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On models of axiomatic systems
O n models of axiomatic systems. BY
A. it1 o s t o w s I< i (Warszawa).
This paper is devoted to n discussion of various notions of models which appear in the recent investigations of formal systems. The discussion will be applied to the study of the following problem: Given a formal system L3 based on an infinite number of axioms Al,&,A8, ... is it possible to prove in S the consistency of the system based on a finite number A1,& ...,A,, of these axioms?
1. Notations and definitions. We shall consider two systems S and s based on the functional calculus of the first orderl). We shall not describe these systems in detail but give only some definitions which will be required later. S y s t e m s. We assume that the following symbols occur among the primitive signs of 8: 1. Variables: sl,s2,.~3 ,...,z,y,z ,... 3. I n d i v i d u a l c o n s t a n t s : fi ,....fa. 3. F u n c t o r s (i. e. symbols for functions from individuals t o individuals): gl,...,gp. We denote by qj the number of arguments of 91 ( j = l , ...,/!I). 4. P r e d i c a t e s (i. e. symbols for relations): rl, ...,r y . We denote by p l the number of arguments of ri ( i = l , ...,y). 5. P r o p o s i t i o n a l c o n n e c t i v e s a n d q u a n t i f i e r s . We use the symbol I for the “stroke function” and define other connectives in terms of the stroke. Quantifiers are denoted by symbols ( 3 4.and (si). Among expressions which can be constructed from these signs we distinguish the following: 6. Terms. Variables and individual constants are terms. If I‘,,...;Iyq, are terms, then so is gI(Tl,.,.,Tq,),j=1, ...,p. Terms will be denoted by the letters F7T‘l,I‘2,... l ) For the functional calculus of the first order see e. 9. C l i u r c l i [I]. Chapter 11.
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7. E l e m e n t a r y
193
f o r m u l a s are expressions of the form
rt(T,,...,P,,) where I',,...,I',,are terma. Elementary formulas will be denoted by the letters E, El, E,, ...
8. P r i m e f o r m u l a s are elementary formulas in which no variables occur. Prime formulas wili be denoted by the letters P,Pl,pz,* * 9. Matrices. Elementary formulas are matrices. If 31, and i?f2 are matrices, then so are i?f,lN2and (3q)X1 for i = 1 ,2 , ... Matrices will be denoted by the letters M , iffl, H,, ... Note that matrices containing propositional connectives other than th'e stroke are easily definable by means of matrices containing only the stroke symbol. We shall occasionally use the following abbreviations : J P for ilf, H1 for -$I,
-
2~~for i=l 0
0
11 &.Ii i=l
for
JI,V...V~BI~.
x ~..A,I ~.
10. F r e e a n d b o u n d v a r i a b l e s . S u b s t i t u t i o n . The distinction between free and bound variables is assumed as known. The formula which results from a formula A by the substitution for the variables ~ , . . . , L cwill ~ be denoted by of the terms T1,...,Tn Subst A(xl/rl, ...,g&). The operation of substitution is always performable when A is a term. If A is a matrix, it is sometimes necessary to re-name the bound variables occurring in A in order to make sure that the operation S&st can be performed. We shall always assume that the necessary changes in the bound variables of A have been performed before the operation Subst has been applied,). 11. &-matrices. These are matrices in which no bound variables occur. 12. Axioms. We assume that the axioms of s are finite in number and have the form of &-matrices in which no constants, functors, or predicates occur besides those which were enumerated in 2, 3, and 4. The axioms will be denoted by ai or a l ( q y,...,z), i=l, ...,6. _____ _-
2 ) An exact definition of the operation Subst is @en in C h u r c h [l], pp. 56-58.
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13. The r u l e s of proof admitted in 8 are the usual ones. We adjoin to them the rule of explicit definitions and the &-rule8), We shall add a few words to explain the &rule. To this end we define recursiveLy the notions of &-termsand &-matrices. The terms and matrices defined in 6 and 7 are &-terms and &-matrices.I f P,Pl,...,Pk are &-terms,then flubst m
t , / q , . .,X i k / r d
is an &-termprovid'ed that (i) zfl,...,sf&are the free variables of r; (ii) the bound variables of P are not free in rl,...,rk.If r,,...,r,, are &-terms,then r t ( r l,...,Pp,)is an &-matrix( i = l ,...,y ) . I f Ml and M8 are &-matrices,then so are HIIMa and (3x,)M1. I f M is an &-matrix, then ( m f ) Nis an &-term. The &-rulestates that for every &-matrix M the matrix
M 3 Sabst M ( x ~ / ( E x ~ ) M ) can be assumed as a theorem of s. The question arises whether the assumptions concerning the form of axioms and rules of proof (cf. 1 2 and 13) are general enough to cover the cases of standard formal systems based on a finite number of axioms. The answer is affirmative. To see this we remark that the &-ruleenables us to get rid of quantifiers in the axioms provided that we introduce a sufficient number of &-terms*). Since the explicit definitions are allowed in s, it follows that we can bring t h e axioms to the form of Q-matrices provided that we add a sufficient number of symbols to the symbols enumerated in 2 and 3. The resulting system then satisfies our assumptions and is equivalent to the given one provided that suitable definitions are introduced into the latter. E x a m p l e . Let one of the axioms have the form
0)
($1 (3Y)M@,Y ) We introduce a new functor g(s) and an axiom (ii)
M(x,g(s)). Clearly (i) is deducible from (ii). Conversely (ii) can be obtained from (i) by means of the &-ruleand the explicit definition g(s) = E y J m , Y ) . 3) The ordinary rules of proof for the functional calculus are given e. g. in Church [l], p. 40. For the &-rulesee H i l b e r t - B e r n a g s [6], qp. 9-18. 4 ) See H i l b e r t - B e r n a y e [5], pp. 16-17.
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ON MODELS OF AXIOMATIC SYSTEMS
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S y s t e m 8. The structure of 8 will be assumed similar to those of 8 . We shall, however, not assume that the number of &oms of S is finite. Furthermore we shall assume that 8 contains an arithmetic of integers to an extent which allows UR t o arithmetize in B the syntax of s and to prove in S the basic theorems of qathematical logic, e. g., the completeness theorem of CfBde16). We shall use freely in 8 the arithmetical notions such as identity, sum, product etc. 14. A r i t h m e t i c a l c o u n t e r p a r t s of s y n t a c t i c a l n o t i o n s will be denoted by words printed in spaced italics. For instance matrix is short for a matrix of S with one free variable which is satisfied exclusively by the Godel numbers of matarices of s. The actual construction of such a matrix is not needed; it suffices to h o w that it can be constructed. To simplify the notation we shall often use in 8 the non-formal language and write, for exainple, 3
is a m a t r i x of s
instead of matria, (2). Translations of such non-formal formulas to the official language of 8 will always be possible6). The following constants will be used in 8: 15. 3 for the functor of fl such that 3t,3s,3n... are the variables of s. I n other words 3f is the Godel number of the variable xr ( i s l ,2,. ..). 16. f, ,...,fl for the in d ivid u a l co n s t a a t s , gAl...,ga for the
fzclzctors, and r,, ...,r,, for the predicates of 8 . I n other words fi is the Godel number Of fi, g1 that of $1, and rk that of rk, i=l, ...,a,
j = l , ...,/?,k = l , ...,y. 17. Arbitrary terms will be denoted by letters tltl,tz, ..., arbitrary elementary formulas by letters e,e,,e2, ..., arbitmry prime formulas by letters ,p7p1,p2,...,and arbitrary matrices by letters m, m,,m2, ... 18. Let TI, ...,rq,be terms and 1, ,...,f9, their Godel numbers. ...,Pq,) will be denoted by The Godel number of the term gj(F1, [g,(tl, ...,f,)]. Thus the square brackets symbolize here a functor of 8. G o d e l [2]. *) More details concerning aritlimetization are given in U b d e l 131 a n b f i i l b e r t - B e r n a y s [5], $ 4 . 6)
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The square-brackets notation explained in the foregoing paragraph will be used consistently in many other similar situations. So c. 9. if miJ is the Godel number of the matrix MiJ, then
[fi %mfJ] is the f = l *l
Godel number of the matrix
y
fl 2 0
1=1 j=1
MI,,. If m
is the Godel number of B,then [ml] is the Godel number of MI (i. e. of -dl) and so on. 19. The following lemma is provable in 8: I n order that e be an elementary f o r m u l a it is necesamy and sufficient that e have the form [rf(t17...,fp,)] where i < y and t,,..., tpfare terms. The integer i and terms tl,...7tptare deterniined by e. We put i=Ind(e),
ti= Compj(e),
j=1,
...,p i .
Functors Ind and Comp are definable in 8. 20. The a r i t h m e t i c a l c o u n t e r p a r t of t h e f u n c t i o n &b8t will be denoted by the symbol Bb. Thus if a is the Godel number of an expression A, and tl, ...,t,, are the Godel numbers of terms r,,...,I‘,,then Bb a( 31/tl,..., 3,,/t,,) is the Godel number of the expression dubst A(xl/Tl, ...,xn/Tn). 21. If r,r1 ,...,I’,, are terms of s, then the expressions
. ;,I(sk/r)
,stcb.yt g, (TI,. ., I
and
..., #%bat Tq,(~h/T))
gj(Subst TI(Xk/T),
are identical. The equation
.,
f i b [ 9/(ti, *. b j ) ] ( 3k i t ) = [?ji(Sb tl(3k it),
-*-, Sb f 9 j ( 3k It ))I
is provable in S . 22. The Godel n u m b e r of t h e i - t h axiom of s (see12) will be denoted by [at] or by [ai(m,y,...,z)]. 23. A r i t h m e t i c a l sentences expressing t h e oonsistency of 8 a n d of s will be abbreviated as N ( 8 ) and N(8). 2. Models of the first kind. Let Bo(z),$?(XI,...,spl)be y + l matrices of B with the indicated number of free variables and lot !Pi($,,...,s9,,y) be p matrices of B such that matrices I j ( ~ l , . . . , ~ 9 , , y ‘ ) ’ T j (, ‘. c. ~ 17 s 9 , , y ” ) ~ y ’ = ~ ’ ’ ,
( 3 Y ) W~,,...,%,,Y)
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are provable in S. We define in S functors G (where j = l 1 2 , . . . , j 3 ) in the following way ') G,(:/C.l,..., sqt = (CY)q a , ,...,zqj
1
!/I.
Finally let PI, ...,Fa be 01 constants definable in s'. The a + p + y + l tuple eonsisting of a constants P I , of B functors Gi,and of y + l matrices RL will be called a pseudo-model of the first kind of s in 8. I n order to define when a pseudo-model is a real model we shall introduce some auxiliary definitions. To every term T of s we let correspond a term !Z'r of AS' in the following way. If F is a variable, then T p = T . If r=ff,then T r = F f . Finally, if P has the form gj(F1, ...,Tqj), then we put T r =Gj(Tr*,...,Trq). To every elementary formiila E=ri(I'll...,Tp,) of s we let Trl,...,Trp,) of S. We extend this correspond the matrix E' =Ri( definition to all &-matrices of s by putting (Ml\ J f 2 ) ' = J f ~ l J f ~I n. particular we put A t = a ; ( i = l 1 . . . , 6 ) . D e f i n i t i o n . A pseudo-model (1)
PI,...,F a , GI,. ..,Gp, RO,121,...?R,
is a real rnodd of the first k i n d of s i n S if the formulas
no(%) .R&y) .... .K&)
3 A t ( $ , y, ...,z), i = I, 2 , ...,6 ,
are provable in S. Models of the first kind are the ones with which one has to do in the usual proofs of consistency and of independence of axiomatic systems 8 ) . For comparison with other notions of models to be defined later, we shall note the following general facts concerning models of the first kind: 1. The general notion of models of the first Bind is defined not in X but in the syntax of S. 2. Every particular pseudo-model is a finite set of matrices of S and can therefore be defined within S. The problem whether it. is or is not a real model can be formulated and in particular cases also solved in 8. 7 ) ( 1 % ) [ . . . x . . . ]denotes the zwhicllsatisfies the condition ...s...Cf. H i l b e r t B e r n a y s [5], p. 381. 8 ) C'f. the independence and consibtency proofs in [ 4 ! ('hapter 11.
198
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number of axioms (.independent 3. If 8 contained an of whether their set is or is not definable in f l ) , thsa the problem whether an explicitly given pseudo-model is or is not a real model of s in S would be expressible in the syntax of 8 but not in 8 itself. The following theorems concerning model% of the first kind are well known but are given here for the sake of comparison with other notions of models 0 ) : I. If (1) i s a real model of the firat kind of 8 ilz S and if a Q-matrix A ( s ,y , ...,z ) is provabte i m s, then the matrig
.
Ro(LE).R,(y ) .... * R,(x)3 4 ' ( x y, , . .,2 )
is provable in R. 11. If (1)is a real model of the f i r s t kind of s in 8 and S is eonsistent, then so i s s. Let us now assume that a real model of the first kind of 8 has been explicitly defined in 8. Theorems I and I1 are provable in the syntax of S, hence they are translatable into arithmetic and therefore into S. Denoting by N ( S ) and N ( s ) formulas of 8 corresponding (via arithmetization) t o the syntactic statement: S (or s) is self-consistent (cf. section 1, definition 23), we obtain from 11: 111. The formula N ( S ) 3 N ( s )is provable in 8. In spite of this result models of the firat kind are of no use when one is examining the problem whether the formula N ( s ) itself is or is not provable in S. Models of the second and third kinds which we shall discuss in the next sections will allow us to answer this question in many particular cases. We note still the following theorem due t o W a n g [16]: IV. If the formula AT($)i s provable i n 8,then a model of the first kind can be defined explicitly in S . Indeed, the usual proofs of the completeness theorem of Gadel consist in exhibiting a model of the first kind of a (non-contradictory) first order system in the arithmetic of integers 10). Taking s as this system and repeating the argument of Godel in S (which is possible by our assumptions concerning f l , cf. p. 136) we obtain the proof, of theorem IV 11). O)
lo) 11)
Proof8 of these theorems may be found e.g. in my book [7J,Chapter XI. G o d e l [Z] or H i I b e r t - B e r n a y s [5], p. 185. W a n g [lG], p. 287. gives a more detailed proof of this theorem.
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3. Models of the second kind in the axiomatic theory of sets la), We assume in this section that 8 is an axiomatic system of set theory based e. g. on Zermelo's axioms. The following definitions are to be thought of as belonging to 8. Let be an arbitrary set aad Z a finite set of positive integers. An %-function with the set of arguments 2 is defined as a set F of ordered pairs such that v E u runs over all finite sequences 13) satisfying the conditions
a,
D(u)=Z, D*(V)C%, and the following condition of single-valuedness holds:
F and E P, then v'=v". The symbols D ( u ) and D*(u) denote the domain and the counter-domain of u, i. e. if
E
~E.(U)==(3IY)[<~,Y>E
ul,
~ € D * ( M ) = ( 3 y ) [ < y l l r> FU].
If u is a sequence satisfying the condition D(u)=Z and P is an arbitrary set of positive integers, then we denote by w]Y the sequence u restricted to P, i. e.
I
< i , a> € u Y = ( < i , y> E u ) .( i E Y ) .
An %-relation with the set of arguments 2 is defined as a set R of sequences u such that D(u)=Z and D*(u)C%. If Z consists of integers i , j , k , ... and u is a sequence with the domain 2 such that, , , ... are elements of u, then instead of U G Rwe writ;e R(a,b,c,...) and say that R holds for the elements a,b,c,... Similarly, if H is an %-function, then instead of < u , v > d 3 we write H ( u ) = v or H(a,b,c,...)= v. I& PI be,elements of % ( i = l , ...,a ) , let, Gj bo %-functions with the sets of arguments 2,=(1,2, ...,qj>, j=1, ...,8, and let Rk be %-relations with the sets of arguments Z i = { l , 2 , . . . , p k } , k = l , 2 , . . . y . me a + / ? f y f l tuple (2)
%, Pi,...,Fa, GI,..- 7 G ,
. .., Ry
El,
will be called a pseudo-model of the secorcd kind of s in S. 18) Results of this and the next section are due t o T a r s k i [111 and 1121. and t o W a n g [la] and [17]. 18) Sequences are defined as functions (many-one relations) with domains contained i n the set of positive integers. Cf. T a r s k i [ll], p. 287.
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We shall now explain when a pseudo-model is a real model. As in section 2 we need some auxiliary definitions. We shall denote by B(t) the set of f r e e v a r i a b l e s which o c c u r in a t e r m t and by B(m) the set of f r e e v a r i a b l e s which o c c u r in a m a t r i x m of s. With these definitions it, is not difficult to prove the existence and uniqneness of a function H f ( u ) and a relation Stsf which are of fundamental importance in the investigations of the semantic of 8. The exact definitions of the function H and the relation Xtsf are given below in lemmas 1 and 2 together with the proofs of their existence and uniqueness. To facilitate our exposition we explain informally the intuitive meaning of these concepts. Let t be a t e r m , m 5 m a t r i x of 8, and let 1' 1 be a sequence {,, ,...} 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
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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').
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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)
<{7 - - 9I, ! I mu , (- - .,%,)>,
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 {<31,u1>,..., < 3 k , u h > ) with u,, ..., u h . 3 i , then (4)
u 8tsf [a,] = a,(ul,
...,Wk)
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(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
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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
P ( h , w Tmp 6, A a ) = 0,
f where h is the Godel number of the proof of (17) in s. It follows from (18) that if N(6) were provable in 8 (i. e. if 20 were provable in s), then b would be provable in s and hence by (17) 8 would be inconsistent. Lemma 8 is thus proved. Theorem XIX results from lemma 8 by the same argument which was used in the proof of theorem X.
216
FOUNDATIONAL STUDIES
[32], 157
The following result can also he obtained from XVIII. Following T a r s k i and Szmielew*8) we shall call a system s interpretable in S if there exists rz model of the first kind of s in 8. We have then XX n o ) . For every finitely axiomatizable sub-system s1 of the arithmetic of iwtegers 8 there is a finitely nxiomatixable szrb-system s of S such that s i s not interpretable i n sl. P r o o f . Define s as a system obtained fram 6, by the adjunction of the sentence N ( s l ) to the axioms of s,. According to XVIII s is a finitely axiomatizable subsystem of X. If 8 were interpretable in sl, then the implication N ( s l ) 3 N ( s )would be provable in s1 and hence the sentence N ( s ) would be provable in s . B u t this is impossible since N(a) is provable in s only if s is inconsistent (cf. lemma 8, p. 156). To complete our discussion we remark that the results established in section 7 hold not only when S is the system of the arithmetic of positive integers but more generally for all systems 8 which contain arithmetic and have the property th a t inductive definitiond of the form (7) and (8) are expressible by single matrices of the system S . Bibliography. [l] Alonzo Church, Introduction to ilfathenuiticd Logic. Part I , Annals of Mathematics Studies 13, Princeton 1944. [2] Kurt Giidcl, Die Vollstiindigkzit iler Axiome des logisel~enFunktionenZalkuls, Slonatshefte fur Mathematik unit Physik 37 (1930), pp. 349-360. [3] - ober fornnnl unentscheidhnre Satze der Principia Xathematica und wmuandter Systenie I , Ibidem 38 (1931), pp. 173-193. [4] Da& Hilbert, Grzcndlngoa der Geometrie, 7-th edition, Leipzig-Berlin 1930. [ 5 ] David Ililbert and Paul Bernays, Grundlagen der Zathenaatik, r o l . 2, Berlin 1939. [6] Jerzy Zok, 0 inntycnch logicznych, Prace Wroclawsliiego Towarzystwa Naukowego, seria B, Nr 19, 1949. [7] Andrzej Mo s t o w s ki, Logika nzaternntycxa, Monografie Matematyczne, TO^. 18, Warssawa-Wroclaw 1948. [S] Ilse L. h'orak, A Conslructmz for Jfodels of Goasistent Systems, Fundamenta Mathematicae 37 (1950), pp. 87-110. [9] J. Barkleg Rosser, Review of [15], The Journal of Symbolic Logic 16 (1951), pp. 143-144. 2*)
20)
T a r s k i - S z m i e l e w [14]. Theorem XX a i d its proof were coinmuiiicated to me by A. T a r s ki.
[32], 158
O N MODELS OF AXIOMATIC SYSTEMS
217
[lo] Ceeslaw R y I I - N a r d z e w s k i , The Bole of the Axiom of Irtductios is the EZemestcrry Arithmetic, Fundarnenta Mathematirac 39 (1953). [ l I] Alfred T a r s k i , Der Wahrheitsbegriff L I P den formalisierten Spracherc, Stutlia Philosophics 1 (1936), pp. 261-405. [12] f f b e r den Begrifj der logisehnn B'o'olgerzoq, Actes tlu C,!ongri.s International do Philosophie scientifique, VII h g i q u e . Actiinlit6fi Scientifiques e t Industrielles 384, Paris 1936, pp. 1-11. [ 131 - Einige Betrachtungen iiber die Begriffe der w-Widerspruchsfreiheit uiul der w-Vollstundigkeif, Yonatshefte fur Mathematik und Physik 40 (1933), yp. 97-112. [14] and Wanda Snmielew, Xutual Interpretability of Borne Essentially Undecidable Theories, Proceedings of the International Congress of Mathematics, vol. 1, Cambridge 1950, p. 734. [I51 Hao W a n g , The Non-finitizahility of Tmprediccllive Priiwiples, Proceedings of the National Academy of Sciences of the USA 98 (1950), pp. 479-484. [IS] - Arithmetic Transkztions of Axiom. Syetems, Transactions of the American Mathematical Society 71 (1951), pp. 283-293. [171 The Irreducibility of Impr&icative Primiplee, Yathematische Annalen 125 (1952), pp. 56-66.
-
-
-
A. it1 o s t o w s I< i (Warszawa).
This paper is devoted to n discussion of various notions of models which appear in the recent investigations of formal systems. The discussion will be applied to the study of the following problem: Given a formal system L3 based on an infinite number of axioms Al,&,A8, ... is it possible to prove in S the consistency of the system based on a finite number A1,& ...,A,, of these axioms?
1. Notations and definitions. We shall consider two systems S and s based on the functional calculus of the first orderl). We shall not describe these systems in detail but give only some definitions which will be required later. S y s t e m s. We assume that the following symbols occur among the primitive signs of 8: 1. Variables: sl,s2,.~3 ,...,z,y,z ,... 3. I n d i v i d u a l c o n s t a n t s : fi ,....fa. 3. F u n c t o r s (i. e. symbols for functions from individuals t o individuals): gl,...,gp. We denote by qj the number of arguments of 91 ( j = l , ...,/!I). 4. P r e d i c a t e s (i. e. symbols for relations): rl, ...,r y . We denote by p l the number of arguments of ri ( i = l , ...,y). 5. P r o p o s i t i o n a l c o n n e c t i v e s a n d q u a n t i f i e r s . We use the symbol I for the “stroke function” and define other connectives in terms of the stroke. Quantifiers are denoted by symbols ( 3 4.and (si). Among expressions which can be constructed from these signs we distinguish the following: 6. Terms. Variables and individual constants are terms. If I‘,,...;Iyq, are terms, then so is gI(Tl,.,.,Tq,),j=1, ...,p. Terms will be denoted by the letters F7T‘l,I‘2,... l ) For the functional calculus of the first order see e. 9. C l i u r c l i [I]. Chapter 11.
[32], 134
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7. E l e m e n t a r y
193
f o r m u l a s are expressions of the form
rt(T,,...,P,,) where I',,...,I',,are terma. Elementary formulas will be denoted by the letters E, El, E,, ...
8. P r i m e f o r m u l a s are elementary formulas in which no variables occur. Prime formulas wili be denoted by the letters P,Pl,pz,* * 9. Matrices. Elementary formulas are matrices. If 31, and i?f2 are matrices, then so are i?f,lN2and (3q)X1 for i = 1 ,2 , ... Matrices will be denoted by the letters M , iffl, H,, ... Note that matrices containing propositional connectives other than th'e stroke are easily definable by means of matrices containing only the stroke symbol. We shall occasionally use the following abbreviations : J P for ilf, H1 for -$I,
-
2~~for i=l 0
0
11 &.Ii i=l
for
JI,V...V~BI~.
x ~..A,I ~.
10. F r e e a n d b o u n d v a r i a b l e s . S u b s t i t u t i o n . The distinction between free and bound variables is assumed as known. The formula which results from a formula A by the substitution for the variables ~ , . . . , L cwill ~ be denoted by of the terms T1,...,Tn Subst A(xl/rl, ...,g&). The operation of substitution is always performable when A is a term. If A is a matrix, it is sometimes necessary to re-name the bound variables occurring in A in order to make sure that the operation S&st can be performed. We shall always assume that the necessary changes in the bound variables of A have been performed before the operation Subst has been applied,). 11. &-matrices. These are matrices in which no bound variables occur. 12. Axioms. We assume that the axioms of s are finite in number and have the form of &-matrices in which no constants, functors, or predicates occur besides those which were enumerated in 2, 3, and 4. The axioms will be denoted by ai or a l ( q y,...,z), i=l, ...,6. _____ _-
2 ) An exact definition of the operation Subst is @en in C h u r c h [l], pp. 56-58.
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FOUNDATIONAL STUDIES
[32],135
13. The r u l e s of proof admitted in 8 are the usual ones. We adjoin to them the rule of explicit definitions and the &-rule8), We shall add a few words to explain the &rule. To this end we define recursiveLy the notions of &-termsand &-matrices. The terms and matrices defined in 6 and 7 are &-terms and &-matrices.I f P,Pl,...,Pk are &-terms,then flubst m
t , / q , . .,X i k / r d
is an &-termprovid'ed that (i) zfl,...,sf&are the free variables of r; (ii) the bound variables of P are not free in rl,...,rk.If r,,...,r,, are &-terms,then r t ( r l,...,Pp,)is an &-matrix( i = l ,...,y ) . I f Ml and M8 are &-matrices,then so are HIIMa and (3x,)M1. I f M is an &-matrix, then ( m f ) Nis an &-term. The &-rulestates that for every &-matrix M the matrix
M 3 Sabst M ( x ~ / ( E x ~ ) M ) can be assumed as a theorem of s. The question arises whether the assumptions concerning the form of axioms and rules of proof (cf. 1 2 and 13) are general enough to cover the cases of standard formal systems based on a finite number of axioms. The answer is affirmative. To see this we remark that the &-ruleenables us to get rid of quantifiers in the axioms provided that we introduce a sufficient number of &-terms*). Since the explicit definitions are allowed in s, it follows that we can bring t h e axioms to the form of Q-matrices provided that we add a sufficient number of symbols to the symbols enumerated in 2 and 3. The resulting system then satisfies our assumptions and is equivalent to the given one provided that suitable definitions are introduced into the latter. E x a m p l e . Let one of the axioms have the form
0)
($1 (3Y)M@,Y ) We introduce a new functor g(s) and an axiom (ii)
M(x,g(s)). Clearly (i) is deducible from (ii). Conversely (ii) can be obtained from (i) by means of the &-ruleand the explicit definition g(s) = E y J m , Y ) . 3) The ordinary rules of proof for the functional calculus are given e. g. in Church [l], p. 40. For the &-rulesee H i l b e r t - B e r n a g s [6], qp. 9-18. 4 ) See H i l b e r t - B e r n a y e [5], pp. 16-17.
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ON MODELS OF AXIOMATIC SYSTEMS
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S y s t e m 8. The structure of 8 will be assumed similar to those of 8 . We shall, however, not assume that the number of &oms of S is finite. Furthermore we shall assume that 8 contains an arithmetic of integers to an extent which allows UR t o arithmetize in B the syntax of s and to prove in S the basic theorems of qathematical logic, e. g., the completeness theorem of CfBde16). We shall use freely in 8 the arithmetical notions such as identity, sum, product etc. 14. A r i t h m e t i c a l c o u n t e r p a r t s of s y n t a c t i c a l n o t i o n s will be denoted by words printed in spaced italics. For instance matrix is short for a matrix of S with one free variable which is satisfied exclusively by the Godel numbers of matarices of s. The actual construction of such a matrix is not needed; it suffices to h o w that it can be constructed. To simplify the notation we shall often use in 8 the non-formal language and write, for exainple, 3
is a m a t r i x of s
instead of matria, (2). Translations of such non-formal formulas to the official language of 8 will always be possible6). The following constants will be used in 8: 15. 3 for the functor of fl such that 3t,3s,3n... are the variables of s. I n other words 3f is the Godel number of the variable xr ( i s l ,2,. ..). 16. f, ,...,fl for the in d ivid u a l co n s t a a t s , gAl...,ga for the
fzclzctors, and r,, ...,r,, for the predicates of 8 . I n other words fi is the Godel number Of fi, g1 that of $1, and rk that of rk, i=l, ...,a,
j = l , ...,/?,k = l , ...,y. 17. Arbitrary terms will be denoted by letters tltl,tz, ..., arbitrary elementary formulas by letters e,e,,e2, ..., arbitmry prime formulas by letters ,p7p1,p2,...,and arbitrary matrices by letters m, m,,m2, ... 18. Let TI, ...,rq,be terms and 1, ,...,f9, their Godel numbers. ...,Pq,) will be denoted by The Godel number of the term gj(F1, [g,(tl, ...,f,)]. Thus the square brackets symbolize here a functor of 8. G o d e l [2]. *) More details concerning aritlimetization are given in U b d e l 131 a n b f i i l b e r t - B e r n a y s [5], $ 4 . 6)
196
[32], 137
FOUNDATIONAL STUDIES
The square-brackets notation explained in the foregoing paragraph will be used consistently in many other similar situations. So c. 9. if miJ is the Godel number of the matrix MiJ, then
[fi %mfJ] is the f = l *l
Godel number of the matrix
y
fl 2 0
1=1 j=1
MI,,. If m
is the Godel number of B,then [ml] is the Godel number of MI (i. e. of -dl) and so on. 19. The following lemma is provable in 8: I n order that e be an elementary f o r m u l a it is necesamy and sufficient that e have the form [rf(t17...,fp,)] where i < y and t,,..., tpfare terms. The integer i and terms tl,...7tptare deterniined by e. We put i=Ind(e),
ti= Compj(e),
j=1,
...,p i .
Functors Ind and Comp are definable in 8. 20. The a r i t h m e t i c a l c o u n t e r p a r t of t h e f u n c t i o n &b8t will be denoted by the symbol Bb. Thus if a is the Godel number of an expression A, and tl, ...,t,, are the Godel numbers of terms r,,...,I‘,,then Bb a( 31/tl,..., 3,,/t,,) is the Godel number of the expression dubst A(xl/Tl, ...,xn/Tn). 21. If r,r1 ,...,I’,, are terms of s, then the expressions
. ;,I(sk/r)
,stcb.yt g, (TI,. ., I
and
..., #%bat Tq,(~h/T))
gj(Subst TI(Xk/T),
are identical. The equation
.,
f i b [ 9/(ti, *. b j ) ] ( 3k i t ) = [?ji(Sb tl(3k it),
-*-, Sb f 9 j ( 3k It ))I
is provable in S . 22. The Godel n u m b e r of t h e i - t h axiom of s (see12) will be denoted by [at] or by [ai(m,y,...,z)]. 23. A r i t h m e t i c a l sentences expressing t h e oonsistency of 8 a n d of s will be abbreviated as N ( 8 ) and N(8). 2. Models of the first kind. Let Bo(z),$?(XI,...,spl)be y + l matrices of B with the indicated number of free variables and lot !Pi($,,...,s9,,y) be p matrices of B such that matrices I j ( ~ l , . . . , ~ 9 , , y ‘ ) ’ T j (, ‘. c. ~ 17 s 9 , , y ” ) ~ y ’ = ~ ’ ’ ,
( 3 Y ) W~,,...,%,,Y)
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are provable in S. We define in S functors G (where j = l 1 2 , . . . , j 3 ) in the following way ') G,(:/C.l,..., sqt = (CY)q a , ,...,zqj
1
!/I.
Finally let PI, ...,Fa be 01 constants definable in s'. The a + p + y + l tuple eonsisting of a constants P I , of B functors Gi,and of y + l matrices RL will be called a pseudo-model of the first kind of s in 8. I n order to define when a pseudo-model is a real model we shall introduce some auxiliary definitions. To every term T of s we let correspond a term !Z'r of AS' in the following way. If F is a variable, then T p = T . If r=ff,then T r = F f . Finally, if P has the form gj(F1, ...,Tqj), then we put T r =Gj(Tr*,...,Trq). To every elementary formiila E=ri(I'll...,Tp,) of s we let Trl,...,Trp,) of S. We extend this correspond the matrix E' =Ri( definition to all &-matrices of s by putting (Ml\ J f 2 ) ' = J f ~ l J f ~I n. particular we put A t = a ; ( i = l 1 . . . , 6 ) . D e f i n i t i o n . A pseudo-model (1)
PI,...,F a , GI,. ..,Gp, RO,121,...?R,
is a real rnodd of the first k i n d of s i n S if the formulas
no(%) .R&y) .... .K&)
3 A t ( $ , y, ...,z), i = I, 2 , ...,6 ,
are provable in S. Models of the first kind are the ones with which one has to do in the usual proofs of consistency and of independence of axiomatic systems 8 ) . For comparison with other notions of models to be defined later, we shall note the following general facts concerning models of the first kind: 1. The general notion of models of the first Bind is defined not in X but in the syntax of S. 2. Every particular pseudo-model is a finite set of matrices of S and can therefore be defined within S. The problem whether it. is or is not a real model can be formulated and in particular cases also solved in 8. 7 ) ( 1 % ) [ . . . x . . . ]denotes the zwhicllsatisfies the condition ...s...Cf. H i l b e r t B e r n a y s [5], p. 381. 8 ) C'f. the independence and consibtency proofs in [ 4 ! ('hapter 11.
198
[32], 139
FOUNDATIONAL STUDIES
number of axioms (.independent 3. If 8 contained an of whether their set is or is not definable in f l ) , thsa the problem whether an explicitly given pseudo-model is or is not a real model of s in S would be expressible in the syntax of 8 but not in 8 itself. The following theorems concerning model% of the first kind are well known but are given here for the sake of comparison with other notions of models 0 ) : I. If (1) i s a real model of the firat kind of 8 ilz S and if a Q-matrix A ( s ,y , ...,z ) is provabte i m s, then the matrig
.
Ro(LE).R,(y ) .... * R,(x)3 4 ' ( x y, , . .,2 )
is provable in R. 11. If (1)is a real model of the f i r s t kind of s in 8 and S is eonsistent, then so i s s. Let us now assume that a real model of the first kind of 8 has been explicitly defined in 8. Theorems I and I1 are provable in the syntax of S, hence they are translatable into arithmetic and therefore into S. Denoting by N ( S ) and N ( s ) formulas of 8 corresponding (via arithmetization) t o the syntactic statement: S (or s) is self-consistent (cf. section 1, definition 23), we obtain from 11: 111. The formula N ( S ) 3 N ( s )is provable in 8. In spite of this result models of the firat kind are of no use when one is examining the problem whether the formula N ( s ) itself is or is not provable in S. Models of the second and third kinds which we shall discuss in the next sections will allow us to answer this question in many particular cases. We note still the following theorem due t o W a n g [16]: IV. If the formula AT($)i s provable i n 8,then a model of the first kind can be defined explicitly in S . Indeed, the usual proofs of the completeness theorem of Gadel consist in exhibiting a model of the first kind of a (non-contradictory) first order system in the arithmetic of integers 10). Taking s as this system and repeating the argument of Godel in S (which is possible by our assumptions concerning f l , cf. p. 136) we obtain the proof, of theorem IV 11). O)
lo) 11)
Proof8 of these theorems may be found e.g. in my book [7J,Chapter XI. G o d e l [Z] or H i I b e r t - B e r n a y s [5], p. 185. W a n g [lG], p. 287. gives a more detailed proof of this theorem.
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ON MODELS OF AXIOMATIC SYSTEMS
199
3. Models of the second kind in the axiomatic theory of sets la), We assume in this section that 8 is an axiomatic system of set theory based e. g. on Zermelo's axioms. The following definitions are to be thought of as belonging to 8. Let be an arbitrary set aad Z a finite set of positive integers. An %-function with the set of arguments 2 is defined as a set F of ordered pairs
a,
D(u)=Z, D*(V)C%, and the following condition of single-valuedness holds:
F and
E
~E.(U)==(3IY)[<~,Y>E
ul,
~ € D * ( M ) = ( 3 y ) [ < y l l r> FU].
If u is a sequence satisfying the condition D(u)=Z and P is an arbitrary set of positive integers, then we denote by w]Y the sequence u restricted to P, i. e.
I
< i , a> € u Y = ( < i , y> E u ) .( i E Y ) .
An %-relation with the set of arguments 2 is defined as a set R of sequences u such that D(u)=Z and D*(u)C%. If Z consists of integers i , j , k , ... and u is a sequence with the domain 2 such that
%, Pi,...,Fa, GI,..- 7 G ,
. .., Ry
El,
will be called a pseudo-model of the secorcd kind of s in S. 18) Results of this and the next section are due t o T a r s k i [111 and 1121. and t o W a n g [la] and [17]. 18) Sequences are defined as functions (many-one relations) with domains contained i n the set of positive integers. Cf. T a r s k i [ll], p. 287.
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FOUNDATIONAL STUDIES
[32], 141
We shall now explain when a pseudo-model is a real model. As in section 2 we need some auxiliary definitions. We shall denote by B(t) the set of f r e e v a r i a b l e s which o c c u r in a t e r m t and by B(m) the set of f r e e v a r i a b l e s which o c c u r in a m a t r i x m of s. With these definitions it, is not difficult to prove the existence and uniqneness of a function H f ( u ) and a relation Stsf which are of fundamental importance in the investigations of the semantic of 8. The exact definitions of the function H and the relation Xtsf are given below in lemmas 1 and 2 together with the proofs of their existence and uniqueness. To facilitate our exposition we explain informally the intuitive meaning of these concepts. Let t be a t e r m , m 5 m a t r i x of 8, and let 1' 1 be a sequence {
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
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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
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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)
<{7 - - 9
where ml,...,mq, vary independently in 3f. Further let be X-relations defined by the equivalence {
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 {<31,u1>,..., < 3 k , u h > ) with u,, ..., u h . 3 i , then (4)
u 8tsf [a,] = a,(ul,
...,Wk)
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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
FOUNDATIONAL STUDIES
[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
P ( h , w Tmp 6, A a ) = 0,
f where h is the Godel number of the proof of (17) in s. It follows from (18) that if N(6) were provable in 8 (i. e. if 20 were provable in s), then b would be provable in s and hence by (17) 8 would be inconsistent. Lemma 8 is thus proved. Theorem XIX results from lemma 8 by the same argument which was used in the proof of theorem X.
216
FOUNDATIONAL STUDIES
[32], 157
The following result can also he obtained from XVIII. Following T a r s k i and Szmielew*8) we shall call a system s interpretable in S if there exists rz model of the first kind of s in 8. We have then XX n o ) . For every finitely axiomatizable sub-system s1 of the arithmetic of iwtegers 8 there is a finitely nxiomatixable szrb-system s of S such that s i s not interpretable i n sl. P r o o f . Define s as a system obtained fram 6, by the adjunction of the sentence N ( s l ) to the axioms of s,. According to XVIII s is a finitely axiomatizable subsystem of X. If 8 were interpretable in sl, then the implication N ( s l ) 3 N ( s )would be provable in s1 and hence the sentence N ( s ) would be provable in s . B u t this is impossible since N(a) is provable in s only if s is inconsistent (cf. lemma 8, p. 156). To complete our discussion we remark that the results established in section 7 hold not only when S is the system of the arithmetic of positive integers but more generally for all systems 8 which contain arithmetic and have the property th a t inductive definitiond of the form (7) and (8) are expressible by single matrices of the system S . Bibliography. [l] Alonzo Church, Introduction to ilfathenuiticd Logic. Part I , Annals of Mathematics Studies 13, Princeton 1944. [2] Kurt Giidcl, Die Vollstiindigkzit iler Axiome des logisel~enFunktionenZalkuls, Slonatshefte fur Mathematik unit Physik 37 (1930), pp. 349-360. [3] - ober fornnnl unentscheidhnre Satze der Principia Xathematica und wmuandter Systenie I , Ibidem 38 (1931), pp. 173-193. [4] Da& Hilbert, Grzcndlngoa der Geometrie, 7-th edition, Leipzig-Berlin 1930. [ 5 ] David Ililbert and Paul Bernays, Grundlagen der Zathenaatik, r o l . 2, Berlin 1939. [6] Jerzy Zok, 0 inntycnch logicznych, Prace Wroclawsliiego Towarzystwa Naukowego, seria B, Nr 19, 1949. [7] Andrzej Mo s t o w s ki, Logika nzaternntycxa, Monografie Matematyczne, TO^. 18, Warssawa-Wroclaw 1948. [S] Ilse L. h'orak, A Conslructmz for Jfodels of Goasistent Systems, Fundamenta Mathematicae 37 (1950), pp. 87-110. [9] J. Barkleg Rosser, Review of [15], The Journal of Symbolic Logic 16 (1951), pp. 143-144. 2*)
20)
T a r s k i - S z m i e l e w [14]. Theorem XX a i d its proof were coinmuiiicated to me by A. T a r s ki.
[32], 158
O N MODELS OF AXIOMATIC SYSTEMS
217
[lo] Ceeslaw R y I I - N a r d z e w s k i , The Bole of the Axiom of Irtductios is the EZemestcrry Arithmetic, Fundarnenta Mathematirac 39 (1953). [ l I] Alfred T a r s k i , Der Wahrheitsbegriff L I P den formalisierten Spracherc, Stutlia Philosophics 1 (1936), pp. 261-405. [12] f f b e r den Begrifj der logisehnn B'o'olgerzoq, Actes tlu C,!ongri.s International do Philosophie scientifique, VII h g i q u e . Actiinlit6fi Scientifiques e t Industrielles 384, Paris 1936, pp. 1-11. [ 131 - Einige Betrachtungen iiber die Begriffe der w-Widerspruchsfreiheit uiul der w-Vollstundigkeif, Yonatshefte fur Mathematik und Physik 40 (1933), yp. 97-112. [14] and Wanda Snmielew, Xutual Interpretability of Borne Essentially Undecidable Theories, Proceedings of the International Congress of Mathematics, vol. 1, Cambridge 1950, p. 734. [I51 Hao W a n g , The Non-finitizahility of Tmprediccllive Priiwiples, Proceedings of the National Academy of Sciences of the USA 98 (1950), pp. 479-484. [IS] - Arithmetic Transkztions of Axiom. Syetems, Transactions of the American Mathematical Society 71 (1951), pp. 283-293. [171 The Irreducibility of Impr&icative Primiplee, Yathematische Annalen 125 (1952), pp. 56-66.
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