) and (11). Lemma 6 is thus proved. Observe that this lemma is not 8 theorem of X but a theorem-schema. The statement of this lemma must be proved separately for every r. Definition. I f is a term and H a @matrix of 8 and if h is the largest integer such that sh occurs in r or in M',then we put
r
= Bwbst
r (S@(
...,@,/@(a( h, a))), 1,a)), ...,~*n/@(u( h, s))).
U( 1,s)),
M ( x ) = #&st M(@(u( is a term of B with one free variable x. SimilNote that arly X(x) is a matrix of X with one free variable s. 0
.(v)
Lemma 6. If X = x fleilwhere the Ev,l are ekmentary v = l 6=1
fornauh and t t e ?,,(
are equal to 0 or 1, then
The proof is obvious.
212
[32], 153
FOUNDATIONAL STUDIES
We shall now define a valuation Q, of which we shall show later that it is a real model of 8 in rS.
E { ( I d (PI
t= 1
=
W P ) = 4=
i) ( T i ( @ (comP1 (PI),...,@ (COmPp,(P)))(s = v 1 *
*
V “ ~ / ( @ ( C O ~( P )I ) , . . . , @ ( ~ o ~ P , *(s= , ( P )A ) )1))-
(Cf. section 1, def. 19, p. 137, for the definition of the functions
I n a and camp).
It is evident that Q, is a functor of 8. The formula
@P(p)=V or @ ( p ) = A is provable in 8. Indeed, it can be proved in 8 that if p is a prime formula, then Comp,(p) is a constant term and hence @(Compj(p)) is a perfectly defined term of 8. Lemma 7. Let r,(rl,...,rPi) be an elementary formula and e=[ri(t,, ...,tpr)]its GdaeZ number. The follozoing equivalences art3 then provable i n #,for A = O , l : @(W(e,m))=vL==e(Tp) ,...,Tg)). Proof. By definition W(e,m)= [rXW(t1,m),...,W(tpi,m))] whence it follows that equations Ind ( W(e, m ) ) = i ,
ConzPl,(WS(e,m))=B(tl,l;),
. . . . .
..
. . .
Comlpp,(8( 63%)) =fl( ‘&I 4
are provahle in 8. Now observe t h a t the formula S(e,m) is a prime formula
is provable in 8. Upon using the definition of @ we obtain therefore the formula provable in 8 @(Ste,z))=
v L= ? ( @ ( W I , @ ) ,
..*,Q(4$,,4N-
Since equations Q(W(t,,m))=Tf? are provable in W (cf. lemma 5 ) , we get the desired result directly from the last formula. Observe that lemma 7 is not a single theorem but a theoremschema of 8.
[32], 154
213
ON MODELS OF AXIOMATIC SYSTEMS
is provable in S. By lemma 6 the right side of the equivalence can
i. e. by a?). It follows that the formula
VaZs([u,],s)=v 5 rap)
is provable in 8. Since up) is
8 substitution of an axiom of provable in 8. Hence the equation
8,
it is
V w [ a f l l ~ )v=
is provable in 8 and theorem XVII is proved. From theorems XVI and XVII we obtain XVIII.If 8 .is a finitely miomatizable 8ub-8~8te~n of 8, the% the formula N(8) i s (provable in 8. XIX"). 8 i s not finitely axiomatizable. ") This theorem is due to R y l l - N a r d z e w s k i who obtained it in [lo] by a different method.
214
FOUNDATIONAL STUDIES
[32], 155
To prove this theorem we need the following L e m m a 8. There esists a finitely axiomatizabl4sub-system 8, of 8 such that if s is a finitel?! adomatizable sub-system of S which c w tains sol then the sentence N ( s ) i s aot provable in s. We shall content ourselves with a sketch of the proof. We shall take for granted the arithmetizstion of the system 8 along the lines indicated by Godel [3]. The arithmetical counterparts of the metamathematical notions will be denoted by symbols used by Godel although strictly speaking the symbols should be modified because the system arithmetized by Godel is different from 8. Let F be ft primitive recursive function such that F(m,n,p)=O if and only if n , p are sentences of 8 and m is a p r o o f of the i m p l i c a t i o n p Imp n in the functional calculus of the first order. Let F’ be defined at8 follows
Let, f‘ be the Godel number of the equation F ’ ( n ~ , i z , p ) = O . H i l b e r t and B e r n a y s [5], pp. 310-323, have shown that the following implication is provable in S :
Analysing their proof we find that lothe x whose existence is stated in (13)is a primitive recursive function Q of m, n , p ; 2O the axioms of S which occur in the proof d t h the Godel number Q(m,n,p) are finite in number and independent of m, n , and p. If we denote by & . . . , A k these axioms and by ax the Godel number of their conjunction, me obtain instead of (13) the folloming implication provable in 8:
H i l b e r t and Bernaj-s have further shown (cf. [ 5 ] , pp. 307-308) that the implication (15)
(Ba)F‘(x,l’TGen i , . p ) = ( , 3 ( ~ ) ( 3 n . ) F ’ iLb ( 2 ,I , (
is provable in S.
W19Y ) ) , p ) = O
[32], 156
ON MODELS OF AXIOMATIC SYSTEMS
215
Now let so be a sub-system of S based on the axioms A41,...,Ak and on those axioms of S which are necessary to prove implications (14) and (1.5) as well as the recursion-equations for the functions P, Sb, Imp, Neg, Z, and Q. We shall show that so has the property stated in the lemma. Indeed, let s be a finitely axiomatizable sub-system of S containing so and denote by Ax the Godel number of the conjunction of the axioms of s. It follows that (14) and (15) (when expressed in the symbols of 8 ) are theorems of s. From (14) we infer that the following implication is provable in s: We put
I t is easy to see that 8=17 Gen Neg Sb p
*
Repeating the argument of Godel [3], pp. 187-189, we can prove that 6 is unprovable in s, provided that s is consistent, i. e. that N ( s )3 (m)P(s, 8, dlr) 0. (17)
+
This proof can be repeated word by word in the system s owing to the circumstance that (15),(16), and recursion-equations for the functions P, Q, Sb, Imp, Neg, and Z are available in s. Hence if we denote by w the Godel number of the sentence N ( 6 ) and observe t
for almost all a and for j = 1, . . . , 9. If therefore e does not satisfy this condition, the weak power is an utterly trivial theory. This remark shows at the same time that the dependence of the weak power on e is very essential.
2.3. Examples. We begin with some examples of strong powers. 1. Let F be the set of real numbers, R the ternary relation such that R ( x , y, z ) = x =y z, Z a set with n elements, e. g. the set of integers 1, 2, . . . ,n. Functions f of F ' can be identified with the n dimensional vectors with the componentsf(l),f(2), ..-,f(n). The relation R' holds between vectors f , g, and h if and only if f(i) = g ( i ) h(i) for i = 1,2, . . ,n, i. e. if the vector f is the sum of g and h in the sense of vector algebra. Hence if T = T ( R ) , then the strong power T ' is the elementary theory of additjon of n dimensional vectors with real components. 2. More generally, if G is a group and X denotes the group multiplication and if we define R by the equivalence R(x, y, z ) 3 x = y X z, then T(R)' is the elementary theory of multiplication in the direct product
+
+
GX G
-.*
X G = G".
3. If T is defined as in example 1 and Z is a denumerable set, then T ' is the elementary theory of addition of infinite-dimensional vectors. 4. Let F be a set consisting of two numbers 0 , l and let R be the relation S in the set F. For an arbitrary Z the set F' consists of characteristic functions of subsets of I . We shall identify subsets of Z with the& characteristic functions and prove that R' is the relation of inclusion between the subsets of Z. Indeed i f f and g are characteristic functions of sets X and Y and R'(f, g) holds, then f(a) 5 g(a) for each a e Z and hence f(a) = 1 . 3 g(a) = 1 which
224
[331, 7
FOUNDATIONAL STUDIES
proves that a t X 3 e Y . Conversely if a e X 3 a e Y , then f(a) = 1 implies g(a) = 1 which entailsf(a) 5 g(a). Hence R'U, g) = X C Y .
Since the Boolean operations on sets are all definable in term of inclusion, we may say that if T = T ( R ) ,then TI is the Boolean algebra of subsets of I . We shall now give some examples of weak products. 5. Let F , R, and T be defined as in example 1 and let I be the set of nonnegative integers. Define e as the function of F' such that e(a) = 0 for ail a e I : We let correspond to each functjonf e *F: the polynomial Pf(2) = c7-'-o f(i)ex8 (the number of terms in the sum is finite since almost all numbers f(i)yanish). The correspondence f ~ r Pf ? is evidently bi-unique and we have
R"U, 9, h)
(i)v(i)= g(i)
+ h(i)]
Pf
=
Pa
+ Pn .
Hence the weak product *Td is in this case identical with the elementary theory of addition of polynomials with real coefficients. 6. Let F be the set of non-negative integers and R the ternary relation such that R(z, y, z) = z = y z. Define e and I as in example 5 and let pl , p z , . * * be the sequence of primes. We let correspond to each function f e *F: the integer nf = n7-1pi'" (the number of terms in this product is again finite because f(i) = 0 for almost all i). We have thus obtained a one-one mapping f + nl of *F: onto the set J of positive integers. R" is in this case isomorphic with the relation z = y . z in the set J . Indeed
+
n OD
4-1
p!'" =
n oy
I-1
p?")
.
n m
i-1
pl'"
nf = na.nh.
Hence *T:is identical with the elementary theory of multiplication of positive integers. We could have taken instead of p , , pz , . . . the sequence of prime ideals in an arbitrary number field or the sequence of irreducible polynomials over any such field. ' T : would then be the elementary theory of multiplication of ideals' or of polynomials. 7. Replace the relation R of the foregoing example by a binary relation S such that S(z, y) =z 5 y. S" is then isomorphic with the relation of divisibility of integers and consequently the weak power *T: is identical with the elementary theory of the relation of divisibility. 8. Replace F in example 6 by the set of all integers (positive and negative). The same argument as above proves that *T: is now identical with the elementary theory of multiplication of rationals different from 0. Using the final remark of example 6 we see that *T: can also be considered as the elementary theory of multiplication of rational functions over a number field or as the elementary theory of multiplication of fractional ideals of such a field. Cf. Skolem 161, $4. Skolem showed also that 'Tf can be considered as the elementary theory of multiplication of algebraic integers of any number field.
WI, 8
225
ON DIRECT PRODUCTS OF THEORIES
9. Let F, e, and R be defined as in example 6 and let Z be the set of ordinsls less than w.. To each function f c *F: we let correspond the ordinal
Xf
=
c
m.>tZO
f(€)J
where the summation is taken in the decreasing order of - . > Zr, then
+ -- +
4- I < € d . W t 2 * f(&>4. The formula f $ XI establishes a one-one correspondence between functiom of *F: and ordinals less than w. . R” is in this case isomorphic to the relation = t~ 0 f where t, 7, f run over ordinals less than w. and 0 denotes the so-called natural addition of ordinals: To prove this we simply observe that
b
= f(€l).wfl
+
R”V, g, N = W(€) = d-9 WI
{Cf(E)*w‘
=
c b(€)-k h(010‘}
(Xf
8 A,).
Hence *T: is identical with the elementary theory of natural addition of ordinals less than w a . 10. Let F consist of elements 0 and 1 and let R be the relation = in F. Let Z be an arbitrary infinite set and let e have the constant value 0. Elements f of *F: are characteristic functions of finite subsets of I and R“ is the relation of identity between such functions. Since the set of finite subsets of I has the same cardinal number i as I, we infer that the elementary theory of identity in an ‘infinite set with the cardinal number i is the i-th weak power of the theory
TW).
11. Taking F, e, I as in example 10 and denoting by R the relation in F we obtain as *T(R):the Boolean algebra of finite subseta of I. Proof is the same as in example 4. T)e number of such examples could be multiplied indefinitely. It must be said, however, that only exceptionally does one come across a really interesting example of a power theory. We mention finally that there exist theories which cannot be represented as powers. In such “indecomposable” theories the basic relations are isomorphic to no relations of the form RI or R:. As examples one can cite the elementary theories of arbitrary types of order, the elementary theory of addition of integers, and many other theories. In the rest of this paper we shall be concerned with the proof of a theorem which says, roughly speaking, that the evaluation of the truth-value of formulas of TI is effectively reducible to the corresponding problem for formulas of T. Similar result holds for the theory if e is a constant function whose (unique) value is definable in T. & a corollary we shall derive a theorem which says that the decidability of T entails the decidability of TI and, if the above as-
*e
7
Compare e. g. Hausdorff
PI.
226
FOUNDATIONAL STUDIES
.
WI, 9
eumptions are satisfied, also of *T: It follows in particular, from this corollary, that the power theories considered in examples 1-11 above are all decidable: @3.Auxihry theories C, S, and S*. We shall consider in this section an arbitrary theory T = T(RI, * - , R,) with the field F and an arbitrary nonvoid set I. Most definitions and theorems will be given in pairs, one holding for the weak and another for the strong power of T. Dealiig with weak powers of T we always tacitly assume that I is an infinite set and e an element of F'.
-
3.1. Theorg CO . Let 11be the set of all subseta of I and let C be the relation of inclusion between the elements of 11 The elementary theory of relation C wi l l be denoted by C. For typographical reasons we shall use the letter 2 for the predicate corresponding to C and shall write Zt.1 instead of Z(xt, XI). The field of C is evidently I1 . We shall now define a series of formulas of C which will be needed later. In writing them definitions we adopt the convention that Merent small Latin letters denote diflerent integers. Thie convention is particularly important in the case of formulas containing bound variables. Usually it does not matter which particular bound variable occurs in the formula but it is important that it be Merent from the free variables contained in the scope of the binding quantifier. Hence the indices of the bound variables in the formulas to be given below may be fixed quite arbitrarily, provided that the convention that diflerent letters denote ditrerent numbers is fulfilled. The letters i and j with or without subscripta will always denote one of the indices 0 or 1.
.
~
8 Other proofs of decidability of most of these theories exist already in the literature. a m p . Szmielew [S] for the theories 3, 5,8, Skolem [4] for the theory 4, Skolem [6]for the theory 6,Tarski 181 for the theory 10. It could have easily been guessed that no deep result8 can be expected from an application of a very general theorem. The fact however that 80 diverse proof8 of decidability can be shown to spring all from the same source i a p e r h a p not without Borne interest. Theory C was discussed first by Skolem [4]. His results are quoted without proofs by Tarski (71, $3. @
1331, 10
221
OF DIRECT PRODUCTS OF THEORIES
The meaning of these formulaa is explained in the following theorem, the proof of which is obvious:
---
THEOREM 3.11. I f zk , 2 1 ,zkz, ,zk,,are e h k of a n d i j w e p u t z o = z,zl= I - z f o r z c I ~ , t i m FC
zi,l(zk,23 3zk
kc &(Zk)
(i.e. aubclassea of I )
c 2: ,
(Zk itt V o i d ) ,
...,zkm)= (the intetsedion 22 n -. n 2:
kc I>Z:"'*h ....&kl, Fc
I1
is upid},
ZL:::: ,:: ,:&, *
...,
2kJ
= (the interseclion xi: n .-.n z$ &aim . ,zk)
1 il i. kC A k a : . ..3.(zk1
d bat h ete?nents),
* *
= (the inkreection zf; fI . . n zt contains aaclly h elements). a
The theory C waa shown decidable by Skolem." More specifically Skolem's m l t can be expressed aa follows:
, XS,
- -,
THEOREM 3.12. For every form& q5 of C maththe free vanizbles xtl, there exist non-negative intqers h, 1. ,and and indices if), j::: ( v = 1,2, h, p = 1,2, - - ., 1., p = 1, 2, ,n ) svch that
c)
*
3.2. Theories S and S*. W e shall now imbed theories C and T into more comprehensive theoriea S and S*. S will be the elementary theory of the following relatione Ii , C,F, Ri
, - - * ,R, , F', I , C, Q
where Q is a ternary relation defined by the equivalence
&Cf,u,z)=CftF').(ucI).Cf(a)
' 2 )
and c is the membership relation between the elements of I and those of S* will be the elementary theory of relations
4 , c,F,Ri,...,R,,*F:,I,r,Q
I1
.
.
We shall denote by F1 the predicate of S corresponding to the set F' and by F: the predicate of S* corresponding to the set *F: . We denote further by t the predicate of both S and S* corresponding to the relation c and write xt c XI instead of s(xk, xl). Finally we m u m e that S and S* contain a variable, my G , which does not occur in T.
8.3. Formulas T, $*, ( 0 ;i, h ) ,and ($;it h)*. We now let correspond to each formula 9 of T certain formulas which belong to the theories S and S*. We begin with the inductive definition of 3 and $*. 10
Skolem [a], p. 36.
228
P31, 11
FOUNDATIONAL STUDIES
If 4 is the elementary formula Rh(xk, ,
- - - ,x b A ) ,then
If#istheformula41jh,then6 = & I & a n d ? Tf 6 is the formula ' ( a X k ) & , then
5
=
-
(aXk)Fl(Xk)
* -61 - -* l #-*z .
- 6?1*
4*= (%k)F:(xk?
6119
' the
6 and 6*are thus defined Ly induction. We now denote by an accent relativiration of quantifiers to 21 (compare section 1.1 in $1) and put
.
.&':I . nFib,), x. = 6*).&'a .II F;h,), ( I
(6;i, h ) = (~x.)[~I(x,)(XO)(XO E X. =6)
.
.-1
I
(6;i, h)* = ( 3 x , ) [ I ~ k , ) ( X O ) ( X O E
-.1
where it is supposed that xk, , * * , xk,, are the free variables of 4, s # 0 and 8 # k. ( v = 1, 2, * - ,n). I n order to explain the meaning of these definitions we shall prove the following theorems : +
f
THEOREM 3.31. If 6 is a formula of T with the jree variables xk, , ,xk,, , then S and 6* ?formula of S*. The free variables of 6 and of 6* are a
6 i s a formula of XO
xk,
' *
Xk.
.
Proof by a straightforward induction from the definition.
THEOREM 3.32. Let # be a formula of T with the free vuriabh and assume that a t I , f k , , . , f k , a F', , , t *F: Then
-
jti . - fl"
.
7
Xk,
,
*
-
*
, xi,,
k8 &(a,f k i . * f k n ) k T & f k i ( a ) , * .. f k n ( a ) ) , b 0 6 * ( a ,fti, - * * ,f3= I-TdXi(a), * * - ,f:n(a)). PROOF. w e proceed by induction. If 6 is the formula R h ( x k , , . * . ,xiph),then Y
2
9
, . ,f k p h ) = (there ezist elements ztl , - . , such that &(f,k, , a, Zt,) fOr v = 1, 2, ,ph and
ts&a, Ct,,
'
fkl
*
* * *
Rh(ztl , *
*
, z t P , )= ) {there ezist elements zrl, . . - ,ztph
such that f k , ( a ) = 21,for Rh(Zti IT
, '.
* 1ztPh)]
6(fki(a),
'*
v =
1, 2,
*
Rh(fki(a), * '
fki
. . .fd '
{not k6 6l(%f k i ,,'
* *
fk.)
,p h and
fk,,(a))
,f k p , ( a ) ) -
Suppose that the theorem holds for formulas have
1 8 &a,
*
$1
and & . For 4 = or
16 &(a, fki
3
41
I & we 9
1.
fh)
WI, 12
229
ON DIRECT PRODUCTS OF THEORIES
From the inductive assumption we infer that the right aide of this equivalence is equivalent t o not t-T+lUk,(a),
-
*
. ,fk,(a)) or not
I-r &(./&,(a),
.
*
,fk.(a))
which, in turn, is equivalent t o FT+Cfkl(a),... ,,jkm(a)). If = ( 3 x r ) g 1 , then we use the inductive assumption and the definition of satisfaction and obtain the following equivalences
+
t8 &a, f k , , . ,fd = W d f t c F' *
1
(1)
and
I-8
( a f t ) [ fhi
&(a, fi , 5 ,, . . . ,fdJ
F' and kT+l(./t(a), fki(a), . . *
fk.(a>)l
from which we infer that I-8
6 ( a ,f k ,
,
*
-
*
,f J
2 (3zt)br t
F and
FT+ 1 h ,S,(a), . . . ,fk.(a))l = I-T +(./k,(a), .. ,k ( 4 ) . *
Suppose conversely that F T + ( f k , ( a ).,. . ,fkn(a)).There exists then an element xr of F such that IT &(zr ,f k , ( a ) , . . . , f k , ( a ) ) . Denoting byfi any function of F' such that f l ( a ) = zl we obtain the right side of the equivalence (1) and hence the desired result follows. (We have tacitly assumed here that +1 contains the variables xL freely; otherwise the equivalence which we have to prove is trivial.) Proof for the formula &* is entirely similar. Theorem 3.32 which we have thus proved discloses the following relationship between the formulas and 6:If + says that elements 5 6 , , . . . X k m of F have a property P , then 6 says that functions fk, , . . . , fkn of F' and element a of I are such that f k , ( a ) , . . . , fk.(a) have the property P . As a corollary we obtain from 3.32 the following theorem:
+
+
THEOREM 3.33. If i s true in T , then every a from I satisfies 6 in S and S*; if+ i s false in T , then no a from I satisfies 6 in S and 6*in S*.
'6 in
We pass now t o the formulas {+; i, h ) and {+; i, h ) * . From the definitions we obtain immediately:
THEOREM 3.34. For each formula + of T (+; i, h ) i s a formula of S and a formula of S*. The three formulas have the same free variables.
\+; i , h}*
THEOREM 3.35. Zf + i s a formula of T with the free variables xil , . . . , xk, , if f k , , . . . , f k " t F', j:, , . . . jk,, t *F: , ami i f A or A* are sets of those a t I for which
. , fk.(a)) or ~-T+Cfk*,(a), . . . , fkn(a)),then Fs { $ ~ ; O , h ) C f k, ,. . ( A hasatkasthelements], Fs.(+; 0 , h)*Cfzl, ,fkJ = {A* has at least helements), Is ( + ; l , h ) ( f k ;.. , ] f k , , ) = ( A hnsexact2yhebments), Fs.($Z; 1, h)*(f:, , . * . ,I;,,) I (.4*has exactly h elements).
krb(fkl(a), * .
1 . .
PROOF. We shall consider only the first equivalence since the proof of the remaining three is entirely analogous.
230
FOUNDATIONAL STUDIES
--
WI,
13
From the definition of satisfaction we obtain Fa (4; 0, h )&, * ,fh) ( M i e a s e t X , c I s u c h t h a t a c X , = , Fa6(a,fkL,* - . ,jh)and t.aZ':(X,)J. From theom 3.32 we infer that
.
[a e X. = Fa &a, B, , * and from theorems 1.31 and 3.11 that l a ~L':(x,)
*
,fs>l3=(X,= 4
= (x.ha^ ai hast h ekment.8).
The two equivalences entail the daaired result. Theorem 3.35 may therefore
be considered ss proved.
If 4 hm no free variablea, then A and A* are either void or equd to I according as t$ ia fake or true (see theorem 3.33). From 3.35 we thus obtain the following corolhuy: THEOREM3.36. If 4 is a formula of T Saiuloutfree variablea, then (4; i , h ) and (4; i, h)* are tnre only in m e 8 ezha3ifedin the following table:
true (6 66
false
I
finite with n elements 66
66
66
infinite arbitrary
66
I
0
4n
0 arbitrary
arbitrary 0
1
n
WI, 14
231
O N DIRECT PRODUCTS OF THEORIES
THEOREM 3.47. If 61 , 43 , . - . , +,, are pairwise contradictory to each other and
= 1, h = 0,1,2, . . . ,then there ezist integers k , Zf) and indices j F ) ( v = 1,2, * . . , k , p = 1, 2, * - * ,n)suchthat
i = 0 or i
Theorem 3.47 follows directly from 3.46 by induction on n.
THEOREM 3.48. If h > 0, then
h--1
-{+; 1, h~
Moreover
+-+a*
mi+; 0,OI
C (+; h O
*a
1, .I* v {+; 0, h
{+; 1 , O I
- {+;
-{+;
O,O]* +-+a* {+; 1,O)*
-(+;
1,Ol H a 1,Ol* -a*
-(+;
i+;0,11,
-
+ 11.
1,11, {+; 1, I]*,
{+; 0, 11.
I n the next theorem, which will be the last of this section, we shall use the accent to denote the operation of relativization of quantifiers to ZI (compare section 1.1).
232
Wl,
FOUNDATIONAL STUDIES
...
15
THEOREM 3.49. Let 41, , +,, be fomnulus of T , x,, , -.-, x,, all varinbles which ate free in at least one of the fonnulas 4, ,and X k l , . ,Xk,, variables different from Ule variabfes x., (p = 1,2,. , 8 ) . Under these (LBsLcmptio7Es
--
--
PROOF. We shall consider only the first part of the theorem because the proof of the second is exactly the same. Because of the presence of the factor
I
2-1
Fl(x,,) the left side can be satisfied
- -
only by sequences whose r,,-th terma (p = 1 2, . , s) are elements of F'. Also the right side can be satisfied only by such sequences. Let fil , . ,fr, be elements of F' which satisfy in S the left side of the equivalence given in the theorem. This holds if and only if there exist subsets x k l , * * * X, of I such that a P Xb,
(2)
u t-8
&(a, fr,
j
* * *
j
k8 XL ti1.-..,i,, k1,-...km(Xk1
(3)
fr,)
= 1,2j
(V
'.
*
a),
Xk.).
*
--
Denoting by A, the set of all a t I for which kr +.(jrI(a), ,f,,(a)) and using theorems 1.31, 3.11, and 3.35 we infer from (2) and (3) that Xb, = A, for Y = 1, 2, . . , n and that the intersection Af' n A: has at least h or exactly h elements according as i is 0 or 1. Conversely (2) and (3) follow from these cmditions. Observe now that the intersection A f l n . . n A'^ , is . precisely the set A of +f'(j,l(a), .- . f,,(a)). Hence (2) and (3) are equivthose a t I for which FF nI"-I alent to the statement that A has a t least or exactly h elements, i.e. according to theorem 3.35 to the condition 18 { i, h ) . Theorem 3.49 is thus proved.
n ---
-
-
nr-l+fp;
Theorems on elimination of quantifiers." 41. Classes X and P.We denote by X the smallest clam of formulas of S which contains all the formulas {+; i, h ) and contains I e whenever it contains and e.
+
+
11 For the origin of the method consisting of successive elimination of quantifiers CODpare Tarski 191, p. 50, note 11.
WI, 16
233
O N DIRECT PRODUCTS OF THEORIES
Replacing S by S* and (+; i, h } by {+; i, h)* we obtain the definition of the Clsss THEOREM 4.11. Each f o r n t h of the c h s X or X* is equivalent (in S or in S*) b a to&d sum of prodwta of the form
x*.
fi b ; i , ,h.1
fi {+,;i,) h.l*.
or
.-I
u-1
PROOF. Using the well-known theorems on the conjunctive normal form of expressions of propositional calculus we prove easily that each formula of the class X is equivalent to a sum of products of the form nI”-1 ;( i.)4h.}”, . where the indices j , may be either 0 or 1 (the upper index here replaces the negot .on sign; compare section 1.1 of $1). Using lemmas 3.41 and 3.48 and the distributivity of logical multiplication over addition we can transform each product of the given form into a s u m of products in which all the j , ‘s are 0. Proof of the second half of the theorem is similar. Throughout the rest of this section we shall consider n arbitary but fixed formulas +1, . . * , of T and denote by $1, . J/*the m = 2“ products with i, = 0 or 1 .
+.
--
THEOREM 4.12. Formulat, $1
)
. . * , $,,
n:-l&’
)
are contradictory to each other.
n:-l
THEOREM 4.13. Each logical product (4, ; i , h,) is either contravalid or equivalent in S to a sum of products of the jorm (G,, ;j,, ,k,,}. Each logical product {$,I ; i, h.)* is either contravalid or equivalent in S* to a sum of prodvets of the jorm ($,, ;j,, , k,,)*. PROOF. Each 4, is equivalent in T to a sum of some #,,’s, namely t o the sum of those $,, which contain the factor 4; . Applying theorems 3.47, 3.41, and the laws of distributivity we transform the product (4. ; i . , h,) into a sum of products whose factors have the form {$,, ;j,, , k,,).Each of these products may contain several factors with the same $, , e.g. ($,, ; j: k; I ( $, ; 2; k: ) , . . . Theorems 3.42, 3.44, and 3.45 allow us to reduce the numbers of these factors to one or to infer that their product is contravalid. If all products are contravalid, then the whole sum is also contravalid. Otherwise we may omit the contravalid products and obtain a sum of the desired form. The proof of the other half of the theorem is similar.
n:-l
)
m--l )
)
)
)
4.2. Theorem on elimination of quantifiers in the theory S. We assume in this section that the formulas 41, . . * 4. all contain x, as a free variable. I t is evident that xI is free in each product ;$ : let XL, , * . , Xk, be the remaining free variables of these products. We shall prove the following theorem: )
n-,
THEOREM 4.21. Let 0 5 h 5 m and let ll , . . . , 1, be integers >= 0 . Then there exists in X a formula A with the free variables x k , , . . . Xk, such that )
234
FOUNDATIONAL STUDIES
PROOF. We put (2)
e.
=
(ax,)+. for
Evidently we have
Y
= 1,2,
.- .*,
m,
=
Y = 1,2,'... ,m. (11) X." YP" Since by theorem 4.12 the formulas (I. are contradictory to each other, we obtain furthFr
(12)
X,,and X., ape disjoint for Y Z p,
I
S m,p 5 m.
Wl, 18
From (7) and theorem 3.35 we obtain (13)
235
ON DIRECT PRODUCTS OF THGORIES
- ,It), ( p = h + 1, . - - ,m).
X., has at least 1, elements
~
(14)
(Y
X,, has exactly 1, kbments
= 1,2,
* *
Finally we prove that (15)
the set
Indeed, if
-
Z-
c
n:d+l(Z - X , J n (I - W,)
W,
, then
is void.
no b c F satisfies the condition k r
+
m
IT
h + l
(-$,,)
&(a,), ,fkp(@), b) and hence for b = f,(@) thereexists a p ( h 1 5 p 5 m) such that $,,cfk,(%), . . , f&d, f.(aa)>, i.e. aa t X,, . Formulas (11)-(15) entail that sets (8), (9), and (10) satisfy in C the formula in the brackets ( } on the right side of (4). Hence k c B(Y,, , . . , Y , , W,) and therefore in virtue of theorem 1.31 a
FSB'(YP1
(16)
9 * *
Y,, W J -
3
3
From (2), (9), and theorem 3.32 we obtain equivalences (17)
a0 c
Y,,
j-8
a,(&,
fk,
,-
*
,fkJ
(Y
= 1,2,
- . ,m ) *
for each a,, t I. Similarly from (3), (lo), and theorem 3.32 follows the equivalence %6 w q k S fi(% 6,, * * * ,fkp)* (18) Since evidently Fa I1(Y,,) for Y = 1, 2, ... , m and k e 11(W,), we infer that the sets (9), (10) and functions f k , , . . * ,fkp satisfy in S the formulas j
II(X,,), !xg)tI(xo) 3 (xo E xp. 3 adl, Il(Xq), (xo)[I(xo) 3 (xo E X , This result together with (16) entails that
r of equation (5).
fk,,
- .. ,
fk,
= Ql.
satisfy the formula
Suppose conversely that f k , , . . ,6, satisfy r. We infer first of all that f k , c F' for p = 1, 2, . . , p and furthermore that therg exist sets Y,, and Wp which fulfill the conditions (16), (17), and (18). From (17) and (18) we infer that Y,, and W, are identical with the sets defined in equations (9) and (10). From (16) we infer the existence of sets X., which fulfill the conditions (11)-(15). X , , ,then by (11) aa e Y p ,and hence by (9) there exists in F an element If b = b.(aa) such that
-
+
(19) We have (20)
k r $'*cfk:(@),
a,(%)
* * *
,ft,(%),
# b,(aa) for Y #
P, Y
b*(%)).
6 m,~.rS m
because of the theorem 4.12. If a,, belongs to no X., , then by (15) aa t W , and therefore by (10) there exists a b = b*(aa) such that (21)-
not
I T $'#cfk,(%),
' *
'
9
fk,(%),
a*(%))
236
FOUNDATIONAL STUDIES
+ ...
forp = h 1, , p. Define now fr aa foltows: fr(aO) = b,(ao) for a0 c X,,
(22)
f,(a0) = b*(&) for
(23)
c
,
Y
= 1,2,
.-.,m,
n?(I - X,,).
Note that this definition is correct because of (12). We shall now show that There exist at least 1, elements a0 t I such that (24)
F T hCfk,(@),
.’ ’
f*(%)) for
fk,(@>,
” = 1,2, . *
Thcre exist exactly 1, elements a0 t I such that (25)
IT
. ,fkJao>,
+*Cfiil(ao>,
for P
fr(a0))
* *
=
h
+ 1,
*
h.
. ,m. *
(24) follows immediately from (19), (22), and (13).
+ - -
As to (25) we fir& infer from (19), (22), and (14) that for each p = h 1, * ,m there are at least 1, elements with the required property, namely all the elements of X., .To prove (25) it is therefore sufficient to ehow that if a0 does not belong to X,, , then not I-T $,Cfk,(ao), .. * ,fk,(ao), fi(aO>). Indeed if a0 t X,, with Y # p , then
I-=
Mfk,(a~),
*
* 9
f k p ( ~ > ,fr(N>)
and therefore not Ic;(fkl(aO), . * * , fkp(ao), f,(a0)) because $4 and h are contradictory to each other. If a0 does not belong to any X,, ,then we apply (23) and (21) and obtain directly
not
FT
!J%d.fkl(@),
* * *
9
fkp(@),
fi(a0)).
(25) is thus proved.
Using theorem 3.35 we infer now from (24) and (25) that satisfy the formula
fkl
, - - ,f ~ ,,fr *
and hence that fk, , . . . ,fkp satisfy the left side of (6). Equivalence (6) is thus proved. In order to abbreviate the following formulas we shall put p,+1 = q and &,,+I = 0.We can then rewrite equation (5) in the more concise manner:
Observe now that theorems 1.31 and 3.12 entail an equivalence of the form
WI, 20
ON DIRECT PRODUCTS OF THEORIES
237
Substituting the sum on the right side of this equivalence for B' in I' and using the laws of distributivity we obtain the equivalence
Theorem 3.49 ,&OWE that the right side is equivalent in S to
Taking this & for the formula A we get the equivalence (1) and theorem 4.21 is thus proved since A evidently belongs to the class X. A s an easy corollary of theorem 4.21 we obtain the following theorem on elimination of quantifiers:
THEOREM 4.22. Each formuh of the fm
[
fi {b;i,, h.) ]
(34 FdxJ .
-1
i a either contravalid or equivalent in S to an expression of the class X .
PROOF. According to theorem 4.13 the given formula is either contravalid or equivalent in S to a sum of formulas of the same form as the formula on the left side of (1). Applying to these formulas the theorem 4.21 we get the desired reault. 4.3. Theorem on elimination of quantifiers in the theory S*. Dealing with the theories *T: and S* we shall from now on assume that e is a constant function with the (unique) value eo which is definable in T . This assumption will be made once for all and we shall not mention it in the formulation of the subsequent theorems. The proof of the theorem 4.21 does not work in the case of the theory S*. The chief obstacle is the construction of the function jrsatisfying the conditions (24) and (25). The function defined by the formulas (22) and (23) does not, in general belong to *F: because there is no remon why the elements b.(ao) and b * ( ~ should ) be identical with eo for almost all values of G . Only in case h = m d o e the proof given in section 4.2 work for the theory S* as well. Indeed in this case there is no set W , and we can prescribe the values of fr(m) quite arbitrarily apart from the arbitrarily chosen 1, elements of X,,(Y = 1, 2, - . ., m). Hence we can in this case arrange the definition so that f, c *F: . I n the general case we formulate the theorem as follows. We denote by the double accent I' the relativization of quantifiers to F. This operation applied to expressions of the theory T yields expressions of the theory S*. THEOREM 4.31. Let 0 5 h 5 m and let ll , . . * , 1, be integers L 0. There exists
238
WI, 21
FOUNDATIONAL STUDIES
x*
then a j o r m u h A* o j With the free variabks x k l T without free variables, such that
,
-
* *
, x b ,and a
fm& P Of
I
.
PROOF. Denote by Z a formula of T defining eo We may evidently suppose that Z does not contain the variables x, , x k , , .. , 4, Further we denote by s, and Z k # formulas resulting from by substitution of xr or x k , for the free variable of Z. Since in the case h = m the proof given in the section 4.2 can be carried over without change to the theory S* we &all consider only the case h < m. Let P be the formula
-
.
Define O., Q, and B exactly as in section 4.2 (equations (2), (3); and (6)). Further define r as in section 4.2 but adding an asterisk over 6,, h, and F1 We shall show that
.
-
The proof that if j k , , . . ,j k , satisfy the left side, then they satisfy r does not differ from the proof given in section 4.2. We have only to show that P” ie also satisfied. Since PI‘ has no free variables, it is satisfied by a sequence if and only if it is true in S*. Using theorem 1.32 we reduce our problem to the proof that P is true in T.This will be shown indirectly. Suppose that P is false, i. e., that there is a p such that h 1 6 p 6 m and that for each 5, , a&, , . . , zkp in F the formulas tT Z,(z,), C-T E k I ( Z k I ) , * , bT EkP(zkp) imply kT #,,(%I, , . , %kp , z,). Because of the meaning of E we obtain therefore IT+ p ( e ~, . . , eo). Since there are infinitely many a0 in I such that j k l ( a o )= . . . = j k , ( a o ) = ji(ao) = eo (for each j r t *F:) we infer that the set of ao t Z such that bT +,,(jk,(%), . . . ,j k p ( a o ) , j,(ao)) is infinite for each j I This however contradicts the hypothesis that there exists in *F: an fr which together with j k , , . . ,j k psatisfies in S the formula (+,, ; 1, L)*.Hence P” is true. Suppose now that j k , , . . ,f b satisfy in S* the conjunction pr’ I’, i. e., that P” is true in S* and f k , , , j k psatisfy r. By theorem 1.32 P is true in T. From the second assumption we infer, as in section 4.2, that j k , , * ,f k , are elements of *F: and that there are subsets X , , , Y,, ( v = 1, 2, * * * ,m), and W, such that the conditions (9)-(15) of section 4.2 are satisfied. We define b,(@) for t X , , ( v = 1, 2, * * , m) exactly as in section 4.2. The definition of b*(ao) will, however, be slightly different. We define namely b*(acj as equal to eo if eo satisfies the conditions
+
-
-
-
.
-
.
--
not I-T + P U k I ( a o h ’ ,j*,(ao), eo) for p = h 1, . . . , m,and as any element such that (21) is satisfied in the remaining cases. The existence of b*(ao) follows as in section 4.2.
+
WI, 22
239
ON DIRECT PRODUCTS OF THEORIES
We now choose inX., an arbitraly subset . ,m ) and put
( v = 1,2,
x., with exactly 2,
f.(ao> = b . ( ~ for ) Q Q Z. , ( v = 1,2, fr(ao) = eo for t X,,,- X,,(v = fib) = b * ( d for aot n : ( I - X.,).
elementa
. ,m), *
1,2,
*
- - ,m),
We prove, aa in section 4.2, that the function f r defined in this way satisfies the conditions (24) and (25). All that remains to be shown is that f , L *F: , i. e., that fr(ao) = eo for almost all Q . Since the sets are finite, an infinite number of different ao's with I,(&) # 6 could exist only if a*(&) were different from eo for an infinite number of a0 . Since, however, P is true in T,we have kr -#,(eo , . . . , eo), and hence for each a, such that fk,(@) = -. = f k p ( a c ) = eo the value of b*(ao) is eo There) eo must be finite since for almost all a0 we fore the number of with b * ( ~ # have the equalitiesf k , ( @ ) = * = f k p ( & ) = eo . Equivalence (27) is thus proved. The transformation of I? into a formula A* of X* is then effected aa in the proof of 4.21, and we arrive thus at the eqiiivalence (26). From theorem 4.31 we obtain now easily the following theorem on elimination of quantifiers for the theory S*:
a,,
n:+ '~
.
THEOREM 4.32. Each formula of the form
.
is either contravalid or equivht in S* to a sum C EP: ~ A: where A: ate formulas of X* with the free variables xkI , . . , Xk,, ,P. are formulas of T withaut free variables, and the double accent denotes the operation of relativization of quanti&rs to F. Proof is the same as for the theorem 4.22. $36. Corollaries to the theorems on elimination of quantifiers. 6.1. Relations between the theories TI and S. Although neither the theory S is an extension of T' nor the theory S* an extension of *T:, it is nevertheless possible to expresa in S each proposition expressible in TI and in S* each ,reposition expressible in *T:. This will be shown in the theorems proved in this and in the next section. THEOREM 5.11. To each formula 6 of TI there exists a formula A of X such that (1)
4
TI-8
A
PROOF.We proceed by induction on 4. If is the elementary formula = (-4; 1, 0) and have then equivalences which hold for arbitrary functions f k l , * . ,f k , , :
R:(xLI, . * . , xLJJ then we put A
$J
-
240
1331, 23
FOUNDATIONAL STUDIES
Suppose that the theorem holds for formulas $1 and & and let A1, A, be two formulas of X such that +i
6 TI * a 4.
Ai
rr *s
It is then evident that A = Al I A2 has the same free variables as $ = 41 I & and satisfies the equivalence (1). It remains to consider the formula $ = (3xr)$1. From theorem 1.35 it follows that 4 TI ++a Qxr)[Fl(xr) * &I. Using theorem 4.11 we obtain an equivalence of the form
(2)
u
~1 *a
0.
C II
u-18-1
~6
,h = . ~ )
where 8.8 are formulas of T. Using elementary logical transformations we obtain (3)
(3xr)[Fl(&).AJ
++a
2 (Ix,) [Fl(xr). I?
6-1
u-1
(&p
;LB, hu.~}].
Factors [ ~ 9 ~ ;, 8i . , ~bud) , not containing the variable xr as free can be taken outside the scope of the quantifier ( 3 ~ ~The ) . remaining expression, according to theorem 4.22, is either contravalid or equivalent in S to an expression of X containing the same free variables. Since X contains contravalid expressions with arbitrary free variables (e. g. expressions of the form (6=+; 1, 0 ) )we conclude that the right side of (3) is equivalent to an expression of X with the m e free variables as 4. Theorgm 5.11 follows now immediately from (2) and (3).
We can now formulate our main result concerning the theory of strong powers:
THEOREM 5.12. To each formula + of T‘ there exist integers h, 1, h,,, , indices i.+ ,and mutually contradictory formulas 8, of T ( v = 1,2, . ,h, p = 1,2, ,1) such that IT 8, and
- -
--
Each 9, has the same free variables as 4. PROOF. By theorems 5.11 and 4.11 there exist integers u, v o ,w.,b, and indices j . , ~aa well as formulas $#, such that (4)
n.,s
Consider the I = 2 D 1 w * ~formulas ~~w’U where each i o . 8 is either 0 or 1; denote them by 81 , 8, , . . . , el . Each $.,b is equivalent to a sum of some of the formulas 8i and since these formulas are contradictory to each other we can apply the lemma 3.47 and represent each formula ($a,8 ; j a , p , w , , , ~ a]s a sum o€products of the form n7{07 ; i, , m , ) . Using the laws of distributivity we finally tram-
f331, 24
241
ON DIRECT PRODUCTS OF THEORIES
form the right side of (4)to a ~ u m of producta of the same form. I n general each product will contain €actors with the same e, but a succeeive application of theorems 3.42-3.45 allows us to get rid of such repeated factors. If not all e,, occur among the terms of any product, we may add valid fsctors of the form (e,, : 0, 01. Since the formulas 01, , el evidently sstisfy the condition ka 8, , we see that the ~ u m of products thus obtained satisfies the contention of the theorem. From (4) we infer that the totality of the +.# has the same free variables aa 4. Since each 0, is a product of these formulas or of their negations, we infer that each 0, has the same free variables as 4. Theorem 5.12 is thus proved in all details. It is important to observe that the syntactical and semanticd existential statements made until now in the paper were all &ective. Theorem 5.12 admits therefore the following methodological amendment:
-.-
THEOREM5.13. The intqers h, 1, h,,#, indim i., and formulas e, of the lheotem 5.12 are effectively calculable if 4 i s egectively given.
*e
6.2. Relation between the theories and S*. We shall now derive resulte analogous to those of section 5.1 for the weak powers. To formulate our theorems conveniently we shall use the double accent ' I with the same meaning as in section 4.3.
THEOREM5.21. To each formula 4 of *T: there exist formuha A:
S*and formulas PI, - - . , Pb of T without free variables such that
, . . , :A
of
--
PROOF. If 4 is the elementary formula R [ ( x k , , ,x k p , ) then we put K = I, :A = { -$; 1,0)*,P1= l' v -r where I' is an arbitrary formula of T without free variables, and prove exactly as in the proof of 5.11 that equivalence (5) holds. Suppose that the theorem holds for two formulas 6 and A . We have then
41 *T!*-)B.
c P:.A?,
9s *rf*m
C G-E:
k
-1
I
P-1
and it follows by elementary logical transformations
n (-Q," WE:). The right side can evidently be brought to the form r -1P:. the theorem. k
k
41142*l'fH8*
II (-P: -1
V
-A:)
V
V
lul
A? required in
Finally if the theorem holds for a formula 4, then it holds also for the formula (3x,&. This is proved as in theorem 5.11 but using theorem 4.31 instead of 4.22. Theorem 5.21 is thus proved.
242
WI, 25
FOUNDATIONAL STUDIES
From theorems 5.21 and 4.11 we obtain aa in section 5.1 the following two theorems: 5.22. To each formula #I of *Ti there ezist integers h, 1, h., THEOREM i,, ,and formulas , P., of T such that:
P,,hasnofreevan'ables
( v = 1,2,
..-, h ,
The formulas 0, are contradictory to each other,
p = 1,2, /-r
, indices
... , I ) .
ep, and
Each 0, has the same free variables as 4. THEOREM 5.23. Zdgers, indices, and f o r n u b whose existenee is stated in the theorem 5.22 are effectively calc212able for each given 4.
6.3. General theorems on power theories. Theorems 5.12 and 5.22 together with their methodological amendments 5.13 and 5.23 make it possible to prove some general theorems about power theories. We begin with the following theorem : THEOREM 5.31. If T is a &&able theory, then TI and *T: are also decidable. PROOF. Let 4 be a formula of TI without free variables and let h, 1, h., , i.,,, , 8, be determined according to the theorem 5.12. I n order that 4 be true in T' it is necessary and sufficient that at least one of the products E-1{$, ; i,.,,,A,,] be true in S(cf. theorem 1.34). This condition is satisfied if and only if all factors (e, ; i,,, , h,,,,) are true in S. Using theorem 3.36 we reduce this condition to a number of equalities and inequalities between the integers h,,pand the cardinal number iof I and to a number of conditions of the form 6, i8 true or 6, is false. Since T is decidable we can check whether these conditions are or are not satisfied, and since the numbers h,,,, are effectively calculable we can decide whether they are greater than, less than, or equal to i.Hence we can decide in a finite number of steps whether 4 is true or not. Proof for the theory *Tiis analogous and makes use of the fact that the truthvalues of formulas P:, can be calculated in a finite number of steps according to the theorem 1.32 and the assumption that T is a decidable theory. THEOREM 5.32. If 4 is a foTmula of TI OT of and if for each Pnite set I 4 is true in TI, then 4 is true in TI and in *T:for aTbitTUTy Z. PROOF. Determine h, 1, h",,,,i., , and 8, as in theorem 5.12 and choose a finite set Z such that the number of its elements be greater than h,,,, for all possible values of Y and p . According to the assumption of the theorem we have (e,, ; i",,,,It",,,]for p = 1, 2, . , l . t r r 4 and hence there is a Y S h such that From theorem 3.36 we infer that this is possible if and only if each p satisfies CY of the following two conditions:
-
0, is false and h,,,, = 0,
e, is true and i.,,, = 0.
WI, 26
243
ON DIRECT PRODUCTS OF THEORIES
From the same theorem 3.36 we infer that if these conditions are satisfied, (0, ; i,,,, , h,] holds for arbitrary infinite I. This proves the theorem then for the strong power TI. Proof for the weak power is exactly the same. We shall now determine relations and elements of F' and of *F: definable in
T' and in *Ti.
We shall call partition of a set A any decomposition of A into a 8um of disjoint sets (some of which can be void). A partition A = A1 A , A I is said to satisfy the condition [0, h]j if A j has at least h elements and the condition [I, h] if A { has exactly h elements. The class of all partitions of a set I into I G( = 1, 2, .-., I) will be denoted sets satisfying the conditions [i,,, by PE:[i,,, &I. L e t I - Z l u I , u . - .UIIbeapartitionIIofZandF" = F; U F ; UF; a partition r of the n-th power F; = F X F X .- X F of F . We denote by [ n : ~the ] set of n-tuples cfi , . ,fn) of functionsf i e F' such that a t I j always implies Vl(a), -..,fn(a)) c Fj"b= 1, 2, , 2). Similarly [n:& denotes the set of n-tuples Cfi , * ,fn) of functions fi c *F: with the same property. After these definitions we can formulate the following theorem:
u u -- u u
-
--
-
THEOREM 5.33. A necessay umditzhn for an n - k m d relation R between the elements of F' (or between the elements of *F:) to be definable in TI (or in *T:)is urat there ezist a partition r F" = F ; U * * * U F ; , integers h and h,, ( v = 1,2,.
i.,$
(v =
. - ,h, p = 1,2,.. . , Z),
1,2,
such that R is the union of sets
..-,h,
p =
and indices
. - -,I )
1,2,
[ n : ~(or ] of sets [n:r]:)where lI runs over the ael U L ck,, h,,I.
PROOF. Suppose that R is definable in T ' and that #I is a formula with n free variables which satisfies the equivalence
..*
~ R I-r~dCfi, S *.. ,fn). Let h, 1, h,, , i,, , and 0, have the same meaning 88 in theorem 5.12. Define F,,as the set of (zl , * , 2), e F" such that IF O,,(z, , ,z,,). Since kF ELl e, and 0, , .. ,e1 are contradictory to each other, we see that F; . u F; is a ,fn) e R, then there is a Y d h such that partition T of F". If (fl ,
cfi,
-
(7)
,fn)
- -
---
-
l-n (0,
;i.+,h.,)(f1,
... ,fn)
for cc = 1,
-..
u --
.*.
,I.
. u --- u
Denote by I,,the set of a E Z such that Vl(a), ,fn(a)) e F," Since the sets F," are disjoint and exhaust the whole F it is clear that Il It is a partition n of I. From (7) and theorem 3.35 we infer that n.satisfies the conditions [&.,, , h.,L for p = 1, 2, *.. , I and hence n e =;[& , h,,]. , Finally from a c I,,it follows that &(a), Conversely, assume that ( j 1 ,
- - ,fn(a)) e F,"and hence - - ,fn) [n:r)where II *
*
t
* * ,fn) c [n:r]. c;[i,,, La].The
(f1,
t
244
WI, 27
FOUNDATIONAL STUDIES
-
set I,,of those a E I for which IT O,,(fi(a),* ,fn(a)) satisfies then the conditions [i.,,, , h,,,,],,for p = 1, 2, ... , 2 which proves, accordmg to theorem 3.35, that (e, ; i.,,, , hv,,,)for p = 1, 2, . . . , 2 and hence that k p r &fl, .. ,f,,), which proves that (fl, - - ., f,,) e R. This proves the theorem for the case of the strong powers. The proof for the weak powers is similar. We have only to observe that in the equivalence (6) of the theorem 5.22 each true P:,,, can be dropped and that if (b is not contravalid, then for at least one Y S h all formulas PE, , ... , P:, must be true. As a corollary we obtain now:
-
THEOREM 5.34. If a function f e F' is definable in TI,then f has a constunt value definable in T;iff o *Ft is definable in *T:, then f = e. PROOF. Let K be a unit class consisting off alone. Iff is definable in TI,then K is also definable in T' and hence K has the form described in theorem 5.33 for n = 1. Since K has only one element, it follows that if ll is in the union U l l PEZ$,,,, , h,,,,],then [ n : ~contains ] exactly one function, the same for all n. If n contained at least two summands Ij , I& for which the corresponding summands Fi , F: were non-void, then [n:F] would evidently contain more than one function (because it would be possible to construct a new function from a given one interchanging two elements of I: and It). Hence each ll contains exactly one non void Ii such that F: is non void. Since, however, each function ] be defined on the whole I , it follows that Ij = I. If Fi were not of [ n : ~must a unit class, it would be possible to construct a t least two functions fl ,f 2 such ] fz t [I:.]. Hence Fi contains exactly one element z which is that fi t [ n : ~and therefore definable in T, and the function f has the constant value x. The proof for the theory *Tiis similar. Since each f e *F: has the value eo for some arguments, it must be identical with a function which is identically equal to eo and hence f = e. We shall now show by means of a counterexample that the necessary condition given in theorem 5.34 is in general not sufficient. This result implies in particular that the theorem converse to 5.33 is false in the general case. Our counterexample is particularly simple. We take as T the theory of a cyclic group G of order 2 and as I a set with 2 eIements. TI is then the theory of the four group G x G. If we denote by 0 and 1 the elements of G, by the groi~poperation in G, and by 0 the group operation in G X G, then the elements of G x G are pairs (O,O), (O,l), (l,O), (1,l) and the relations defining @ are
+
(0,O) 0 (0,O) = (1,l) 0 (1,l) = (O,O),
(0,l) 0 (1,O) = (1,O) 0 (0,l) = (1,l).
+
The element 1 is definable in T since (x = 1) = (z z # 2 ) . However the element (1,l) which can be considered as a function of G' which takes on identically the value 1 is not definable in TI. Indeed the four group G X G admits automorphisms moving (1,l) into any other element of the group different from (O,O), and hence if (1,l) satisfies a formula (b, then also the elements (0,l) and (1,O) satisfy (b.
WI, 28
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ON DIRECT PRODUCTSOF THEORIES
The theorem converse to 5.33 would evidently be true if the following condidition were satisfied: for each formula of the form {4; i,h] or (4; i,h)* there exists a formula 11 of TI or of *Tisuch that
+
or
11 TI++# (4; i, h )
(+; i, hj*.
*=!-a*
The counterexample given above proves that in general power theories do not satisfy this condition. It is fulfilled however in many important cases, e. g., in cmes considered in examples 4 and 6 in $2. We give still another application of theorems 5.12 and 5.22.To obtain it we must first discusa a little more closely the equivalences formulated in these theorems. Suppose first that one of the 0, is contravalid. If h.,, > 0, then evidently no functions cfi , * * ,fn) satisfy the formula { 0, ;i,, ,h.J or (0, ;i,, , h.,,]!. The products (0, ; i,,, , h,,,) or ~ L(0, I; i,,, , h,,l* corresponding to such values of p and Y can therefore be omitted and the equivalence remains true. We show similarly that a product ~ LP:, I (0, ;,.i , h,,]* can be omitted in (6) provided that one of the sentences P,.l, PV,¶, . . * , P,,t is false. If, on the contrary, all these sentences are true, there is not need to write them out as factors in the product. Finally we can omit the products corresponding to such values of Y for which h,,, is greater than the cardinal number of I, as well as those for which = a,,) = - .. = i,,r = 1 and h.,, is less than the cardinal number of I. It follows, namely, from theorem 3.35, that if these conditions are satisfied, then no n-tuple of functions satisfies the product in question. We see therefore that equivalences proved in theorems 5.12 and 5.22 can be replaced by the equivalences,
-
El
.
ELl
ELl
where the following conditions are satisfied:
(10) If 0, i8 contravalid, then h,,
= 0 fOT p =
1,2, * *
- ,1;
Y
=i
1,2, * * * ,h.
I
(11)
Ifi..l=i..¶=
= i.J = l,then~h,,,= i.
c h.,
P-1
1
(12)
P-1
rS;
t
(where t is the cardinal number of I). Observe however that equivalences proved in theorems 5.12 and 5.22 were effective, i. e. their right sides are finitely calculable for each given 4, whereaa the right sides of (8) and (9) are in general not calculable unless T is decidable. Note that the proof of the formulas (8) and (9) works also in the case where all the products considered there are to be omitted. 4 is then contravalid in T’(or in *Tt).
246
WI, 29
FOUNDATIONAL STUDIES
We now prove the following theorem:
TFIEOREM 5.35. If TI and Tzare two theories with the same jiefiekE, then the basic relaticma of the theory T:(or *T:) are dejnabk in the themy T:(w *Ti*) if and a l y if the basic relations of the theory TI are &$nab& in the theoty Ts . PROOF. Sufficiency of this condition is evident. Necessity will be proved only for the strong powers because the case of weak powers can be treated analogously. Let R be a basic relation of the theory TI. Suppose that this is an n-termed relation. The relation R' between the elements of F' is defined by the equivalence
..
R'cfi
fn(.))l.
3 RCfi(a), * *
(a)[a
fn)
and it is a basic relation of the theory Ti. Let us assume that this relation ia definable in the theory Ti.According to (8) we have then (13)
(a>Nfi(a), . * * ,fn(a))
c h
f
t
k~h l -1
(0, ;&,
,h,,1(It,
* *
-
,fn)
where el , * , 61 are formulas of Tt, contradictory to each other, satisfying the condition k T t e,, , i., , h., are integers satisfying the conditions (10)(12), and S is a theory corresponding to Tt according to definitions formulated in $3. If R is a void or a full relation, then it is evidently definable in Ts . Assume now that R is neither void nor full, i. e., that there exist elements 21 , . * . ,z,,of the field F of TI such that R ( z l , * , Zn) and elements y1 , * , yn of F such that - R ( ~ I * * ,Yn). Consider those formulas 0, for which there exist at least one n-tuple y~ , * * ,y,, O,,(t/t, * ,yn). We can suppose that these such that -R(yl , . , y.) and ,y,,) formulas have indices from 1to s where s 6 1. Since each n-tuple (gl , must satisfy in Tpexactly one formula 0, it follows
ELl
a
-
1 .
3
kTt C e,(yx, .- . ,v.). PI
We shall show that h,, = 0 for
(15) i,, = 1,
--
a
--- ,
-R(Y~,
(14)
-
-
-
p
= 1,2,
-..,s,
v
-
1,2,
,h.
Suppose the contrary and let vo , h be integers such that JLC, 5 a, vo 6 h and either i.,,,, = 0 or h.,,, > 0. Let (yl , * , yn) be an n-tuple such ~,,,(YI, ,y-1. that -R(YI, - ,3-1 and Conditions (10)-(12) entail the existence of a partition II
-
-
z = ZlU
--
uz,
such that n c P:$,,,,,, hve,,,]and such that I,, = 0 for such values of p for which e,, is contravalid. If h.,,,,, # 0, then I,,, # 0 and if h.,,,, = 0, then i.,,, = 0. If all Z,(p = 1, 2, , 1) have exactly h.,,, elements (which is possible only for finite I) and the set Z 0 is void, then we see easily that the value 0 for i.,,,,, can be replaced by 1 without altering the validity of (13); hence (15) is satisfied automatically. If not all I, have exactly h,,,, elements, then we can always
--.
247
ON DIRECT PRODUCTS OF THEORIES
suppose that I,o is non-void, because we can take one element away from an I# which h a more than h.,,, elements and add it to I,, ; the modified partition remains evidently in PGi[i,o,,,he,,]. Let b be an element of I,, . The formulas 01 , . * , 81 define a partition r
F" = F; U
U F;
of F"; we define namely F," as the set of n-tuples (21, , 2%) such that t r x O,(zl, . , 2"). Since F," = 0 implies I, = 0 according to (lo), we see that there exist functionsfl , . ,fn such that (j1 , * * * ,$1 t [II:r ] .In particular we can put fj(b) = y j 0' = 1, 2, * . * ,n ) since ( y ~ *, * , yn) t F;o . that It follows from the assumption n e Pg:[i,,,, 7
-
--
-
and hence that the functions fl , * * * ,fn satisfy the right side of (13). They do not however satisfy the left side of (13) because for a = b we obtain
wR(y1,
wR(j~(b),. . *
Yn)
+
3
fn(b)).
This contradiction shows that conditions (15) must be satisfied. ,f,, , znare such that R(zl , * , 2,) and let f1, Suppose now that z1, be constant functions such that fi(a) = zjfor each a e I. Evidently we have (a)R(jl(a), ... , fn(a)) and it follows by (13) that there is a vo 5 h such that
-
---
-
It follows in particular
t - d I I ~~,c;~vo4,hvmI(f1, ...,fJ; )I-1
i: e.;by (15), I-8 (Or ; 1, O ) ( j l , can satisfy the condition t r xe,cfi(4,
whence
FTx -Or(zl , - * - ,2,)
- - - ,fn) for p = 1,2, * *
,f&))
B =
for p = 1,2,
~ ( 2 1 ,* * * $2,)
3 j-rs f
n
-
0
.
1, 2,
,s. Hence no value of u * *
-,
8
-.. , s. We have thus shown that . . ,Zn).
dfi(~1, *
M-1
Combining this with (14) we get the equivalence
R(zI,
.
* ,Zn) 3
brx
P-1
.
-O,(zi,
* * *
,Zn)
which proves the definability of R in Ts Theorem 5.35 is thus proved. Application. Let Tz be the elementary theory of relation S in the set of positive integers and TIthe elementary theory of addition of such integers. It is easy to see that the basic relation of Tl is not definable in TIand hence in
248
FOUNDATIONAL STUDIES
[331, 31
virtue of theorem 5.35 the basic relation of Ti1 is not definable in T t 2 .Take in particular I = set of positive integers and aa e a function identically equal 0. In view of examples 6 and 7 of $2 we can express the result which we obtained as follows: multiplication of positive integers is not elementarily definable in terms of the relation of divisibility.“ The notion of a power of a theory is but a special case of the more general notion of a direct (Cartesian) product. The reader will have no difficulties in constructing this more general notion imitating the notion of direct product of groups or of rings well known from algebra. Many of our previous theorems can be extended so as to become applicable to product theories as well &s to power theories. Since however the most important result, namely the theorem 5.31, cannot bB so extended, and since no interesting examples of product theories seem to exist, we shall content ourselves with this brief remark and shall not develop the theory of direct products of mutually dzerent theories. BIBLIOGRAPHY
(11 Alonzo Church. Infroduction to mathematicat logic. Parf I. Annals of mathematics studies, number 13, Princeton 1944. [a] Felix Hausdorff. Mengenlehre. 3rd edition. J. Springer, Berlin 1935. [S] Julia Robinson. Definability and decision problems in arithmetic, this JOURNAL, vol. 14 (1949), pp. 98-114. [4] Thoralf Skolem. Untersuchungen riber die A x i o m des Klassenkalkuls und Bber “Produktations- und Summcrtionsprobleme”, welche gew’sse Klassen von Aussagen betreffen, Skriffer ufgit av Videnskapsselskapef i Krisfiania, I. klasse, no. 3, Oslo 1919. [6] -. Uber gewisse Satzfunktwnen in der An’thmetik, Ibid., no 7,1930. [S] Wanda Szmielew. Decision problem in group theoty, Proceedings of f h e Tenth Infernational Congress of Philosophy, vol. I, Amsterdam 1949, pp. 763-766. [7j Alfred Tarski. Der Wahrheitsbegriff in den formalisierten Spmehen, Studio philosophica, vol. 1 (1935), pp. 261-405. [8] -. Grundztige des Systemenkalkiils, zweiter Teil, Fundamenfamafhemaficae,vol. 26 (1936), pp. 283-301. [9] -. A decision method for elemenfarg algebra and geometry. Rand Corporation; Santa Monica, California, 1948. [lo] -. Cardinal algebras. Oxford University Press, New York 1949. UNIVERSITY OF W A R S A W
1:
This remark solves a problem proposed by J. Robinson [S],p. 101.
On a System of Axioms Which Has no Recursively Enumerable Arithmetic Model BY
A. M o s t o w s k i (Warszawa) According to a well-known result of L o w e n h e i m , Skolem, and G o d e l every consistent axiomatic system 8 based on the functional calculus of the first order has an interpretation in the set of positive integers'). Hence if A is the eonjunction of the axioms of sa) and R,,R,, ,..,I f p are the predicates 3, which occur in A , then there are relation 11)
R171 '1-
quaatiticrs
D e f i n i t i o n 1. .?$'=Ex Y
17 [ y ( m , y ) = O ] .
xo x z x o
l) Each y, in this end the immediately preceeding formula o h be replaced by a complex y,,,~,, ylk,.
...,
[48], 261
SETS D E F I N A B L E B Y M E A N S O F TWO A N D T H R E E Q U A N T I F I E R S
277
THEQREBI 1. The family {@)>,zo i s +dentioat with Pr'2). Proof. It immediately follows from the definition that z'," E PF). It remains therefore to show that for each p in CP with three arguments a function y in 0 can be found such that
zn [ @ ( P , S , Y ) = o l ~fl~[ r ( 3 , ? l ) = o ] .
(1)
P
xo x>xo
S
We define y along with two auxiliary functions taneous induction:
Y(O,Y)=e(o,o,Y),
AT + 1,Y)= ~ 4r-t 1,!/)
't(O,y)=O,
+
( 7 4 % ~ 1~ 0 )
= E't(n,y)
+1
1 7 Y) = [ U( S: ,Y) f
U(m+
z
and u by a simul-
o(O,y)=O,
+
,Y)*[I (1 r ( ~ ) ) ] @("(W) 74$,Y 1 1,Y) "r(m,y)l,
+
1 p (~-Y(w))]+ 11 11 'Y(m,?/)I-
+
743,y) .l-1~ r ( a , ? / ) l ,
'
To explain the meaning of these definitions let us arrange all the pairs RU infinite system
( i , j ) into
(0,O),(0, I), (072) 7 ...
(170)7(171),(1,2), ...
............
and consider a variable point Py which moves over the bystem in such a way that in the xth moment its coordinates are (n(a,~),u(m,y)).It is clear from the definitions of functions x and u that i? the ( a + l ) s t moment the point Py either moves one place to the right in the same row or jumps to the initial point of the next row. The first move occurs if y(s,y)=O and the second if y(s,y)#O. Furthermore, it is clear from the definitions that y(m,y)=&z(x,y),a(m,y),y), i. e., that y(m,y) gives the value of p(p,s,y) calculated for the coordinates (p,s) of the point Py. Hence, if e(p,s,y)=O, then Py moves from the point ( p , s ) to the point ( p , s + l ) and if e(p,s,y)fO, then Py moves from the point ( p , s ) to the point, (p 1,0). NOMlet us assume that zfl[e(p,s,y)=O]. Let p , be the smallest
+
integer such that p(p,,s,y)=O for s = 0 , 1 , 2 , ... For each i < p o there is a (smallest) integer s, such that e ( i , s , , y ) f O . I n this case the moves of the point P,,may be described as follows: starting at the point ( 0 , O ) it moves so places to the right, then jumps to the point ( 1 , O ) and moves s, P
S
-
*) A theorem rqiiivalent to theorem 1 has been proved independently by
nald Ill.
Mark-
278
[48], 262
FOUNDATIONAL STUDIES
places to the right, jumps again to the point (2,O) and so on; eventually it reaches the poth row and moves on it indefinitely. Hence y ( m , y ) = O for m >so+ s, ... spo. Let us non assume that [e ( p ,s ,y) # 01 and let sp denote the smitlP S lest integer such ‘that e ( p ,sp,y) # 0. Repeating the previous argument we see that the point P, passes through all points ( p , s , ) and hence y ( z , y) # 0 for .infinitely many m.
+ +
nz
Formula ( 2 ) is thus proved. R e m a r k . A theorem similar Co theorem 1holds also for the class Pik). In order to obtain this more general theorem we replace the 9”in theorem 1 by “(y1,...,yk)”. D e f i n i t i o n 2.
x:)=E[Iz Y
10-Xy(t,y) (11.
x-
THEOREM2. The famiry {Z:)),,
is identical with Pr).
P r o o f . For each y , Z ~ ) C P since ~)
Now let us assume t h a t Z C ~ ’According . t o theorem 1 there is a primitive recimive function 6 such that Z=-Z‘,l‘. P u t
y(m, y) = lOx[l2-(1-
a($, y))].
If y € 2 ,then b(.r.y)=O from a certain m on, and hence ~ l O - * y ( z , y ) = O . X
If ~ $ 2then , S(z,y)#O for infinitely many m and lO-’y(z,y)=l for infinitely man;r x. It follows that li~1O-”y(m,y)=l and hence 2-2:). X
D e f i n i t i o n 3 Z~’=E[limlO-xy(z,y) exists and is equal to DJ. Yx-ux,
THEOREM3. The family {ZF)}7eo is identical with Proof. For each y, c Q$) since limlO-”y(z,y)=O= X
S o i i let us assume that 2
I6 = Since the set
a primitivr
n 2 [] [ p y ( z , y )‘7
Qf), i. e.
that
n 2 n re(P ! l , r , y ) =Ol, P
P
~
Qf’.
7
,o 6 @.
E Z ( / [ e ( p , q , r , y ) = O ] is in Pp’,there exists by
(Y.P) 4
I ~ C C I I I ’ S ~ Vfunction ~
y ( p ,z,y) such that
theorem 1
[48], 263
SETS DEFINABLE BY MEANS O F TWO A N D THREE QUANTIFIERS
219
Here (po),[...I denotes the least integer xp.Let p be arbitrary and x > m a x q . If .z(s,y) m a x s j it follows that K P
K
y’(v,z+l,y)=O
P
for w p for all s > x o . This is evident for z=zo. Let us assume that n ( s , y ) > p for an x>xo. If n ( z + l , y ) is defined by means of (4), then evidently n(a+l,y)>p; if n ( z f 1 , y ) is defined by means of ( 3 ) , then again n ( s + l , y ) > p since y ’ ( n ( z + l , y ) , m + l , y ) # O by formula ( 3 ) , whereas y’(v,x+l,y)=O for rT
z,,. and hence y ( r ,r -t1,y)=O = y’(w ,s+l,y).
280
[48], 264
FOUNDATIONAL STUDIES
We have thus proved that
fl [n(x,y)>p] and, since p was arbitrary,
x2-3
this proves that lim n(x,y) = 00. X--roo
Next we assume that y 4 Z . By (2) there exists a smallest p , such that there are, infinitely many values of x such that y(p,,x,y)#O. We ... and have therefore denote these values by x17xe, for
y ' ( p o , $1, y) # 0
(6)
i =1 , 2,3 ,...
Exactly as before we can show that n(s,y) >p,, from a certain soon. We can assume that x, is chosen so large that y(v,x,y)=O for v < p o and a:>xo. If s,>xo+l, then the relation between n ( s j 7 y ) and n(q-1,y) IS expressed by formula (3) and the smallest w < n ( s ~ - l , y ) satisfying y'(w,sj,y)#O 'is equal to p,. Hence a(irj,y)=p, for infinitely many j, which proves that 1i.m 4"i,Y)=Pa.
(7)
J
Fomiula (5) is thus proved. From the definition of y' we easily obtain the inequalities 48a)
y'(p,s,y) < 10x/p
(8b)
- - l S I O X I P ~ Y ' ( P , ~ , ~ )if Y'(P+,Y)fO
for arbitrary PlX7Y7
(if p = 0,then fractions with the denominator p must be taken equal to 1). Now let &(sly)=y'(n(s,y),s,y). If y E 2, then from ( 5 ) and (8a) we obtain lim 10-"8(x,y)=O. If y 4 Z, then we use (6), (7) and (8b) and X
lipo for infinitely many s, ; hence infer that 10-xf6(s,,y)> lim1O-"8(m7y) either does not exist or is different from 0. This proves X
that Z=Z?), q. e. d. THEORW4. The family Proof. If y
@*, then Z):
{.@?}7c,p is
E
identical with
Qf).
Qf) since
Let us now assume that Z E Qrf' and let Z' be the complenient of Z . Since 2' E pg),we can apply theorem 1 and obtain a function 8 such that y E Z'EZ'
Let.
I]
xg x>xg
[s(x,!/)=o].
[48], 265
281
SETS D E F I N A B L E BY MEANS OF TWO A N D T H R E E Q U A N T I F I E R S
I t is evident that
NslY)~P(s+1,Y)l y E Z=[lim~(s,y)=oo], X
-10-”+1/B(s,y) 1 and p(s,y) X
is constant from a certain s on. 2. Let us call a real number 1 recursive if there are funct,ions a , @E.@ such that L = l i m a(s)/B(s).Theorems 3 and 4 remain true if we replace 0 X
in the definition 3 by a recursive real number 1. D e f i n i t i o n 4. Let @** be the subclass of @* containing all the functions ‘p such that among the numbers lim 10-x&z,y) , y =O ,1 ,2, ... X-+CS
there are only finitely many different numbers. THEOREM 5 . The family {Z:)Jyrrg..is identical with P ~ ) ,@I.T Proof. First let us assume that y E @**. We denote by y,,y,,...,yk integers satisfying the conditions limlO-Xy(slyl,)#limlO-xy(m,y~) for j f h , X
X
lim10-”y(s,yj)#O
for j = 1 , 2 , . . . , k ,
X
2 [lim lO-”y(s,y)=lim 10-xy(s,yj)]. The existence of integers y, ,...,Yk follows from the assumption y a**. if lim lO-”y(s,y)fO, then X
i4
which proves that @? @. We assume now that Z E Qt). By theorem 3 there is a function y such that Z=Zy). Putting B@S, Y
we obtain a function
1=y(sc, Y ),
B such that Z=@:)=@).
T~~omu~md 8. The family
r,,.
W e denote by H, the
((
/=4
where
(2)
h=l
stands for repeated alternations.
HERBHAND'S (I)
I=1
Tmoneni.
-
The following conditions are equivalent
:
I'isproonble in T(,?); there is an n such thal II!, isprovable in T(A).
Proof of ( I ) + (2). Assume the contrary and denote by %' the smallest class containing all the substitutions of axioms of T(X). Further pul T'=T (3'u { - H,, - H a , - H3, . . . )). If T* were inconsistent, there would be an s such that HiV H2V.. .V H, were provable in T(3').Hence fi, were provable in T(%) against our assumption. Hence T' is consistent. All substitutions needed in the derivation of Y from % being already performed in the passage from 4to s', we easily see that Y is ti theorem of T'. Two matrices :If and M of T' will be called equivalent if the matrix M= N i s provable in T*. The set of all matrices equivalent to M is
294
[W, 21
FOUNDATIONAL STUDIES
denoted by /MI and the set L of all equivalencc-classes I M J were M runs over matrices of T*is referred to-as the Lindenbaum algebra of T' where (cf. [43, p. 295). L is a lattice with respect to the ordering I MI & I NI is an abbreviation for the matrix M D Nis provable in T')). W e denote by e and - e the unit element and tlie zero element of I,. Thus e is the set of all theorems of Tfand --e is tlie set of matrices which are refutable in T* Since Yis a theorem of T', we have Y I = e# - e. Since the existential quantifier corresponds to the operation of forming I he s i i n i in L, we obtain the formula ((
I
I ~ ~ = ( ~ ) C 1 ( y t ) ( 3 y l ) ( y r ) ( 3 y s ~. . ~. t (~ ss )i l,#~- e2 ( jZ-1
from which we infer that thcre is a ,i such thal
I (yi)( 3 . ~ 1 )
(YA
)(
3y.s)z(zji,Y * * . .
*~
z
1 f) - e .
Since the general quantifier corresponds to thc operation of forming
a product in L, we obtain further
(~d~i~~~3ys~~y~~( 3 y s ) * ( s i ,= f . y s + + ) ~ + e
I 2 I
aiid hence by choosing the value o f t such tllilt s L ' I = y h l j ) we gel
I( 3.~:,)
t 3 ~ 2 Z) ( z j , Yktj,?Y:! Y&u.)/#-
( ~ 4 )
c.
Continuing in the Sam6 way we obtain an 1 such tlial.
1
3
~
5
Z( ) zir YhClli
~
1
Y9 u , / Y) S )
and finally an h such tlmt
I z ( s j ?Yh[jj?
yl(j,/*
I #-
If-
e'
j'iij,/jt
*A)
e
O n the other and we have
1 - Hn I L I - z(*zjlyhlj,,
Since -H,,is provable in T', we obtain therefore
I - - Z ( x j , y&(j~&y~/,p~ ZL) J =e wich contradicts our former result.
I
WI, 22
I,
'd
295
A PROOF OF H E R B R A N D S THEOREM
x'(
3u:)%(.C'/r
]'k(j
I,
1
.cI,
y((i,/l, Yh) h' (3YJi (YA) 3 yi) z~~:,~.y,,,,l,YJ,YC,Y5').
j ./
In each surnniaitd of the lirst sum we replace J,, b y y , and add the ciuantifier ( 3jr1). The resulting matrices ;Ire absorbed by terms occurring in the second sun] ( i l i = n or I = n ) or by terms occurring W e thus obtain i l l C,,+, . ( i ) t-
c,+,v
x'(
31';)z ( x j >)'kl,)?
ysl
yLij,/,,
j.1
v ( 3y3) t y + )( 3 y : )
z(.L.n,
Yk[ni?y3, J ' L t Y3).
From tlic definitioii of I ( j , I) it follows that. the variable y t ,,11 occurs only once in (i). Hence we can place the quantifier (yl,! , l ) ) in the front of the summand of ( i ) which contains this variable and then rename the variable ylij,l)into y t , In the resulting .expression we ) front. Terms and place the quantifier ( 3 . ~in~ the replace L ( : ~ 1 hictr arise in this way from the terms occurring under the x'-sigii in ( i ) are either absorbed by summands already occurring in C,,+, or are identical with the last term of (i). The first, case occurs n and the second when j =n. In this way we come to the when j formula )'k[~~l. 513, yr, J s ) . k crt- I v ( 3 Y 3 ) (,YL)( 3.)'~)
. T h e implication ( I ) --f ( S ) is obvious. T o prove the converse implication we assume ( 3 ) and deny ( 2 ) . Denoting by L the same Lindenbaum algebra as in the proof of(^) + (2) and by I a n arbitrary (dual) prime ideal of L, we define a model of T' in the following way. Two terms x;,xj are called equivalenl if I l J ( & i > 3 F ( x j ) I is in / for each F . T h e congruence class containing .zjis denoted by 2;. T h e congruence classes ;Z,, .i2,. . , are taken as individuals of' the model 1 0 be constructed. A function-symbol f of T' with p arguments is interpreted in the model as the function which correlates with classes - .c. x. . . . , xi,, h e classJ'(zj,, x;,,. . . , x;,,).T h e individual constant y j of T' is interpreted accordingly as y j . A predicate r of T' with q arguments is iriterpreted i n the model as the proposi?ional lunction which correlates with classes S j , , Zj,, . . ., Zj,, the truth value (( truth if and only if 1 r ( z j , , sj,, . . . , xjq)1is in I . It is known that we obtain in this way a model of T" (c,f. [2], p. 175 and [S], p. 199). By ( 3 ) Ji is true in the model. This means that the following condition is satisfied : There exists an 2,sirch that for ewry .,i,, there is O R Y 1such thut jor every ,%.,. diereis an .rl,srich t h I : c l a m s Zj .it,, Zl, FC,, ZIL( i n this o d e r ) salis,6 thc malrix Z(yl, y2,y : { ,y , ,y s )in the model. Since it is arbitrary, we can choose it so that, z,,= y k c j r . This determines the value of 1. Again we can clioose t ' so that ~ ' , = y l , wlierehy ~,~) tlis ~-;:Iucof I/.is determined. Thus \ye obtain the result that the seiiie~ice% ( z ykilr, j , zl, .Y,,~,~,, n.,,)of' T' is i r u e in the model and hence so i s U,, for i i L I n a x ( j , I , 1)). Tliis hov-evcr is a ,$,
))
1501, 24
A PROOF OF H E R B R A N D S THEOREM
291
contradiction because - H,, is provable in T' and hence H, must be false i n each model of T'. 11. The above results suggest tlre followiiig simple method of establishing the fundameiital meta theorems of the first-order logic. W e start with the €-theorems for which a transparent proof has been found recently ( q / : [(;I) and thus. reduce the study o f arbitrary theories l o the study of theories whose axioms are open matrices. For such theories we can prove the implications ( I ) -+ (a) + (3) + ( I ) i n the manner indicated above and obtain thus the theorem of Herbrand, the completeness theorem of Godel, nnd the theorem of Skolem -Lowenheim.
HI. The equivalences ( I ) F
caii be proved for theories based on modal logic S; ( C f . [ 5 ] , p. 80) in exactly the same way as above. Because not every matrix of S; is reducible in S;to the prenex normal form, we do not obtain for theories based on s; the necessary and sufficient condition for provability of arbitrary matrix but only n condition pertaining to matrices in normal form, The proof of ( I ) + ( 2 ) is also applicable to theories based o r t Heyting's functional calculus S; (cf. [ 5 ] , p. 8 4 ) and yields the following theorem : If Y is a m a t r i x in normal f o r m a n d Y is provable i n T@), then there is an n such chiill 11H,, in provable i n T ( 3 ) . (2)
BIBLIOGRAPHY.
[ 11 J .
H F . H B H A K I 1 , Aecherches sur la thdorie de la dt!monstration ( Travau..c de Societ,: des Sciences et des tettres de Varsoriie, classe 111, Warszawa, 1930). (21 J . Lo;, The Algebraic Treatment of the Methodology of Elementary Deductive Systems (Strulia Logica, vol. 2, 1955, p. 151-212). [ 3 ] H . RASIOWAa n d R. SIRORSLI, A Proof o j the Completeness Thtorem oaf Gtidel (Fund. Jfuth., vol. 37, 1950, p. r$-zoo). j 'c] 11. RASIOWA,Algebraic Models of Axiomatic Theories (Fund. Math.. vol. 81, 1955, p. 29'-310). Is] H. R~sicts.4and 11. SIKORSKI, Algebraic Trentmmt of the h-otiorr of Sat& fiability ( F u n d . Math., vol. 60: 1953, p. 62-95). I C ) €1. R a s ~ o w k . A proof of tlu! :-theorems, t o nppaa' in h e Fttrlrinrizcritn .Wut/ternu;ic0 and the implication ( 5 ) + (6) holds for lesser values of k, , then from (8) and (9) we obtain u = k, - 1, whence by (10) and the inductive assumption v = f , ( k , - 1, k,, . . . , kpi, n) and finally by (11) and the inductive assumption m = h ( k , - 1, v, k,. . . ., kp,, n) = f i ( k , , k,, . . ., kp4,n). The lemma is thus proved. Observe now that the formula ( 5 , y ) [F#(Z, y ) V A (y)] is true in p whence it follows that if p, (a,, ,a,,,), then a,,,E a. According t o the lemma this is equivalent to the statement that if m = f ( n ) ,then m = 0 , i. e. t o the equation f(n) = 0. We have thus proved that if a model of 8 has n 1 elements, then f ( n ) = 0.
+
[55],213
309
C O N C E R N I N GA PROBLEM OF n . s c n o L z
Let now 5 be a set with n + 1 elements a, ( j = 0 , 1 , . . ., n ) and let f ( n )= 0 . Define e, a, b, p, (i = 0 , 1 , . . ., 8 ) in the following way: ap e aq if and only if p < q , ap E a if and only if p = 0 , ap E /3 if and only if p = n , pa(a,, , . . . , akp,,a l , a,,,) if and only if 2 = n and ( 6 ) holds.
It can be verified without difficulty that ( E , e, a, p, pl, . . ., q,) is a model 8 . This establishes the validity of our theorem. From the theorem we immediately obtain the following Corollary. Iff is in K and k is any integer, then tkre i s a {irst order formula 0 such t h a t 0 h a s a d e Z w i t h n + l elementsit a n d o n l y i f n > k a n d f(n)=O. It is a n open problem whether the theorem converse to that stated in the corollary is also true. 2. With the help of the theorem proved in the previous section we can easily show that the problem of SCHOLZ admits a positive solution for the set of primes, for the set of powers of a given integer, for the set of integers having the form t t ! etc. (cf. [ l ] ,p. 262). As a slightly less obvious application we shall investigate here the set ( n :nz + 1 i s pim} of which it is not known whether i t is finite or infinite. I n order t o obtain our result we shall define 10 auxiliary functions gl, . .,gl0. It will be immediately seen from the definitions that these functions ’belong to the class K. After each definition we add in parantheses a few words of explanation. of
.
+
gl(O, n ) = S ( 0 , n ) , g l ( z 1,n)= 0 . 1) = 1 , g1(z (gl(z, 0 ) 0 , gl(O, n
+
+ 1 , n + 1) = 0.)
g,(o,y,n) = G ( y , n ) , g2(z + ~ , Y , R )= S ( q 2 ( z , ~ , n ) , n ) . (gz(z, y, n) = m.in(z + y, n ) ; we shall write z y instead of ga(z, y, n).)
+,,
+
g3(0, n ) = 0 , g3 (z 1 , ; ) = u:(z, n ). ( g 3 b 1 , n) = min(z, n ) . )
+
+
WY,
~ ~ (Y.0n), = n ) , g 4 ( z 1 , Y, n ) = g3(g4(z, Y , n ) ,n ) . (g4(z, y. n ) = min(y - z,n ) . We shall wgite z -,,y instead of g4(y, z,n).)
+
gS(O,4 = 0 , g6(z 1 , n ) = S(O,4. (gs(z, n ) = min(sgnz, n ) . We shall write sgnnz instead of gs(z, n).)
+-
Y, n ) = 0 , 96(2 1, Y, n ) = W y , n). (go(”, y, n) = min(y, n ) s g n z . We shall write y
g,,(O,
0,
x instead of g6(z, y, n).)
g7(0,Y, n ) = 0 , g,(z + 1 , . ~ , n )= ( g 7 ( z , ~ i n+nSgnn(y-ng7(z,y,n)))on(y-n ) (g7(z,y, n) is the rest of division of z by S(y, n ).)
g7(z1
~>n)).
9, (0, Y, 2, n) = 97 (?Az, n) ga(z+1, ~ , z , n =gs(z, ) Y , z , ~+)n s g n , ( z - - , g s ( z , y , z , n ) ) o n ( z - n g s ( z , ~ , z , n ) ) . (g,(z, y, z , n) is the rest of division of y by S(z, n ) . ) 9
+
3 10
[55],214
FOUNDATIONAL STUDIES
+
gJ0, ? z ,I n, ) = 0, gs(z 1 , Y, 2, n) = gs(ge(Z, Y, z , n ) , Y, 2, n). (gr(x,y, z, n) is the rest of division of x y by S(z, n).)
Y, z, t , n) = 9s ($79 (x, Y, t, n ) , z , t, n) . (gl0(x,y, z , t, n) is the rest of division of x y
$710(2,
+z
by S(t, n).)
We shall say that a relation e ( x l , . . ., xk, n) is a K-relation if its characteristic function j , belongs to K. If e and u are K-relations, then so are the relations non-e and e V u for their characteristjc functions are g1(f, ( x l , . . ., xk, n), n) and u:(f,(xl,. . ., xk?n),n) +nU:(fo(x1, . . xk, n),n). If f(x, xl,. . ., xk, n) is a function of class K whose values do not exceed 1, then ' 9
the function (12)
g(x, z l , . . ., xk, n) = min(
n f ( j , x l , . . ., xk, n),n)
iSz
belongs to K. Indeed, g ( O , x l , . ..,9:k , n ) = f ( 0 , x l , . . . . x k , n ) ,
g(x + 1 , x l r . . . , x k r n L ) = ~ ( x , x l ,. . , x k , n ) O n f ( x+ 1 , x l , . . . , X k > n ) . It follows from this remark that if e(x, x l , . . ., xk, n) is'a K-relation, then so is
the relation (j)=e(x,xl,. . ., xk, n) where ( j ) , means: for every j not exceeding x. Indeed, the characteristic function of this relation is given by the formula (12). We can now show that the set n composed of 0, 1, and of all primes is in K. Indeed, n E n if and only if (j)n(g,(n,j , n) 0 or S ( i ,n ) = n or i = 0). From this we obtain our final result: the set {n: = 0 or n2 Indeed, n belongs to this set if and only if
+ 1 En} is in K.
(j)n(glc(n, 12, S(0, n),j , n)) =I= 0 or S ( i , n) = n or i = 0). I was not able to decide whether or not the set of FERBIATprimes 2'"
+1
belongs to K nor whet,her the SCHOLZ problem for this set has a positive solution.
[l]
Literature G.ASSER.Das Reprasentantenproblem im Priidikatenkalkiil der ersten Stufe mit Identitat.
This ,,Zeitschrift", vol. I (1955), pp. 252-263. [2] H. SCHOLZ.The Journal of Symbolic Logic, vol. I7 (1952), p. 160. (Eingegangen am 6. August 1956)
On a generalization of quantifiers by
A. M o s t o w s k i (Warszawa) In this paper I shall deal with operators which represent a natural generalization of the logical quantifiers l ) . I shall formulate, for the generalized quantifiers, problems which correspond to the classical problems df the first-order logic. Some of these problems will be solved in the present paper, other more interesting ones are left open. Most of our discussion centers around the problem whether it is possible to set up a formal calculus which would enable us to prove all true propositions involving the new quantifiers. Although this problem is not solved in its full generality, yet it is clear from the partial results which will be discussed below that the answer to the problem is essentially negative. I n spite of this negative result we believe that some at least of the generalized quantifiers deserve a closer study and some deserve even to be included into systematic expositions of symbolic logic. This belief is based on the conviction that the construction of formal calculi is not the unique and even not the most important goal of symbolic logic.
1. Propositional functions and quantifiers. Let I be an arbitrary set and I * = I X Ix ... its infinite Cartesian power, i. e., the set of infinite sequences (xl,x2,...) with x, I for j = 1 , 2 , ... We denote by V and A the truth-values “truth” and “falsity’?. The Boolean operations of join, meet, and complementation are denoted by V, A and -; we use these symbols for all Boolean algebras which we shall have to consider and, in particular, ior the two-element algebra consisting of the truth-values A and V. A mapping F of I* into {V, A } is called a propositional function on I provided that it satisfies the following condition: there is a finite set K of integers such that if x = ( q , x 2,...) € I * , then F ( x ) = F ( y ) .
y=(yl,y, ,...) r Z * ,
and
xj=yl for j r K ,
1) Parts of the reaults contained in this paper were presented to the Torun Section of the Polish Mathematical Society in Jennary 1934. Other parts were included in my psper [ti]. which, however, contains no proofs.
312
FOUNDATIONAL STUDIES
WI, 13
This condition says, of course, that F depends essentially on a finite number of arguments. The smallest set K with the property stated above is called the stipporl of F ; if it has only one element, then F is a function of one argument and can be identified with a subset of I * ) . Let p he a one-one mapping of I onto a set I' not necessarily different from I. If a=(sl,xz,...) E I*,then we denote by q ( s ) the sequence (p(xl),p(x2), ...); if F is a propositional function on I , then we denote by Fv the propositional function on I' such that Fv(q(c)) =F(z). A quantifier limited to I is a function Q which assigns one of the elements v, A to each propositional function F on I with one argument and which satisfies the invariance condition for each F and each permutation q of I. The first part of this definition generalizes the elementary fact that quantifiers enable US to construct propositions from propositional functions with one argument. The second part expresses the requirement that quantifiers should not allow US to distinguish between different elements of I 3). Let (mt,nE)be the (finite or transfinite) sequence of all pairs of cardinal numbers satisfying the equation me+nt=F4). For each function T which assigns one of the truth-values to each pair (mt,iii) we put ~ _ _ -~ F-'( A )) '1 .
Q T ( F ) =T(F-'( V
The following theorem is easily provable: THEOREM1. Qr i s a quaiitifier limited to I ; for each quantifier limited to I tkere is a T stick that Q r = Q . If Q = Q T , then we shall say that the function T determines the quantifier Q ; there is evidently exactly one such function for each Q . Let us put T*(mb,ne)=-T(nt,m l and that a, denumerable subset has been defined. We are going to define a set I k + l . To this end we arrange in a sequence (1) v,,v,,... all closed (proper and improper) formulas resulting from formulas X in U by a substitution of symhols a, (a in I , ) for the free variables of X. Each v, determines a set I k j in the following way: Assume that V j is the formula (Qhx)W. We denotc by W(aJ the formula resulting from mr by the substitution of a, for s and consider the sets
(2)
J,=E ( v a h , ( W ( a , )=) V ] . J,=E [ t & i 1 ( ~ ' ( u o ) ) = A 1. or1
or1
Let m,,m, be the cardinal numbers of these sets. If rn,,rn, arc both infinite, then we take as I k j a denumerable subset of I having infinitely many elements in common with both J , and J,. If m, is finite and m, infinite, then we take as I k j a denumerable subset of I having infinitely many elements in common with J , and containing all the elements of J,. If rn, is finite and m, infinite, then we take as I k j a denumerable subset of I having infinitely many elements in common with J , and containing all the elements of J,. cc
We now put
I k + l = u I k j . ,=1
fined by induction. We now put
The Sets
m
I k
( k = 1 , 2 , ...) We thus de-
I, = U 4 and obtain a denumerable subset of I . w e k=l
shall show that Z is satisfiable in I,,.
t591, 22
321
ON A GENERALIZATION OF QUANTIFIERS
For each I-valuation M we define an I,-valuation N o . M, differs from X by assigning to x, an arbitrary element of I, whenever [ x , h is not in I, and by assigning to Fj the propositional function [FJIMrestricted to I,. If L C , , ~ , , are . . . in I, we have therefore (3)
[FJIhfo($i
7 $2
-..) = [FJIM(%
$2
*.
.) .
LEMNA(b). Lei M be an I-valuation satisfying the conditions
[ x i l M ~ I 0 for i = 1 , 2 , ..., [Fj]~=[Fjl% for j = 1 , 2 , ...
(4) (5)
i 8 a proper formula and v‘ results from v by a substitution of symbols U, ( a in I,) for 8ome or for all free variables of V, then ualMI(V’) = valMoro( V’). If V is an atomic formula F ~ ( x ,..., ~ ,xik)or x i = x J , then our assertion follows immediately from (3)and (4). If the lemma holds for formulas V, and V,, then it is clear that it holds for the formula VJV,. It remains thus to show that if the lemma holds for a formula V, it does so for the formula (Qhxl)V. Let (Qhx,)V’ result from (Q*x,)V by a substitution of symbols U, ( a in I,) for some or all free variables of (Qhx,)V. Let yl,...,yk be the free individual variables of (Qhx,)V’. We substitute (I[y,JM (or, what is the same, for yJ in (Qhxxr)V’ ( j = 1 , 2 , ...,k) and obtain a closed formula (@x,)W. We can assume that this formula occurs in the sequence (1)and is identical with VJ. According to lemma (a), p. 21,
If V
ual~d(Q~xdV’1 = valMl[(Q*xi) Wl , valworo[(Qh~xr)v’l = vaLor0[(Q”~i) Wl
-
Thus it is sufficient to show that (6)
valMd(Qhxi)W] = @ a h I,[(Q’x,) o Wl .
We shall first calculate the left-hand side of (6). According to the definitions given in section 2 we have to define a propositional function F on I with the support {i} such that (?At Yz
,...I =aalM(iy,,.I(W)
and then take the value Qf(F). According to lemma (a) u ~ ~ , , , ~ , , ( W ) is V or A according as valMI(W(uy,)) is V or A. Hence if we denote by Tf the function which determines the quantifier @ and put __.
m,=Ea t 1 [Vab(W(o.))= V ] ,
m , =0 6~1 ~ a l ~ ~ ( w ( ~ , ),) = / \ ]
322
FOUNDATIONAL STUDIES
we obtain waL~[(Q~xt)Wl = T;(ml, m,)
(7)
.
Similar considerations show that (8)
where
~ a l ~ ~ i , [ ( Q ~ ~ T;o(m:,m:) i)Wl=
e0is the function which determines the quantifier Q;o
and
We observe now that if a is in I,, then W(uo) results from V by a substitution of symbols a, (a in I,)for all free variables of V and that we can therefore apply the inductive assumption to the formula W(u.). This gives walMoIo(W (u,,))= walM,(W ( u,,)) for a E I,. Furthermore W ( ua) is ‘a closed formula and thus vaZMI(W(u,,))depends only on [ F j I M , which in view of (5) proves that = vallril(W(oo)). walMI(~(aa))
This equation holds for arbitrary a in I, not only for a in I,. Taking these observations together we obtain the equations
We now use the definition of sets Ikl given above. Remembering that (Qhx,)Wis the jth term of the sequence (1)we see that ml,mz are the cardinal numbers of sets (2). If m,,m, are both infinite, then Ikl has infinitely many elements in common with both J , , J , and hence 0 0 ml= mr = no. If ml= n is finite and m, infinite, then .J,C I, and hence m:=n, and mi=no. Similarly if m z = n is finite and i n , infinite, then mi= n and m: = so. This proves that
~2ml,m2)=~~o(m~,mi) since the quantifier @ does not distinguish infinite p o ~ z r s . Comparing the last equation with ( 7 ) and (8) we obtain ( G ) , which proves lemma (b). We can now conclude the proof of theorem 6. Since Z is a closed ) not depend on [xi]%. Hence w e can formula, the value of v a l ~ , ( Zdoes assume that [xi]% is in I, for all i. Using lemma (b) for JI=@ and V-= %, we obtain waln;?l(Z)= V U Z ~ ~ ~and ~ ( hence Z ) waZ%olo(Z)= V since v u l ~ ~ (=ZV) by the definition of @. This proves that Z is satisfiable in I,.
1591, 24
O N A GENERALIZATION OF QUANTIFIERS
323
We shall now prove that conditions given in theorem 6 are not only srifficient b u t also necehsary for the validity of the Skolem-Lowenheim theorem. THEOREM7 . I f 9 and V occur among the quantifiers Q', ...,@ and if at least one of these qicantifiers distinguishes infinite powers. then there are closed formulas satisfiable in non-denumerable sets but not sutisfiable ~ I Ldejiumerable sets. P r o o f . Let us assume that the quantifier Q1=Q distinguishes infinite powers, i . e., that there are infinite set, I , , I , such thot one of the following cases holds: (a) there is an n such that ! Z 1 , ( n , ~ ) # 2 ' , ( n , ~ , ) , (b) there is an n such that T 1 ( ~ , n ) # T , ( ~ , , n ) , (c) there are infiilite cardinals m,,n, ( i = 1,O) such that n~,+n,=Z; ( i = 1 , 4 ) and Tl(ml,nl)#T,(m,,n,). Here T , i b the function which determines the quantifiers Q,,, i=l,4.
Case (a). Let m, be the least infinite cardinal such that if z = m 2 , then there is an illfinite set I I C I , satisfying the conditions given in (a). Evidently m2> K ~ . ' We can assume that I ; ( n ,m2)= V and T,(n ,in,) = A for all cardinal iiunibers m, satisfying the inequality no <m, O. Since p > 1 we have also n o = 23' with t > O . Evidently no >np whence 3' >3' and t - 1 >s. Since n o Qpn, , we obtain p >no/np= 23'-3' >!P'--~'-'- 2*-'4. On the other hand p <no= Z3' and, since p = 23". we 1*2 < 23"< which entail a contradiction. obtain the inequalities Let A be a set such that H ( A ) is not recursive and let Q, be a quantifier limited to a denumerable set I such that Q l ( X ) = V if and only if X A . We take as Q1 and Q2 the quantifiers V and 3 and as Q3,...,@ such quantifiers that Q:= ...=Q ; = Q I . We consider now an infinite sequence of monadic functional variables Fl,F2,... We shall write F:(x) for F,(x) and Ff(x) for -F,(x). The 5" formulas F;'(x)AF;?(x)A...A F:(x) we call constituents of order n. The lexicographical ordering of the n-tuples ( i , ,..., i n ) determines the similar ordering of the constituents. The first n constituents of order n will be denoted by S,(x),...,S,,(x). Let W, be the formula of (SM)
+ + +
[ ( ~ X ) s i ( X ) lA
[(Q"X)%(X)l A *** A A [ ( Q 3 x )s n ( X )I A [( Q3x )( %(X)
V SAX)V
-*. V S A X ) ) 1 .
10) The proof of this lemma has been kindly communicated to me by J. Mycielski. I t remains an open question whether there are recursive sets A such that H ( A ) is not recursive.
328
WI, 29
FOUNDATIONAL STUDIES
It is evident that this formula is satisfiable in a denumerable set I if and only if n is in H ( A ) ; this proves that the set of formulas of (SM) satisfiable in I is not recursive. Theorem 9 is thus proved. It follows of course from this theorem that there are quantifiers Q', ...,Q' such that the set of formulas of (SM) true in a denumerable set is not recursive. The characterization of quantifiers for which the set of true formulas of (SM)is recursive remains an open problem. In the next theorem we give examples of non-trivial quantifiers satisfying this condition: THEOKEM10. Let m, <mz < ...<ms be s cardiitals such that m,, ...7m, are infinite and m, either is 1 or is infinite. Let Q',Qa, ...,@ be quantifiers such that for each I {Q!(F) =
v 1 = (B> m,),
j = 1,2,
..., S .
Then the set of true formulas of (SM)is recursive''). R e m a r k . If r n l = l , thenQ1=3; if mz=H,,, then Qz==-S; if m3=nl, then Q"= P. The proof of theorem 10 will be based on some lemmas: (a) Quantifiers @ satisfy the equations
-
@Fv@=&F)v@(G)
(ditivity),
@(A)=A
-
(b) If Zl,...lZn are formulae of (S) containing the free variabb xi and if Wl, ...,Wn are formulae of (S) not containing xi, then the formula (@Xi)[(ZiAWl)V...V (Zn Awn)]
i s true.
= [w, A (@xr)z11 v ...v [Wn A (@xi)zn~
We abbreviate the left-hand and the right-hand side of this equivalence as L and R respectively and denote by I an arbitrary set and by M an I-valuation. Finally we denote by F the propositional function on I with the support {i} such that
F (Yi,Yz,...)= valM~~,y,d(Zi A WI)V
a**
v ( z n Awn)]
and by Fh and Gh ( h = 1 , 2 , ...,n) propositional functions on I with supports {i} such that Fh(yi ,%,...)=vazM(i,~s.l(zh) 9 '%(?/I
tYoi
-.a)=
valM(i,y,).l(Wh).
11) This theorem gresenta a generalization of the classical theorem of Lowenheim. For our proof sea Hilbert-Ackermann [2], p. 101.
[W, 30
329
ON A GENERALIZATION O F QUANTIFIERS
Since wh does not contain x i , ahhas a constant value Now we have P ( Y I , Y z , . - ) = [ WAI ~
and hence, by (a),
~ ~ M u . Y ,V...V ~ ~ I [)%]A
= [WIA PI(?/, 9 Ye, **.)Iv
V
[an
A
?oh= v a l M , ( W h ) .
vahw,~,),r(zn)]
Fn(Yi
Y2,
)I
valMf(L) =d,(F)=d,(m,A F ~ ) v . . . v ~ , ( w , A P ~ ) .
Similar calculations yield v a l d R ) = [ W , A Q : ( F J l V...V
[WnAd,(Pn)I.
It remains ttius to verify that wh A @ ( s ) = @(FhA wk). But this is evident if w h = V and follows from (a) if W h - A . Lemma (b) is thus proved. Let Fl, ...,F,, be n monadic functional variables. We put A ~ , i , . , m = ( @ x ) [ F ~A( x...) AF?(x)] where, as before, F:(x)=F,(x) and F:(x)=-F,(x) and denote by %(xl,...,xm)the least class of formulm that contains the formulas Fk(x,) and A{p..,m( h = 1 , 2 ,...,n, j = 1 , 2 ,...,8, 1=1,2 ,...,m, i , = O or 1 for t = 1 , 2 ,..., n ) and that satisfies the condition: if 8,,Z2 are in R(x,, ...,x,), then 80 is ZllZ2. We do not exclude the case m=O. In this case we denote the class simply by R; of course R contains only closed formulas by means of the stroke. built from formulas (c) For each formula Z of (SM)containing xl, ...,xm as it8 unique free individual variables and F,, ...,F,, as its unique functional variables there is a formula U in R(xl, ...,x m ) auch that the formula Z - U is true. Fmmula U can be found explicitly if Z i s explicitly given. If Z contains no symbols Q', ...,Q', then the assertion of (c) is erident. Let us assume the validity of (c) for formulas containing at most p - 1 of these symbols and let Z contain p of them. It is clear that the lemma will he proved in general if we show that it holds for the case when Z has the form (Q/x)Z,with Z, in %(x,,...,x,,x). From the definition of this c l a s it follows that Z, can be represented in the form of a logical sum of formulas F:'(x) A ... A Flf.(x)A W with W in R(x,, ...,x,,,). Using (b) we obtain therefore an U in R(xl, ...,x,) such that the formula Z = U is true. We now have to find the criterion of truth for formulas in %. This criterion will be expressed by means of some auxiliary notions.
330
FOUNDATIONAL STUDIES
Let @ be a Boolean polynomial in the variables a:,,..i, (j= 1 , 2 ,...,s, i,= 0 , l for t = 1 , 2 ,...,m). A function y assigning the values V, A to these variables is called an allowable valuation if j+1
y(aip..im)
I Y(ail...in)
for j = 1 , 2 ,...,8 - 1 , i , = O , l for t = 1 , 3 ,...,12. The number of allowable valuations is obviously finite. We shall say that @ has the property (T) if each allowable valuation gives it the value V. (d) A formula Z in % is true if and only if it results by a substitution for u ; ~ . from . ~ ~ a Boolean polynomial with property (T). of I I n order to show this lemma we first assume that @ has the property ( T ) and that the formula Z resulting from @ by the substitution described in the lemma is not true in a set I. If M is an I-valuation such that v d ~ ~ ( Z ) = rthen \ , the function y (a;,...i,) = vatwI(&l...in)
is an allowable valuation. Indeed, if v a Z M f ( A { ~=. ~V,m ) then the set x tlhe formulo of elements 2 in I such that [ F , ] M , . . . , [ F n ] M ,satisfy F>(X)A ...AF>(x)in I has the cardinal number > m j + l ; hence this car= V. The allowable vadinal number is >mi and hence VUZ,,,~(A:,,,,~~) luation y gives t.0 @ the value caZMf(Z)against the assumption that @ has the property (T). Let us now assume that Z results by the substitution described in the lemma from a polynomial @ without the property (T). We are going to define a set I and an I-valuation such that v a Z M f ( B ) = A . To this end we consider 2" disjoint sets X;l.,.jm each of power nr,. If j=ji,...i, is the great,est integer < s such t h a t y(a;l,,.in) ==V, then we remove from Xil,..jmas many elements as to leave it set Xi,...i of power mi. Now we take as I the union of all set,s Xil,,.imand define an I-valuation M by t,aking as [ x j l Man arbitrary element of I ( j = 1 , 2 , ...) and as [F& the union of those Xil...infor which i 4= 0 ( k = 1 , 2 , ...,n ) . It is easy to see that and hence
{valM(h,ydF:(xh)~ ... A F > ( x ~ )=] V 1 - ( y -
{~~Ld&,...iJ
= V } = ( X i,...is E
Lemma (d) is thus proved.
{j.
11...1"
> mj> >j} 5
t:
Xi,..,i,,)
{mjil,,,im 2 mj} =
v.
[591, 32
331
ON A GENERALIZATION OF QUANTIFIERS
Theorem 10 results immediately from lemmas (c) and (d) since the set of polynomials with the property (T) is recursive. We conclude with a discussion of the systems ( S r ) with n = 1 , 2 , ... In contrast to theorem 9 we have the following THEOREM11. For eaoh n > l the set of h e formulas of ( S r ) and the of formulas of (Sf) which are true in any given set I are recursive. Proof. Let the m= 2" constituents F:'(x)A... AF?(X) be denoted by S,(x), ...,S,(x). Each Boolean polynoniinl W ( x ) in F,(x),...,Fm(x) which does not vanish identically has a %anonical representation" Sk](X)v Skp(X) v
...v S k , ( X )
which is unique up to the order of summands. We put and
(a) Z f A and I) are formulas not containing the free- variable x and M(x) and N ( x ) are fwmulas in which x occurs free, then the equivalences (Q'x)[( AA M ( x))V (B A N( x))]= {(-A v (A A -B
A
A
-
B A A:)
(Q'x)M ( x))v (A A
(Q'x ) N ( x))V
V
(-A
A
(Q'x) [M( x ) V Nix)])
A I) A
are true ( j = 1 , 2 ,...,s). Proof of this lemma in evident. Let us now consider a formula Z of the form Mo(X) V
[ci A Ni(x)l V .-.V [ c p A N p ( X ) I
where C,, ...,C, do not contain x. For each set i,, ...,i, of indices ( =O ,1) we denote by . 2. (x) the sum M0(x)vNi,,(x)v...vNik,(x) where k,, ...,k, .... are all integers k < p for which i k = O . Denoting by . V . the Boolean 1,.
1,
11.
sum over the sets of 2, indices we have ( b ) The equivalence (Q'x)Z= V [ C ~ ... AA@A(@x), il. ....i,
I,.....
....1,
(x)l 1,
is true. We show this by induction on p . If p = 1, then we take in (a) A = ( Q ~ x ) F , ( x ) v-(Q'x)F~(x)=A,,B = c , , M(x)=M,(x), N ( x ) = N , ( x ) . If the lemma holds for the number p - I , then we take in (a) A = A , ,
332
WI, 33
FOUNDATIONAL STllDlES
B = C,, M (x)= M,(s) v [C, A N,( Y ) ] v ...v[CP-,A NP-,(x)], N ( x )= Np(x) and obtain the (true) equivalence
( Q ' x )=~C ~ (Qk){M,(x) A V [C,AN,(X)]
V ...V [C,-i
A NP-i(x)]}V
...V [ ~ , - I
V CiA (Q'x){[M,(x)VNp(x)]V[C]AN,(x)lV
A N,-I(X)]}
.
Lemma (b) results now immediately if we use twice the inductive assumption. We introduce now the class %(xi,...,8 , ) in much the same way as in the proof of theorem 10 (cf. definitions preceding lemma (c) on p. 30). The only difference is that we require from the present class %(xl,.-., x m ) that it should contain the s formulas A: and the s(2*"-1) formulas Ah instead of the former 2"s formulas A ~ l . . . iLemma ~. (c) of the proof of theorem 10 holds in the present case and will be referred to as lemma (c,,). I n order to prove this lemma it is sufficient to show its validity for Z having the form (Q'x)Zl where 8, is either the formula F I ( x ) ~ - F l ( x ) or the formula, [C,A skl(x)]V ...v[Cpr\Sk,(X)] with Ch independent of x. I n the former case it is sufficient to take U = A i and in the latter (c,) results immediately from (b). We introduce now the concept of a valuation allowable for a set I . This is a function y which assigns the truth-values to the s.22" formulas A; and A& in such a way that there exists an assignment of cardinal numbers mk to constituents sk(x) (k= 1 , 2 , ... , m )satisfying t'he conditions: 10 ml -tm 2 ...i-tll, = I , 20 if W ( x ) i s a Boolean poh~ibo~iial in F,(x),...,FJx) and
+
ILW
:=
the cnizonicul representations
l'(
P
4
2'nik,, 2
i=l
r=l
nil,)
where T J
i.9
of
W ( x ) uncl
-W(x),
then ? ( A & )
the fzinrtion uhic*h determines the qrtan-
tifier Q { . Of coiirse one of the sunis (15) disappears if W ( Y ) or -W(x) vanishes ident,ically; the corresponding sum of cardinals is then 0. Our clefinition w v t m the case when W is identically A if we agree that -4& is then to bt: iritoryreted as A:. This we do t c i t l y in the rest of the proof. (d) lf M is (in 1-valuation, then the functivn y ( A L ) = aaZ(A&) is nn nllo~~~cilile vuluation for I .
[591, 34
333
ON A GENERALIZATION O F QUANTIFIERS
Proof. We set
&= E [ valhm,d,/(&(Xt))= v 1 7 rr/
k= 1 2
... ,m
and assign to &(x) the cardinal aumber mk=gk. The condition l o is obviously satisfied since the Rk are disjoint and I is their union. Now let (15) be the canonical representations of W ( x ) and - W ( x ) . We have then E [nal,w(t.x).r(W(Xi)) = V ] = R r , u ...w Rk, 7 xr/
E [~aZ~(M(,.x,,,(WCx,))=r\]=R~l~ ...w&,
xr/
and hence, by the definitions of section 2, y (A’w) = v a h d A’w = v a h I (( Q‘xJW
( xA)
This proves that y satisfies 2”. (e) For each valuation y allowable for Z there is an I-valuation M such that y(A&)=walM,(A’,) for each W . Proof. Let mk be the cardinal number correlated with the kth constituent S k ( x ) in accordance with conditions 10 and 20. Let further Z=R, u . . . ~ R , be a partition of I into m disjoint sets such that R k = mk (k= 1 , 2 , ...,m). W e take as [x& an arbitrary element of I ( i = 1 , 2 , ...) and define [F,IM as the union of sets Rk corresponding to constituents & ( x ) contained in F i ( x ) ( i = 1 , 2 , ..., n). It is then easy to show that (16)
(va~M~h,,,./(sk(xh))=v} {Y
Rk}
.
If W ( x ) is a polynomial in F l ( x ) ,...,F J x ) and (15) are the canonical representations of W ( x ) and of - W ( x ) , then it follows from (16) that (DazM(~,y)./(w(X*))=v} -{YCRklu...uRkp} 9 ( z ) a z ~ ( h . y ) . , ( w ( x h ) ) = r \ }3
{Y R I , .*. ~ uR/,I 7
and hence, by the definition of the function val,
Since y is an allowable valuation, the right-hand side of this equation is equal to y(A&). Lemma (e) is thus proved.
334
FOUNDATIONAL STUDIES
P91, 35
We are now abie to prove theorem 11. Let I be a set. The set of formulas in R which have the value V for a single fixed valuation y is of course recursive and so is the set of formulas which have the value V for a finite set of such valuations. Since the set of valuations allowable for I is finite, it follows by lemmas (d) and (e) that the set of formulas in R which are true in I is recursive. By lemma (c,) for each closed formula Z of (S:) we can find effectively a formula in iTI which is true in Z if and only if Z is true in I. Hence the set of formulas which are true in Z is recursive, which proves the second half of the theorem. Now denote by R, the set of valuations allowable for I. The number of such sets is of course finite (since so is the set of functions assigning truth values to formulas A: and AL). Let 3 be the union of all different 3;s. Replacing in the previous proof the words “valuations allowable for I” by “valuations which belong to R”, we obtain the proof of the first half of the theorem. I n spite of its generality (or perhaps just because of its generality) theorem 11 has no practical applications. We illustrate this by means of the following example: let Q1,Q* be quantifiers such that for each infinite set I and for XCZ
{ Q : ( X )= V } =
-
{x 2s a prime},
{ Q : ( X )= V} = {*Thas the form 2*“ f l }
.
The problem whether the formula
I[(zx ) F W ~ (x ) F2( x )
[(Q1x)F(x!~(QSx)F(x)l
(3)
(5)
is true in a n infinite set Z is equivalent to the famous number-theoretical problem whether there are more than 5 Fermat primes. It) is, however, impossible t o solve t,his probkm on the basis of theorem 11 in spite of the fact that theorem 11 asserts the existence of a finitary method for testing whether an individually given formula is or is not true in I. Such a test would indeed be possible if we knew effectively the recursion equations for the characteristic funct,ion of the set of formulas which are true in I. Unfortunately our proof of theorem 11 does not provide ur( with those equations. We have merely proved their existence (in a non-effective wag) and cannot therefore draw any practica,l consequence from our result. I*)
Symbols
(*)
2 occurring
in this forninln denote quantifiers defiiied in tiectioii 2 ( b ) .
O N A GENERALIZATION O F QUANTIFIERS
335
References
B. P. Halmos, Algebraic lo& (XI), Fund. Math. 43 (1956), p. 255-325. [2] D. Hilbert und W. Ackermann, Grtcndziige der thtoretieclren Logik, Die Grundlehren der mathematiwhen Wieeenschaften in Eimeldaretellungen, Bd XXVII, Berlin-Giittingen-Heidelberg 1949. [3] A. Lindenbaum und A. Tarski, tfber die BeschrcEnktheit der Auedmcekemitlel deduktiver Theorien, Ergebnisse eiuea mathematischen Kolloquiums 7 (1936), p. 15-21. [4] F. I. Mautner. A n ezkneion of Klcin’e Erlanger program: l h g i c ae invariant-themy, Amer. J. Math. 88 (1946). p. 345-384. [5] A. Mostowski, Development and applicatwne of the “projective” claseificatim of e d s of ilttegere, Proceedings of the International Congress of Mathematicians 1954, vol. 3, Amsterdam 1956, p. 280-288. [6] Th. Skolem, tfber die Nieht-charakterieie7barkeil der Z a h h r e i h e ntitlels endlich oder abziihlbar u d l i c h vieler Aueragen mit aueechliePlich Zahhvariablen, Fund. Math. 23 (1934). p. 150-161. [7] A. Tarski. Der Wahrhcitebegrifj in. dcn formalieiden Sprachen, Stud. Philos. 1 (1936). p. 281-405. [8] - Introduction to 2ogic and to the methodology of deductive sciences, New York 1946. [Q] - Udecidable theoraes. Studies in Logic and the Foundations of Mathematics, Amsterdam 1953. [l]
Repu par la ReXaction le 23.10.1955
On computable sequences by
A. M o s t o w s k i (Warszawa) A real number a (0 < a elk, then d ( k , n ) = p " - l + $- 1 + pn-l-*l k = p" + p"-l-Q and hence a k = l / p + l/p"l k + % . If k E XP,we similarly find l l p -l/p*k+* where 8sk is the least 8 such that %(k,8%k)=0. If k # X,WX,, then p,(k,y)=p,(k,y)=l for all y and hence Qk=p-'. ThuR = p-' + &kP-'k-' where &k = 1,0, -1 aCCOrding a8 k E XI, k x , ~ x ,k,c X , and where t k is an integer 20. We shall now show that { a h } E C,. From the definitions it follows that
i s no
+
,
ak - 6 ( k n)/pn+l= &kp-'r-P -p-n-1-
,
[1 - pl( k n)][p-*-1+ p-*- [1
If
Ek=O,
and
&k=l
- 6( k ,n)/p"+' = - p-"-l. then tk=8ik and
n<Sik,
ak-8(k,n)/pn+l=p-rir-P+p-n-1
If
and
&k=l
ak-6(k,n)/pn+1= p - s i r s - p-n-1-p-n-1 If
Fk=
- 1 and
Sik,
then
7 '8 < & k ,
tk=8ik
-p-s,k-P=
-3
dP
-88-1
and
ak-6(k,n)/pn+1= -p-%t-z+p-n-1,
whence If E k = - 1
0 < a k - 6 (k7 n)/pn+' 8ak,
then
ak-6(k,n)/pn+1= - p - m r * -
p-n-I -p-m-l
+p-slk-8= -2p-"-1.
Thus for all k and n
I ah - 6 ( k 7n)/pn+11i tLp-n-I, which proves that
{ah} E
c,.
-
- p2(k ,n ) ][p-n-l -p-z-~dkfl)].
then ,t?,(k,n)=,t?,(k,n)=l and hence Clk
If
yl(k++J
*
342
[601, 43
FOUNDATIONAL STUDIES
The assumption that {ak}c C,, l e a to a contradiction. Indeed, if OD
ak=
z p ( k , n ) / p n , then y ( k , l ) = O for k
n-1
c
X, and p ( k , l ) = 1 for k X,.
The sets XI,& are thus separated by the set E[p(k,l)=O] and hence p cannot be B recursive function. TIiEomM 5. I j 7u) power of q ie di&ible
k
by p , then Cr,c#Cs,.
The idea of this proof is as follows. We develop l i p on the ecale q
n-1
and show first that 0 < x ( n) < q for infinitely many n. Let nl,n,, ... be a sequence of (not necessarily JI)values of n which satisfy this inequality. We now consider the same sets X , , X , and functions a,,% as in tho prea
vious proof and put a*= zy(k,n)/qn where y ( k , n ) = x ( n ) except when n-1
is the first term nh of the sequence nl,ns, ... such that a 1 ( k , h ) = O or % ( k , h ) = O . In this exceptional case we put y ( k , n ) = x ( n ) F l according as al(k,h )= 0 or %(k ,h) = 0. It follows that a k < l / p for k E X,and a* > l i p for k E X,,and hence the development of a k on the scale p is not recursive. The exact proof runs as follows. We may assume that for infinitely many n x ( n ) > O . Since the development of l / p is periodical, we can represent l / p in the form 7)
m
...+ x ( n , + s ) / ~ + ' i
iip=x(i)/q+...+x(n,)iq"o+ Zlq-IS[x(n,+i)iplo+l+ I-1
where % ( n o + ] )..., , x ( n , + s ) form the period of the development. In view of the assumption made above not all integers % ( n o + ] ) ..., , x(n,+s) vanish. If All of them were = q - 1 we should have l / p = x ( l ) / q + . ..+x(n,)/q"o+l/q*o
where x(n,) Im
p ,s) I 'P"0 -
- J.t(n0,
m
-v I m - I,(n ,p ,s)1 'P" L
"=,Io
1I / p *
i
I
n
2
1,P"O - (m,Ip%+l)
= 1 2p"o ~
"=O
whence n,>(s-l)'lg,p. Now we put
If k d Z , then &(n,p,k)=yi(n,p) for all n and hence m
(4)
If k c Z and sk=min[a(k,uj=O], then according to (3) F , ( n , p , k ) = y i ( n , p ) for 8k>Itlgpi+1.
Formula (2) proves that for these values of n we have also y,(n,p)=I,(n,p,ek). Hence
E , ( n tp k) = l i ( n , p y s k ) . The same equation holds for the remaining values of n as we immediately 8ee from (3). Formula (1) proves therefore that (4) holds also in the present case. From (4) we immediately obtain {at'} c C, and theorem 6 is thus proved. THEOREM7. C , - G, # 0 # C, - C, . Proof. Let e be a primitiQe recursive function such that the set
belongs to the class &? but not to the class 62;' for all k and z'). Put,
and that e ( k , s ) G I
m
n-1
The existence of a function e with these properties has been established by Markwald [2] and Mostowski [ 5 ] . 7)
346
1601, 47
FOUNDATIONAL STUDIES
We shall show that
{ak} E
c5. Indeed,
m
2
( q ! / n ! ) e ( k , n&} = { [ ( p , q , k ) = 1) where
6(p,rl,k)=sgn[p(q-1)--(*!/n!)p(k,n)]. n-1 If {ak} were in C,, we should have the equivalence {ak
is mtional) - C [ ( p / q < a ~ ) ( p / q ~ a L ) 1 - C [ [ ( p , q t k ) = 8 ( p , p , k )= 01. P.4
1.4
Since k E z if and only if ak is rational, the above equivalence would prove that 2 is a recursively enumerable set. Hence { a k }E C, - C,. Now we put a;= 1- a*. Since
{plq a k } = {C(y - p , q , k ) = I } , we obtain {a;} E C,. On the other hand {ai} 6 C, since { k . Z}={a; ~ is rational}. Theorem 7 is thus proved.
8. Let
2f(k,n)/p 01
ak=
n-1
and let g and h be functions such that
{ P / q < a k } { g ( p , q , k ) = 1) and { p i n > ak}= { h ( p , y ,k)= 1). w e know from theorems proved in sections 1 and 2 that if { a k } E C,, then the functions f , g , h do not need t o be general recursive. We shall prove here two theorems which characterize to a certain extent the nature of these functions.
[601, 48
347
O N COMPUTABLE SEQUENCES
THEOREM 8 defined by means
{ak} 6
of the
C, if and only if the ternary relations R, and Requivalentes
R + ( p , q l k ) :P/q are recursively enumerable. tl/tZ, then Proof. If l~k-q(~&lk)/tZl
=
ak
R*(p,q,k)-=C[piq> c c p ( ) z l k ) : t t * lin] n
and hence relations Rk are recursively enumerable. Assume now that R*(P , q , k ) " C A * ( P 1q 1 LlZ) n
where A & are recursive relations. We shall construct a general recnraive sequence of intervals I n k = (qk(n)/3",yk(n)/3") whose lengths tend recursively to 0 m d such that ak61nk for each n and k. The plan of this construction is as follows: We start with the interval I,k= ( 0 , l ) . Let us now assume that I n k is defined for a value of n. We subdivide it into three equal parts lyi= l . Since the next equation is evident, it remains only to prove that C$ = C$. If a= p / q , then a C$ and a E C$, we can therefore assume that a is irrational. The development of a on the scale p is given by the form
mula a= ZCF(n,p)/p"where E ( l , p ) = [ p a ] and 5 ( n , p )= [p"aI- p[p"-la] n-1
350
FOUNDATIONAL STUDIES
[601, 51
for n > l . If a c C,, then [p%] is a primitive recursive function of p and n and hence a s C,. If a C,, then l (1 , p ) is a primitive recursive function and hence p / q > a is a primitive recursive relation since { p / q > a } = { P >lqal) = { P > € ( l , q ) I . THEOREM 11. If p , q > l and a power of q i s divisible by p , then
c,c cp.
I n order to prove this theorem we repeat the proof of theorem 3 supressing the argument k in all the functions considered and assuming tp to be primitive recursive. It remains an open question whether a theorem converse to theorem 11 is also true. Another open question is whether C, is the common part of Cep ( p = 2 , 3 , ...) and whether C$ is the common part of C:,( p = 2 ,3 ,...). References [l] S. C. Kleene, A symmetric form of G6del’s theorem, Pndag. Math. 12 (1950), p. 244-246. [2] W. Markwald, Zur Eigenechaft primitiv-rekureiver Funktiomn, unendlich viele Werte anrunehmen, Fund. Math. 42 (1955), p. 166-167. [3] S. Mazur, Recursive alaalyeia (to be published). [4] A. Most owski, Development a d applications of the “projective” classification of sets of integers, Proceedings of the International Congress of Mathematicians, Amsterdam 1954, vol. 3 (1956), p. 280-288. [5] - Ezamples of sets definable by means of two and three qzcantifiers, Fund. Math. 42 (1955), p. 259-270. [6] R. Myhill, Criteria for constructibilily of real numbers, J. Symbolic Logic 18 (1953). p. 7-10. [7] R. PBter, Recursive Funktionen, Budapest 1951. [8] H. G. Rice, Recursive real numbers, Proc. Amer. Math. Soc. 5 (1954),p. 784-791. [9] R. M. Robinson, Review, J. Symbolic Logic 16 (1951), p. 280-282. [lo] E. Specker, Nicht-tonstruktiv beweisbare Salze der Analysis. J. Symbolic Logic 14 (1949), p. 145-158.
Recu par la R&zction le 17. 11. 1955
On Recursive Models of Formalised Arithmetic by
A. M O S T O W S K I Presented
OR
Yay 9, 1957
The aim of this paper is to prove a theorem on recursive models for axiomatic systems of arithmetic. It will be shown that under certain assumptions every such model is isomorphic with the s. c. classical model. 1. Let S be a consistent system of arithmetic with the primitive 2 ,j u 4 , y s , ..., where v j is the symbol for a function notions =, 0, 1, with p, arguments. The number of yj’s can be finite or infinite. We shall instead of 0, 1, f, 2 and abbreviate E - q = O a8 t < q , m i t e juo,pl,pz,p3 (6< q ) - ( t = 7j) as 5 (7, and - ( t = q ) as t f q . Let f 4 , f s , ... be primitive recursive functions such that f, has p j arguments. Let f o = O , f l = 1, f z ( a , y ) = x + y , f 3 ( a , y ) = z ~ y . We assume that the set of functions fj is closed, i. c., that all the functions which occur in the recursive equations for f j occur among the functions fk ( k # j ) . The system S is said to be adequate for the functions f j (j=0,1,2, ...) if it satisfies the following four conditions: I. All the tautological formulae of the ordinary first order logic with identity are theorems of S . 11. The following 9 formulae are provable in S :
+,
0is a denumerable model for the classical set theory and
X,={E(n, I), Ebb 21,
...>,
then the function 6 is not in I7:. Thus, the elements of a denumerable model cannot be enumerated by means of a I7:-function. Thus far we have considered only arithmetic of numbers. I n the rest of this speech I shall deal with some topics concerning constructivistic foundations of analysis and set theory. Instead of attempting a systemating exposition of this highly developed and difficult subject, I shall limit myself to a comparison of modern tendencies in this field with the older ideas of the French semiintuitionistic school. The starting point of the new theories is invariably the set of integers. All other notions are derived by dehitions from this fundamental notion. I n particular, the notion of a real number is not taken as granted but introduced in this way into the theory. Real functions are defined not everywhere but only on the constructivistic continuum. In older theories one took the notion of a real number as intuitively clear and did not analyze this notion. Limitations were imposed only on real functions which, however, were defined on the whole line. The new theories go deeper in the matter than the older ones as they start with a simpler basic notion. It is striking, however, that both the old and the new theories develop along parallel lines : in the new theories we have a classification of sets of integers into arithmetical and analytical hierarchies and in the old theories the classification of sets of reals into Bore1and projective hierarchies. The analogy between these pairs of notions was discovered by ADDISONin his thesis [l]; it goes much deeper than the analogies between arithmetic and projective hierarchies which I have stated in [15].
370
FOUNDATIONAL STUDIES
(691, 189
(la) A set which is recursively enumerable together with its cornI'lement is recursive (POST[21], KLEENE[5]).
( l b ) A set (contained in the Baire zerospace) which is open together with its complement is a union of a finite number of intervals.
.: (2a) A set which belongs to Z together with its complement is hyper-arithmetic (KLEENE[ 8 ] ) .
(2b) A set which is analytic together with its complement is Borelian (LUSIN,see 1121).
(3a) Each set X in IT: is rcpresentable aa the union UC, where u runs over denotations of constructive ordinals and the relation n E C, is recursive (KLEENE[7]).
(3b) Each analytic complement
X is representable aa the union
U C , where a runs over well orderings of integers and the relation a E C , is Borelian (LUSIN[12]).
It is not yet dear how far these analogies go and whether there exists one general theory which contains as its particular cases the t,heory of arithmetic and analytic sets on one side and the theory of Borel and projective sets on the other. There are theorems concerning projective sets of which satisfactory translations t o the theory of analytic classes of Kleene are not known. An example of such a theorem is furnished by the theorem of LUSIN-SIERPI~SKI [23] according t o which analytic complements are representable as unions of HI Borel sets. A particularly striking example o t parallelism between the old and new theories is g h e n by a series of theorems known as separation theorems. These theorems discovered first by LUSIN[12] have permitted him to solve great many problems concerning Bore1 and projective sets. Similar theorems for arithmetic and analytic hierarchies are of paramount importance in modern researches on recursive sets and their generalizations. An example of application of these theorems was mentioned above in connection with i-recursive sequences. I shall mention another example which has found interesting applications in metamathematics. Let K be a class of sets such that if X E K , then - X E K and also f - ' ( X ) E K for each general recursive function f . Let us call a pair A , B of two disjoint sets of K weakly universal for K if for every set X in K there exists a primitive recursive function 0 such that, X = @-'(A) and - X = 0-1 ( E ) .Sets forming a weakly
[69], 190
ON VARIOUS DEGREES OF CONSTRUCTIVISM
371
universal pair are not separable by means of sets which belong to R together with their complements. Indeed, assume that there is a s e t X , s u c h t h a t A C X , E K a n d B C - X , E K . L e t O,(x) b e a general recursive function universal for primitive recursive functions and put X=ri[O,(n) E- X,]. It follows that X E K ; hence, X = @il(A), - X = Oil ( B ) for suitable q. From these formulas we infer that 2 E X = O,(x) E A 3 O,(x) E X, and x non E rX = = O,(z) E B 3 O,(z) non E X, whence 2 E X = O,(z) E X,. Putting here x = q and observing that q E X = O,(q) non E X , we obtain a contradiction. It can be easily shown that one obtains a weakly universal pair for the class Z y (of recursively enumerable sets) by talung for as A and B sets of (Godel numbers of) provable resp. dsprovable sehtences of Peano arithmetic. This proves the essential undeaidability of this arithmetic. Other applications of weakly universal pairs are given in [4]. I n former considerations concerning the notion of effectivity an important role was played by the notion of definable sets or functions [lo], [ 2 5 ] . A mathematical object a is definable, if there exists a formula F with one free variable such that a is the unique object which satisfies this formula in the absolute model, i.e., in the model containing all objects whose types occur among the types of the variables of F . The notion of definability is essentially impredicative, similarly as the notion of analytic sets of Kleene: in the formula F which defines a may occur quantifiers whose range contains a. This impredicativity is responsible for the fact that arithmetic or analysis based on the notion of definable numbers or functions would not correspond to any constructive thesis. The relations between the notion of definability and Kleene’s analytical classes are also evident from the remark that the union D = (J Z : coincides with the class of sets which are definable by n
means of formulas all of whose variables have types < 1. Assuming the axiom of constructibility we can show that D is a model for the classical (impredicative) theory of sets of integers, i.e., that the comprehension axiom is true in D. It follows that axioms of classical arithmetic are true in D . Similar results hold also for the class of definable sets of higher types. I n order to prove these results we remark that if F ( X , , ...,X,)is any formula and
372
FOUNDATIONAL STUDIES
1691, 191
FD(X,, ..., X k )is obtained from F by a relativization of all quantifiers to D, then the formula is true. The proof proceeds by induction on the number of logical connectives and quantifiers contained in F and uses the fact that if 3 is a defimble well ordering of constructible sets, then the earliest (with respect to 3 ) set X , satisfying the condition P ( X , , ..., x k ) is definable, provided X,, ..., x k E D. The above theorems provide us with simple examples of denumerable models of the classical set theory. They can also be used to establish the existence of models K ; (see p. 185). At the same time these theorems confirm the impredicative character of the class D. In order to avoid misunderstanding let us observe that mat'hematicisns who have introduced the notion of definability did by no means want to take it as a basis for a new constructivistic foundation of matherhatics. They were interested in existential statements of the form ( E X ) F ( X ) and inquired, whether they can be strengthened to statements of the form ( E X ) [ ( X E D&) F ( X ) ] (if they had a constructivistic tendency they would have aimed at statements of the form ( E X ) [ ( XE D ) & P J X ) ] , where the meaning of the index D is the same as above). Using the terminology of KREISEL[9] we can say that the problem which these mathematicians faced was to determine, whether definable sets constitute a basis for all existential stathments. If we assume the axiom of constructibility, then a positive answer t o that problem follows immediately from (5); probably it would be extremely difficult to answer this problem without assuming the axiom of constructibility. To finish this review of older problems and ideas I shall devote a few words to the axiom of choice. 50 years ago this axiom played a prominent role in discussions ton effectivity in mathematics. There were even mathematicians who defined a proof as effective if it does not use the axiom of choice. In the modern research the axiom of choice seems to have lost its central role: constructivists of today take scarcely any notice of it. It is, of course, rejected by them but along with it many other axioms of the classical theory are rejected too, because they are found guilty of not conforming with constructivistic principles. In particular, the axiom of exten-
(691, 192
O N VARIOUS DEGREES OF CONSTRUCTIVISM
373
sionality which states the existence of an element in the symmetric difference (X- Y ) u ( Y - X) of any two different sets is clearly nonconstructive because it gives no method of constructing this element. It appears probable, although very difficult to prove, that there exist definable sets X, Y of real numbers such that (X - Y ) u ( Y - X ) contains no definable element. It is doubtful at present, whether one can give any precise meaning to the notion of a “set whose existence can be proved without the axiom of choice” and we must state that works on constructivistic basis did not bring us nearer to a solution of the now burning problem of independence of the axiom of choice. Curiously enough, the theory of constructible sets obtained by Godel by an extension into transfinite of the constructivistic ramified theory of types has resulted in a complete solution of the consistency problem of this axiom. I want to conclude with some purely subjective remarks on constructivism in general. Its most promising feature is that it wants to inquire into the nature of mathematical entities and to find a justification for the general laws which govern them, whereas platonism takes these laws as granted without any further discussion. The general aim of constructivism seemed therefore always very attractive to me. The solutions proposed by constructivists are, however, by far not so attractive. It is most disturbing that to one question we do not get one answer but a multitude of them. This is a natural result of the existence of a multitude of constructivistic programmes some of which we described in this paper. In my opinion there are no compelling reasons to accept any of these programmes as definite and to reject all the others. I do not see, for instance, in what should consist a superiority of recursive analysis over the ramified type theory. Notions which were created in the course of constructivistic discussions of the foundations of mathematics (like, e.g., the notion of a recursive number of computable functionals) are, of course, interesting and deserving study quite independently of the philosophical programme of constructivism just as the notion of Bore1 sets which is important for every mathematiaian quite independently of his sharing or not sharing the philosophical theses of half-intuitionists. But as far as a philosophical program is concerned, I am of the opinion that one should try to develop a philosophical analysis of the classical
3 14
FOUNDATIONAL STUDIES
[69], 193
foundations of mathematics. If we do not limit this analysis to pure mathematics only, but take into Bccount its applications, then it is not a t all impossible that we shall finally reach a more satisfactory theory than the various constructivistic trends have been able to produce thus far.
BIBLIOGRAPHY [l] J. W. ADDISON(jr), Analogies in the Borel, Lusin and Kleene hier-, archies. Thesis; University of Michigan 1955. [2] P. BERNAYS,A system of axiomatic set-theory, part I. The Journal of Symbolic Logic, 2 (1937), 65-77. [3] K. GODEL,The consistency of thc axiom of choice and of the generalized continuum hypothesis with the axioms of set-theory. Annals of Mathematics Studies, No 3, Princeton 1940. A. MOSTOWSKI,Cz. RYLL-NARDZEWSKI, The [4] A. GRZEGORCZYK, classical and the w-complete arithmetic. To appear. [5] S. C. KLEENE,Recursive predicates and quantifiers. Transactions of the American Mathematical Society, 53 (1943),41-73. [el , Arithmetical predicates and function quantifiers. Ibidem, 79 (1955),312-340. [71 , On the form of predicates in the theory of constructive ordinals (second paper). American Journal of Mathematics, 77 (1955), 405-428. PI , Hierarchies of number theoretic predicates. Bulletin of the American Mathematical Society, 61 (1955), 193-213. [9] G. KREISEL,A variant to Hilbert's theory of the foundation of arithmetic. The British Journal for the philosophy of science, 4 (1953), 107-129. [ 101 K. KURATOWSKI, Topologie I. Monografie Matematyczne No 3, second edition, Warszawa 1948. Einfiihrung in die operative Logik und Mathematik. [ll] P. LORENZEN, Die Grundlehren der mathematischen Wissenschaften, 78,Berlin 1955. [12] N. LUSIN,Legons sur les ensembles analytiques et leurs applications. Collection des monographies s u p la thhorie des fonctions (Collection Borel), Paris 1930. [ 131 W. MARKWALD, Zur Eigenschaft primitiv-rekursiver Funktionen, unendlich viele Werte anzunehmen. Fundamenta Mathempticae, 42 (1955), 166-167. [14] S. MAZUR, Computable analysis. To appear. [15] A. MOSTOWSKI, On definable sets of positive integers. Fundamenta Mathematicae, 34 (1946), 81-112. , A claesification of logical systems. Studia Philosophica, 4 [I61 (1951), 231-274.
[69], 194 [17]
ON VARIOUS DEGREES OF CONSTRUCTIVISM
375
--,
A lemma concerning recursive functions and its applications. Bulletin de l’Acad6mie Polonaise des Sciences, Classe 111,
1 (1953), 277-280. [18] -___ , Contributions t o the theory of definable sets and functions. Fundamenta Mathematicae, 42 (1955), 27 1-275. , On computable sequences. Ibidem, 45 (1957), 37-51. ~ 9 1
(201 S. OREY, On w-consistency and related properties. The Journal of Symbolic Logic, 21 (1956), 246-253. [21] E. L. POST, Recursively enumerable sets of positive integers and their decision problems. Bulletin of the American Mathematical Society, 50 (1944), 284-316. [22] H. G. RICE, Recursive real numbers. Proceedings of the American Mathematical Society, 5 (1954), 784-791. [23] W. SIERPI~SRI, Sur une propri6t6 des ensembles (A). Fundamenta Mathematicae, 8 (1926), 362-369. [24] A. TARSKI,Der Wahrheitsbegriff in formalisierten Sprachen. Studia Philosophica, 1 (1935), 261-405. [25] , Sur les ensembles definissables des nombres r h l s I. Fundamenta Mathematicae, 17 (1931), 210-239. I261 H. WEYL, Daa Kontinuum. Kritische Untersuchungen uber die Grundlagen der Analysis. Leipzig 1918.
A generalization of the incompleteness theorem BY
A. Mostowski (Warszawa) The aim of this paper is to prove the following generalization of the Godel incompleteness theorem (of. [l] and [9]): Let a formula @ with one numerical free variable be called free for a system S if for every n formulas @ ( A o ) ,@ ( A , ) , ...,@(An) are completely independent (i. e., every conjunction formed of some of the these formulas and of the negations of the remaining ones is consistent; @ ( A , ) denotes here the formula obtain d from @ by substituting the j-th numeral for the variable of a). We shall prove that free formulas exist for certain systems S and some of their extensions. In fact we shall prove for a class of formal systems 8 a slightly more general result: given a family of extensions of S satisfying c d a i n very general assumptions, there exists a formula which i s free for every extewion of this family. The following circuFtance deserves perhaps mentioning and justifies to a certain extent the length of the paper. Our considerations prove the existence of free formulas not only for systems based on the usual rules of proof but also for systemrJ based on the rule w. Thus they furnish another illustration df the parallelism noted already in [2] between these two kinds of systems. Our discussion of systems based on the rule w rests on the remark due to J. R. Shoenfield that the decomposition of the ZI: sets into constituents (cf. Kleene [a], theorem I, p. 417) can on several occasions be exploited in the same way as the recursive enumerability of the sets. Thus our paper can be considered as a test of this useful heuristic principle. From aresult noted a t the end of the paper it follows that no similar phenomenon ocours for Z I l sets. In view of these remarks the author hopes that his paper, in spite of its rather special subject may throw some light on a more important and broader topic, to wit the constructive analogue of the theory of projeptive sets.
P
1. We consider a consistent theory IT with standard formalization and infinite sequence do,A,, ... of its terms without free variables. The G a e l number of a formula @ will be denoted by r@l. A k-ary relation R (i. e., a subset of N t = No x ... x hT,where No is the set of integers > 0)
[74], 206
A G E N E R A L I Z A T I O NOF T H E INCOMPLETENESS THEOREM
is weakly represenhble in T if them i s a formula such that (1)
(4,
...,nk)
R E /- @(Anl.
Cg
377
with k free variables
...,AnJ .
R is strongly representable in T if besides (1)the equivalence (2)
(911,
nd $ R
F-@(Am,
-*-)
AnJ
is true for arbitrary n19...,mk. A function f : N t - + N o is representable in T if there is a formula @ with k + l free variables such that /- @(&,
--)
Ank,
X)
,...,
0 = df(n1
nk)
-
A relation R is weakly or slrongly representable relatively to a set Q C N ; if (1)and (2) hold for arbitrary (a, np,..., mk) in Q. We shall assume that I. Every primitive recursive fundion is strongly representable in T. 11. There are primitive recursive fasnctiom Neg, I m p , Con, Alt, Bb, Ex, a such that Neg(rCP7)= r-@l,W l I r n p r V = r @ 3V, r @ l C o n r ! V =r8ubst(si/Ai)@l, = r@ & V, r C P l A l t r V = r C P v V , B b ( i , j , r@l) E z ( i , j , r@1) = ~ ( E s m,)@l, ~, B(n)= rA,,l. We shall also assume that there are: a set P, called the set of proofs of T , and a quaternary relation .( which satisfy the following conditions: 111. The relation ( m ,n)< ( p ,q ) is refkxive, trarasitive, and well-founded in P x N o . IV. There is a formula n(x)which weakly represent8 P in T and a formula M ( x , y ; u , v) with the free variables indicated which strongly represents < in T relatively to P x No x P x N o . Moreover, these formulas satisfy the conditions: (i) F n ( c )& n ( z )& M ( s , y ; z , t ) & X ( z ,t ; x,y ) 3 y = t ; 3 “(3, Y ; A P , A,) v M ( A P , 4; x,y ) l ; (5) if p c p, then En(%) (iii) i f p , P~ and k- @(A,,dq)for every pair ( p , q ) in P x N o such that (P,q ) Q ( p o ,qo), then I - ~ ( x )& M ( x , Y ; A,, dQ0)3 @($, Y ) . Let {A,}, j = 0,1,2 , ..., be a family of sets each of which consists of formulas of T. We shall say that {A,} is a representable family of c m sistent extemwm of T if, for each j, (a) /- CP implies d c Aj; (b) @ A , implies -@ $ A,; (c) @ E A , and 0 3 Y EA, imply Y EAj; ( d ) (sk)@z Ai implies @(An) E A , for k, n z No; ( e ) t k e is a ternary relation. C such that @ E A ] = (Ep)[(p E P)& & C ( P ,i , ‘@l)I;
378
FOUNDATIONAL STUDIES
[74], 207
( f ) these are formula8 T ( x , y , z), r * ( x , y , z ) with the free variablea indicated which drongly represent relatively to P XN; the relations C ( p ,j, n) a d C*(P,i , n) = C ( p ,i, N e d n ) ) ; (g) Fn(4 n(s’) 6% Nb,Y ; s’, Y‘) & X ( s ’ ,Y’i s, Y ) 3 m,1 , = Y’, $)I; I- n(8)Bt n@’)82 J f b , y ; s‘, Y’) & J f w ,Y’; s, Y ) 3 [T*(s,y , 2) 3 T*(s’,y‘, z)]. We shalldenote by 4 the relation (m,n)(p,q), strongly and by g ( s ,y j z , t ) the formula X ( s , y ; z , t ) $ - M ( z , t ; s , y ) . represents 4 relatively to the set P xN, x P x N o .
m‘,
2. I n this section we generalize Rosser’s proof [9] and obtain THEOREM1. If {A,) is a representable family of corwistml e x t m b z s of T,then there is a closed formula 8 such that for any j neither 8 l u y ~-8 i s in A j . Proof, Let ,a(r@l)= Subst(y/dr,p-~)@ and let z ( x , y ) be a formula which strongly represents u in X. Consider the relation R(1, m,n) defined thus: c ( 4 m , n ) 3 ( E P , q ) { ( PE P ) B t “ ( P , q ) ~ ( l , m ) l & C * ( P , q , n ) } and the formula @(u, w , y ) :
(3)
n u , w , Y ) 3 (&, t)rnb)L B ( 8 ,t ; u,.a) & m a , t , Y ) 1 -
We shall show that @ strongly represents R relatively to P x x . Indeed, if 1 6 P and R(1, m , n),then either non-C(Z, m,n ) or there are p , q auch that P E P , (p,q) 2 , I- a ' k - 2 '
(d:r(f,na.n),t l ,
***,
tk--2)
B'k'(Af, t,,
& ( j , m , n ) such that,
(1)
...
7
tk--2, A m , An)
*
(1) This function need not be recursive in k although in the examples which we shall discuss later this is actually the cam.
[74], 210
381
A G E N E R A L I Z A T I O NOF T H E INCOMPLETENESS THEOREM
Let {Ai} be a representable family of consistent extensions of T and h an integer such that, for every j < lh(h) the integer (h)i is the Godel number of a closed formula F,,,i. Let &,j = Fh,o for i > lh(R) and let Ajh) = {Y: Fh,K(j) 3 Y E ALCi)}where K , L are functions inverse to the pairing function J ( m , n) = i ( m + n ) ( m + n - l ) + n . LEMMA1. If neither ph,j nor -Fh,j belong to Ak (i, k = 0 , 1, 2 , ...), then {A;?)}is a representable family of consistent extensions of IT. P r o o f . Conditions (a)-(d)are obvious. To prove (e) we denote by c'h the relation
< zh(h))
c(P7L(j)7( h ) K ( d m p t 2 ) (K(j)
>-w))
v q p , L ( i ) ,(h)oImPn)& (W)
and easily verify that (e) is satisfied. Finally, to prove (f) and ( g ) n-e denote by F h the formula
(E~~7v,~)[r(~7 ~7 v) & A ( Y ,w ) & B ( A h i y , z , v ) & D ( y , A n ) V r ( S , W , W )&B(&,
&, 2 , W ) & w D ( ~ A7h ) ] ,
where A , B , D are formulas which strongly represent the functious L ( j ) ,(h)=(,)I m p n and the relation K ( j ) D(a, p, 9i(LlQ1(kll i(s, t»), ,,(t») = 0 , f-., Dt«,
p, e , t):> D(a, p, 9i(Ll
Q1(k),
i(S, t»), I(t») = 0 •
Obviously in both formulas we can replace t by lh(s). From these formulas we infer that (19) implies (in A.,) the following formula
(EP)(s, t)D(a, P, s, t) :> ip(Yllll+1 (9i(L1 Q' (k)' i(S, lh(s»)) ,
x))
'9i(Yz",il+I{9i(Ll Q1(k),i(S, lh(s)l) ,
x)) = O.
Conversely this f!,lrmulaimplies (19), as we easily see using the theorem f-.,(E!s)D(a, P, s, t). We can simplify the formula obtained above by observing that f-., its, lh(s») = 9i(L1 q , s) for a suitable q and hence f-., 9i(L1l1/(kll its, lh(s»)) = 9i(Llil(kll s) with e.(k) = 12J(l?i(k) , q) + 8 (see axiom (017 », Thus (19) is equivalent to
(EP)(s, t)[tu», P, s, t):> 9i(Yll il+t{9i(Lli,(kll s), xl) '9i(Yz, il+t{9i(L1 il(k), s),
x))
=
0].
Finally, we notice that in view of axioms (016H Q ts)
h. 9i(Yll il+t{9i(L1i!(k), s), xl)·q;(yz, i':I(Llil(kll s), x)) = 0
== 9i(nk(Yll Yz), il+l(S, xl)
=
0,
where nk(Yll Yz) represents the function 12J(12J(mll &(k») +8, 12J(m21 ez(k») +8)
+ 7.
LEMMA. 21. For every Z there are primitive recursive functions gt(m, n), 9t(m, n) such that if a, u are elementary numerical formulas representing gl and iiI' then
f- .. H(k,l)(a(Yll Yz),
a,
x)
== [Hlk.I)(Yll'"
xlv H(k,l)(yz,
a, x)),
-(k,l)() _ -(k,1) -(k,1) f-., H a(Yll Ya), a, x = [H (Yll", x) & H (Yz,", x)].
[74],224
395
A GENERALIZATION OF THE INCOMPLETENESS THEOREM
Proof. .Again we shall prove only the second equivalence. Let be the formula (z)[y(z) = i(PI(Z), P.(z)ll and 8·(8,8',8") the formula [lk(8) = lh(8') = lh(8'!)] & (z) lz ~ lh(8) :) [(8)" = i((8')", (8"),,)1). It is obvious that 8(y, Pll P.)
and I- .. D(y, 8, t) & D(Pu s', t) & D(PI' s", t) & If m, m', m" satisfy 8· in
si», Pll
PI) -:J 8·(8,8',8").
mo, then m' = 2K (lm)o) . 3 K (lmh ) ... P:it~7)1It("I)
= f'(m), m" = 2 (lm)o) ,sL«m p~I~~)Ih("') = t"(m). The function f', i" aTe primitive recursive; let r ll r. be integers such that iji(Ll..., e) represents f' in A .. and iji(Ll r . , w) represents t" in A ... We then have 1-.. 8·(s, iji(Ll... , s),iji(Ll..., 8») and 1-..(s)(E!s', 8")8·(8, 8',8"). It is now easily seen that the right-hand side of the second formula. in the lemma is equivalent to L
(20)
h ) ...
(Ey)(8, t){D(a, y, 8, t):) [iji(Yll i,+I(iji(Ll... , 8),
x))
+iji(YI' i,+I(iji(Llra, 8),
x))
= 0]
I.
We reduce this formula to the desired form as follows: since the I K1I+ ...,..(I+1)(»)' imitirve recursive, . ) ... , Ai+! • ) ( p, P IS prirm there is an integer r' such that J 1+1(f'(K(I+I)()) ' f unction I p, -(Ll 1", Z) 1-.. II'
»)
l)()) -(I+l)() - {-( = 'Hl II' "-I~ rll -(I+ "1Z, "I Z, ... , -IHl)( "1+1 Z
.
This gives
Similar equation is provable with r, replaced by r. and r' by r", Thus (20) is equivalent to j](k,l)(ii(YI' y.), a, x) with
Hence UI(m, n) = 12J (l2J(m, r') +8, 12J(n, r") + 8) + 6. LEMMA 22. There are primitive recursive functions gil U. such that if l ;;. 1 and
T,
T are numerical formulas' representing g. and U., then
1-.. H (k,l,(Y,
a, X ) =_
H(k,I-I,(T ( ), Y, W,-I
- (k,I)(Y, 1-.. H
a, X ) =-
j](k,J-I)(-( ) T y, WI-I),., W., ... , W,-2 •
a, W., ... , W,-2) ,
396
FOUNDATIONAL STUDIES
Proof. By axiom (&)
[74], 225
we obtain
Lmn&i 23. For every 1 there are primitive recursive functions g3, & euoh thut if u,S are elementary term representing them i n A,, then for j = O,l, ...,I - 1
Proof. The right-hand side of the second formula is equivalent to
fl
pFmVn). We shall show (informally) that ( 2 1 ) is equiv-
pb(("')jdm*R))f
oe=k
dent to (22)
( E W ) ( D ( a Y, ,5 , $1
j.
3 ( s f ) [ x f ~ t ~ @ ( y , ~ ~ + l ( @ ( A , , ~ ( =o] xj,s))),z) If
Assume (21) and choose s , t , q so that D ( b , y , s , t ) and x f , < t . B(z) = y ( A z f ( A g + + ) ) , then @ ( A , , r ( x f ,8 ) ) satisfies the condition
D ( a , B , @ ( A , , t ( q , s ) ) ,t') for a suitable t; and hence, by (21), we obtain
F(Y, F*+l(@(Ar,i(xjci,s ) ) , z))= 0 . The converse implication is proved similarly. It remains to reduce ( 2 2 ) to the form indicated in the lemma. By the technique already used we find a primitive recursive function g, such that if E is an elementary formula representing g,, then
1741, 226
397
A G E N E R A L I Z A T I O NO F T H E INCOMPLETENESS T H E O R E M
It follows by axiom (Q,) that I-,
@(&m7
di)+d117
r(wL
h(87
..., Q-]))) = 0
x0, ..., sf-],
(xj)[zj < l h ( s ) I @ ( t ( Ydj)7 , i ( h ( 8 , xu,
***f
zj-1,
xi+l7 e e . 7
q-11,
s f ) )= 03
-
The left-hand side of this equvalence can obviously .be represented in the form @ ( v ( y ,dj), lz(s,zO7 ~ j - ~q, f l ,..., = 0 where ii is an elementary term representing a primitive recursive function. Since kw D ( a , y , s, t ) 3 t = lh(s), we see that (22) is equivalent to
...,
B'k,'-"(~(y, A ! ) , a , so,...,$1-1,
...,
L E ~ 24. A There are primitive recursive functwlur gs, g6 such that
if
v , i j are elemeutary terms representing them in A,, then for j = 0 , l ,...,1-1
,a ,
+"
jpk.l)(y,
%)
~
%)
H ( k J + 1 )( V ( Y ,
di,A t ) , a7 so2 " - 7
xi-17 27
xi,
zz-1)
9
fl(kJ+l) ( ~ ( ~ 7 A j 7 ~ ~ ) , a 7 x o , . . . ~~, x i"-7 l 1 -) .-7~~-1)
-
The proof is similar to that of previous lemmas. LEMMA 28. For every 1 there are primitive recursive functiom ge, g6 such that if 5 , ase elementary terms representing them in A,, then fw
j = 0,1, 0
...,1-1
H(kJ-1)
( U y ,A , ) , a , $ 0 )
k,* p J - 1 ) ( f ( y 7dj)7 a )$ 0 )
"'7
q-1,
%+I,
"'9
$i-17
%i+l,
*..,
$1-1)
"'f
(k I )
= (q)H' ( Y , a , 4,
*-I)
(&f)B'k'"(y7
%) *
Proof. Let us again consider only the second formula. The righthand side of it is equivalent to (EB)(s7 t ) [ D ( a 7 8 7 8 7 t ) 3 @ ( Yc 2 +,l ( s 7
$09
o..?
"i-1,
B(Oo),
*-*)
Q-I))
= 01-
We can replace B(do)by ( ( s ) ~ ~and ) ~ then ~ use the technique of the preceding lemmas to reduce the right-hand side to the desired form. L E ~ 26. A Let r be a n elementary numerical formula and & a n elevnentary propositional formula and led the variables (free and bound) occwing in them be s o w of the variables a+, ..., a k V l , xo7...,Q - ~ . Then there exist integers j , f , f , e , Z such that
F o r =~
" ( L Ia ~, 2) , ,
~ ~ ~ ~ = l $ ~ J ) (
&,-t = H ( ~ * ' ) ( LaI,~x, ) , ~ t j- ,,~ a =B('J) , z ) , (&, a , 4 *
Proof. From lemmas 23 and 25 it is obvious that i f lemma 26 holds for the formula C: then it does so for the formulas ( q ) E and (&j)&.
398
[74], 227
FOUNDATIONAL STUDIES
Lemmas 19, 20 and 21 show that if lemma 26 holds for formulas el, &, then it docs so for the forinulas -El, &,v &, and &, & (5%. Let us assume that the lemma holds for numerical formulas r,, r, and that fi, f t are the corresponding integers, i = 1 , 2 . For convenience of notation replace 1 by 1 1 and assume j = 1 f 1. Then
+
z =
ri -= H‘””+’~(LI~,,n, x ) gt ( t ) [ ~ ( ~ , ‘ + ~ ) ( v a, ( ~ s, j , )t ), v z = t ] a,
(t)pl+l)
(4407
0 7
( z = 4)
s,t )
and we easily see, using lemmas 19, 20, 25, that there are integers f: ,ji such that
t- zi = r, =_ @’+1)(At;,
(*)
Since
a,
x7 zi 1 7
I-- z.I -- r. t -= B(kJ+’) (4;, a7 x7 %)
k m r= l r, = (mi7z 2 ) w
(k,Z+l)
and since the formula z,
m;?
a , ~7
~ 1 )
& H(k.’+l)
= z,
(4,a, x7 22)
& ( Z l = %)I
can be brought to the form
~‘k,l+Z)
z17
s9
?
we easily infer, using lemmas 20, 23, and 24, that there is an integer e such that rl = rgzz H(k**)(A,,a , n). Similarly we find an integer E, thus showing that lemma 26 holds for the formula rl = r,. Replacing in the above proof the formula z1 = x, by z = z,+z, or z = zl.zz,we show similarly that lemma 26 holds for the numerical formulas rl+r,and r, x r,. Since xi can be represented as PD(At, a, x) and in a similar form with P$k*z) instead of I$“*”,we see that lemma 26 is true for the numerical formulas q ,j = 0 , 1 , ..., 1- 1. It is also obvious that if lemma 26 holds for the propositional formula t,it does so for the formula (q)&. Hence ) the assumption it remains to prove lemma 26 for the formula a f ( r 1 under that it holds for the (numerical) formula r1. Assume for simplicity that Farl = ( L Z ) H ( ~ * ~ + a ~ ,) x, ( L Iz)~ , ,and toI‘, = (LZ)B(ksz+l)(Aj,, a , n, z). Then by (*) FoaArl)= (11))(E4(B)(b7 t ) ( D ( a ,B , & .[
8,
=((8)4)0]
l-oaf(G) = ( 4 ( E z ) ( E B ) ( 8t)(D(a, , B,
8,
t ) &, (We) 2 X )
& IF(& TZ+&, t)3
s9
4) = 01) ?
((W-= 2)
v [v = ( ( 8 ) A j ) S ] & I@(d?:, 6+de7
s9
=O]))
-
[74], 228
399
A G E N E R A L I Z A T I O NO F T H E INCOMPLETENESS THEOREM
It is obvious that we can determine integers f : ,
I-*(W) > z) L [ v = ( ( 8 ) A , ) C ]
fi
[+(~l:,~ltP(~,= x,4 01)
= +(A,:, &+&, I-a@(s) -=c z ) v [ v
+=
such that
x,
8,
v))= 0 ?
((44J.1 8z [@(A%,h t 2 ( 5 , x , $1) = 01 = @ ( A j ; , t l + s ( s ,2 , B , w))
=0.
We thus obtain t-aaf(Tl)= ( ~ w ) ( E ~ ) H ! ~ , ' +a, ~ x) (, Az,~v;), ,
F m a f ( T l= ) ( w ) ( E z ) ~ ( ~ . ' + ' ) (a, A ~x,; , z , v ) ,
whence we obtain the desired result using lemmas 23 and 25. LEMMA27. For every k , 1 there are primitive recursive functions g 7 , gT such that if S,s are elementary numerical formulas representing them in A , and a' = ( a l , ..., ak-l), then
Fa f p - 1 . 0 (6 (
?
4,a ' , 2 ) = (ao)@,')(Y,
a,
4
9
.
IFk-'*')(tT(y,dk),a ' , x) = (E%)P*Q(Y, a, x) Proof. We shall prove the first formula. The right-hand side of this formula is equivaledt to (23)
(V)(Eao,B)(Es,t)[S(Y,ao,B)~~D(a,B,8,t)~+(Y,O+l(s,~)) =o] ,
where B is the formula used in the proof of lemma 21. Let h ( m , k) be the primitive recursive function h ( m , k ) = 2'((m)*).p n ) t
(m)r-x ...pk-1 pk
l"((m)r)
where f' and f" have the same meaning as in the proof of lemma 21 and let r be aa integer such that +(Ar, I ( $ , y ) ) represents h ( m , k ) in A , . Then ~ ( 0B, , @(dry ~ ( 8 A , & ) ) ,t ) ] t - - m ~ ( ~ ,66, B) 3 [ ~ ( a 'Y, , s,t) and we infer that (23) is equivalent to
(Y)(fi,1 ) [ D ( a ' , Y ,8~ t , & @ ( Y ,rl+i(@(dr,
dk)),
.)) = O]
*
This formula is reducible to the form required in the lemma in the way used several times in the preceding proofs. LEMMA28. Formulas H(O*"(y,x ) sdisfv condition VI. Proof. (i) follows from lemma 21. To prove (iii) we notice that + ( A ! , i t ( t o , ...,tt-s,dm, An)) =(P(Ac(,,rn,w), ~ - z ( t o , - * ) tl-s)), where t i 8 prim-
400
FOUNDATIONAL STUDIES
itive recursive (cf. axiom
-
-f
p - 2 )
(dC(l,m,n),
$09
*-?
(am)), and hence
[74], 229
km H(o*t)(di, q,,...,Q - ~A,,,, , dn)
Q-8).
Formula (ii) has i i ~our case the form
n(4)3 {H(0*4)(4,), tl, .*- 7 $4) (Eu,a ) [ n ( u & ) ( r (")-5r(ti, ~ , %I) &
(24)
& ~ ( o ' 4 ) ( ~ ru, , v,
SinCe l?m
~(r(G9 t z ) A r ( ~a!) )
t3, 4)1)-
n(4)& n ( u ) I ( [ ( r ( U , f W N l , t A ) & -(m1, t z ) 3 r ( $ h , q] = (Y)[".T+, Wl, t e ) 2 P ( u ,41)
(cf. lemma 15 and the remark following lemma 13), we can give to (24) the form (cf. lemma 10)
n(4)3 (~(0s4)(&w, t1, - - ?
t4)
(Eu,v ) ( a ,B, Y)
(&w(a,8,u)L"I11Z(Y,Il(tl,ta,,r(U,"))] &B'o'4'(4, % Q , W 4 ) j ) *
It follows from lemmas 26, 27, 23, and 20 that there is a primitive recursive T for which this formula is provable in A,. Lemma 28 is thus proved. It remains still to give examples of families {Aj} of extensions of A,, which satisfy the assumptions of theorem 2. Let B f be a recursive family of sets of closed formulas. Hence there is a recursive relation R such that 9 c Bj
3
R ( j ,rW).
We denote by r an integer such that the formula @ ( d , , i ( z , y ) )= 0 'strongly represents R in A,. Let us further assume that the sets A! = Cn,(Bf) are consistent. We shall show that the assumptions of theorem 2 are satisfied f6r the family {At]. Let &It,, be defined by transfinite induction on 6 as follows:
u closed forncuh)& (ICY)[(!?'&I a formula with e d l y &It+.,= {@: (@ one free a u r i a ~ ez)&(N)(Y(A,,) c s ~ & , ~t-((a)[Y(z) ) 3 @I)])
.
Thus 8t.j is the set of (closed) formulas which can be derived from B f by f applications of the rule w. Spector [lo] proved that A f = CN,(B,) = &,,j. Let FZm(zj, Zh(z) be elementary formnlss which (strongly) represent in A, the set of formulas with one free varimble st and the set of M e 1 numbers of closed formCP such that t-9.
(741, 230
401
A G E N E R A L I Z A T I O NOF T H E INCOMPLETENESS’THEOREM
Further, let s b ( z , y), gen(a), x i m p y be elementary numerical formulas which represent in A, the primitive recursive functions B b ( 1 , 1, r@T), I m p r!Pl (cf. 11, p. 206). N e g E s ( 1, 1, N e g ( r @ l ) , We consider the following formulas: Z o ( u ,2): ( y ) [ + ( u ,i ( r ,y)) ZAU, g , a‘): (Y)(+(%
(a is the minimal element of GU),
= 01
t ( Y , $1)
=
03
I+(%
;(a,
(a is the successor of Zdt474: ,-Z,((C,
4 85 (Y)([+,
t(Y,
Y)) = ov+(,,
3 (EY’) (Y f- Y‘ 8z m # Y‘ & I@(., ;(Y,Y’))
$7) = 01
in the ordering &),
3c’
9)= 01
i(Y,
(Y # 4 =
01 & [ + ( u ,i ( y ’ , 4) = 01))
(x is a limit clement of the ordering & ) . Let Z ( a , u , a) be the formula
(4(ZO(%$1 3 ( Y ) [ a ( i ( 3 , y ) = ) 0 = +(&,i ( v , Y))
= 01)
& ( a ,~ ) [ Z , ( U5 ,, d)3{ ( y ) I ? i ( i ( y)) ~ , = 01 = ( E t ) ( c ) l P l m ( t ) Li3
(4!tzz(qc,3) 3 (Y)([&.,
T h ( s b ( z , t ) ) &Th(!?en(timpy))]j)
Y)) = 01 3 (E$’)[pl(u,i($’, 3)) = 0 (m’ # $1 & a ( i ( z ‘ ,y)) = 01)) .
The following lemma explains the meaning of this formula: L E ~ 29. A Let p be in W ai,d let tp, p , j satisfy Z in %-, Thew [ v ( J ( n k)) , = 01 = [ k i s the Godel number of a formula @ i n Be*.j where En i s the order type of a segment of N o determined by n in the well-ordering <JP r o o f by induction on En presents no difficulties. LEMMA30. 1 - - , ~ ( u ) 3 ( E ! u ) Z ( a , u , v ) . P r o o f . by the formalization in A, of the usual existence and uniqueness proofs of functions defined by transfinite induction. LEKWA31. I-.,n(u) & n ( u ’ )& [T(u, v ) %l‘(zr‘, v’)] & Z ( a , u ,v) & Z ( a ’ , up, v ’ ) 3 { ( & ) [ a ( t ( ay)) , = 0) = ( W ) [ a ’ ( t ( dy)) , =
01) .
P r o o f (informal). The antecedent implies (cf. lemma 1 7 ) that v = v’ and 1c w u’. Let # be I a similarity mapping of N,, onto itself carrying , s2))= 0 over into @(u’,[(s;, si))= 0. We then the relation @ ( u [(al, prove by transfinite induction on the order type of the segment of the , = 0. first relation determined by s that a ( i ( z ,y)) = 0 = a ( i ( / l ( m ) y))
402
[74], 231
FOUNDATIONAL STUDIES
LEMMA32. The formulas
r ( u ,v , u9: ( a ) ( Z ( au, , v ) 3 ( E @ l a ( W w))= 01) I
I-*(%&,v , w): ( a ) ( Z ( a u,v) , 3 (E+(T(T,
.(W)))= 03
(where Y ( W ) is an elementary numerical formula which represents the function Neg in A,) strwgly represent ilz A , relatively to W x N: the relations
U st,j)l (E@)[(TL = r- @l) 8~(@ E U b'k.,)l
C ( p , j , l a ) : (E@)[(th= r@l) &.(@
C * ( p ,j , 98):
E
WPl
B n, and no variable V,, t > m, is free in CP and also that no qllision of bound and free variables occurs as a result of the substitution. We assume as known the Godel numbering of formulas. The Godel number of a formula CP will be denoted by W1and the formula with the Godel number n will be denoted by X. If X is a set of formulas, then we put 'X' = { 'CP' : @EX}.
1.2. Semantics. A frame for T is a relational system K = (K,X,ll, A) consisting of a set K and of two ternaryrelations X,II and one binary relation A with the field K. A K-valuation is a mapping f of the set {uo,u1,u2, ...} u {V,,V,, ...} satisfying the conditions: f(u,) is an element of K,f(vi> is a finite subset of K. For every K-valuationf we define the relation f StsfR CP as the smallest relation satisfying the following conditions :
In the last two definitions it is understood that f* runs over K-valuations. It is well known that conditionsf StsjKCP and f' Stsj=@are equivalent iff and f' coincide for arguments ui, V, which are free in CP. If CP has exclusively the free variables ni,uj,...,Vp,V*,... ( i < j < ...,p < 4 < ...), then
412
[76], 272
FOUNDATIONAL STUDIES
instead of f Stsf,@ we shall write briefly I,@[ f (q),f (uj),. .., f (V& f (V,),. ..].A closed formula @ is true in K (symbolically I-#) iff Stsf,@ for an arbitrary K-valuation$ In view of the above remark, if @ is closed, then (I-,@) = (Ef)(f Stsf,@). If X is a set of closed formulas, then C n ( X ) is the set of all closed formulas @ such that kK@ for every frame K satisfying the condition: kY , for every Y in X . Tarski’s problem referred to at the beginning can now be formulated as follows: if K O is the field of real numhers, does there exist a set X of formulas such that ‘X’ is recursive and C n ( X ) is the set of all formulas true in KO? We shall show that there is no such set even if we replace the word “recursive” by the word “analytic” 2. Arithmetization df the semantics of T
2.1. Auxiliary notions. We shall use the customary notations of the theory of recursive functions (cf. Kleene [l]). Greek letters p,$, ... stand for infinite sequences of integers. The j-th term of p i s denoted by $r or p(j). A relation R(p, ,..., pk, n , ,...,n,) with k functional and I numerical variables is called arithmetic (or anulytic)(’) if it belongs to the smallest class of relations containing relations
-
where S is a primitive recursive relation and i,, ( j 2 k , s 5 q j ) are integers 5 I , and closed with respect to the logical operations V , as well as the operations (Enj) (or closed with respect to these operations and the operations (Epj)), j = 1,2,... . Let J( j,n) be the one-one primitive recursive function 2j(2n 1) which maps pairs of integers onto integers and let n,(cr) be the function B such that B(n) = cr(J(j,n)).
+
*2.1.1. If R(pl ,..., pk,nl,...,nl) is an arithmetical relation, then so is R(nj,(p,),..., nj,(VJ,nl,. ..,n!)* . Let us denote the relation R(njl(pl),..., njk(Vk), n , , . . .,n,) b y R‘(pl ,..., p)k, n , ,..., n,, jl,..., j k ) . If R has the form 2.1 (I), then PROOF.
Cf. Kleene [2]. Our definition of “analytic” deviates slightly from the one (I) given by Kleene but is equivalent to his.
[76],273
AXlOMATlZABlLlTY O F THE FIELD O F R E A L N U M B E R S
413
and hence R’ is arithmetical. If R , , R , are two relations such that the corresponding relations R t l , R’, are arithmetical, then the same is true for -R,, R, v R , and (Ens)R, since ( -Rl)’ = - R r 1 , ( R , V It,)’ = R’1 V R‘2, ((Enj)Rl)’ = (Enj ) R’ .
,
*2.1.2. If R ( p , ,..., pk, nl ,..., n,, j ) is an arithmetical relation, and m < k, then there is an arithmetical relation“ S(+hl,..., $ k , n l ,..., n,) such that
PROOF. In view of 2.1.1 it is sufficient to show that
If the left hand side of (1) is true, then for every j there are functions $ k , j such that R ( Y ~ , .., v m , $m + 1 ,j ’ . .,$k. j , nl ,..- 9 1 1 , , j ) . Defining Q m + r such that 9m+r(2j(2n+1)) = $m+r,j(n) for j , n = 0,1,2, ..., r 5 k - m , we have nj(9,,,+J= $ m + r , j , and hence the right hand side of (1) is satisfied. Conversely, if the right hand side of (1) iv satisfied, then so is the left, since for every j we can take as $,,,+r,j the function ~ ~ ( 9 ~r +=~1,2, ) ,..., k - m . Theorem 2.1.2 is thus proved. 2.2. Frames K,. For every sequence p we denote by K , a frame (No,Z,,II,,,A,) where N o is the set of integers and Z,,lI,,A,, are defined as follows.(2)
IL, + 1,j,*
(2) We use the symbol n(,) for the exponent ofp,-th prime in the expansion of n into a product of primes. The symbol means (u(j))ck,.
414
FOUNDATIONAL STUDIES
(761, 274
2.2.1. There are arithmetical relations R(v,+,n) and S(p,+,n) such that for euery closed formula @
PROOF. We introduce the following abbre~iationd~)
The following properties of this relation can easily be established. (3) (Ej),, ( ( j ) “ )is an abbreviation for: for at least one (for every) j which is less than n or equal to n. Ih(n) is the “length function”.
[76], 275
AXIOMATIZABILITY OF T H E FIELD O F REAL NUMBERS
412
(1) If T(p,p,a), then a(o)sa(l),..., a(lh(a)) are Gadel numbers of formulas 'ii(O), ...,&,(o)) each of which is either (1") atomic or (2") can be obtained from one or two of the preceding formulas by joining them with a stroke or by prefixing with one of the quantifiers (Eud, (EV,). (2) If T(p,p,a), then for every j 5 I&) and every q
wherefq(v3 = 4 ( 2 , ) andf,(vi) = :nEq(2,+1,). (3) If (DO,...,@"is a sequence of formulas satisfying (1") and (2"), Ih(a) = n, and a is such that a(j) = r@jl for j 5 n, then there is a p such that T(p,p,a). Indeed, let pj(q) = 0 or 1 according as f,Szsf 0, or f q non Szsf CPj 5 KC and define p by the equation y(2'(2q+ 1)) = pkq). We have then n,(p,q) = pj(q) and the equivalence (i) is satisfied for every q and for j = 0, 1,. .., n. From this we easily obtain T(p,p,a). We define now relations R and S as follows. = n>&(Eq)(%h(a)(F,q) = o)],
R(Y,&n)
(Ea)[T(~,p,a)di(a(lh(o))
S(9#,n)
(a) [T(p,&a)& (a(lh(o)) = n,
= (&) (nlh(a)(p,q)
= O)].
Let @ = 0" be a closed formula such that k @; choose a sequence ...,(Dn satisfying (1") and (2"). If a,p are Krsuch that T(p,p,a) and W1, then by (i) for j = Ih(a) we obtain (qh(,,)(p,q)= 0) = 0 ; since I- 0, the right hand side is satisfied for an arbitrary K# (Eq) [ q h ( o ) ( f p , q ) = 01. This proves that
(DO,
[FK,.l
= (w(p,4r@1).
If (p)S((p,p,W1),then by (3) we choose a p satisfying T(p,p,SCP1) and obtain (Ep)R(p,p,'@'). Hence
Finally,if R(p,p,W1), then by (1) and (2) there is a q such thatf,Szsf and hence k CP. This proves that
s
(Ep)R(v,~,r@l) 3 kK,,@
and the theorem is proved.
=I4
CP
416
[76], 276
FOUNDATIONAL STUDIES
2.2.2. Every f r a m e K = (K,X,ll,A) with a denumerable K and with non-void Z,ll,A i s isomorphic to a f r a m e K,,.
PROOF. We can assume K = N o . Enumerate (if necessary with repetitions) triples (aj,bi,ci) and (a;,b;,c;)
satisfying Z(ajyb;,c;) and
ll(a;,by,c;) respectively, and pairs (aJ:, b y ) such that A(aJy,cJy).Define ,u such that ,u(3:-&l)
$4) = a;,
= c;t
P( 3 (1) i + 2 ) = a;,
= b‘j ,
p C( 3 )j ) = CJY
pcL‘3($2)
P( 3(1)j + l )
=
a;,
(3j+I) (2)
P
= bw
i’
= bJ7. For any such ,u we ob-
viously have K,, z K.
2.2.3. If A is a set of closed formulas such that in no finite f r a m e all formulas of A are true and if ‘A’ is analytic, then so is ‘Cn(A)’.
PROOF. If 0 # Cn(A), then there is a frame K such that kKY for every Y in A but non kK@. By the generalized Skolem-Lowenheim theorem (Tarski [4]) there is a denumerable frame K with the same properties. Using theorem 2.2.2 we obtain (1)
E
C n ( A ) = (,u){(j)(j E A
2
kKPj’)
3
kKP@)
and the theorem follows from 2.2.1.
* A set A of closed formulas is ll-definable if there is an arithmetical relation P(pl,,...,plk,n) such that
*2.2.4. For every ll-definable set A of closed formulas there is an arithmetical relation Q(pl, ,...,plm,n) such that O E E n ( A ) = (pli ,..., 9”)
.
Q(vi,. . , ~ m , ~ @ ’ ) .
PROOF. Formula (1) above together with theorem 2.2.1 gives
Using theorem 2.1.2 we reduce the right hand side to the desired form.
[76], 277
A X l O M A T l Z A B l L l T Y OF T H E FIELD O F R E A L N U M B E R S
417
3. Definability of recursive functions 3.1. Axioms for the theory of ordered fields. We denote by X o the set consisting of the following axioms :
THEOREM 3.1.1. K i s a frame such that kKYfor Y e X Oif and only if K i s an ordered field. PROOF. Obvious. 3.2 Abbreviations. We shall introduce abbreviations for a number of formulas :
418
[76], 278
FOUNDATIONAL STUDIES
+
+
In the last definition, m = max(j,k,l) + 1, n = rn 1, p = m 2, q = m + 3 . In the following theorems K denotes an ordered field, K its set of individuals, 0 and 1 its zero and its unit, a + b and a b are results of addition and of multiplication of K and c is its ordering relation. The inverse of a is denoted by l / a . The element 1 + 1 ... + 1 (n times) is denoted by n. Proofs are omitted for the most part, as‘ they are very elementary.
+
3.2.1.
I-,Z,[a]
3.2.2.
I-,Sc[a,b]
3.2.3.
’
= (a
+ 1). = (m + 1 = n).
(b = a
3
kKS[m, 1, n]
3.2.4.
I-H,[A,a]
3.2.5.
kKN[a]
= n).
= (OEA)&(b)[(bEA)&(b n for every n then A’ contains all elements n which contradicts the finiteness of this set. Hence there is an n such hat n 6 a < n +- 1. Since kKH[A,,,a] it follows U E A ,whence a = II. 3.2.6. I-,C[A,a]
(ic
a
cj
E
(En) ((a
=
n)&(j){(f < n)
3
( E ! a ) [ ( a € A )&
+ 1>1>>.
3.2.7. kKW[A,a,b,c]
=
(En,p)[kKC ( A , n)]&(b = p)&
( p < TI)&(U = n)&(p
+ l / ( c + 1) E A ) ] .
3.2.8. Euery set A satisfying k,C[A,n] determines a sequence co,cl,...,c,-I of positiue elements of K ; cj is the unique element of K satisfying j + l / ( c j + 1) E A ; it can also be characterized U S the unique element of K such that kKW[A,n,j,cj].
[76],279
AXIOMATIZABILITY OF T H E FIELD OF R E A L N U M B E R S
419
3.2.9. For arbitrary positive elements cO,...,c.-, of K there is a set A such that t-gc[A,n] and t-KW[A,n,j,cj] for j = OJ,.,.,n-1. 3.3 Definability. A relation between integers R(n,, ...,nJ is called definable in K if there is a formula 0 with k free variables such that
3.3.1. Relations R,, R,, R, defined as follows R,(n,)
= ( n , = 01, R$’(nl,
R2(n1,n2) = (nl = n2
+ 11,
...,n& = (n, = n j )
are definable in an arbitrary ordered field K.
PROOF.In the case of R , we take as CP the formula Z,(ul),in the case
of R , we take as CP the formula Sc(u,,u,) and in the case of Rk,.’ we take as CP the formula (u, = uj)&(u2 = u,) ...&(uk=u&.
3.3.2. I f f , g are functions such that the relations p =f ( n l ,...,ns) and q = g(m,, ...,mJ are definable in K, then so is the relation
We outline the proof only for the case s = t = 2. Let CP and Y be formulas such that
LeLO(x,y,z,t) be the formula (Eu)[Y(u,y,z)&N(u)&@(x,u,t)]. The condition t-,0[p,m,,m2,n,] is equivalent (cf. 3.2.5) to the existence of an integer q such that t,Y[q,m,,m,] and kK@b,q,n,], i.e. q = g(m,,m,) and p =f(q,n,), or equivalently p =f’(g(ml,m2),nl).
3.3.3. Zfg, harefunctions,f(O,n, ,...,n,) = g(n,,...,n,),f(n+l,nl ,...,n,) = h(f(n,nl ,...,n,), n, n , ,...,n,) and if rhe relations p = g(n,,...,n,) and q = h(m,n,nl,...,n,) are definable in K, then so is the relation q = f(n,n ..,n,).
420
(761, 280
FOUNDATIONAL STUDIES
PROOF. For simplicity we assume s = 1.Let @,Ybe formulas such that
and denote by O(X,y,z) and Z(x,y,z) the following two formulas:
+
+
lynl]. Hence k,C[A,n 13 Let A be a set such that k,O[A,n and the sequence co, ...,c, corresponding to A (cf. 3.2.8) satisfies the condition: every c j has the form mi, where m j is an integer, kK@[mo,n,], if j < n then kKY[mj+,,mj,j,n,]. From this we infer that m , = g ( n l ) , m i + , = h(mj,j,n,) for j < n, and hence m j =f ( j y n l ) ,j = O , i , ...,n. By running the argument backward we see that if A is a set such that the corresponding sequence coy...,c, satisfies the equations c j = mi, where mi =f(j,n,), then k,O[A,n+l,n,]. Assume now that q =f ( n , n , ) and put mi =f ( j , n l ) for j S n. If A is a set such that the corresponding sequence co,c,,...,c, satisfies c j = mi, I' =< n , we have k,Sc[n,n+l], k,O[A,n+l,n,] and k,W[A,n l,n,q] and hence k,Z[q,n,n,]. Similarly from k,Z[q,n,n,] we obtain q = f ( n , n l ) ,
+
Q.E.D.
3.3.4, If f is a primitive recursive function, then the relation m = f ( n , , ...,n,) is definable i n K . This foIlows from 3.3.1 - 3.3.3 by induction. 4. Representability of analytic predicates 4.1. Formulas representing sequences. We say that a formula cf, with
three free variables represents sequences in an ordered field K if for every a in K and every integer n there is exactly one integer n7 = P("'(n) such that k,@[a,n,m]. The sequence p(a) in general depends on K and cf, and would be denoted more accurately by We shall construct an example of a formula representing sequences. We obtain it by expressing in T well-known definitions of the theory of continued fractions.
Bg,,.
(761, 281
4,21
A X I O M A T l Z A B l L l T Y OF T H E FIELD OF R E A L N U M B E R S
In the definitions 2-4 we assumed m = i + j + 1, n p = ;+ j + 3 and similarly in the definition 5.
=
i+j+2,
The next theorems explain the meaning of these formulas; K denotes, as usual, an ordered fietd. 4.1.1. tKZ[a]E (0 < a < l)&(a is not representable as m/n f o r a n y integers in, n). 4.1.2.
k,F[a,b]
4.1.3.
k,A,[u.h]
E (En)[(b = n)&(n
= t-,F[l/a,b]
Ia < n + l)].
or b
=1
and a=O or no c satis3es
tKF[l/a,cl.
= ( b = l / a - c)
4.1.4.
k,Ro[a,b]
4.1.5.
t,A[a,b,c]
= ( E n ) ( ( b = n)&{ there
t-KAO[d,&],
kKRo[a,do]and k,&[dj,dj+ l ] f o r ewry j < 1 2 ) .
such that
where t-,&[a,c]
or a=O andb=1.
is a sequence do,d, ...., d,
4.1.6.
tK(E!C)AO[n,c],t-,(E!d)R0[a,d], tKAo[a,c]3 ( E t l ) ( c =n).
4.1.7.
t,(E!c)A[n,n,c], t,A[a,n,c]
2
( E m ) ( c = m).
Proof by an easy induction on n. From 4.1.7 it follows that the formula A represents sequences. The meaning of this formula is particularly obvious when K = K O is the field of reals :
422
4.1.9.
FOUNDATIONAL STUDIES
7 =B”K,
[76],282
For every sequence y of integers there is an a c K , such that and kKol[a]*
This is a corollary of 4.1.8 and of a well-known theorem of arithmetic. 4.2.
Theorems on analytic predicates. In this section we assume that
K is a fixed ordered field and A a formula representing sequences in K. Instead of /3:q)K we write simply PI. 4.2.1. Let S be a primitive recursive relation of 1 + q l + ...+ q p arguments, let i,,* (r = 1,2, ...,p, s S 4,) be positive integers 5 1. There is: a formula CP with p 1 free variables such that for arbitrary al,...,a p the following equivalence holds:
+
PROOF. For simplicity we assume p
= 2, q1 = 4 , = 2, 1 = 2 and put
i,, = i,, i,, = i,, i,, =j,, i22= j,. Thus il,i2,il,j2 are a formula such that
2. Let Y be
and let CP be the formula (with free variables v1,u2, z(=uj), w(=v4)):
then for arbitrary c,, c,, d,, d 2 , from
[76],283
AXIOMATIZABILITY O F T H E F I E L D O F R E A L NUMBERS
423
see that (3) is verified for ck = mk, dk=pk, k=1,2, and hence (4) and (1) give S ( m l , m z , ~ l , ~ z , n l , ni.e., z), (5)
Conversely, if (5) is satisfied, then by (l),
with the same meaning of m,, P f (k = 1,2) as above. If cl,cZ,dl,dzsatisfy (3), then by 3.2.5 there are integers m;,m;,p;,p; such that ck = m;, dk = p; (k = 1,2); since A represents sequences in K we infer that mi =p l ) ( n k= ) m, and p; = /3@*)(nk) = p k , k = 1,2, which by (6) proves (4). Thus (3) implies (4), and (2) is satisfied. 4.2.2. For every arithmetical relation R(tpl,...,g~,, n, ,...,n,) there is a formula CP with k I free variables such that
+
PROOF. Let C be the class of relations for which there is a formula @ satisfying (1). It is obvious that if R,, R z are in C , then so are non-R1 and R1 v R2. I f R is in C, then so is (En,)R, since from(1)it follows that (En,)R(j?@'),...,p(ak),nl,...,n,) = tKY[nl,...,n, -l,n,+l,...,nl,al,...,a~] where Y is the formula (Ev,)[N(v,)&CP]. By 4.2.1 C contains all relations of the form 2.1 (1) and hence it contains all arithmetical relations. *4.2.3. Let K be an ordered field and A a formuia representing sequences in K. For every arithmetical relation R(q1,...,p)k, nl ,...,n,) there is a formula Y with 1 free variables such that
If, in addition, K satisfies the condition: every function recursive in one (functional) quantifier predicates for an a in K,
(w) is representable in the form
r hen
424
FOUNDATIONAL STUDIES
(761, 284
PROOF. Take as Y the formula ( u , + ~..., , v , + ~ ) @ where , Q, satisfies 4.2.2 (1). If (p,,...,pk)R(p,,...,vk,n , ,..., nl), then for arbitrary a , ,...,ak in K t,@[n,,. ..,n,, a , ,..., ak] and hence tKY[nl, ...,n)]. This proves (1). Now assume that ( W ) is satisfied and that there are pl ,..., p?k such that non-R(p, ,...,pk,n , ,..., n,). It is known (cf. Kleene [2], p. 324) that if such p exist, we can assume them to be recursive in one quantifier predicates. Hence, by ( W ) , there are a , ,..., a k in K such that non-R(B("l',..., /3'""), n , ,..., n ) and the result follows by 4.2.2 (1). 4.3. De3nability of analytic predicates in K O . I n this section, K O is the field of real numbers, KO is the set of all real numbers and A is the formula defined in 4.1. 4.3.1. For every analytic relation R ( 9 , ,...,p k , n , ,...n,) there is a formula CD with I - + k free variables such that f o r arbitrary a,, ...,uk in KO
(1)
R(p("l),...,/3("k), nl ,...,n,) = tKo@[n,,...,n,,a, ,...,u,].
PROOF. Let C, be the class of relations for which there is a CD satisfying (1). From 4.2.2 it follows that C, contains all arithmetical relations. We prove as in 4.2.2 that if R , , R , are in C,, then so are the relations non-R,, R , v R , and ( E n j ) R , . Hence it remains to prove that if R is in C,, then so is R' = ( E q j ) R . Let Y be the formula ( E u , + ~[Z(U,+~>&@]. ) I f there is a pj such that R(/3("1),...,/3("j-l), pj, /3("1+') ,...,/3(""), n , ,...,n,), then by 4.1.9 there is an uj such that t K 0 Z [ a j ] and qj =/3("?. Hence from (1) we obtain t,,Y[n, ,...,n,, a , ,...,a j - , , a j + ,,...,a k ] . Conversely, if the last formula is satisfied, then there is an a j such that tKoZ [ a j ] and tKoCD[n,,...,n,, a , ,...,uk] which gives by (1) R(p("l),..., pCok), n 1,...,n,) and hence R'(/?("l),...,p("j-l),p(uj+l) ,...,P("k),nl,...,n,). Theorem 4.3.1 is thus proved. 5.
Theorems on axiomatizahility
5.1. Solution o j Tarski's problem. In this section we make the same assumption as in 4.3 that K O is the field of reals. Our proof does not differ from the well-known undecidability results of Godel and of Tarski on indefinability of the notion of truth.
[76], 285
AXlOMATlZABlLlTY O F THE FIELD OF REAL NUMBERS
425
5.1.1. There i s a primitive recursive function s(n) such that f o r every formula Y with exactly one free oariable vo
PROOF. Concatenation of formulas is reflected as a primitive recursive operation on their Godel numbers. 5.1.2. T h e set U
= {W’: kKo@}
is not analytic.
PROOF. Assume the contrary. Hence the set ( n : s(n) 4 U> is. analytic and by 4.3.1 there is a formula Y with exactly one free variable vo such that s ( n ) 4 21 = ~K,(oO)[Zn(vo)= ‘YI. Putting n = Y1and using 5.1.1 we obtain
which is a contradiction. 5.1.3. There is no set B of closed formulas containing axioms 3.1 such that ‘B’ is analytic and @ E Cn(B) = kKo@.
PROOF. If there were such a set, then U = Cn(B) would be analytic by 2.2.3, which contradicts 5.1.2. (The assumption of 2.2.3 is satisfied since (x) ( E y )[L(x,y)] is true in KO but not in any finite frame.) *5.2. A related result. In this section we assume that K is an ordered field, A a formula representing sequences in K. *5.2.1. If U is a II-definable set of closed formulas, then there is an arithmetic relation Q(v1,..., vm,n) such that for every formula 0 with exactly one free variable:
Proof. By 2.2.4 and recursivity of s. *5.2.2. If K satisfies condition ( W ) of 4.2.3, then there is no II-definable set U such that (1)
@ E Cn( U) = kK a.
426
FOUNDATIONAL STUDIES
1761, 286
PROOF. Assume that (1) is true for a ndefinable set U and choose Q according to 5.2.1. By 4.2.3 there is a formula Y with exactly one free variable uo such that (971,...,~,)Q(~1,...,~~,n)= FKY[n] and hence
The right hand side is equivalent to FK(Euo)[Z rel(uO)&Y(uO)], i.e. to non- kX(Vo)[Zre,(Vo) =I -Y(uo)]. Hence taking 0 to be -Y we obtain
i.e. by (1) Z(r-Y1) thus proved. -pc
E
Cn( U)
E
Z(r-Y')$
Cn(U). Theorem 5.2.2 is
REFERENCES
KLEENE,S. C., 1952, Introduction to metamathematics, North-Holland Publishing Company, Amsterdam. [2] 1955, Analytic predicates and function quantifiers, Truns. Amer. Math. SOC., 79, 312-340. [31 TARSKI, A., 1959, What is elementary geometry? The axiomatic method with special reference to geometry and physics, 16-29, North-Holland Publishing Company, Amsterdam. [41 1958, Some model-theoretical results concerning weak second order logic, Notices Amer. Marh. SOC.,5, Abstract 550-6. [I]
-
Definability of Sets in Models of Axiomatic Theories by
A. GRZEGORCZYK, A. MOSTOWSKI, C. RYLL-NARDZEWSKI Presented by A. MOSTOWSKI on January
23, 1961
In the present paper we introduce and discuss the notion of dehability of sets of integers in models of first order theories. This notion is a natural generalization of the notion of representability in a model of type theory [I]. *) Our main result states that under very general assumptions the family of sets which are definable in all models of a theory coincides with the family of recursive sets. **) 1. The space of models. Let T be a consistent first order theory with a finite or denumerably infinite number of predicates and with a denumerable number of variables xl, x2, ... The propositional connectives are noted as -, +, v, A , and the quantifiers as v and A. A frame %I of T (in the set of all integers) is a family of relations indexed by the predicates of T, each relation having the same number of arguments (running over integers) as its index. A frame %I is a model if all theorems of Tare true in %R. We denote by M the family of all models of T. From now on until section 4 the word “model” will be synonymous with the expression “an element of M“. If rp is a formula, then t T a, means that p is provable in T. Obviously we have then k~ A xc xj ...X k ‘p for arbitrary i,j, ...,k as well as kT Sb (xI/x~)qj, where Sb (xg/xj) ‘p is a formula obtained from p by replacing the free occurrences of xi by xj and the bound occurrences of xj (if any) by Xj+k where k is the largest index of a variable occurring in v. For %I in M we write 1-m p if a, is satisfied in %I by an assignment which correlates the value j to the variable XI, ,j = 1,2, ... We put [p] = {%I: I=m y } . It is known that set theoretical operationsonclasses [pl correspond to logical operations on formulas, e.g., [(PI u [v] = [ y v y]. We shall consider M as a topological space by taking sets [p] as a basis for open sets; p runs here over all formulas of T. *) Parts of our discussion (es&y
those concerned with the o-standard models) were stimu-
of this paper. **) A part of our results was announced in [4].
ated by the reading
428
[77],164
FOUNDATIONAL STUDIES
THEOREM 1. M is 0-dimensional separable space which admits a complete metric (i.e., M is an absolute G J . Proof. The only non-trivial part of the theorem is that which states the existence of a complete metric. We shall sketch a short but indirect proof based on ideas of [6]. Formulas of T form a Boolean algebra under the usual operations, formulas mutually equivalent in T being treated as a single element of the algebra. The family of all prime filters of this algebra (i.e., its Sfone space) is a bicompact 0-dimensional separable topological space. The open basis of this space consists of sets {p} of prime filters containing p. Our space M is homeomorphic with the set of those prime filters which preserve denumerable meets corresponding to general quantifiers. More exactly these filters f have the property: whenever p is a formula and Sb (xt/x,) 97 is in f for n = 1,2, ..., then so is' Axt p. Since the set of all formulas is denumerable. filters f with the above property form indeed a G . set. (In [6] it is proved, moreover. that this set is residual which immediately yields the completeness theorem). In order to conclude our proof it is sufficient to use the well-known fact that every G , set contained in a complete metric space possesses a complete metric (see e.g., [3]. p. 316). 2. to-models. Let n n , n = 0, 1, 2, ... be a sequence of formulas with exactly one free variable x1. We denote the formula Sb (sl/x,) n n by nn(xj) and assume that t T n n -+ no for n = 1,2, ... We call T a n w-closed theory (with respect to the sequence n,) if for every formula the condition that kT nn -+ p for n = 1, 2, ... implies that tTno + p. We call YJl an w-model (with respect to the sequence ns)if for every formula (1 with at most free variable x, the condition I=w v x l ( Z ~ A Q ) )implies the existence of an integer n 2 1 such that 1-w V x l (nnh p). We call an w-standard model (with respect to the sequence nn) if for every integer i 2 1 there is an integer n 2 1 such that bw no (xi)-+ nn (XI). The following obvious theorem and the (slightly less obvious) Corollary 3 given below clarify the connection between w-models and w-standard models. Write YJl1 ri, W2 if for every p without free variables the conditions I=w, p and !=a2 qj are equivalent (i.e. Wl and W2 are elementarily equivalent, [7]). THEOREM 2. IfWm,ICI W2 and Wl is an w-standard model, then W2 is an w-model; in particular every w-standard model is an w-model. THEOREM 3. If T is o-closed, then the set M , of its w-standard models is a residual (;,$-set in M. Proof. Since M ,
=nU [no(xi) 0 3 m
i=ln=l
--f
zn(xt)] it is sufficient to show that each
of the open sets U [no(xiv) --z nn(xi,)], io = 1, 2, ... is dense in M , i.e., that for n=l
no p this set is disjoint with [p] unless [p] = 0. Otherwise we would have 0 no (xi,)] u [p A n n (xtJ] for n = 1. 2. = [p] n [no(xt,) -> n n (xio)] = [p A whence we would obtain for every W in M '-
no ( ~ j , ) ,
~
-%
nn (uiJ
-- - p for n
=
I , 2, ...
=
[77],165
429
DEFlNABlLlTY OF SETS IN MODELS OF A X l O M A T l 8 THEORIES
-
-
Using completeness theorem and the assumption that T is oJ-closed we would further obtain k T q -> rr, (XI,) and k T zo( ~ $-+3 q, whence kT q and [q] would be void. 1. If T is o-closed, then M , # 0. *) COROLLARY Indeed, the space M being complete and hence of second category on itself, the void set is not residual. 2. An to-closed theory has a complete consistent and o-closed extenCOROLLARY .rion. Proof. Take the set of formulas without free variables which are true in an a>-standardmodel of T. COROLLARY 3. For every o-mode/ of T there ir an w-standard model !Ut' such rhot m m'. Proof. The set of formulas q without free variables which are true in !Dt IS ;I consistent w-closed extension of T. According to Corollary 1 this extension has 411 to-standard model m'. Obviously Iuz m'. 3. Representability and definability. A set Z of integers is representable. in T (with respect to a sequence zn)if there is a formula q with exactly one free variable XI wch that
-
-
n E Z implies kr v x1 ( n ,A q), I I non E Z implies FT VX-1(nnhy ) .
-
2 is definable in a model by a formula q (which may contain arbitrary free variables) if n E Z is equivalent to J=w Vxl ( Z n h q) for n = 1,2, ...
A theory which is consistent and which results from T by adjunction of a finite number of individual constants and of a finite number of axioms is called afinite e-rtension of T. TnEoREM 3. A representable set is definable in every model of T. Less obvious is the following LEMMA. Given an arbitrary set Z of integers, the set M z of models PRINCIPAL in which Z is definable is an F, set in M ; if Mz is not of first category in M , then there is a finite extension of T in which Z is representable. Proof. M z is an F, since
=u0 { ~ : n & z ~ ~ = w V X ~ ( ~ n A ~ ) } = U ~ 9 . m
9 n-1
9
If Mz is not of first category, then one of the sets ,'A must have a non-void interior. Let [y]be a non void element of the open basis of M which is contained in X,. For every n and every !I3 satisfying /=ary we have either n E Z and *) This corollary was first proved in [5] in essentially the same way. We believe that the use of the topological language makes the proof more conspicuous.
430
FOUNDATIONAL STUDIES
-
1771, 166
I=% V x1 (nnA p) or n $ Z and I=% v x1 (nnA p). Applying the completeness theorem we obtain t T y v x l (nflAV) for n E Z ,
--
kTy+
v x l ( n f l A p )for n $ Z .
Now add to T as many constants as there are free variables in the formula v x l (nflhp) and an axiom resulting from y by replacing its free variables by the constants. In the resulting finite extension of T the set Z is representable. Note that the extension is consistent since [ y ] is not void. This concludes the proof of the principal lemma. +
4. Final results. A model 1112 of T (which may or may not be an element of M) is called w-standard with respect to a sequence n,, pf formulas with exactly one free variable x1 if the interpretation 3f no in 1112 is the set theoretical union of the interpretations of n,, in 1112. A set L of integers is deJinable in 1112 (with respect to the sequence nfl)if there is a formula p and an assignment f of values in 1112 to the free variables of y such that f satisfies in 1112 the formula v x l ( Z ~ Ap) if and only if n E Z. It is easy to see that if Iu7 is in M, then these definitions coincide with those given in sections 2 and 3. We denote by 5 (by 5,) the family of sets Z which are definable in all models (all w-standard models) of T. By 8' (resp. 5:) we denote the family of sets Z which the family are definable in all models 1112 in M (resp. in M,). Finally we denote by of sets representable in T. LEMMA.5 = 5' and 5, = 3 : . Proof. The inclusions 5 E 5' and 5, E 5: -being obvious it is sufficient to show that if Z is definable in all denumerable models (w-models) it is definable in an arbitrary model (w-model) 1112. Let 1112' be a denumerable sub-model of 1112 such that Iu7 is an arithmetical extension of 1112' (cf. [S]). The lemma follows from the remark that if 1112 is an w-standard model, then so is 1112', and that if Z is definable in 1112', then it is definable in 93. COROLLARY 4. If T is axiomatizable and all recursive sets are representable in T , then the families B0 and 5 coincide with the family R of recursive sets. Proof. It is obvious that Go = 3. By the principal lemma and Theorem 1 the condition 2 E 5' implies that 2 is representable in a finite extension of T and hence (by the axiomatizability of T) recursive. 5 . If T is o-closed, all hyper-arithmetic sets are representable in T COROLLARY and if the set of theorems of T is a Ill set, then 5o and 3, coincide with the family HA of hyper-nrithmetic sets. Proof. The equation ?j= 0HA is well-known (cf. [2]) and the inclusion 5-E 5o is proved as above using Theorem 3 instead of Theorem 1. Call a theory T persistent with respect to representability if sets representable in a finite extension of T are representable already in T.From the Principal Lemma we directly obtain
[77], 167
43 1
DEFlNABlLlTY O F SETS IN MODELS OF AXIOMATIC THEORIES
so
COROLLARY6. If T is persistent with respect to representability, then = 8. is o-closed, then 5o= 5., These corollaries can be applied, e.g., to theories A and A, of [2], if we modify them so as to render them theories of first order. To achieve this we drop the distinction between number and function variables and introduce two’ new predicates Po (x) (“x is im integer”) and Fo (x) (“x is a function”) as well as one new operation symbol denoting the application of a function to an argument. Axioms of A are modified by appropriate relativizations of quantifiers. As no we take the formula Po (XI) and as n n (n 2 1) the formula XI = n. In conclusion we mention an application to an axiomatic system To of set theory. To fix the ideas let us assume that To is the Zermelo Fraenkel axiomatic system. Let no be a formula of To expressing the property of being an integer and z n (n 2 1) the formula expressing the property of being the n-th integer. Modifying slightly the definition given in [l] we shall say that a set Z of integers is representable in a model 9Jl of To if there is an element m in 9Jl such that n is in Z if and only if the formula v x l (nnh(xl E xz)) is satisfied in 9Jl by m. From the Corollaries 4 and 5 we obtain. COROLLARY 7. Sets which are representable m‘all models of To are recursive; sets which are representable in all o-standard models of To are hyper-arithmetic.
If moreover T
INSTITUTE OF MATHEMATICS, POLISH ACADEMY OF SCIENCES (INSlYTtJT MATEMATYCZNY. PAN)
REFERENCES
[l] 0. Gandy, G . Kreisel, W. Tait, Bull. Acad. Polon. Sci., %r. sci. math.,astr. et phys., 8 (1960), 577-582. [2] A. Grzegorczyk, A. Mostowski, Cz. Ryll-Nardzewski, Jour. Symbol. Logic. 23 (1958), 188-206. [3] K. Kuratowski, Topologie I, Monogr. Matem., W-wa-Wroclaw, 1948. [4] A. Mostowski, Cz. Ryll-Nardzewski, Jour. Symbol Logic, 23 (1958), 458459. [S] S. Orey, Jour. Symbol. Logic, 21 (1956), 246-252. [6] H. Rasiowa, R. Sikorski, Fund. Mathem., 37 (1950), 193--200. [7] A. Tarski, Roc. Int. Congress of Math., 1 (1950), 705-720. [8] A. Tarski, R. L. Vaught, Compositio Math., 13 (1957), 81-102.
A Compact Space of Models of First Order Theories by
A. EHRENFEUCHT and A. MOSTOWSKT Presented br A. M O S T 0 WSKI on March 22, 1961
Let T be a consistent first order theory whose primitive notions are exclusively relations and which does not have finite models. We assume that the identity predicate occurs among the primitive notions of T . The aim of this note is to prove the There exists a family G of models qf T satisfying (1)--(4) below: THEOREM. (1) The domain of each M in G is the set of non negative integers; ( 2 ) B is a bicompart topological space; ( 3 ) f o r everyformula rP of T and every assignment f of integers to the free variables of @ the set { M : @ is satisfied in M by the cissigninent f } is open and closed in G ; (4) for every denumerable niodel of T there is an isomorphic model in G
The introduction of topology to the set of denumerable models of a theory is of course nn novelty (cf. [1]--[3]).*) Under these topologies models do not. i general, form a compact space unless T satisfies certain additional conditions (cf. [4]). The authors believe that compactness of G together with the continuity property (3) and the universality property (4) may find some applications. The construction of G will be carried out in two steps. In the first step we construct an auxiliary space @ whose elements are not models but relational systems in which the identity symbol is interpreted as a relation in general different from the relation =. In the second step we map $3 onto a new space G which has all the properties required in the Theorem. **) 1. W e f i r st e x t e n d T t o a n a u x i l i a r y t h e o r y TO; this new theory is obtained from T by adding to it all possible “Skolem functors” for formulas of T. *) The first step towards the use of general topology in the theory of models was made by Blake [l] in a proof of Glide1 completeness theorem. His topological proof of this theorem was presented and further developed in [2]. Rasiowa and Sikorski [3] approached the problem from a d i k n i and very fruitful point of view. **)The first part of the proof was found by the second author who also formulated the problem discussed in this note. The second part of the proof is the work of the first author.
[78], 370
A COMPACT SPACE OF MODELS OF FIRSTORDER THEORIES
433
TO is defined as the union of theories T,, where To has the same language as the given theory T and as axioms only the axioms of identity. T,+lis obtained from T, by the following process. For every formula @ of Tr which is not a formula of Tr-l and for an arbitrary integer k we add to T, a new functor (i.e., a symbol for a function) fo,x with as many arguments as there are free variables in @ different from the variable ox. Relational symbols remain unchanged. Axioms of T V e lare those of Tr together with all axioms of the form
.... V f p ) 3 ' @(fo, x (w,,..., WfP),
(Wk, Di,,..., V t P ) [ @ ( W E , Zk*,
of,, ...,7J@)lr
where @ is a formula of T, which is not a formula of T r - , and o k , rt,, ..., ot, are all of its free variables. We next define a Skolem resolvent QSk of a formula @ of TO. If @ has no quantifiers then !DSk is 0.If @ is the formula (EQ) Y and ot,,q,..., of,, are all of its free variables then askis the formula Sb (or/frsk, x (q, ..., ofp))Ysk where Sb denotes the operation of substitution. Finally (I@* @#k is @?.*) @ and QSk have the same free variables. I . 1. @ = GSL is provable in TO for every formula @ qf TO.
[@Sk
2. T h e space 'p of p s e u d o models. Let C be the set of terma of TO, i.e. the smallest set which contains the variab!es and has the property that whenever f is a functor of TO with, say, k arguments and t , , ..., t k are in the set, then so is
f (tr, ..., t k ) .
Let I be set. A family of relations and functions indexed by the predicates and functors of To is called a pseudo model of TO over Z if this family satisfies the following conditions: Every relation from the family has the field I and as many arguments as its index; every function from the family has the domain Z and the range contained in I and the number of its arguments is the same as the number of the arguments of the functor which serves as its index. C-pseudo models are pseudo models over C such that for every tunctor f of To the function with the index f coincides with the function Ff ( t l , ..., t k ) = f ( t l , ..., t x ) where k is the number of arguments o f f . We denote by [@I, where @is an open formula of TO, the set of &pseudo models P such that 0 is satisfied in P by the assignment ,f :z'k --> v k (k == 0, I , 2, ...). Note that c k F C and thus .f asigns an element of the domain of P to each variable. 2.1. The family of &pseudo models is a compact separable Hausdnrff space when sets [@] are taken as the open basis. Proof that Hausdorff axioms are satisfied is obvious. Separability follows from the denumerability of formulas. Compactness is proved as follows: Let [Tj-,,[QjJ # 0 for n = 0, 1, 2, ... . Consider the Boolean algebra B whose elements are sets [@I, where @ runs over *) jis the Sheffer's stroke. Our forrnu1.i ,.iys o f course that the formation o f resolvents is distributive over the operations of the propti\i(ional calculus.
434
FOUNDATIONAL STUDIES
[78], 371
open formulas of TO. Sets [a], for which there exists an n such that [@I z / l j s n [@j], form a non trivial filter in B. Let be its extension to a prime filter. If r is a relational symbol of TO and k is the number of its arguments then we define a relation Rr in 8 by the equivalence Rr ( t i , ..., tr)
[r (ti, ..., tr)] E @.
We denote by PO the &pseudo model thus obtained. We can now show by an easy induction that for every open formula @ with the free variables wl,, ...,wt, the following conditions are equivalent:
@ is satisfied in Po by the assignment f : et -+01. Hence POE [@9] which concludes the proof of 2.1. 2.2. The family Cp of &pseudo models in which are true all axioms of TO as well as all formulas askwhere @ is an axiom of T is closed in the space of all &pseudo models. This follows at once from the fact that axioms of 7 0 and formulas GSkare open. 2.3. Cp considered as a topological space with topology induced by that of the space of all pseudo models is a compact separable space and has the properties: (3') for everyformula @ of TO and every assignmentf of terms to its free variables the set {P :P E Fp and is satisfied in P by the assignment f } is open and closed in $; (4') for every pseudo model Q over an at most denumerable set Z in which all the axioms of T and P are true there is an &pseudo model P in Cp such that Q reduced modulo the equivalence relation E, which interprets = in Q is isomorphic with P reduced modulo Ep .
Proof. 'p is compact as a closed subspace of a compact space. (3') Because of 1.1 it is sufficient to consider only the case when @ is an open formula. If its free variables are wg,, ..., or, and f assigns a term ti to 01 ( i= 0, 1, 2, ...), then the conditions: @ is satisfied in P by f and P E [@ ( t {&..., , ti,)] are equivalent, which proves (3'). (4') Let Q be a pseudo model over an at most denumerable set Z and assume that all the axioms of T and of TO are tnie in Q. Let f be any mapping of the variables onto 1.w e extendfto a mapping of 8 onto Zby puttingf (f ( t l , ..., tk)) = Ff( f ( t d , ..., f ( t x ) ) for arbitrary t l , ..., t i in 6 and for an arbitrary functor f of 7 0 with k arguments. If S r is the interpretation of a relational symbol r of 7 in Q then we define a relation Rr C by the equivalence Rr ( t l , ..., tr) z Sr( f ( t l ) , ...,f (tk))for arbitrary t i , ..., t k in 8.The &pseudo model determined by the relations R, satisfies the condition (4'). 2.4. Zf P is in $3 then the number of equivalence classes of & under Ep is infinite. Proof. Otherwise T would have a finite model obtained from P by reducing it modulo E p .
[78], 372
A COMPACT SPACE OF MODELS OF FIRST ORDER THEORIES
435
3. P r o o f o f t h e Theorem. For every P in Fp we define a mapping qp of & onto integers. Let to, t l , ...
be a sequence corlsisting of all elements of
C and put
.yp(to) = 0,
In view of 2.4 pp maps ~5onto integers. From the definition of y p we immediately obtain the following three lemmas: 3.1. pp(tn) = pp(tm)
3
tnEptm;
3.2. vp(tn) ,< n ; 3.3. The necessary and sufficient condition for the equation qp(tn) = k (0 < k < n) to hold is that there exist k f l integers ao,.... ak such that (i) 0 = a0 < < a1 < ... < ah; (ii) tn Ep f u k ; (iii) tui non Ep fa,, for i # j ; (iv) for every b < a* there is a p < k such that tb Ep tap. 3.4. For arbitrary integers k , n ( k < n) there is a formula R , k of T o such that fo r every P in ?, the conditions
~ ~ ( t=nk) and P E [Qn,kl are equivalent. Proof. In view of 3.3 it is sufficient to take as a , * the disjunction of formulas (tn=tak)& / I o q t < j < t - ( t a i = t a j ) & AO,b:sak
VOGpGk (tb=
fa,,)
extended over sequences satisfying (i) of 3.3. Let R, be the relation which interprets in P the relational symbol r of T. Define a relation Qr between integers as follows (k is the number of arguments of r ) Qr (nl, ..., n k )
3
{there are ml,
..., mk
such that
(qp(tm,) = nl) 8~... & ( v p ( t m k ) = n k ) 8~Rr(tm,,
...,tmk)).
From 3.1 we obtain 3.5. If r is the symbol = then Qr is the relation of identity. Let us denote by p (P)the family of relations Qrwhere r runs over the relational symbols of T. 3.6. p ( P ) is isomorphic to PIEp and the isomorphic mapping is given by t / E p - + y p ( t )for t s C . Proof. A relational symbol r of T is interpreted in PIEp as the relation R: which holds between congruence classes Cr mod Ep if and only if there are tml in Ci (i = = 1,2, ..., k) such that R r (tm,,..., fmk). If we let correspond to C g the integer
436
[78], 373
FOUNDATIONAL STUDIES
cpp (Cf) = rpp ( t m i ) which is independent of the particular element CI we obtain R: (CI,..., Ck) Qr (VP (Cd, ..., 9~(Ck)).
fmi
chosen from
Since the mapping C + pp (C) is one to one we obtain 3.6. 3.1. If P is in 'p then p (P)is a model of T. P r o o f . If @ is an axiom of T, then Pkis true in P and hence, by 1 . 1 , @ is true in P hence also in P/Ep and finally by 3.6 in p(P). 3.8. Any denumerable model of T is isomorphic to a model p ( P ) with P in 'p. Proof. The lemma follows from 2.3 (4') and the remark that if Q is a model, thcn Ep is the identity relation. 3.9. Conditions ti,,
..., tik satisfy disk in P
q p (ti,). ...,
cpp
(ti,)
satisfi, @ in p (P)
are equivalent for any forniula @ of T with exactlj. k free variables and any terms
ti,, .... ti,.
Proof follows at once from 3.6 and 1.1. Let now 6 be the set of all p (P) where P runs over !$ and let a topology be introduced in 6 by taking sets W e ,f = { M : @ is satisfied in M by the assignnient .f ] as neighbourhoods in G. Every neighbourhood is thus determined by a formula '3 and an assignment f of integers to its free variables. 6 obviously satisfies condition (1) of the Theorem. By 3.8 it satisfies condition (4). From the definition of the neighbourhoods it is obvious that it satisfies condition (3). Finally 6 is a separable Hausdroff space. Thus it remains to show that G is compact. Since a continuous image of a compact space is itself compact it will be sufficient to show that the mapping is continuous. Let p (Po)E W e ,f and let the integers correlated (via f) to the free variables ..., Bik of @ be n l , ..., nk. Denote by tj,, ..., f j k terms-such that y p o ( t j s )= n, for s = 1, ..., k . Hence PO E [QjS,ns J for s = 1, 2, ..., k and, in view of 3.9, PO belongs
Ci,,
to [QSk ( t j , , .... fj,)]. Thus, n s G k [Qj,, ,&,I n [QSh ( t j , , ..., t j k ) ] is a neighbourhood U of PO in 'p such that p (P) F We, for every P in U. REFERENCES
111 A. B l ake, Canonical expressions in Boolean algebra, The Univcr\i[y of Chicago Librarie,.. Chicago, 1938. [21 A. M o s t o w s k i , LogiLn rriatemorycma. Monogr. Matem., 18 (1948) Warszawa-WrocIew. [31 H. K a s i o w a and R. S iltors ki, A proof of the coriipleteness theorem of Godel. Fundam. Mathem., 37 (1950), 193-200. I41 R . S i k o r s k i , A topological charactcr.i:ation of open rheor;~s,Bull. Acad. Polon. Sci., Ser. Sci. math., astr. et phys. 9 (1961),
An addition to the paper “A proof of Herbrand’s theorem” by
J. L o S , H. R a s i o w a and A. M o s t o w s k i (Warszawa) (Translated from French original Addition au travail “ A proof of Herbrand’s theorem” by W . Marek)
Mr. Solomon Feferman of Stanford University kindly pointed out to us a mistake in our proof of the theorem of Herbrand published in volume 35 (1956), pp. 19-24 of Journal de MathCmatiques Pures et AppliquCes.
The purpose of this note is to correct this mistake. In our proof of the Herbrand theorem we considered a formula Y provable in an open theory T. Reasoning by “reductio ad absurdum” we supposed that none of the Herbrand disjunctions H,, is provable in T and adjoined to the theory T as new axioms - H , (n = 1 , 2, ...). Our further demonstration was based on the following facts: (A) The theory T* obtained from T by adjoining of the axioms - H . is consistent; (B) In the Lindenbaum algebra L corresponding to T* the quantifiers correspond to infinite joins and meets in L, i.e. I(Ex)F(x)l and ((x)F(x)( are respectively equal to the infinite join and the infinite meet of elements F(xj) (j = 1 , 2 , ...) where xi is an infinite sequence consisting of all the “terms” of T*. Now each of these two facts separately is true if an appropriate interpretation is given to thesymbols xj. For (A) one has to consider the terms x j in the formulas - H n as constants, whereas for (B) it is necessary that the sequence xi should consist of all the terms, in particular the variables of T*. Since it is not possible to interpret the symbols xi so as to have both (A) and (B) true, it is clear that the proof given in Section I1 of our previous paper is erroneous. In order to correct our proof one can use the method sketched in Remark 11 of our paper. The method is based only on (A). In order to clarify the situation completely we describe this proof in full detail and modify it slightly. Let T be a consistent theory with the following primitive symbols: R , , R l , ... (relational symbols); x , , x 2 , ... (variables); v, & , 2 , (sentential
-
438
1791
FOUNDATIONAL STUDIES
connectives); (xi), (Exj) (quantifiers). The axioms of T are: (1) the axioms of classical propositional calculus; (2) the formulas A(xi) 3 (Exi)A(xi), (Xi)A(xi) 3 A ( x i ) ; (3) an arbitrary number of open formulas G , , G2,... As the rules of proof we adopt (4) the substitution rule; ( 5 ) the detachment rule; (6) the generalization rule (i.e. A 3 B(xi) implies A 3 ( x i ) B ( x i )provided x i is not free in A); (7)the particularization rule (i.e. B(xJ 3 A implies ( E x i ) B ( ( x i )3 A ' under the same assumption as in (6)). Let S be an extension of T obtained by adjoining infinitely many constants c,, c 2 , ... We add to the axioms (1)-(3) formulas which may be obtained from the axioms of T by replacing an arbitrary number of variables by constants. We also add infinitely many new axioms (3') G ; , G ; , ... which are closed formulas without quantifiers. The choice of those formulas are not changed and the rule will be established later. The rules (947) (4) is enlarged by the admission of the substitution of constants for the variables. 1 the set of closed forFor each closed formula q of S we defined 1 ~ as mulas y of S such that I- p = y. The sets form the Lindenbaum algebra L of the theory S. This is a Boolean algebra with respect to the following operations :
IplnlYI = I&Yl, -Id = 1-dThe unit element of L is the set e of closed theorems of S and the formula Iyl < JpIis equivalent to k y ~ p and 1 so also to Iy3pI = e. Let L* be an extension of L to a complete Boolean algebra (for instance L* may be the minimal extension of L, cf. H. M. MacNeille, Transactions of the American Mathematical Society, vol. 42 (1933, p. 416). We identify L with the subset of L* isomorphic to L. For every sequence 6 = [6(1), 6(2), ...I of positive integers we define inductively a mapping va of the set of all the formulas of S into L* as follows:
I ~ l ~ l =Y Il P Y I ,
Ck,, ck *..)I, ..., ck,,ck2,...)) = IRi(ct9(jl),% ( j , ) , %(--PI = --ya(p), %(PI Y ) = %(dO % ( Y )
Y6(Ri(Xj,, xj
9
9
0
(where denotes an arbitrary binary propositional connective and-simultaneously-a corresponding Boolean operation), 0
~ ~ ( E x j ) p ( x=j )U v a ( d c j ) ) ,
the symbols U and
n denoting the infinitejoin and the infinite meet i n L*.
[79]
A N ADDITION TO THE PAPER 'A PROOF O F H E R B R A N D S THEOREM"
439
Lemmas (8)-(l 1) easily follow from the definitions by induction with respect to the length of the formulas: (8) /f xi is not free in 9; and 6 ( k ) = [ ( k ) for k # j , then v,(q) = vc(q). (9) va(v(xi)) = va(~)(ca(d). (10) If y is obtained from q by a substitution of x& or c&for x i , then va(y) = vaik(q)where 6&) = S(j) for j # i and a i k ( i )= k . (1 1 ) If q is in a prenex normal form
ve(q) =
... ( x z p - 1 )
(Ex,)
(XI)
then
(Ex,p)y(x1
9.
..., x z p ) ,
n U ... fl U Iy(cj,i c j 2 , ..., c j 2 p - l , jl
CjZJI.
ia
j,,-c
izp
(12) uq is an axiom, then v q ( q ) = e for each sequence 6. For the axioms ( I ) the lemma is obvious. If q is one of the axiom (2) we find: va(~= )
-vtr(A(Ca(i)))
or
v@(y)
= -
u
Ui v,g(A(Ci)) = e
nv,(A(ci)) u vg(A(c+,(i)))= e. I
If q is one of the axioms (3), (3') we find, by (9), v d ~ )= ) ISubstoql,
where Subst, is the operation of the substitution of cO(&)for x&( k and consequently va(q) = e, since +Subst,(q).
=
1 , 2 , ...),
(13) if13 is obtained from q and y by one of the rules (4)-(7) and $itey(p) 6,then v,(8) = e,for each sequence 6.
= v,(y) = e for each sequence
Case of rule (4). If y is obtained from q by the substitution of x&o r c& for xi, then, using (lo), we find va(y) = vaik(q)= e. Case of rule (5). In this case q = y 3 8 and so
e
= v e ( ~= )
-ve(y) u ~ ~ ( =8 -) e n ~ ~ ( =8 ~) ~ ( 8 ) .
Case of rule (6). In this case = A I> B(xi),
8
=
A
3
(~i)B(xi),
xi being not free in A. Let 6 be an arbitrary sequence and defined in (10). By an inductive assumption:
e
= voi,(q)
= -vOik(A)
u v+ji,(B<xi>).
6ik
a sequence
440
[791
FOUNDATIONAL STUDIES
Lemmas (8) and (9) imply that and
= vB(A)
v8(tk)(A)
v8,k(B(xi))
=
vBik(B(Ck))
= vB(B(Ck)),
since the variable xi is not free in none of the formulas A and B thus obtain e = -v B ( 4 u % ( B ( C k ) ) whence %(A) G %(B(CL)) and so v8(A)
.. ,cDp, belong to K , then so does &CJl ... GP,; (iii) if 0 < s < b and CJ Pelongs to K , then so does Qx@, p = 0,1,...
[80],166
AXlOMATlZABlLlTY OF M A N Y V A L U E D CALCULI
443
The distinction between free and bound variables of a formula is assumed to be known. A formula without free variables is called closed. The result of the substitution of c for xq in @ is denoted by Sb(x&)@. Besides So we shall also consider systems obtained from Is, by adjunction of coilstants co, c,, ... whose number may be finite or infinite of any power. The “rules of formation” (i)-(iii)remain the same with the amendment that each xis in (i) can be replaced by a constant. We choose a Godel numbering of expressions of H0 and denote by r@l the Godel number of @; the expression with the Godel number n is denoted by Z. We assume that the functions rG%l and T-xnl are recursive and increasing. Fiom this assumption it easily follows: 1.1. The following functions are recursive:
( a ) jy(n) = 0 , 1 , 2 , 3 according as ii is a n atomic fornirLla, a formula which begins with a connective, a formula which begiits with a quantifier or ii is not a forrimla or is undefined. (b) f:(n) = j , k, 0 according as fy(n)= 1 and G begins with gj, I:(%) = 2 and ‘i begins with Q k , or f i ( n ) # 1 , 2 .
+
(c) f : ( j , n ) = o if f 3 i L ) 1 or j = o or j > pt;(,,; f!&, n ) = qj if f:(n) = 1‘and ‘i has the form &q1q2 ... qp, . (d) f:(n)= q , f!(n)= r if f:(n) = 2 and % has the form Qfluy; j:(n) = o = fi(n) in the rernainiiig cases. 2. Semantics. Lot Z be a set, v8 a mapping of Z x ... x Z = zp‘ into Z , Qt a mapping of 2” (1) into 2 (0 < s < a , 0 < t < b). Let D be a subset of Z.We call elements of 2 truth values, those of B distinguished truth values; y s are interpretations of connectives and Q finterpretations of quantifiers. A model of So (or of a system resulting from So by the adjunction of constants) in a set X is a mapping p satisfying the following conditions. px is defined if x is an individual constant or a predicate variable; in the former case px E X, in the latter px E Zx’= Z x x x x * ” x xwhere j is the , of p such number of arguments of x. A valuation of p is an extension G that the domain of ,lconsists of all individual constants, predicate variables and individual variables; if x is an individual variable, then ,is E 2. Whenever ,lis a valuation of p we denote by Wq,#the set of all valuations Y of p which are identical with ji except possibly for the argument xq. Let v be a valuation of p and let @ be a formula. We define by induction the value of @ a t v (denoted by Val,@): (1)
22 denotes the set
of all non-void subsets of Z.
444
1801, 167
FOUNDATIONAL STUDIES
if @ is Fitl ...t, (where each tk is either an individual variable or an individual constant), then Val,@ = v ( P { )( v ( t l ) ,..., v ( t j ) ) ; if @ is
... a,
then Val,@ = pl~(Val,@~, ...,
if @ is .ClrxqY,then Val,@ = Qf(Val, Y: e
c
Wq,,}).
The following lemmas are easily proved: 2.1. lf p is a model and v a valuation of p, then Val,@ E Z for every formula @. 2.2. If v’, v” are valuations of a model p and if v‘xq = v”xq for all p such that xq i s free in @, then Val/@ = Val,,f@. 2.3. If @ i s closed, then Val,@ depends only o n the model p of which v is a valuation. Val,@is denoted in this case by Val,@. 2.4. If c is a n individual constant, v a valuation of a model p, v’ c WP.., v‘xq = vc, then Val,Sb(xq/c)@= Val,#@. A formula @ is called satisfiable if there are a set X , a model p in X , and a valuation v of p such that Val.@ E D ;@ is valid if Val,@ c D for every set X , every model ,u in X and every valuation v of p.
3. AT-valuedlogics. I n this section we assume that Z = (0 ,1,... ,N - 1 ) where N is a positive integer and that D = ( 0 , 1 , ... ,111-1) where M is an integer < AT. We define a sequence of systems Sn; Sois the system described in Section 1, Sn+l results from Sn by adjunction of constants Ah,q,o where h = 0 , 1 , ..., N - 1 , p = 0 , 1 , 2 , ... and @ is a formula of Sn which is not and has a t most one free variable xq. Let S, be the a formula of Sn-l union of all systems 8,. It is not difficult to see that a Godel numbering of expressions of S, can be chosen so that rAh,q,olis a recursive function of h , p, r@l. It follows that there exists a recursive function g which enumerates the Godel numbers of all individual constants of S , and recursive functions f : , f : , f:, f: satisfying conditions analogous to 1.1(a)-(d) but with “formula” replaced by “closed formula of 8,”. Let p , p’ be models in X of systems Sn,Sm,m > n, m = 1 , 2 , ..., 00. If p’c = pc for every individual constant of Sn and p F i = p’Fi for i , j = 0 , 1 , ..., then we say that p’ is an extension of p. The following lemmas are obvious: 3.1. If p i s a model of Snlp‘ its extension, and v , v‘ are valuations of p , ,u’ such that vxq = v‘xq for every p, then Val,@ = Val,#@ for every formula @ of 8,.
3.2. I f pn is a model of Sn and pn+li s a n extension of fin ( n= 0 , 1 ,2 , ...), then there i s a model p , of S, which i s a joint extension of all the k ’ s .
[80],168
445
AXIOMATIZABILITY OF M A N Y V A L U E D C A L C U L I
3.3. Every ntodel p of X, can be extended to a model prn of S, in such a way that for every formula @ of 8, with at most one free variable xq the following eqziation holds (2) {Val,@:
(1)
1’
is a valuation of ,urn} =
,
CVal,wSb(xe:4jA0.q,aJ@, ... Val,olSb(xe/AN-1,e.6)@}
P r o o f . Put po = p and assume that an extension constructed such that pn is a model of X, and that
/ln
of p has been
for every formula @ of S,-,, nith a t most one flee variable zq. This assumption is clearly satisfied for n = 0 for AS',,-^ in this case is empty. \Ye shall extend ,tin to a valuation of Sn+l and we therefore have to dcfirir pn+lAi,e,o for j = 0 , 1, ..., N-1, q = 0 , 1, ... and such @ which are formulas of S n but not of Xn-l and which hare a t most one free variablc xe. Let @ be such a formula. Since the set {Val,@: e i s a valuation of pn} is.contained iri 2, we may assume that it coiisists of integers sl, ... , sm < N where 1 < m < A’. Choose valuations ei of p, such that Val,@ = si, and put p,+1Ai,q,8= eixq for i = 1,2 , ..., m, pn+lAi,P,O = emxq for j = m + 1, ..., N - 1 . The mapping ,u,+~ thus defined is an extension of pn and hence of p. If @ is a formula of with a t most one free variable me, and e is a valuation of P,+~, then e restricted t o symbolsof X n i s a valuation of p, and hence we have equation (2) from which, in vicw of 3.1, we obtain {Va&@:
e
i s a valuation of P,+~} =
IVal,,,+lSb ( x ~ / A o , ~ , ... ~ ),@ Val,n+,Sb(~ecpiA~-l,,,O)@} , .
If @ is a formula of 8, but not of IS,-,, then the same equntioii holds true in view of the construction of ,u,+~. Thus WP obtain a scquence of successive extensions pn of p satisfying ( 2 ) for each n. If ,urn is a joint extension of the models p,, then clearly equation (1) holds for eveyy formula @ of X, with a t most one free variable xq. We shall now express arithmetically the notions of satisfiability and of validity. We put f & , n) = ‘Sb(x,:,,,/Ai,,:c,,,T:cn~)~~( )L)T if f : ( n )= 2 and 0 < j < N t n d f & , n ) = 0 otherwise. Furthermore we put f : ( k , n ) = r S b ( ~ , : , , , / ~ ( k ) ) f , ‘ ( n )ifl f t ( w ) = 2 and f:(k, n) = 0 otherwise. Functions / : , f : are iecursive. (2)
(a,b ,
(f(t):
...L...)
denotes the set of all f ( t ) where t satisfies tho condition set consisting exclusively of a , b , ..., m.
..., m } d e n o t ~ the .~
...t...;
446
[go],169
FOUNDATIONAL STUDIES
Let a be a function from integers to integers. Wo call a an A-model if the following conditions are satisfied:
< f:(n) < 3,
then 0
(3)
if 0
(4)
if
f:(N = 1, then
(5)
if
f:(4= 2 ,
then
< a(n) < N ; a(iz) = 0 for (a (
a(n) = (3)
(!lo,
f h .I)
!lN-l)N
**',
7
-*a
9
f:(n) >, 3 ;
a (f:(Pf:(n),%I));
((j)"a(f&,
N ) = a,]
*",
3 [a(n) = &,:(,)({!70,
aN-lHI];
if f:(m) = 2, then (k)(EI)N[a( f : ( k ,m ) ) = a ( $ ( j , n ) ) ].
(6)
3.4. A closed formula @ of So i s satisfiable (valid) if and oizly if a ( r @ l )< M for an (every) A-model a.
P r o o f . Let p, be a model of 8, oxid put a(n) = Val,+=(%)if 0 < f:(n) a(%)= 0 otherwise. First we show that if p, satisfies (l), then a is an A-model. Condition (3) is obviously satisfied. -1 If f:(n) = 1, then = 8,;(,)f;(l, 12) ... fs(p,;(,), n ) and hence
< 3,
-
= P),i(,)
This proves (4). If f : ( n )= 2 then
G
-1
(a ( f a 1 7
n ) ," ' 3 wImf*(Pt:(,),
.I)
9 **- 9
= D,;(n)s,i(n)Ti(m) and
a(%)= ~al,,(G) =
a (f:(Pt:(,)t
.I)
N))
hence
({VaI,fl(m): v is a valuation of p,])
whence by (1) a(%)= &,:w
-
= vt;(JV~4,fS?1,
a(*) = Vdl,,G)
,
-
( {Val,,Sb (",:(n)/Ao,,:,n)~~(,))f:(n) 7 ..., Val,,Sb
= &,;(,((V~L,f:(O,
($,:(n~AN-~,,:(n)~~(n))~(~)})
n ) ," ' 9 Val,,f:(N--l,
41) *
This proves ( 5 ) . If f:(n) = 2 then j?'(k, n) = S b ($,;(,&jk))E(n) and hence by 2.4 u ( j : ( k , n ) )= Val,+$(k, n) is equal to Val,f,'(n) where e E Wf:(n)&m and @a,:,,, = p , y ( k ) . By (1) there is a j < N such that Val,T;(m) = Val,,Sb (",:(n)/A,*,:(,).T:(n))f:(n) = VGIo,f:(i 7 f i ) = a (f2i9 N ) . This proves (6).
-
( 8 ) ( 2 , j, ....pjN means: for arbitmry integers i , j , means: there are integers i , j, ..., p < N .
..., p < N ; similarly ( E i , j,...,I ) ) ~
[SO], 170
AXlOMATlZARlLlTY OF M A N Y V A L U E D C A L C U L I
447
Xow let a be an A-model. We define a model p, of 8, in the set X , of all constants of 8, as follows: for c in .&., we put pmc = c and we let p , f l t o be a function y such that tp(ul, ..., u j ) = a ( r F l u , ... u j l ) for u l , ..., uj in X,. For any formula @ of 8, we denote by 6 the set of closed formulas of 8, which can be obtaincd from @ by substitutions of individual constants for free variables; and by the set of formulas of 8, which have a t most one free variable and which result from @ by substitutions. We shall show that if @ is a formula of 8, and Y is in 6 then
6
Val,, Y
.
= a( r Y 1 )
Case 1: @ is an atomic formula. I n this case any Y in 6 has the form Ffu , ... u j where u l , ..., u i E X , and hence by the definition of p, Val,,Y/
= a(r&u1
... u j l ) = a ( r ~ 1. )
Case 2. @ has the form SjG1 ... QP,. I n this case any Y in 6 has the form 5,Yf, ... !Ppjwhere !Pcis in 6i for i = 1 , 2 , ..., p j and hence Val,, Y
... , Val,, Yp,) .
= q~~(Val,,!PI,
Using inductive assumption and (4) we obtain VaI,mY = qJf (a(rY',l),
..., a ( r Y p i l ) = ) a(r!V)
because f i ( r Y 1 ) = 1,f $ ? P l ) = j , f i ( i , r W ) = r Y c l for i = 1,2 , ...,p f . Cafie 3. @ has the form Qix& I n this case any Y in 6 has the form Qjxq17 where I7 is in g or in 2 according as xg is or is not free in E. S u b c a s e 3&.xq is not free in E. I n this case I7 is closed and Val,, !P = QA{val,,n>) and hence
r171= f i ( i , r v )
for
7 (4)
i = 0 , I , ..., N - i
Val,, Y = Q , ({v~I,,J~(o, r v ) ,...,V~I,,%(N-- 1 ,r ~ ) ,) )
whence by the inductive assumption and by ( 5 ) Val,,~=Qj({a(j:(~, ru/l)), ..., a ( f : ( N - - l , r ~ l ) ) ]=) a ( r m ) . S u b c a s e 3b. xq is free in 5. I n this case n has just one free variable xq and Val,,Y = Qj({ValR17: e is a valuation of p,}) . If ValJI (')
Q
is a valuation of p, then exg = 6 is in X , and hence, by 2.4, If r c l = g(k),then by the inductive assumption
= Val,,Sb(xg/c)n.
(a]is the unit set with the sole element a.
448
FOUNDATIONAL STUDIES
[80],171
and the remark that Sb(xq/c)17is in 8 we infer that the right hand side isequal toVal,,~~(k,ryrl)=a(f:(k,ryrl))andhence, by(6),to a ( f ; ( j , r y r l ) ) where j is an integer < N . Conversely a ( f & , ryll))is an element of the set (Val,17: e is a valuation. of pm} for a ( f ; ( j , "P)) = Val,IZ where This proves that e E Wq+, and exq = ~ , A I , ~ , ,= Val!I,p
= &I
( { a( f h 'V)),...1 a ( f i ( N - 1 , r m ) ] )
and hence by (5) that Val,m!P = a ( r Y r ) . Now let @ be a closed formula of So. If @ is satisfiable then there is a model p of So such that Val,@ < 111.We extend p to a model ,urn of 8, satisfying ( 1 ) according to 3.3 and obtain thus an A-model a such that a(.) = Val,,% whenever 0 < f:(n)< 3. In particular, a ( T @ l )= Val,,@ = Val,@ < M . conversely if there is an A-model a such that a ( r @ l )< M , then there is a model pm of S, in the set X , such that Val,,,@ = a ( r @ l ) < M . Restricting p, to symbols of So we obtain a model p of So in X , such that Val,@ < M and thus @ is satisfiable. If @ is valid and a is an A-model, then (as shown above) there is a model p of Soin X , such that Val,@ = a ( r @ l )and hence a ( r @ l )< M . Conversely, if this inequality holds for every A-model a and p is a model of Soin a set X then there is an extension of p to a model p, of S , in X satisfying (1). We proved above that there is an A-model a such that Val,,@ = a ( r @ l )and hence Val,@ = Val,,@ < M which shows that @ is valid. Theorem 3.4 is thus proved. 3.5. The predicate W i s a closed satisfiable formula of S:' i s expressible in the form ( E a ) H ( x ) R( Z ( X )r@l) ~ where R i s a recursive binary relation, H = { a : (z) ( a ( % )< N ) ) and ( E a ) H means: there i s a n a in H . The predicate , V ~ I , , , , S ~ ( X ~ / B ~ ~,, ~ , , , ) @ z’non < ValpnSb(xg/Cz,A,B)@
for every z’ in Z’, y = 0 , 1 , 2 , ... a i d evciy foimula @ of with at most oiie free variable xg. This assumption is satisfied if n = 0 since in this cnsc is empty. We shall IIOW extend /tn to a model of SIL+l and have thcwfore to define pn+lBzt,g,,, and p,+, C,,,,, for every z‘ in Z ’ , p = 0 , 1 , 2 , ... and every formula @ of S, which is not a forinula of SnT1 and which has a t most onc frcc variable x q . Let @ be such a formula and put Y = {Val,@: e E Wq,,, z‘, then we chooso p l L + l B z ~ , qaibitiarily. ,O I f g.1.b. Ynon >, z’, then there is a 0 in Wg,,, such that Val,@~ion2 z’. We choosc again a (r of this sort and put CZ:,,,,= uxq. If g.1.b. P >, “u’, then we choose , U , + ~ C ~ , arbitrarily. ~,@ The mapping ,u,+~ thus defined is an extension of pn and hence of p. If @ is a formula of Snw1 with at most one free variable xq, then VaI,,,,+,Vxg@ = Val,,VxqCJ, and V ~ ~ , , , , S ~ ( X ~ / B ~ ~ , ~ , ~ ) @ = Val,,Sb ( X ~ , ’ B ~ ~whence ,~,~)@ in , view of ( 9 )
z’ 2 Val,,+,~~xg@ or
z’iioii
~ ~ l , ” + l S b ( z g / B z ., , q , ~ ) ~
The same formula holds true if @ is a formula of Sn which is not a formula of as we immediately s ~ fiom e the definition of ,u,+~ and 2.4. A similar relation is also provable. for the formula Axg@. We lhus sde that the sequence pn of niodels satisfies (9) for every 12 and every forwith at most one free variable xg. It is now obvious that (8) mula @ of is tiue if we chuose as p, the joint extension of models p,,. Remark 1. Theorem 4.1 holds under the assumption that 2 is a completo lattice and 2’ an arbitrary subset of 2. We denote by 5 a fixed function which onumeratcs the elements of 2 . It is easy to see that a Godel numbeiing of expressions of B, can be
[80],174
45 1
A X l O M A T l Z A B l L l T Y OF M A N Y V A L U E D C A L C U L I
so chosen that rBc(r),q,e17 r C ~ ( ~ ) be , ~ recursive ,~l functions of r , q , r@l; indeed we can choose as these Godel numbers any integers uniquely and r. It follows that there exists a Godel numbering determined by q , r@l of formulas of 8,such that the Godel numbers of closed formulas and the Godel numbers of formulas with at most one free variable form recursive sets. From this it follows again tha,t it is possible to enumerate the expressions of & in such zt way that the Godel numbers of the constants Bl(r),q,e,and CC~r),q,8 of are recursive functions of r , q , r@l. Continuing in this way we infer that there is a Godel numbering of &, such that the Godel numbers of the constants BC(r),q,e, CC(r),q,e are recursive functions of Y , q , r@l. Hence there is a recursive function g which enumerates the Godel numbers of the constants of am.We continue to denote by r@l the Godel number of @ and by ?i the expression with the Godel number of n. A further easy consequence of the construction of the Godel numbering outlined above is that there exist recursive functions f:, ...)fi satisfying conditions similar to conditions 1.1 (a)-(d)but with “formula” replaced by “closed formula of ~ 9 ~ ” . We put (8) f ; ( k , n ) = r S b ( z , ; ( , l $ ( k ) ) Z ( n )ifl f:(n) = 2 and f ; ( k , n ) = 0 otherwise. This function is obviously recursive. We also put fi(r,n ) = rSb(m,~,,/B,(r),,:(,,~:,,,)fs2(nn)l if f:(n) = 2, fi(n) = 0 and fi(r712) = 0 otherwise. Similarly we put $ ( r , n) = r S b (z,;cn,lC,r,,,:,n,..i:of52 (n)l if f:(n) = 2, fi(n) = 1 and f t ( r , n ) = 0 otherwise. Both functions fi and fi are recursive. Using these notations we shall express arithmetically the not,ions of satisfiability and of validity. A mapping x of the integers into Z is called a B-model if it satisfies the following conditions:
-
(10)
if
fm
= 1,
-
x ( n ) = B,:@)
(11) if fi(n) = 2 and f:(n)
=0,
( x (r31, .I)
7 **- 7
x (faP,:(n), n ) ) )7
then. ( k )( x ( n )2 x ( f % k ,n))),
fl(n) = 1, then. ( k )( ~ ( 1 2 )< x(f@, and ti(.) = 0 , then (Y) [ ( ( r ) 2 x ( n )
(12) if f i ( n )= 2 and
( 1 3 ) if f:(n) = 2
,
n))]
v S(r)non > x (fi(r, n ) ) ],
(14) if ff(n)= 2 and f:(n)= 1 , then ( r ) [ ( ( r ) < x(n)
v C(r)non < x (f;(r, n ) ) ].
to
1;
(4) f: is omitted to preserve analogy with Section 3; f; will play a role analogous wheress in place of the former f; we ahall have two functions ff and 1:.
452
[80],175
FOUNDATIONAL STUDIES
4.2. A closed formub Q, of So is satisfiable (valid) if and only if x(r#l) D for a (every) B-model x. Proof. Let p be a model of No; construct an extension p, of p satisfying (8). We shall show that any function x such that x ( n )= Val,,,% if 0 < f:(n) < 3 is a B-model. Condition (10) is obvious. Condition (11) follows from the fact is the 1.u.b. of a set P that if f:(n) = 2 and fi(n) = 0 then Val,n of which Val,,,~~(k,n) is an element. Proof of (12) is similar. To prove (13) let -us assume that j:(n) = 2 and fi(n) = 0 and c(r)non 2 x ( n ) = Val,,% = Val,,Vx,:,,,f~(n). It follows from (8) that c(r)non Y
-
'
12) = x (f&, .I) . Hence t(r)non > x ( g ( r ,n)). Proof of (14) is similar. Now let x be a B-model and let X , be the set of all constants of 8,. Define a model ,urnof 8, in X, by taking p,c = c for c in X, and by letting pmFf to be a function y such that y ( c I ,..., c i ) = ~ ( r f i c , ... c r l ) for arbitrary c l , ...,ci in X,. We shall prove that if @ is a formula of 8, then (15) val,,Y = x ( r ~ 1 ) for every Y in 6 .
> ~ a l , s b ( s , : ( ~ ) / ~ ~ ( , ) , , : ( ~73.)1. ~ ~=~ ~~al,fs(r, ) -??
Case 1. @ i s an. atomic formula. I n this case Y has the form ... cj with c,, ...,cf in X, and hence (15) follows from the definition of pea. Case 2. @ has the form ... I n this case (15) follows from (10) and the inductive assumption. Case 3. @ has the form vxqZ. In this case Y has the form Vxq,17 where 1 7 6 9 or 17~9 according to whether or not xq is free in 9.
I&
sjQ,l
S u b c a s e 3'. xQ is not free in 3. In-this case V € !,'a,l = 1.u.b.(Val,17} and hence ValPwY= x ( r I I 1 ) by the inductive assumption. > x(rSb (sa,'gV(k))171)= x(rT1).If From (11) it follows that ~("yl) x ( r U l 1 ) were # x ( r Z P ) , then by the density of Z 'there would be an r such that x(rT1)< t ( r ) and c(r)non > x(rY1) Khich contradicts (13) since fi(r, W l )= r P . S u b c a s e 3b. xg free in. E. I n this case VaI,,!P = 1.u.b. {Va1,17. e is a valuation of p,); hence, by 2.4 and the inductive assumption, = Val,-,17
Val,, !P = 1.u.b. Val,,Sb csx,
=
(s,&)n
1.u.b. x(Qb (xq,'ij(k)),177)= 1.u.b. x ( f ; ( k ,J-m)).
k=0,1,2,
...
k=0,1,2....
From (11) u*a obtain therefore Val,!P < x ( " P ) . If x(rY'l) were [(r) and different from Val,,Y there would be an r such that Va!,l€ X ( f d k t n))
Val,,@.
If
is not
x
limit number, then, by (25),
x (LSb(%&,@)HJ) = %(h(L%@J, L@J) < 1;;$.x(fs(LcJ,
%(L@-I) = (f*(L@J)) =
L@J)
which proves (27). I f x ( ~ @ J )is a limit number, then again by (25)
(t)[ E < X ( L @ - J )
3 [+I G x (fll(L~&lz.@J, L@-l))] *
If we had x ( ~ @ J )> Val,,,@, then we would obtain
x (fe(LQ***@J, L@J) Val,@+
1GX
( f 6 ( L 4 , P . W , L@J))
which is a contradiction. (27) is thus provea in Case 3. Case 4. Y has the form AxqB. Hence @ has the form A x g H where H €8or H € 2and Val,,@ = g.1.b. (fe(Lc_l, L@J)). Fiom (24) we obtain crc
and from (26) X(L@-J) = x
X(LG-l) G Val,,@
(fdL@J))= x (fdL%@-l,L@J))
> g,.’f.,b.x ( f a L C.J,
L@-l)) = Val,@
9
whence we obtain (27). 6.2 is thus proved. Our next task will be to express the conditions for validity and satisfiability so as to make evident their recursive character. To this end we shall consider an arbitrary but fixed finite set of relations R,, ..., Rk defined in the set Z and denote by Z the elenwiitary theory of relations R,, ..., &, 1=1: 0111 : [13 , bu "" btu' nil , n wu' (;u (;i, qi, m,), (qH1' mi+1) ,
= '" = f,(m llu) = 2) = ... = fz(m1lJ = 1) qll m,) , ... ,
, (qllu' m llu)]}'
We now use Lemma 6.4 and replace
!='ZoM:[e, bu ... , btu, n u ... , n wu' (~"
(~u
qu m ,), ... ,
q" m,), (qi+I, m/+1) , ... , (qllu' mllu)]
by an equivalent condition
with M** depending recursively on the indices. Further we denote the formula
[80], 187
FOUNDATIONAL STUDIES
464
= 0) & ... & ..., q )[T,(m,) = 2) & ... & (Ti(nli)= 21 (Urn,)
(a,,
...
(?.(mi) = 0) & (fa(mj+l)= 1)
(C(mv,,)= 1 ) 3 W i ~ l , . . . , q u u ]
by -M~3ul.....*u". (33) is thus equivalent to
~#"M%, nwu)N&l(41 ..., ..., -..,
(u)db,, " ' 7
"'7
4u)ch-l
"'7
i=z.M:re, b,, bfu, nl1 nwUld,, ..., 4,,1 & (u)-(b,, bt,,)m(ni, nw,,, mi1 .*.,mv,,)ivn-,(i, j ) i < j G v u - - a )
(ni, "'9
***
~ v U ) - l = 2 * ~,,,~i,q ul , . . . , q u ~ e tb i i
b:,,i
...i
m,,
? ~ i i* * * in w u i
...i
mvuI
which, after contradiction can be brought to the same form as the loft hand side of (30) but with h replaced by h - 1 . In order to prove (30) it remains to prove it for h = 0. The left hand side of (30) has in this case thc form (uIm(a1,
aa.7
aPu)Q4n1.* * ' , ~ P U ) N o I - p * ~ u r e a1 l7
"'1
"CP,,
n1,
%*I
a " )
*
Performing the same operations as in the firfit step of the reduction of h to k-1 we are left with a condition of the form (u)w(a,1
*.*
, aPu)@J(n:, .*.,
n;u)Xol=Z2CI:,n: ,.....:,,.a1
.....a9,.c
where M' is a formula of Z and X o is the set of Godel numbers of closed formulas of So. Contracting, we finally obtain a condition of the form (p)/=2Ho,p,ewhere H,,,p,c is a closed formula of Z depending recuisively on p and e. Lemma 6.6 is thus proved. We now repent with minor changes the construction carried out on pp. 176-J78. Let sequences of p 6 ordinals be denoted by German letters and their terms denoted by corresponding Roman letters:
+
..., W P )
ul = (w, w1,
where
P
W I ,
= maX(P,,
Assume that the relation w
= y,(w,,
W " ) W " ' , W'V,
W)
..., pa) .
...,wpJ is
definable in
Pf( a, a,, ... , up?)be a formula of Zwhich defines this relation, j
Z and
=0
let
, 1 , ...,a.
Notice that relations [ < q , [+I < q and "[ is a limit number" are gofinable in Z (which was supposed to b+>an extension of the elementary theory of the < relation). Let formulas which define these relations be a y } should be closed in Z x Z. No reasonable topology seems to satisfy this condition and this is the chief reason why it is an open question as to whether or not methods similar to those of Section 4 are applicable to our problems. We limited ourselves chiefly to the study of quantifiers whose interpretations were the 1.u.b. and the g.1.b. operations. It is easy to construct examples showing that for an infinite Z, e.g. for 2 = {(: L!< w } , another choice of quantifiers may lead to a “functional calculus”, in which the set of valid formulas is not recursively enumerable. It would be interesting to solve the following problem: E. What is the general characterization of quantifiers which lead to functional calculi with recursively enumerable sets of valid formulas? References [l] Garrett Birkhoff, Jattiee theory, American Mathematical Society Colloquium Publications, vol. 29. New York 1949. [2] Andre e j G rz egorcz yk, Computable functionals, Fundamenta Mathcmaticae 42 (1955), pp. 168-202. [3] S t e p h e n C. Kleene. Introduction to metamathematics. North Holland Publishing Company, Amsterdam and P. Noordhoff, Groningen 1952. [a] - A Note o n computable functionals, Indagationw Mathematicae 18 (1956), pp. 275-280. [5] Georg Kreisel, A variant to Hilbert’s theory of foundations of arithmetic, The British Journal of Philosophy of Science 4 (1953), pp. 107-129. [6] J. Rarkeley Rosser, Aziomatization of infinite valued logics, Logique et Analyse 3 (1960), pp. 137-153. [7] J. Barkeley Rosser and Averell Turquette, Many.valued logics, Studies in Logic and the Foundations of Mathematics. North Holland Publishing Company, Amsterdam 1952. R e p par la. Eddaction le 13. 10. 1960
REPRESENTABILITY OF SETS IN FORMAL SYSTEMS BY
ANDRZEJ MOSTOWSKI The aim of this paper is to advocate a method (due in principle to Gijdel
[2], though never elaborated by him in details) to present in a uniform way
the theories of recursive, hyperarithmetical and related families of sets. The gist of the method is to defim these families using the notion of representability in suitable formalized theories. The techniques worked out by Kleene and other writers yield probably better results when one wants to discuss properties of a single family; the writer believes however that the method developed below is very helpful when one wants to discuss common properties of these families and to detect reasons of their affinities. The method will be presented for families consisting of sets of integers and sets of functions. An extension to higher types has not yet been tried, but seems to present no essential difficulties. The writer had planned to entitle the paper “ Kleene’s theories as I see them.” Although the final title is more conservative, the influence of Kleene’s work on the present paper should be obvious to every reader even moderately acquainted with the literature.
I. GENERAL THEORY CHAPTER
OF REPRESENTABILITY
1.1. Formal systems. In Chapters I and I1 we shall deal with formal systems having a common language and differing from each other by the notion of consequence. The common language of these systems is that of .second order arithmetic 131 with constants for both types of objects (integers and functions). Latin 1.c. letters will be used for variables and constants of type 0 (integers) and Greek 1.c. letters for variables and constants of type 1 (functions). We use the first letters of the alphabet for constants and the last ones for variables. Numerals are denoted by a,, a,, ‘ . . We denote by A x the set of axioms consisting of the usual axioms for the propositional and functional calculus with identity, of Peano’s axioms for arithmetic, of the so-called pseudo-definitions, i.e., axioms of the form (&)(x)[&) = 0 = 01, where 0 is a formula in which the variable is not free, cf. [3], and finally of the special form of the axiom of choice which allows one to permute the functional and the numerical quantifiers, cf. [ll, p.2171. If X is a set of formulas then Cn,(X) denotes the set of formulas which can be obtained from A x u X by the usual rules of proof. We assume that in each formal system which will be considered in our theory there is defined a function of consequence Cns acting on sets of formulas and yielding such sets. Further we assume that this function satisfies the
-
Received by the editor April 6, 1961
1811, 30
REPRESENTABILITY OF SETS IN FORMAL SYSTEMS
469
following axioms:'
(A) X c Cns(X). (B) X E Y implies Cns(X) E C n d Y ) . ( C ) Cno(Cn8(X))E Cn8(X) and Cn8(Cno(X))E C n , ( X ) . (D) If P' is a closed formula and I E CndX U {P'}), then ?F=I I E C n d X ) . (E) There are infinitely many inessential constants of both types. A constant a or a is inessential for S if for every set X of formulas none of which contains a (or a) the condition I E Cn8(X)implies I' E CnXX) where I' results from I by a substitution for a (or a) of a free variable of the appropriate type which does not occur in I. Note that Cno(Cn8(X)) = Cns(Cno(X))= CnAX) by (B), (C), and the obvious properties of Cno. In some theorems we shall assume that Cns is an idempotent operation. These theorems are marked by an asterisk. If 2 is a set of formulas then ExtdZ) denotes the system S' with the function of consequence defined thus: Cn,,(X) = Cns(X u 2). 1.1.1. Zf there are infinitely many constants (of both types) which are inessential for S and do not occur in formulas of 2, then the system Exts(Z) satisfies (AHE). Let 1 be a functional variable or a functional constant and (p a function from integers to integers. We denote by D,(A) the set of formulas A(&) = & P ( ~ ) and call this set the diagram of (p. To maintain the symmetry between both types we denote by &(I) the formula l = 8,; here l is a numerical constant or a numerical variable. 1.2. Representable sets and functions. Let o be the set of integers 2 0,o" the set of all mappings from o into o. Elements of o will be briefly called numbers and elements of om functions. We denote functions by the letters (p, 4, r?, * We denote by Rk,1the Cartesian product omX . . . x om X o X . . * X x w ' ; elements of Rk,, are denoted by German letters p, q, . . . . Thus o= p is a sequence ((p,, . . , ( p k , nl, . . . , nl) consisting of k functions and 1 numbers. Let the German 1.c. letters a, 6, . . . , 0, IU, . . . denote sequences consisting of k inessential constants (or variables) of type 1 and I inessential constants (or variables) of type 0. Such sequences are called briefly k , l sequences and we shall write a = ( a I ,. . ., a k , a,, . . -,at) and similarly for other letters. We Put +
a .
-
D,(a) = DV1(al) u
. . . u Dq,(ak) u Dn1(al)u . . . u D,,(al) .
A set A C Rk,Iis weakly represented in S by a formula I if 0 has k free variables of type 1, I free variables of type 0 and I These are esentially Tarski's axioms [17];we did not include all of Tarski's axioms in order to have a wider range of applications.
410 (1)
FOUNDATIONAL STUDIES
q EA
= @(a) E Cn@&a));
here @(a) is the formula obtained from @ by a substitution of the constants a for the free variables of @. The family of weakly representable subsets of Ri,[is denoted by s k , l ( S ) or briefly by 9 k . l when S is fixed. If besides (1) the equivalence q non E A
=
-
@(a) f Cns(Dq(a))
holds, then we say that A is strongly represented in S by @. T h e family of ) briefly by strongly representable subsets of Rk.[ is denoted by ~ ; , I ( . S or 9”: .1 . It is worth while to remark that families 9 k . l and 9:,[ may well coincide, e.g., if S is a complete system. R,,,,,, is represented in S by a formula @ with k m A mapping f : free variables of type 1 and 1 n free variables of type 0 if for every q in
+
+
Rk.1
here D and a are k , l sequences of variables and of inessential constants and II, and b are m ,n sequences. T h e family of representable mappings of Rk.1 into R,,,,,, will be denoted by &,t;m,,,(S) or briefly by K I : ~ . ~ . If m = 0, i.e., if f maps Rk,i into w or into a Cartesian product of finitely many copies of w , we can replace (3) by (3’)
@(a, h) E CndDQ(du D/(&))
and obtain an equivalent condition. Since (2) and (3‘) imply (for m = 0) (3”)
-
@(a, b) E Cns(DQ(a)u D,,(b))
for n # f h )
we infer that 1.2.1. Iff E A < I ,then ~ ,the ~ relation f(q) = n is strongly representable. I f f : Rk.1 + R0,% and the relation f(q) = n is weakly representable, then f E f i , t ; o , n . No similar theorem holds if m # 0. 1.3. Properties of strongly representable sets and of representable functions. We list below a series of elementary theorems whose proofs can be obtained immediately from the definitions. 1.3.1. S‘,*.[ is a Boolean algebra of sets. 1.32. If A E S ? ~then , I , A x (I) E and A x w’ E 9Z’lL,,~. Every set which arises from A by a “permutation of axes” belongs to s k : along icith A . 1.3.3. If A E g k : + 1 , then the set A” = ((1: ((1, n ) f A } belongs to %?. A theorem similar to 3.3 would be false for the operation A V = (11: (rp, (1) f A}; we only have a weaker result 1.3.4. If A E s k T 1 , l ( s ) and a is a constant inessential f o r S then A’ E .@k?Ext@,(a))). I.3.5*. If f E - % , I ; ~ , % and A E .%:.,‘ then !-‘(A)E .%*L.
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REPRESENTABILITY O F SETS IN F O R M A L SYSTEMS
471
The notion of a recursive mapping fi Rk,! + Rm,* will be defined formally in $11.1. It will be seen there that for k = m = 0, n = 1, this notion coincides with the usual notion of a recursive function with 1 arguments. then 1.3.6. r f f is a recursive mapping of R k . 1 into Rm,,, and A E f-'(A) E S?~,L. Note that no assumption of idempotency of Cns was needed in Theorem 3.6. contains all recursive mappings; permutations and identijca1.3.7. tions of variables do not lead outside the family of representable functions. 1.3.8. Superposition of two fzmctions one of which is representable and the other recursive ieads to representable functions. I.3.9*. The family of representable fulrctions is closed with respect to superpositions. 1.3.10. A E g:,,, if and only if the characteristic function of A belongs to
gc,,
9 m , n : O . 1.
1.3.11. r f A E G?:,I+~ and if for every q in Rk.1 there is an n such that ( 9 , n ) E A, then the function min, [(q,n)E A ] 6eZongs to .$%,f:o,l.
As the last theorem we note that the family of strongly representablesets is not affected by extensions of S. More exactly 1.3.12. I f X is'a consistent set of formulas, then .%';J(S) = S:L(Exts(X)).
1.4. Properties of weakly representable sets. 1.4.1. If A, B E L%?~.I, then A n B E 9 k . l . 1.4.2. I f A E g k , 1 and B E LZ.&,then A u B E .9Pk.I. It is not known whether the union of two weakly representable sets is always weakly representable. Most probably this is not the case, but no counter-example is known a t present. Theorems 1.3.2, 1.3.3 and 1.3.4 remain true for weakly representable sets. The Theorems I.3.5*, 1.3.6, are probably false if &'* is replaced by 9. following weak form of 3.5* survives: 1.4.3. If A E so.^ and f E S5,1:a,~, then f - ' ( A ) E @ a , l . The following example shows that Theorem 1.3.12 does not hold when g* is replaced by @. Let Cns be the function Cne defined in [lo] and IZ a formula such that the set Cn8({n})be consistent and complete (cf. [lO,'p. 1661). In this case the family ,%?a,l(Exts({IZ})) is a Boolean algebra because it coincides with .GPo*.I(Exts({IZ})). In [lo] it has been shown that the family gO.'(s) coincides with the family IZ: and hence is not closed under complementation. We note a weaker theorem (which we may note in passing, holds not only for , ! % ? kbut , l for .GP'l as well). 1.4.4. If 17 is a closed formukc, then ak,l(EXts({R}))S SZPdS). Indeed, if B represents A in Exts({IZ}), then IZ 3 B represents A in S. It is remarkable that under special, but not too narrow assumptions the analogue of Theorem 1.3.12 can be proved for the family S P k , i . We shall discuss this phenomenon (discoved for recursive sets by Ehrenfeucht and Feferman) in later sections. 1.5. Universal functions. Let e,, el, . . . (or more exactly e:"", e:".", . . .) be an enumeration of the Giidel numbers of formulas with k free functional
472
m1, 33
FOUNDATIONAL STUDIES
variables and 1 free number variables. T h e formula with the G a e l number e. is denoted by en. The sequence e. is primitive recursive and logical operations on formulas (including substitutions) correspond to primitive recursive operations on integers en. Put
U ( n ) = U "'"(n)= {q : Z,(a) E Cns(D,(a))}.
1.5.1. g k . 1 coincides with the family of sets U ' k , " ( n )n, = 0,I, . . . . Thus U"." is a universal function for the family 5 P k . I . Theorem 1.5.1 provides us in the usual way with examples of nonrepresentable sets. 1.5.2. The set {(q, n) : (q, n) @ U'k"+"(n)}is not weakly representable. If one wants an example of a subset of &&%+l.O which is not weakly representable one can take the set {(p, PI,
.'
' ,(Pk)
: (p, +'I',
' ' '
,pk)
U'k" o ' ( d o ) ) }.
1.5.3. The set {(q, p) : q E Ui.'"'(p)} is not strongly representable. Let T be the set of Gijdel numbers of closed formulas which are provable in S, i.e., which belong to Cns(0). 1.5.4. There is a recursive function of two variables g ( n , m ) = g.(m) such that U'"'(n) = g;'(T). 1.5.5. [email protected]%':I, W T#.GPo,1. Theorems 1.4.1 - 1.4.3 and the analogues of Theorems 1.3.2 - 1.3.4 for weakly representable sets have strengthened versions showing that if the operations mentioned in these theorems are performed on sets U(p), U(q), . . . then the result is a set U ( x ( p , q , . .)) where the function x is recursive. E.g. the strengthened version of Theorem 1.4.1 reads: +
U(P) n U(q)= U M P , q)), where x ( p , d = min, [e, = rZp 8z &'I Let A R k . l . s 5 1, m = (n,+!,. . . , nl) and let Am,ebe the set
.
{ ( ~ ~ ~ , ~ ~ ~ , p k , n ~ , ~ ~ ~ , n , ) : ( p ~ , ~ ~ ~ , ~ k , n ~ , ~ ~ ~ ,
If A = U " ~ l A 1 ' ( then r ) , Am, e= lJ'"''(r'),where r' depends on m,e, and Y . Using the strengthened versions of Theorems 1.4.1-1.4.3 and of the analogues of Theorems 1.3.2 - 1.3.4 for the family .A? and repeating the proof of Kleene [4] we obtain the recursion theorem (or, in Myhill's terminology, the fixed point theorem). then 1.5.6. There is a recursive function E(m, r ) such that if A = Utk.l+"(r), A m , n ( m , v i = U'k,8'(E(m Y ),) . I.6*. Degree6 of representability. In the whole 56 we assume that Cn,JCns(X))= Cns(X). Let y , + be functions, i.e. elements of R l , o . We say that the degree of representability of p (in S) is not hiqher than that of 4 (symbolically cp 5 ++"I, if p is representable in Exts(D+(a)), where a is an inessential constant for S:
co s s 4 = c o ~ ~ , ~ , ~ . ~ ( E x t s ! D ~ ( a ) ) . This definition is an obvious adaptation of the .definition due to Kleene-
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R E P R E S E N T A B I L I T Y OF SETS IN F O R M A L SYSTEMS
473
Post [S] to the more general situation considered here. Similarly as for recursive degrees we define-cp z S(I, as (cp S s + ) & (4 s.9cp). I.6.1*. The relation S s is reflexive and transitive; the relation z S is un equiva lence relation. The equivalence classes under z S we call degrees of representability. We show similarly as in [6] that I.6.2*. Degrees form an upper semi-lattice whose minimal element is fi,l.Q,l. I.6.3*. If V E 26,1,0,, and 4(n)= 9(cp(n)),C w ) = ds(n)), then 4 Sat? and C S s9 , but in general non = s C . In the next two theorems we denote by %A the characteristic function of a set A and put r = X T (cf. 1.5.4). I.6.4*. If A € then Ssr . Indeed, x,,(n) can be represented as r(V(n)),where cp is recursive. I.6.5*. r f T‘ is the set of Godel numbers of formulas provable in Exts(D,(a)), where a is an inessential constant of S , then r < s x T , and X A < s f o r every
+
A in &Po.l.
The theorem is proved by showing that r = s ~ T , would imply that TI is strongly representable in S’ = Exts(D,((a)) which contradicts 1.5.5 (with S replaced by S’). The formula r S s x T , is a consequence of the following result: for arbitrary cp, if T * is the set of Gijdel numbers of formulas provable in Exts(Dq(a)), then 2 T*S;Scp . Theorems I.6.4* and I.6.5* generalize the basic properties of the “jump operation” of [6].
1.7. Separability and decidability. We call two subsets A, B of Rk.1 separable in S if there is a C in .P:, such that A C C and B n C = 0 . 1.7.1. T and the set Tn of Godel numbers of formulas which are refutable in S are not separable in S. The proof is identical with the proof of 2.5.B in [3]. S is called S-undecidable if T$.G%?’:l;it is called essentially S-undecidable if no consistent system Exts(X) is S-decidable. From 1.7.1 we obtain the result that,under the assumptions made in 51.1, 1.7.2. S is essentially S-undecidable. 1.8. Properties ( A ) ,(C), and (S). Most of the formal systems encountered in practice enjoy one or more of the following fundamental properties: (A) I f - A , BE s k , i , then A U BE 9 k . 1 , k , I = 0 , 1 , 2 , . . . . ( C ) If X is a consistent weakly representable set of formulas then the set
is weakly representable, i.e., Mk.I(X)E s k , l + l ( s ) , k , I = 0 , 1 , 2 , . . . . The phrase “weakly representable set of formulas” means of course a set such that the set of the Gijdel numbers of its elements is weakly representable. A special case of (C) in which we assume x = 0 is called (co). Sets k f k , I ( o ) are denoted simply by kfk.1. (s)For every A, B in. &Pk,lthere is a formula 0 with k f r e e functional variables and l free number variables such that
474
FOUNDATIONAL STUDIES
- -
@(a)f Cna(Dq(a)), if q E A - B @(a) E Cns(Dq(a)),i f q E B - A
@(a) E Cns(Dq(a))or
, ,
@(a) E Cns(Dq(a)),i f q f A
uB.
@ is called a separating formula for A and B.
Condition (S) was formulated for the first time (in a slightly weaker form and for a special system S) by Shepherdson [15]. Condition (C) was used implicitly by several authors. 1.8.1. Each of the properties (A),(C), ( S ) is preserved when one passes from S to a system Exts(D,(a)), where a is an inessential constant and cp a function. 1.8.2. (c,) implies that {(q, p ) : q E u“ “ ( p ) )E g k , l + l ( s ) and T E ~ o . I-( s ) %.:(S, . PROOF.The first part is identical with (Co). In view of 1.5.5 it is sufficient Let 9 be a recursive function such that e$& = to prove that T Ego.l. & ( x =x)’ for every closed formula @. Since’ 101
E
T = Zv‘:&(a)
E Cns(Do(a))= (0, &@l)) f
Mo,l
we infer by 3.6 that T E S ~ . ~ ( S ) . 1.8.3. (S) implies that S k * [ coincides with the family of sets A S Rk.1 such that A E @ , I and Rk.1 - A f gk,l . Obviously every A in g k : belongs to g k . 1 together with its complement (cf. 1.3.1). If A and Rk,[- A are in 9 k . i then every separating formula for these sets strongly represents A in S. 1.8.4. (C) and (S) imply that S is essentially incomplete, i.e., that for every weakly representable consistent set X of formulas the system S’ = Exts(X) is incomplete. PRoof. Condition (C) implies that the set T’ of Godel numbers of formulas provable in S’ is weakly representable in S along with X . If S’ were comwould imply w - T’ E @o,l(S),whence by 1.8.3 we plete, then T’ f .@o,l(S) whence by 1.3.12 T’ f &%:(S’). Since this contrawould obtain T’E go:@), dicts 1.5.5 we infer that S’ is incomplete. Theorem 1.8.4 gives an abstract form of the incompleteness theorem of Rosser [14]. 1.8.5. (C,) and (S) imply that the theorem of reduction [7] holds in %,i(S), k,l= 0,1, . ’ ’ . Hence the second separation principle holds for weakly representable sets and the first separation principle holds f o r complements of weakly representable sets. PROOF.If A , B E Hk,&S) and @ is a sepal‘ating formula for these sets then the reduction of A u B is effected by taking A l = A n {q : @(a) f Cns(Dq(a))), BI = B n { q : @(a) E Cns(Dq(a))l 1.8.6. (C,), (A), and ( S ) imply that f o r every pair A , B of disjoint sets in L%b,i(S) there is a formula 8 such that 8 weakly represents A and -8 weakly represents B.
-
*
‘0’ denotes the Giidel number of the formula Q.
[811, 36
REPRESENTABILITY OF SETS IN F O R M A L SYSTEMS
PROOF.Let Y and u be recursive functions such that v(n) = r.-ZLk."1 Sbki:; = Sb(x/B.) ZAk*'"'. Sets
475 and
A* = ((4, n ) : q A v (q,u(v(n)))E k f k . 1 1 , B* = {(q, n ) :q E B v (q, 0)) E kfk.11 are weakly representable in S according to (A) and (G).Let Zik.'"' be a separating formula for A* and B*; we can assume that the last of its number variables is x. If 8 arises from Z:k"+l' by the substitution of Bq for x, then 8 has the desired properties. Theorem 1.8.6 gives an abstract form of a theorem due to Putnam and and Smullyan [13]; the idea of the proof given here is due to Shepherdson [U]. 1.8.7. (C), (A), and (S) imply that if X is a weakly representable consistent set of formulas, then f o r every pair of disjoint sets A , B in .B'k,I(S) there is a formula 8 such that 8 weakly represents A and -8 weakly represents B in ExtdX). The proof is similar to that of 1.8.6;the only difference is that we replace M k . 1 by kfk,i(X). 1.8.8. (C), (A), and (S) imply that %,i(S) = a.I(Exts(X))for every consistent weakly representable set X of formulas. PROOF. The inclusion S . l ( S ) C 9h.I(Exts(X)) follows from 1.8.7. If A E Hk.[(Exts(X))and Z, represents A in Exts(X), then q f A = ( 4 , p ) f Mk,r(X) whence A EM k , t ( S ) ,since, by (C), Mk,I(X) belongs to S . l ( S ) . Theorem 1.8.8 represents an abstract form of a theorem discovered by Ehrenfeucht and Feferman [l]. It is an open question whether the operation Ext preserves any of the properties (A), (C). For the property (S) the answer is negative as is obvious from the observation that 1.8.3 is false for the system Sp discussed in [lo]; cf. the remark following Theorem 1.4.4.However, the property (S) is preserved under finite extensions: 1.8.9. If S has the property ( S ) then so does the system S' = Exts({17)), where 17 is an arbitrary closed formula. S ' )A,, B E g k , l ( S ) ,cf. 1.4.4. Let b be a separatPROOF.If A , B E ~ ~ , ~ (then ing formula for A and B in S. We easily show that it is a separating formula for A and B in S'.
CHAPTER 11. APPLICATIONS OF THE
GENERAL THEORY
11.1. System So. The function of consequence for this system is simply the function Cno. Thus, apart from the existence of inessential constants, So is identical with the system A of [3]. 11.1.1. So satisfies axioms (A) - (E) and the condition Cno(Cno(X)) = Cno(X). The notions of recursive and recursively enumerable subsets of Ro,lare known. Subsets of R l , oare called recursive or recursively enumerable if they are unions of recursive (recursively enumerable) sets of neighbourhoods in the Baire space R,,o under the usual numbering of neighbourhoods. These definitions can be generalized in an obvious way to subsets of Rk.l for arbitrary k , 1. With these definitions we have:
476
1811, 37
FOUNDATIONAL STUDIES
11.1.2. S ? ~ . I ( S ~and ) gk:(S0)coincide with the families of recursively enumerable and of recursive subsets of Rk.1. 11.1.3. S%.t(Exts,(D+,(a))) and GP~,(Extg,(D&))) coincide with the families of sets which are recursively enumerable (recursive) in bp . 11.1.4. The family ~ , o : o , l (consists S o ) of the muppings f: Rl,o-+ o such that f(p) is a recursive functional in the sense of [4]; the family .9?&.o(So) consists of the mappings f : Rl.o4Rl.owith the following property: there is a recursive functional F with one functional and one numerical variable such that f ( v ) = 9 ( n ) [ W= ) Rep, n)l . The characterization given in 11.1.4 can easily be extended to functions in .9&m,a for arbitrary K, I, m, n. Functions of this family will be called recursive. 11.1.5. So satisfies conditions (A),(C), and (S). PROOF.( A ) is obvious from the properties of recursively enumerable sets. (C) follows from the possibility of expressing the relation of provability in So by an existential statement whose initial quantifier binds a numerical variable and has as its scope a formula which defines a recursive relation. (S) is proved as follows. Let A , B be recursively enumerable sets, let r ,A be formulas which represent them in So and let IT be a formula with the free variables x , y , b such that IT strongly represents in So the following relation P:
-
m is the Godel number of a formal proof of
5 from
Dq(a).
Hence IT(&, a,, a) E Cno(D,(a))if P ( m ,p, q), and n(s,,a,, a) E Cno(Dq(a)) if non -P(m,p , 9). Repeating the proof of Rosser [14] we show that the formula ( E X ) ( ~ (8X q-7 , ,b) & (x’)[z’< x
3
~ L ’ ( x &di, ’ , b)])
is separating for A and B. 11.1.6. Systems Extao(D&)), where a is an inessential constant of So, satisfy conditions (A),(C), (S). This theorem which results from 1.8.1 and 11.1.5 explains why sets recursive or recursively enumerable in arbitrarily given functions have properties similar to absolutely recursive and recursively enumerable sets. System So is known to possess various properties which do not follow from the general theory of Chapter I (e.g. the existence of effectively inseparable recursively enumerable sets). We shall not deal here with these properties since our aim is to show how much can be already obtained from the general assumptions made in Chapter I and not to develop the theory of recursive sets and their generalizations.
11.2. Systems S., These are systems obtained from So by the repeated use of the rule o. The precise definition runs as follows: For a set Xof formulas we put
Here R and
7c
are ordinals and R is a limit ordinal. We define S. as the system
(811, 38
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REPRESENTABILITY OF SETS IN F O R M A L SYSTEMS
whose function of consequence is Cn.. By an easy induction on 11.2.1. S, satisfies conditions (A) - (E).
IT
we obtain
11.3. Constructive dehition of systems S,. Before we can discuss further properties of systems S. we must introduce some definitions. Let 9 be an arbitrary function and a a constant inessential for So. We denote by P a ternary relation which is recursive in 9 and universal for the family of relations (i.e. subsets of R0J primitive recursive in 9. Let W e be the set of e in o such that R,' is a nonempty well-ordering relation. The order type of R,' is denoted by lei'. From [9] it is known that
{lel':eE W'} = { I T IT: < of} = o:, where or is the first ordinal not constructible in cp. Along with the functions Cn, we shall consider auxiliary functions Cnf, where e is in W'. Whereas the definition of Cn, would be unacceptable to a constructivistically minded mathematician, the definition of 6:would be almost acceptable for him. I E a : ( X ) if and only if there is a function 5 whose domain coincides with the field of RP, whose range consists of sets of formulas and which satisfies the following conditions: (a) if no is the minimum of R.?',then 5(no)= C n o ( X ) , ; (b) if n is the successor of n, in the ordering R.?',then B(n) = F(B(n,)); (c) if n is a limit element of the field of RP, then F(n) = F ( j ) with summation extended over j such that RP(j, n ) and j # n ; (d) I E g ( j ) , where the summation is extended over the field of RI. If 'e E W', then there is exactly one function 5 satisfying (a) - (d) for each set X. 11.3.1. &:(X) = Cn,,,,(X)for e in W'. PROOF.By an easy induction on /el'. Let P be a formula which strongly represents R'" in Exts,(D'(a)); thus P has 3 free numerical variables and one inessential functional constant a. Let d(a, t) be the formula
u
u
= t , x , Y ) v P ( a , t , Y, 41) & [ p ( a ,t , x , Y)& P(a, t , Y, x ) = ( x = r)l&[P(a,t , x , y ) 8z P(a, t , Y , 4 = P(a, f, x , 4 1 & (P)(Ex)CP(a,f, P ( x + I), N x ) ) =
( x , Y , z)(CP(a,t , x , x ) & P(a, t , Y. Y)
[ B b ) = P(x + I)]))
.
This formula is obtained by expressing in the language of So the usual definition of well-ordering. 11.3.2. If e E W' and I e I' L 1 , then d(a, a,) E Cn,,,,(De(a)). The proof proceeds by induction on I e 1' and uses the fact that in Sowe can prove a formula expressing the well-known set-theoretical theorem sayin: that a relation well-orders its field if and only if every segment ,If the field is well-ordered. Kreisel and the author (independently of each other) have shown that there
418
[811, 39
FOUNDATIONAL STUDIES
are integers e in W v such that d ( a , 6,) non E Cn.(Dv(a)) for z < I e 1". Kreisel calls such integers and the corresponding ordixials I e Iv "autonomous". Their existence shows that Theorem 11.3.3 cannot be strengthened by replacing I e 1' by a smaller ordinal. 11.3.3. There are formulas r(t,x , y , b, E ) and A(t, y , b, E ) such that for every formula 0 and every e in W v that following conditions are satisfied ( i ) if , c Iv 2 1 and R:(f,f ) , then @ E G X ( o , ( a ) ) = r(&, a/, are-, a, a) E Cnl,lcP(Dq(a)u &(a)) , (ii) if l e l v z 1, then @ E cnZ(Dq(u)) 3 A(&, arei, a, a ) E Cn,.lvp(D,!a) U &(a)) , (iii) if le iv 2 1, then 0 6 Tn:nlP(D,(a))= -A(&, b,a, a) E Cnlslvtl(Dy(a)u Dv(a))), (iv) A is satisfied in the standard model of So under the interpretation of t , y , b, E as e, [@I, q, (p if and only if 0 f &:(Dq(a)) . PROOF (IN OUTLINE). To construct the formula r we express in the language of So the arithmetized definition of the relation 0 E &Y(Dq(u)) . The formula which we obtain in this way has the form
(W)[ZI(t,E, 4) 82 Z*(t,E , 7, b) = 7/(2"(2Y + 1)) = 01 , where Zl and 2,can be described as follows: Zl(t,E, 8 ) is the formula
(u , V)[8(2"(2V + 1)) = 0 = P(E, t , u, u)l (i.e. 2, "says" that p(Z"(2v 1))= 0 implies that u is in the field of R:). &(t, E , 8 , D) is a conjunction of three formulas each of which gives necessary and sufficient conditions for the vanishing of p(2"(2v 1))in cases (a) when u is the minimum of R:; (b) when u is the successor of an element ul; (c) when. u is a limit element. The formulas describe (in the language of S) the three situations described in points (a), (b), (c) of the definition of G : ( X ) . Having constructed the formula r we show that (1") r is satisfied in the standard model of So witli t , x , y , E, n interpreted a s e, f , r01, (p, q if and only if 0 f z y ( D q ( u ) ). (2O; I f f is in the field of E , if r / is the order type of the segment of this field determined by f and if @ E ay(Dq(a)),then
+
+
r(&, 6f,are>, a, a ) E Cnmax ( 1 , r , ~@,(a) u Dda)) . The proof of (1") is straightforward and the proof of (2") proceeds by induction on r j . The implication from right to left in (i) results from (1") and the implication from left to right from (Z").'
* It is rather significant that we proved this theorem oy means of semantical considerations. Probably a syntactical proof would allow us to obtain much stronger results and in particular to characterize the family S T ~ z(Exts,(X)) , with an arbitrary X. We do not know, however, whether a purely syntactical proof exists.
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REPRESENTABILITY O F SETS IN FORMAL SYSTEMS
(ii) results from (i) by taking as A the formula ( E x ) r ( t ,x , y , D, E ) . (iii) is proved similarly as (ii), but uses the lemma A(E, t ) = (E!rl)[Zdt,€, 7)& 2 2 %
E, T , 0)l E Cno(0)
+
and Theorem 11.3.2. (To explain why the lower index in (iii) is I e 1 and not simply I e as in (ii) we remark that - A begins with a general quantifier and so we must apply once more the rule o in order to prove this formula.) (iv) is a direct corollary from (ii) and (iii). The theorems of this section have nonrelativised versions which we obtain by taking as (p e.g. the constant function 0. When referring to these nonrelativised versions we shall simply omit the index (p and the constant a (or the variable E) in the formulas A,r, and A. 11.4. Properties of S., n < ol. 11.4.1. Systems S. f o r 1 In < o1 satisfy conditions (A), (C,)and (S). PROOF.Let e be an integer such that I e I = n. (A) If 0, T weakly represent sets A, B C R k . 1 , then the formula A(&, 8rem3t a) v
-4%wail, a)
represents A u B in S,. (C,) From the nonrelativised version of 11.3.3 and from 11.3.1 (q, fi) E Mk.1 = $”(a) A(&,
4
E Cn,(Dq(a))=
a) E Cn.(Dq(a)),
where q(p) = r2ik “(a)1. These equivalences prove that if 8 represents g in So then the formula (Ex)[@(a, x ) & A(&, x , a)] represents Mk.1 in S,. (S) Let A, B c Rk,t,let 0 , T represent A, B in S, and let 8 be the formula
(Ex){P(&,x , x ) &k r(4, x, &wl, a) &k (x’)[P(&x ’ , x ) & (x’ + x ) 2 -r(&,x ’ , ha,-, a)]} . Using an argument similar to that of Rosser [14] we show that 8 is a separating formula; the difference between this proof and that of Rosser is that our proof uses the well-ordering Re instead of 5. 11.4.2. Systems S., 1 5 n < o1 do not satisfy conditon ( C ) . PROOF.Cn,(O) is representable (weakly) in S,. By 1.8.8 condition (C) would imply that LG3‘k,&S,) = 5%.1(Exts,(Cn,(0)) which is false, because Exts,(Cn.(0)) = Cn,.2(0). 11.4.3. The family .5%?o.l(Sr,)< , ol, consists of hyperarithmetic sets; every hyperarithmetic set of integers belongs to one of the families .5Zo,l(Sx), n <w,. PROOF.The first part follows from the observation that the set of G a e l numbers of formulas in Cn,(O) is hyperarithmetic (as we easily see from the uniqueness of the function 8 used in the definition of r n : ) . The second part is proved by showing that for each e in 0 the set He of Kleene [5] is weakly representable in SI.I. Theorem 11.4.3 reduces the theory of hyperarithmetic sets of integers to the theory of sets representable in systems S,, R < o1 .
480
[811, 41
FOUNDATIONAL STUDIES
11.5. Properties of the syskm Sp. 11.5.1. The family .G%.t(So) consists of IT: sets, i.e., of sets {q :(cp)R(cp,9)) ,
where R is arithmetic. PROOF. Evaluating the predicate %(a) E CnO(Dq(a)) we show easily that it is of type lIi, cf. [14]. Hence so are weakly representable sets. Weak representability of the lI: sets is an immediate consequence of Orey’s theorem 1121 (strictly speaking the proof given by Orey is applicable only to sets of integers, but a generalization to the more general case presents no difficulties). 11.5.2. So satisfies the condition CnQ(Cno(X)) = Cn,(X) for every set of formulas. PROOF. C n p ( X )is the smallest set containing X and closed with respect to the rules of proof of Se and to the rule o. In 11.5.4 we shall show that a cannot be replaced in $5.2 by any smaller 9 ordinal. The proof is based on the following important theorem discovered by Shoenfield (unpublished) and Spector [16]: 11.5.3. If X i s a set of formulas and x the characteristic function of the set of its Godel numbers, then Cnp(X) = Cn,;(X) . 11.5.4. Cno(D&a))= Cnq(D,(a)) # Cn,(D&)) for a < of. PROOF.The equation is an immediate consequence of 11.5.3. In order to prove the second part we consider formulas A(&, a ) defined in $11.3. The set of those of the formulas A(&,a) which are true in the standard model when a is interpreted as 9 is not hyperarithmetic in (p, since otherwise so would be the set W+‘which is known to be false [5]. Hence there is no z < of such that A(&, a)E Cnx(D,(a))for all e in Wq. On the other hand, A(&, a) E CnQ(Dq(a)) by 11.3.2. This shows that the sequence Cn,(D,(a)) is strictly increasing for n where A is a set and 2 is a function whose domain consists of all predicates and is such that 2 ( X ) sAq(‘) for each predicate X , then we call M a model. A valuation of a normal formula F in M is a function u which correlates with each free variable of F a n element of A . A ternary relation t is called an adequate satisfaction relation for dip if the following conditions are satisfied for all normal formulae F and models M: (1) M t F[u] is defined whenever M is a model and u is a valuation of F in M ; (2) If F is a,formula of dp,,,then M t F[u] holds if and only if u satisfies F in M in the usual sense; (3) If F is i G , then MtF[u] is equivalent to non MtG[u] and if F is GI &G2, then MCF[u] is equivalent to MtG,[u,] and MtG2[u2] where ui is the restriction of u to the set Fr(Gi), i= 1, 2; (4) If F is (5) G then MtF[u] is equivalent to the statement: MtG[u’] for every valuation u‘ of G which coincides with u on the free variables of F ; (5) Let F be a normal formula, P a predicate with n arguments and G a normal formula with n free variables. Then there is a normal formula F, with the same free variables as F such that the following is satisfied: whenever M , = ( A , 2,) is a model and M = ( A , 2 ) differs fromM, just by the fact that 2 (P)= { XE AFr(‘): M , t G(x)) thenhl, t F, [y] = M t F[y] for every y~ AFr(F). F, is said to arise from F by a (functional) substitution of G for P (see CHURCH[1956] p. 192 for the actual construction of F‘, in case of the logic 90).
Assumption 1. There is an adequate satisfaction relation for 2. One of such relations t will be selected once for all and all subsequent definitions will be relativised to it. U,sing the relation t we define two notions with which we shall constantly deal : If F, G are normal formulae, then we say that G is a consequence of F and write Ft G if for every model M and for every valuation u of F&G in M the condition M k F[ulFr(F)] implies M k G[vIFr(G)]. If F is a normal formula, then we say that a predicate X does not occur in F if for each pair of models M = ( A , 2 ) , M ’ = ( A , 2 ’ ) such that 9 coincides with 9’except possibly on X the equivalence M 1 F [u]
= M ’ k F [u]
holds for every valuation of F in M . It is obvious that the truth or falsity of M t F[u]depends only on values of
526
FOUNDATIONAL STUDIES
W1, 89
9 ( X ) for such predicates X as occur in F. We shall henceforth assume that in each formula occur only finitely many predicates. Writing a formula with displayed predicates e.g. F(P,Q, ...,S) we assume S are the only predicates which occur in F. The function tacitly that P, Q, q whose domain is the set of these predicates and whose value q ( X ) is the rank of X is called the type of F. A model M of type q is the pair (A, 9 ) where 9 has the Same domain as q and satisfies ~ ( X ) G A ~for ( ~each ) X in the domain of q. We extend in the obvious way the satisfaction relation Mi=P[v] so that M may be any model whose type is an extension of the type of F.
...,
2. Examples
We enumerate some well-known examples in which the assumptions made in section 1 are satisfied. 2.1. The weak second order logic =Yw (cf. MOSTOWSKI [1961]). In this logic there are two types of variables: individual variables as in -Yo and set variables which range over finite subsets of the universe. 2.2. The strong second order logic =Ys. The syntax of =Ys is the same as that of =Yw but the set variables range over arbitrary subsets of the universe. There are various intermediate second order logics which all have the same syntax but differ in the range of the set-variables. We quote as examples the following possibilities: 2.3. The range of the set variables is the family of sets of a power < a contained in the universe. 2.4. We can define an increasing sequence =Yt of logics as follows: =Yo is =Y,;=Yt+l has the same syntax but the range of set variables consists of those subsets of the universe which are definable in =Yt; if 1 is the limit number, then the range of the set variables is the union of all preceding ranges. 2.5. Logics Q,. The syntax of the logics Q, differs from that of -Yo by the presence of a new quantifier Q to be interpre#ed as: “there are at most tl...” (cf. MOSTOWSKI [1957]). 2.6. Infinitary logics (cf. KARP[1964]) with or without identity. 2.7. Sub-logics of Sm,,,, obtained by allowing not all denumerable strings of symbols but only some regular ones, e.g. hyper-arithmetic (cf. BARWISE [1967]). 2.8. The full strong second order logic 2’;has not only the set variables but for each n >0 has infinitely many variables ranging over n-ary relations. The syntax and semanticsof this logic have been described by TARSKI[1956].
P61, 90
CRAIGS INTERPOLATION THEOREM
521
2.9. Full weak second order logic 9 2 has the same syntax as 9: but its second order variables range over finite relations only. Various intermediate full second order logics can be defined similarly as in 2.3. 3. The interpolation property We return to the general case of a logic 9 satisfying the assumptions set forth in section 1. We shall say that 9has the interpolation property if for arbitrary normal formulae F, G satisfying Ft G there is an interpolation formula H such that Ft Hi- G, F r ( H ) s F r ( F ) n Fr(G) and each predicate which occurs in H occurs also in F and in G. Thus 9has the interpolation property if Craig’s theorem is valid for normal formulae of 9. We shall show that no 9satisfying suitable assumptions has the interpolation property. The assumptions will be satisfied in cases 9 = d p w , 9=9: and 9= Q,. On the other hand it is known from the literature that Craig’s with or theorem is satisfied for the full strong second order logic, for 90,,, [1965]) and for some sublogics of without equality (cf. LOPEZ-ESCOBAR 9@,,@ which were mentioned in 2.7 (cf. BARWISE [1967]). For 9=9a,s with equality and with (a, B)#(w, o) and (a, B)#(ol,o) Craig’s theorem is not satisfied (cf. MALITZ[1965]). To the author’s knowledge the problem of its validity for logics 2.3, 2.4 and 2.5 with u > O is not solved. We shall now formulate two assumptions from which we shall derive that 9does not have the interpolation property. Let o be the set of integers and Po, Q, the relations x + y = z , x=yz. The standard model of arithmetic is defined as M , = ( A , d)where d is a function with domain consisting of one predicate N of rank 1 and two predicates P, Q of rank 3 such that d ( N ) = o , d ( P ) = P , , d ( Q ) = Q , . Assumption 2. There is a normal sentence A = A ( N , P, Q ) such that M , k A and each model M of the same type as M , satisfying M k A is isomorphic to M,. 3.1. Assumption 2 is satisfied for 9 = Z w9 , =92 and 9 = Q o . PROOF.We take as A the conjunction of sentences which say that N is the whole universe, that it is ordered by the relation (Ey) P(x, y, z), that it has the first element, that each element has a successor and that each element with the exception of the first has a predecessor. Moreover we include to A the recursive equations for addition and multiplication and the sentence which says that for every x in N there are only finitely many y
528
FOUNDATIONAL STUDIES
which precede x. In case of logics Pw and 9 :this last sentence is
(4(EX)( 4 C(X E X)*= (EY) p (XI Y, 4 1 and in case of logic Q, it is
(4(Qx) (EY)P 6,Y 4. I
We now formulate assumption 3. For k e o and qeo‘ we denote by Y q the k-fold Cartesian product of the spaces P ( o q ( ’ ) where ) P ( X ) denotes the family of all subsets of X . We conceive .SPq as a topological space with the usual product topology. Let Roy..., Rk-l be predicates of ranks q,(O), ..., q ( k - 1). For p = ( p o , ..., Pk-1)Eyq we denote by M , ( p ) the model (0, 9) where 9 has domain {N, P, Q, R, ,..., Rk-,} and S ( N ) = o , 9 ( P ) = P o , 2(Q)=Qoy 9(Rj)=pi for i< k. For every normal sentence F of the same type as M , ( p ) we call the set { P E q :
MO(P)tF}
the spectrum of F. Assumption 3. There is a recursive ordinal p < o y such that for each normal F the spectrum of F is a Borel set of a class < p . 3.2. Assumption 3 is satisfied for 9=dLPw 9=9$ , and 9=Qo. PROOF.Let F be a (not necessarily normal) formula of gW with the free individual variables xl,...,x, and set variables X,, ..., X,. Let u and V be functions which correlate with each xi an integer and with each Xj a finite subset of o.We prove by induction on the length of F that there is an integer n depending only on F such that for arbitrary u, V the set
is Borel of class 0 let D,, = {zc o: z E ei (mod 2n)}. Each Dn is obviously non-void, definable and, since e:-l = ek (mod 2"-l) for n > 0, it is contained in Hence the family X,= {D,, D,, ...} is a basis of a filter Fo.
+
552
FOUNDATIONAL STUDIES
[102], 234
It is easy to determine the distinguished codes of Dn and to prove that they form an A-set. Hence X , is an &-base of Po. Let b, = { J ( m ,n): ( m > 0) & m = a; (mod 25)}. Since b, is primitive recursive, it is definable. It remains to show that whenever P is an ultra; filter and P 3 X,, then Stsf is arithmetical in RF(bO). Thus as&me that each Dn belongs to P. Since
4 E R,(b,) = {i > 0 : i = a: (mod 24)) e P we obtain taking q = 2"+e; 2"+ e;
t
R,( b,) = D, E P
whence 2"+e;e R,(b,). We now show that 2"+e; is a unique element n of R,(b,) such that 2n < m < 2"+l. To see this we notice that if m E R,(b,) and 2" < m < 2n+1, then E = n, a; < 2" and {i > 0 : i
= a; (mod 2")) E P .
This set must intersect with Dn since they both belong to P.It follows that a; = ek (mod 2") and since both a;, e; are 0 can therefore be defined as the integral part of where x is a unique integer < 2n such that 2"+ x Bp(bO). Since n t Stsf = en = 1 it follows that Stsf is arithmetical in RF(bo) and the proof is finished. I n [3]the theorem was proved only for models which are elementarily equivalent to the principal model. It would be interesting to verlfs whether it holds for w-models of the system 8,resulting from A, by omitting the choice axiom. Iteferences [l] U. Felgner, Comparison. of the azipnas of l c a l a&
[2] [3]
[a]
[5] [6]
umiversal choice, Fund. Math. 71 (1971), pp. 43-62. A. Mostowski and Y. Suzuki, On w-modelswhich are not ,9-md.eZa, Fund. Math. 65 (1969), pp. 83-93. A. Mostowski, At note on teratolgy, Mplmv memorial volume, to appear. - Uonstrwtible sets, with applications, 1969. W . V. Quine, On ordered pairs, J. Symb. Logic 10 (f946), pp. 95-96. D. Scott, On constmting models of arithmeth Infinitistic Hetirods, Prooceedings of the Symposium on Foundations of Mathmtim, Warszawa 1961, pp. 236-266. &pu par la R&z&un le 6. 3. 1971
Errata to the paper “Models of second order arithmetic with definable Skolem functions”, Fundamenta Mathematicae 75 (1972), pp. 223-234 by
Andrzej Mostowski (Warszawa) The proof of the main theorem formulated on p. 223 of this paper aoes not establish its claim. The reason is that the formula given in Lemma 1on p. 223 is not provable in A, contrary to what is stated in the lemma. The formula in question becomes provable if one adds to the axioms of A, the following scheme of dependent choices: (4EdEY),A ( 0 , ? --*I (w)x(EdxW0) ) = w )83 (%A
Mn),dn+”)l-
Thus the results of the paper apply to a system A* of second order rtrithmexic based on the axioma of A, and the scheme of dependent choices and the paper can be (formally) corrected by replacing everywhere the system A, by A*. (In the first draft of my paper I called A, the system here denoted by A*; this confusion of symbolism eventually led to the mistake). I wish to thank Mr Stephen G. Simpson for pointing out the mistake to me; Ebr Simpson and several other colleagues drew also my attentioh to a close similarity between the proof given in my paper and the proof due to U. Felgner to estQb1ish a similar result in set theory; see U. Felgner, Comparison. of the axioms of local and global choice, Fundamenta Mathematicae. 71 (1971), pp. 43-62.
A TRANSFINITE SEQUENCE OF w-MODELS ANDRZU MOSTOWSKI
Let A2 be the axiomatic system of second order arithmetic as described in [2]. One of the models of A 2 is the "principal model" M,, consisting of all integers and all sets of integers. Obviously there exist many denumerable w-models elementarily equivalent to M,, and we shall deal in this paper with some questions pertaining to this family which we denote by F. In $1 we define a rather natural relation e between two denumerable families of sets of integers. From the upward Skolem-Lowenheim theorem it follows easily E 9ordered by e in the type w l , but it is not immedithat there exists a family 9, ately obvious whether there exist a subfamily of 9 not well-ordered by C . In the present paper we construct such a family of type 7 . q where 7 is the order type of rationals and indicate some applications to hyperdegrees. We adopt the terminology and notation of [2], with the only change that we adjoin to the language of A2 the constants yo, v l r . . for the consecutive numerals 0, 1 , 2 , . . . and axioms which characterise them: 1
Nvor
"0, "01,
A("*, "1,
lE("0,"l)8L P("1, "19 k = 0, 1, 2, * * * .
"3
"k+l),
Also we modify the axioms of A2 given in [2] by prefixing them by general quantifiers bounded either to S or to N. $1. Partial orderings E and < . Let J ( x , y ) be the usual pairing function (cf. [2]). For r c N we put r(")= { j E N : J(n,j ) E r}. Sets r(n)will be said to be. in relation e to r : s e r = (En)[s = P ) ] .
Let R be a denumerable family of subsets of N. A set r c N is a code of R if for each n in N, fin) is in R and every set in R can be put in this form. If R' is another such family then we say that R' e R if there are codes r'. r of these families satisfying the condition r' e r. Thus R' e R if and only if some code of R' is in R. The e notation can be used also for denumerable w-models of As since those models are completely determined by the families of their sets. Whenever M, M' are two w-models we shall write M < M' if M e M' and M' is an elementary extension of M. As already stated in the introduction we denote by F the family consisting of all denumerable w-models of A2 which are elementarily equivalent to Mpr. Elements of 9 and, generally, w-models of A2 will be identified with families of sets of integers. Received February 2, 1971.
[103],97
A TRANSFINITE SEQUENCE OF UPMODELS
555
We shall need in the sequel some simple observations concerning the relation 8 between subsets of N. Notice first that this relation is not irdexive, e.g. N a N. Put Mod = {r E N : r is a code for the family of sets of a denumerable *model for Ad. We shall show that 8 restricted to Mod is imflexive and transitive: -1. IfrEMod,t~rnonsrMdtasarimpliestsr. PROOR Let r E Mod and s = P)where n is an integer. By the diagonal argument the set X = {m E N J(m, m) 4 J } is Merent from all the sets which have the form dk),k E N. Since the sentence (a)s(Ex)s(m)N(mE x = J(m,m) 4 a} is provable in A,, it holds in the 0-model whose sets are coded by r. Hence X belongs to this model and there is a p such that X = PI. It follows that s # r i.e. s 8 r implies s # r. Now assume that t 8 s s r and let t = S(k), s = P).Hence m E t E J(k, m) E P ) and by the same argument as above t has the form r”)for some p, i.e. t 8 r. @. The relation < restricted to pmodels is obviously well founded; in case of cu-models the situation is completely different as will be seen from the following theorem. Tmom I. For every dmwnerable model Ma4 M,, there exists a f m i l y S, c S such that the order type of the relation c restricted to Sl is and Ma8 X for every X in Sl. Before we start we must introduce some C O ~ C ~ Dand ~ S notation: The formal language of A2 will be denoted by L.We assume that the only propositional connectives of L are & and and that the only quantifier is the general one. Other logical constantsare defined in the usual way. A language obtained from L by adjoining finitely many setconstants c,d, e , . will be denoted by L(c, d, e, .). In order to simplify our exposition we shall consider only the case when just one constant has been added to L. We can define in arithmetic the set of GMel numbers of formulae and other syntactic entities of the language L. Moreover the set GN of these W e 1 numbers can be formally dehed in A2;also relations between Wel-numbers comspcmdixig to syntactical relations between expressions can be so defmed. We shall use expressions like is a variable which occut8 fmly in a formula E”, “All quantifiers of E are bounded” as short for formulae Of A2 which express that the formulae with the GMel number v and E stand to each other in relations indicated in the above expressions. An occurrence of an existential quantifier (Ex) in a formula F is bounded in F if the quantifier is followed by one of the expiessions “N(x)&” or “S(x)&” and suitable parentheses. We define similarly bounded occumnccs of a e e r a l qUantifier. A formula is called restricted if all occurrences of quantifiersin it arc bounded. A signed formula is a restricted formula together with a partidon of its free variables into disjoint classes called respectively number variables and set variables. t m 2. Every formula of L(c) is equioolent on the bas& of A. to a restricted formula. This follows from the observation that formulae
-
are axioms of A,.
-.
556
[103], 98
FOUNDATIONAL STUDIES
In the sequel we consider exclusively signed restricted formulae. By a finite function of integers we understand as usual a finite set X of pairs <m, n) such that if (m, n) and \m, 11‘) belong to X, then n = n‘. We define the code of X as the integer x = C2J(m*n) where the summation is taken over pairs in X. The void sum is taken to be 0. We define the domain of x as Dom(X); if m E Dom(x) then we denote by x(n) the integer n such that ( m , t i ) E X. $3. Formalization of the satisfaction relation. If b is a code of a finite function i E Dom(b)} is denoted by ?(b). Iff is of integers, then the finite function {(i, r(b(i))): the G N of a signed formula F of L and a, b are codes of two finite functions of integers then we say that a and b fit to f if Dom(a) is the set of GNS of the free number variables of F and Dom(b) is the set of GNs of the free set-variables of F. Let M be an w-model of L and,r its code. Consider the relation Stsf which holds betweenJ a, b, r iff is the GN of a formula F of L, a, b are codes of two finite functions which fit to f and F is satisfied in the model M by the function coded by a and the set function Fb). LEMMA 3. There is a signed formula Stsf of L such that for arbitrary integers a, b, f and an arbitrary set r of integers the follouing equicalence holds
M,, k Stsf [f,a, b, r ] = Stsf(f, a, b, r). The formula Stsf is obtained by writing the inductive definition of the relation Stsf in the language L. Since Stsf is a A:-formula we have additionally the following result: LEMMA 4. Lemma 3 remains true ifwe replace M,, by any w-model M which contains r as an element and in which all the axionis of A 2 are valid. Iffis the G N of a sentence then the only codes a, b which fit to f are codes of the void sequence: a = b = 0. We shall write Stsf(f, r ) for Stsf(f, yo, y o , r). Using Lemmas 3 and 4 we can express in the formal language L the concepts of elementary equivalence, elementary extension and of a submodel. Let us consider formulae of L: x lncl y : (U)N(EU)*’(M)N[J(~~’, u) E X = J ( w , r ) E y ] . .Y Eleq (f)”(f is the GN of a sentence of L ) & Stsf(f, x ) Stsf(f, J ) ] & (x lncl y). (b, x, b’, y ) : (b and b’ are codes of two finite functions with the same domain) & (i),\[(iE Dom(b))--f ( x ( ~ ( = [ ) y) ( b ’ 9 ] . .Y El y : (f)N(a),v(b),v(b‘)x[(fis a GN of a signed formula ofL) & (a and b fit t o f ) & (b, s,b’,y) & Stsf(f, n, b, x) -+ Stsf(f, a, b’, y j ] & ( x Incl y). LEMMA 5. Let r, s be codes of tn’o w-models M I , M z (conceiced as denumerable families of’sets) atid let M be an w-model such lhat r, s are elements of M . Then M k (r Incl sj = ( M , E j14~); M C (r,Eleq s) = ( M I is elcnientarily equiralent to M 2 ) ; M C (6, r, b’, s) = = for eacli i ill the domain of the sequence coded by b); M C (r El s) I ( M , is an elementary subniodel of Mzj. Note that the concepts of elementary extension and of elementary equivalence used above refer to the language L. 11:
--f
[103], 99
A TRANSFINITE SEQUENCE OF u,-MOI)ELS
557
$4. The basic lemma. Let M o be a denumerable w-model which is an elementary submodel of M,, and let m, be a code of Mo. LEMMA 6. There exists a transfinite sequence of sets {me},< o1 such that (1) rn, is a code of an w-model M,; (2) me e m,,for E < q < wl; (3) M , < M,, and hence M , i M,j'or 5 < q < wl; (4) m, = mio)for 5 > 0. PROOF. Assume that > 0 and them, are defined for 7 < 5. Consider the union U,
>
+
(1071,194
A CONTRIBUTION TO TERATOLOGY
587'
Let us define by induction a function f :f(0) = 1, 2, f (n) if n& Stsf, f (n 1) = 2n+i f (n) if n e Stsf.
+
(9)
+
+
By induction on n we show easily that 2, f (n) 2n+i.
,(1, n>)l. Let A, be an arithmetical formula with the- free variable q such that n is the unique element of the universe of M , which satisfies A, in M1.We can select A, so that its Godel number be a recursive fpnction of n. The formula G , = (Evl)(An& F) defines D, in M , in the sense that mf3D,-M1 k G, [{(O, m>}I for each rn and it follows that if g n is the Giidel number of G,, then J(g,, 1) is a code of D, . The distinguished code of D , is thus 6 (n) = min {k:(k6 C*) & (m)[J(m,k) 6 Stsf fJ (m,J (g, , 1)) C tj Stsf I}. Since gn is recursive in n we infer that 6 is a'rithmetical in Stsf. 6.3. The set bo = { J (m, n):(n> O)& (meD,)} belongs to
B(M,) and satisfies condition (8). P r o o f. The definability of b, in M1 is obvious. Let us
now assume that 8 is an ultrafilter containing Bn. For n> 0 we obtain from the definition of b, the equivalences: n f j R 8 (b,) =(m:J (m,n)f j bo} 6 8 & {m:m 6 D, } 6 6 =D, 6 8.
Thus if n 6 Rg(fi, then nfj R8(b,) because the elements n of Rg(f) are # 0 and have the property DnCBOGB . Conversely, let us assume that nfj R g ( b , ) . Since the set { m : J ( m , n)fj b,} is non void (as a member of 8 ) it results that n 0 in view of the definition of b,. Thus we can use the equivalence given above and obtain D,tj 8. Now we repre-
+
588
FOUNDATIONAL STUDIES
[107], 195