A Remark on Models of the Gödel-Bernays Axioms for Set Theory

SETS AND CLASSES; On the work by Paul Bernays

@ North-Holland Publishing Company (1976) pp. 325-340.

A REMARK ON MODELS OF THE GODEL-BERNAYS AXIOMS FOR SET THEORY Andrzej MOSTOWSKIT Warzaw ,Poland

1. Notation

We denote the sets of axioms of Zermelo-Fraenkel and Godel-Bernays by ZF and GB respectively. Both these systems are formulated in a first-order language L with identity (denoted by =) and with one binary predicate E. Unlike GODEL[1940], we admit in GB only one primitive notion, viz. the binary relation E and define the predicates Cls(x) (x is a class), M(x) (x is a set) by formulas x = x, ( E y ) ( x E y ) respectively. Axioms A1 and A2 of Godel can then be omitted. A formula cp is called predicative if all its quantifiers are limited to M. A system obtained from ZF by addition of the axiom of choice is called ZFC; similarly GBC denotes a system obtained from GB by adding to it the (set-form of the) axiom of choice. All models for ZF or GB which we consider below have the form (&I,€ where M i s a transitive set of sets. We write simply M instead of ( M , € ) . The extension of the predicate M in a model M is denoted by VM.Thus, V, = M if M ZF but VMC M if M GB. If MI,M 2are two families of sets such that MIC M , and V M= , V,, then we call M , a C-extension of M I and write M , & M , . A language obtained from L by adding to it constants c, for each element m of a set M is denoted by L, We identify cm with a suitable element of M which allows us to consider L, as a subset of M. All models for axioms formulated in LMalways contain M, and c, is always interpreted as m. L(A) or L,(A) denotes a language obtained from L or L, by adding to it a new one-place predicate A. Models for axioms formulated in L(A) or in L,(A) have the form ( N , X ) ,where N is a transitive family of sets and X C N ; the interpretation of the new predicate A is X and the interpretation of E is the relation € of “being an element of”.

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For each family M of sets we denote by Def, the family of sets of the form {x E V,: M cp(c,)} where cp is a predicative formula of L, with exactly one free variable and cp(cx)arises from q by replacing the free variable by c, wherever it occurs. Sets which belong to Def, are said to be definable in M. A slightly more general notion of definability is as follows: let M be a transitive family of sets, X C V., A set S is definable in M with respect to X if there is a predicative formula cp of L,(A) with exactly one free variable such that S = { x E V,: ( M , X ) cp(c,)}. We denote the family of all such sets by Def,(X). It is easy to prove that if M is transitive, then so is Def,(X) for any X C V,. We intend to prove below the following theorem:

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For a given countable transitive model M of ZFC there are 2-1 models L of GB such that VL = M. Moreover these models form a full binary tree of height w 1 when ordered by inclusion ; if two models L,, L1do not lie on the same branch of the tree, then there is no transitive model L of GB such that VL = M and L > L, u LZ. A similar theorem is also valid for the “predicative extension” of Peano’s arithmetic; an exact formulation of this theorem is given in §5. 2. Auxiliary theorems LEMMA2.1 (Marek). If M is a transitive model of GB and C is the family of all transitive sets K which satisfy the conditions

MCKCP(V,),

VK = V,,

KFGB,

then the union o f any chain L C C is an element of C. The proof of this lemma is very easy and we omit it. Marek deduced from this lemma the existence of maximal elements of C, and asked about their number. A partial answer to this question is given in Corollary 4.1 below. LEMMA2.2. If M is a transitive model of G B and X C V,, then all the axioms of groups A and B of GB as well as the axioms Cl-C3 and D are valid in the model Def, ( X ) .Moreover Def, ( X )is a C-extension of M.

PROOF. The verification of Axioms A and B are left to the reader. In order to prove that Def, ( X ) is a C-extension of M , let us assume that x C VD,, (x),i.e. that there is a set S such that x E S CDef, ( X ) .Hence, S is a set definable in M with respect to X and therefore S consists of

MODELS OF THE GODEL-BERNAYS

327

AXIOMS

elements of V, which proves that x C V., Conversely, if x f V, then the unit set (x) is definable in M with respect to X whence x f(x)fDef, (X) which proves that x C VDef,(x). Since Axioms C1-C3 deal exclusively with sets, it follows from their validity in M that they are also valid in Def, (X) since in both these models the interpretations of sets are identical. Lemma 2.2 is thus proved. In connection with Lemma 2.2 it is necessary to point out that the statement “a sub-class of a set is a set” may (and generally does) fail in the family Def, (X). Thus, if we agree to call elements of V, “sets of M”, then Def, (X) may, and generally does, contain “semi-sets”, i.e. classes (elements of Def, (X)) which are contained in sets of M (see V O P ~ N Kand A HAJEK(1972)).The existence of semi-sets is due to the fact that the axiom of comprehension is, in general, false in Def, (X). In the case when X is definable in M, the axiom of replacement and the axiom of comprehension are valid in Def, (X) and thus, no semi-sets exist. We shall see below an example of X when not definable in M and yet no semi-sets exist because the axiom of comprehension is valid in Def, (X). The crux of the whole construction is a determination of a set X M such that Def, (X) is a model for the axiom C4 of replacement. Unfortunately I do not know of any workable, necessary, and sufficient conditions for this to be the case and have to rely on results established by FELGNER [1971] in a special case of a denumerable model. In what follows, M is a transitive denumerable model of GB. Let P E Def, and R € Def, be a binary relation which partially orders P. Elements of P are called conditions. We write p s q instead of pRq and read this formula: p is an extension of q. We define by induction a binary relation (forcing), whose left domain is P and right domain consists of all predicative sentences of L,(A):

It

Whenever cp is a formula, we denote by Fr(cp) the set of its free variables. For y € MFr(C), we denote by cp ( y ) the sentence obtained from cp by substituting cY(”)for 0 throughout cp for each vEFr(cp). With this , notation we have :9

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LEMMA2.3. For each predicative formula cp of LM(A) the set { ( P , Y ) C P x MFr(C): P I/- cp(Y)l

belongs to Def,. A set G C P is generic (more exactly: generic in P over M ) if it has the properties: (i) If pCG and p s q , then q f G ; (ii) If p , q C G, then ( E r ) [ ( rC G ) & ( r S p ) & ( r 6 q ) ] ; (iii) If S CDef, and S is dense in P, then G n S # 0. LEMMA2.4. Each condition belongs to at least one generic set. LEMMA2.5. If G is generic then ( M , G ) cp = ( E P ) ~ (1 1 P cp) for each predicative sentence cp of LM(A). In order to prove the next two lemmas we make the following assumption about P : ASSUMPTION (A) Whenever x C VM,O# x C P and x is a chain with respect to S , then there is a condition p such that p S q for each 4 in x. LEMMA2.6. If M GBC, G is generic, rn C V, and cp is a predicative formula of L,(A) with exactly one free variable v, then there is a set n C VM such that

( M , G )I= {8~cpl).

PROOF. It is sufficient to show that the set

Q = (P CP: ( X ) r n [ P It- c p ( c x N 7 cp(c*)l) is dense in P. Once this is shown the rest is easy: by Lemma 2.3 the set Q belongs to Def,, hence there is a p in Q which belongs to G and it is sufficient to take n = { x C m : p cp(c,)}. The proof that Q is dense closely follows the construction used by FELGNER [1971]; it is therefore sufficient to indicate only the main steps of this proof. We can assume that rn is infinite. Let qoCP. For arbitrary ( x , p ) in V, x P we denote by Z ( x , p ) the set of elements q in P which have the following properties: (9 4 S P and 4 It- cp(cx)v-7 cp(cx); (ii) whenever r has the property (i), then rk (4) s rk ( r ) . We easily see that Z(x,p)E V., If 0 # a Con n V, and g C V, n P" then we denote by W ( g )the set of conditions q which have the properties: (iii) q s qo and q s r for each r CRg ( 8 ) ; (iv) whenever s has the property (iii), then rk (4) S rk (s).

It

MODELS OF THE GODEL-BERNAYS AXIOMS

329

For LY = 0 we define additionally W ( g ) = 0 . We can show easily that W(g)C VMfor each g C V, n P a and each (Y COn n VM. Let p be a cardinal of VMand let f be a function such that f E VMand f is an injection of p onto m. The existence of p and f follows from the assumption that M GBC and m C VM.Using transfinite induction we define two sequences {Pz},{Qz} of type p which are elements of VMand which satisfy the following equations for each 6 < p :

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Qz= { g c( U pa u {so})"' : (g(0) = 40)tk ( a ) t ( g ( l + a)Epa 1 tk l+*(6>l+,[y < 6 + g ( y ) g(6)11, f't= U { Z ( f ( S ) , y ) Y: CU{W(g): gCQt1)-

Using the axiom of choice in M we can select functions g, h in VMof type P such that g ~ ( 1 + 0 ~ C &h (, 5 ) c W ( g f . ( 1 + 5 ) ) and g ( 1 + 5 ) E Z ( f ( t ) ,h ( 6 ) )for w c h 6 < p. Thus g is a decreasing function and by (A) the set W(g) is non-void. For an arbitrary p in W(g) we have then p C Q and p s q,, which proves the density of Q.

LEMMA2.7. If M is a transitive model of GBC, G is generic and y is a predicative formula of L,(A) with exactly two free variables u, w, then f o r each m in VMthere is a set n in VMsuch that (*1

( M , G ) + ( v ) ( w ) { [ ( vEc,)&cpI+

( E w ) [ ( wEcn)&cpl).

PROOF. This is again a repetition of the construction of Felgner and it is sufficient to indicate only its main steps. We put R =(PEP: (En),(X),(Y),(Ez)"P

It

[ 7 cP

(~x,~,)VcPO~x,~*)I~

and claim that R is dense in P. To show this we choose qoin P and denote by p and f a cardinal and a bijection as in the previous proof. We can assume that p is infinite. Let Z ( x , p ) be a set defined as follows: if there are q s p such that (i) q It- (EW)MP(C,,W)then Z ( x , p ) consists of all the conditions 4 s p which have the property (i) and the following property: (ii) whenever r s p and r has the property (i), then rk (4) s rk ( r ) . If there are no q with the property (i) then ( q ) s p(Er)sqrIt ( E w ) ~ ~ ( c ~ , w ) and hence there are q s p such that (iii) q I / - ( E ~ ) ~ c p ( c ~ ,In w ) this . case we let Z ( x , p ) be the set of all conditions q s p which have the property (iii) and are such that whenever r s p has the property (iii), then r k ( q ) s r k ( r ) . It is easily seen that Z k p 1E V M . We define S ( x , q )as follows: if q non Ik ( E w ) ~ ~ ( c ~then , w )S, ( x , q ) = 0. Otherwise, let S ( x , q ) be the set consisting of all the elements y E VMwith

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ANDRZEJ MOSTOWSKI

the following properties: (v) 4 cp(cx,c,); (vi) whenever q It cp(c,,c,) then rk ( y ) G rk (2). It is again easy to show that S(x,q)C V,. Finally, let W(g), P6, QE,g, h be defined as in the proof of Lemma 2.6. Using once more the axiom of choice, we obtain a function k € VMwith domain V, such that k ( c ) C S ( f ( ( ) ,g(1+ 6 ) )for each 6 < p. Then rg(k) is a set n C VMsuch that for any p in W(g) and arbitrary x in m, y in VM there is a z in n satisfying p -,cp(c,,c,)v cp(c,,c,);moreover p s qo.The density of R is thus proved. If now, p CR n G and n is a set whose existence is secured by the fact that p C R, then formula (*) holds for this set n. This proves Lemma 2.7.

It

It

For the benefit of readers not acquainted with Felgner’s paper [1971] we add below some comments about the intuitive meaning of constructions carried out in two proofs sketched above. In order to prove the density of sets Q and R we have to show that each q o € P has an extension which belongs to both Q and R. These extensions are obtained by constructing successive extensions of qo and repeating this process transfinitely many times. In both proofs we represent the given set m as the range of a one-to-one function f whose domain is a cardinal p and assume that p is infinite. The sequence of successive extensions of qo is denoted in both proofs by g. Thus g is a decreasing function with domain p whose values are conditions. The initial term of g is g(0) = qo. If 6 < p and g ( a ) is already defined for (Y < 1 + 6, then g(1+ 6 ) is an extension of all the g(a)’s which satisfies an additional requirement. In Lemma 2.6 this additional requirement is: g ( l + 6) decides ( P ( C ~ ( ~ , ) ,i.e. g(1+ 6 ) either forces this formula or its negation. In Lemma 2.7 the additional requirement is: g(1+ 6 ) forces ( E W ) , ~ ~ ( C ~if( there ~ ) , Ware ) conditions which have this property and extend all the g(a)’s; otherwise g(l + 6) should force (Ew)Mcp(cf(E),w)To explain the notation previously used, we note that g(l+[) is constructed in two stages: first, we construct h ( 6 ) which is an extension of all the g(a)’s, a < 1 + 6 and then g ( l + 6 ) is selected from among such extensions of h ( 5 ) as satisfy the additional requirements. The first fact is expressed by the formula h(6)C W(g 1 (1 + 6 ) ) and the second by the formula g(1+ [ ) E Z ( f ( ( ) , h ( ( ) ) In . case of Lemma 2.7 we construct still one additional function k such that if g(1+ 6 ) ( E W ) M ~ ( C ~ ( Cthen ),W) k ( 6 ) is an element of VMsatisfying g(1+ 6 ) I t ( P ( C ~ ~ ~ , , C ~ ( ~ , ) . Once we have the functions g and k we can take as p any condition which extends all the g ( l ) ’ s , 6 < p , and as n the set rg ( k ) . The existence

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MODELS OF THE GODEL-BERNAYS

AXIOMS

331

of p is secured by assumption (A). In case of Lemma 2.6 we see immediately that p decides cp(c,) for all x in m because each x can be represented as f(5) and p is an extension of g(1+[) which decides c p ( ~ ~ ( ~ )Hence ). p C Q. In case of Lemma 2.7 we see from the definitions of g ( 1 + 5 ) and k ( 5 ) that if p IF ( E w ) ~ ~ ( c , ,then w ) ,p 1 1p(c,,c,) for a z in n. Hence p ER. The essential point is that functions g and k are elements of V,; otherwise we can neither claim that nCV, nor that assumption (A) is applicable for obtaining p . In order to obtain g and k in V,, we cannot proceed in a simple-minded way and for instance define g as any extension of qo which decides cp(cfo,).The reason why this procedure is faulty, follows: the extensions of the g ( a ) ’ s , a < 1 + 6, do not form, in general, a set which belongs to V,; rather they form a “class”, i.e. a definable subset of V,. Thus we cannot use the set-form of the axiom of choice to select a particular extension. Yet we have only this form of the axiom at our disposal if we want to obtain in the end a function which belongs to V,. To overcome this difficulty we consider not all the extensions of the g ( a ) ’ s ,but only those which possibly have a smail rank. These extensions form a set W(g r ( 1 + 5)) which is an element of V,. Also we select g(1+ 5) not from among all the conditions which satisfy the additional requirements because this would involve a choice from a “class”, but from among those conditions which have possibly small ranks and which therefore form a set which belongs to V,. Also k ( 6 ) and h ( 5 ) are selected from sets S(f((),g(l+ 6)) or W(g (1 + 6)) which both belong to V,. In this way we can obtain the required functions by applying the set-form of the axiom of choice which, according to our assumption, is valid in M and so yields functions which belong to V,. Let us note that it is an open question whether Lemmas 2.6 and 2.7 remain valid if we replace the assumption M GBC by the weaker assumption M GB. From Lemmas 2.2 and 2.7 we obtain the following theorem which allows us to construct models of GB:

r

t=

+

THEOREM2.1. If M is a denumerable transitive model of GBC, P, R EDef, and R is a partial ordering of P such that P satisfies (A), then Def,(G) /=GBC for every set G which is generic in P over M. Moreover, V0,, ( G , = V, i.e. Def, (G) is a C-extension of M. PROOF. If x C y E Def, (GI, then y C VMand hence x c VM.Conversely, if x € VM, then there are y €Def, ( G ) such that x c y . Hence, the last

equation stated in the theorem is proved. It follows in particular that the set-form of the axiom of choice is valid in Def, (G). In view of Lemma 2.2 it remains for us to verify the validity of the axiom of substitution.

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ANDRZEJ MOSTOWSKI

Thus, let m C VMand XEDef, (G) and assume that X is a function. Let cp be a predicative formula with two free variables such that for all u, v in VM ( u , v > € X= (M,G) k 40(cu,c,). In view of Lemma 2.7 there is n , C V, such that whenever u C m and there is y in VMsatisfying (M,G) q(cu,cr),then there is a y in n , satisfying the same formula. In view of our assumption y is determined uniquely by u. Hence, all the values which the function X takes for arguments in m belong to n,. Applying Lemma 2.6 we obtain a set n t V, whose elements are exactly these values which proves the theorem.

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3. A finite tree of C-extensions

Let M GBC be a transitive denumerable model and N an integer. We consider subsets S o , Si,.. .,S k - i , To, TI,. . ., TI-i

of (0, 1,. , . , N - 1) satisfying the conditions:

l T h 1 3 2 , T h - S , # O for j < k ,

h
LEMMA3.1. There are N transitive denumerable C-extensions M, of M such that M, GBC for i < N with the following properties :

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(* 1

for each j < k there is a transitive denumerable model + M i k GBC which is a C-extension of all the models M, with iCS,;

(**)

for no h < 1 and no transitive model Mf'kGBC, M " is a C-extension of all the models M, with iCTh.

PROOF. Let On, = On n M, P = { 2 * x Nt M : [Con,),

where N

,..., N

= (0, 1

- l}.

We call elements of P conditions and order them by inverse inclusion: p G q = p > q . F o r p i n P weputdom(p)=Dom(Dom(p));thusdomp is the ordinal [ such that p E25xN.For each non-void set X N and each p in P we put p [XI = p (dom ( p ) X X ) and denote by P [ X ]the set of all p [ X ]where p C P . For each O f X r N and p C P [ X ] we put

r

6 = {(cu,i>Cdom(p)x X : p ( ( a , i ) )= 1); and G C P [ X I ,then & denotes the set U{6 : p C G}. Let 0, be a

if X N sequence of all dense subsets of P which are definable in M. Similarly

MODELS OF THE GODEL-BERNAYS

AXIOMS

333

D, [XI is a sequence of all dense subsets of P [XI which are definable in M , the ordering of P [ X ] being also the relation of inverse inclusion. If p S q ED. then we say that D,, covers p . To simplify formulas we assume once and for all that the letter i with or without subscripts denotes an integer s N , and the letters j and h denote integers less than k and I respectively. Let {a,} be a sequence (without repetitions) consisting of all the elements of On,. We construct a sequence p , of elements of P such that p n for each n and a function p : w --z w such that the following conditions are satisfied: (1)

(2) (3) (4)

D n [ { i } ]covers pn[{i}3for each i; 0, [Sj]covers p n[ S , ] for each j ; am < p(rn)< p(m + 1) for each r n ; r l Dom ( f i n [{ill) = {ap(o), .. . , a,oz-l)l. i C Th

Before proving the existence of the sequences p , and p ( n ) we show that lemma 3.1 can be derived from (1)-(4). Let Gi = {x E P [{i}l: (En)(x 2 p n [{i}l)}, H , ~ = { EP[Sj]: x ( E ~ ) (>xp n [ S , ] ) } . In view of (1) and (2) these sets are generic respectively in P [ { i } ]or P [ S j ] over M. By Theorem 2.1, the families Def, (Gi) and Def, (Hi) are transitive denumerable models of GBC and are C-extensions of M. Let Mi = Def, (G,), M i = Def, ( H j ) . We show that M i is a C-extension of Mi whenever i E Si. To prove this, it is sufficient to show that Gi EDef, (Hi) whenever i C Sj and this follows from the equivalence x E Gi = ( E y ){(YE H,1 &L [x = Y 1 (dom (y x { i } ) l } .

To prove it we note that each x E Gi is obtained from a function pn [ { i } ] by restricting its arguments to a x {i}, where a is an ordinal s dom (p"). If we restrict pn to a x Si, we obtain a function y EHi such that x = y (dom ( y ) x {i}). Similarly restricting the arguments of a function y EHj to dom ( y ) X {i}, we obtain a function x in Gi. Now we prove (**). Let us assume that M','kGBC and M" is a transitive C-extension of M ifor each i E T,. It follows that Gi E M " ; hence, if we abbreviate nicTh Dom (Gi) by X we obtain XCMr'.Each Gi consists of pairs ( a $ )such that x ( ( a , i ) )= 1 for some x EGi. Hence

r

(4)EGi

(En)bn[{i}l((a,i))

=

(En)((ai)Efin[{i}l).

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ANDRZEJ MOSTOWSKI

It follows that a CDom (Gi) = ( E n ) a CDom (Ij,,[{i}]), a CX= (i)Th(En)(a CDom I j n [{i}]).

If a satisfies this condition then for each i C T h there is an n = n, such that a CDom (fin [{i}]). Selecting the largest of the n, and denoting it with n we obtain (in view of the inclusions Ijn [{i}] 2 f i n , [{i}]) the result a C n,cTh Dom (9"[{i}]). Conversely, if a belongs to this intersection, then for each i C Th.Thus it obviously belongs to Dom (Gi) X

=U n

n Dom ( f i n [{i)l>= U

t€T,

..

ayp(l), . ,a q ( n - l , }

c

= {a,(,):i w } .

Since X C M" and the theorem on inductive definitions is valid in M", we infer that there is a function f in M" with domain w such that f(0) = a,(o), f ( k + 1) = min {tC X : 5 > f(k)}. Hence f(k) = a,(!+ Since a,(,,)> an,we obtain the result ( E k ) , ( f ( k )> 5) which clearly contradicts the assumption MkGGBC. Thus, (**) is proved. We now indicate the construction of sequences {p,,}, {p(n)}. We start with the void function pa and assume that p,,, satisfying (1)-(4), has already been constructed. We first extend pn[{O}]to a condition ijoCDn+J{O}]and add to pn all the pairs ((S,O),E) which belong to ijo- pn as well as pairs (([,i),O) for i # 0, 5 Edom (Go) - dom ( p " ) . The resulting condition pan has the properties:

(c)on,

PO"

ss pn,

pan

[{O}I = qoEDn+l[{O}I

n Dom (fion[{i)l) = {a,(O),. ..,a,
lCTh

The first two properties are obvious. To prove the third we note that if a CDom (Ijon[{i}])then pon((a,i))= 1 and this is possible only if either a CDom (fin[{i}]) or i = 0. Since I T h I a 2, the desired equation follows. In the next step we extend pan. We start by selecting a ijl S ponisuch that ijlCDn+l[{l}]and then extend pOnto pln by adding to panu Ljl all the pairs ((t,i),O) where i # 1 and dom (po,) G 5 < dom (ijl). Again, we have

The condition p l n is extended next to pZnby adding to pIna condition

q26 pln[{2}1 which belongs to 0,,+,[{2}]as well as pairs ((&i),O)where i f 2 and dom (pin) S 5 < dom (q2).Continuing in this way we finally obtain a condition p N - l , n= qn satisfying the formulas:

MODELS OF T H E GODEL-BERNAYS AXIOMS

qn

pn,

135

qn[{ill is covered by Dn+l[{i)l,

n Dom (6,[{ill) = {aq(o), aq(l), . . . ,a q ( n - l ) l .

iCTh

In the next k-steps we extend qn so as to obtain conditions qJnsuch that Dn+,[SJ] covers qJn[S,]for each j < k. We start by extending qn[So]to Fa in Dn+,[S,],then add Fn to qn and also add to qn all the pairs ((t7i),0)where i f s o and dom(q,)
r,[SJ1is covered by Dn+,[SJ] for each j < k.

3 rn,

We show that

r l Dom (?n [{ill) = {aq(o), a q ~ l.). , a . q(,-~)}.

lETh

Assume that a belongs to the left-hand side, i.e. r,,((a,i))= 1 for each i E Th.If a 3 dom ( p " ) ,then there exists a j < k such that q,,,((a,i))= 1 and dom ( q l - l , ns) (Y < dom ( q J n(in ) case j = 0 we replace ql-I by qn).This equation is possible only if i E S,. Hence if a belongs to the left-hand side of the above equation, then for each i in Th there is a j , such that i f S,, and dom (qlt-l,n) =sa < dom (q,,,,). From these inequalities it follows that j , is independent of i and can be denoted by j . But then i C SJ for each i in Th which contradicts our assumption that 1 ThI 3 2 and Th - S, # 0. In the last step of our construction we define p ( n ) . Let Pn = dom (r,,). Hence Pn 3 (Y~(,-~). Let p ( n ) be the least integer such that ap(,,)> Pn and aV(") > an.We extend r,, to pn+lby adding all the pairs ((&i),O), where i < N , Pn s < aq(n) and also the pairs ((c~~(,,),i),l)~ where i < N. Formulas (1)-(4) are then clearly satisfied with n replaced by n + 1. Lemma 3.1 is thus proved. 4. Models for the Godel-Bernays set theory In order to express the main results we now describe some special trees. Let N C w, and let the family of all non-void subsets of (0, 1,. . . , N - l} be partitioned into two families A u B such that (1) A contains with any set X all the non-void subsets of X , and B contains with any set Y all the non-void super-sets of Y ; ( 2 ) if XEA and Y € B ,then Y - X f O ; (3) if YCB, then ) Y l 3 2 ; (4) U A ={O, 1 7 . . . , N - 1).

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ANDRZEJ MOSTOWSKI

We denote by So, S1,.. . , Sk-lthe maximal elements of A with more than 1 element. Let A,(A,B) be a tree with N + k + 1 nodes: {W,

0, I,. .., N

-

1, g o , . . ., ak-1).

The initial node w is connected with the N nodes 0, 1,. . . ,N - 1 which are said to lie below w : i 6 0. The node aj is connected with the nodes i such that i C Sj and Iies below these nodes (i.e. uj6 i iff i C S j ) . Moreover, we assume that w 3 uifor each j . No other pairs of nodes are connected. Nodes which lie below no other nodes are called the terminal nodes of AN(A,B). We also consider infinite trees of height w1 which arise in the following way: there is just one node w of height 0. If nodes of height < 5 and edges connecting them are already defined then we correlate a tree Ag = ANg(AR,Bg) to each branch g consisting of nodes already defined and place this tree below all nodes on the branch g. Thus, the initial node wg of Ag has height 5 and nodes of A8 lying below w, have heights 6 + 1 or 5 + 2. Trees of this kind are called A,,-trees.

4.1. For every denumerable transitive model M GBC and THEOREM every tree h = AN(A,13) with the initial node w there is a mapping 4 of nodes of A into a family of transitive denumerable C-extensions of M satisfying the following conditions : (1) 4 ( w ) = M ; (2) + ( w ) F G B C for each node w ; ( 3 ) if w 1=z w z , then +(wl)> +(wz); (4) 4 (wI ) and C#J ( w2)have no joint C-extension M ' G B C unless there is a w such that w s w 1 and w =z w 2 .

+

THEOREM4.2. For every M a s in Theorem 4.1 and every A,,-tree A there is a mapping 4 of nodes of A into a family of transitive denumerable C-extensions of M satisfying the same conditions (1)-(4) as in Theorem

4.1. Theorem 4.2 results immediately from Theorem 4.1 and Lemma 2.1. To prove Theorem 4.1 we denote by So,. .., the maximal elements of A, and by To,... , T,-l, the minimal elements of B, construct models Mo, MI,.. . ,MN-,,M & ... ,M;-, as in Lemma 3.1 and put 4(0) = M , 4 ( i ) = Mi, +(q)= M:. Conditions (l), (2), (3) of Theorem 4.1 are obviously satisfied. To prove (4)we note that if w l= i l , w 2= iz and there is a w such that w s w l ,w < w2 then i l , i,CSi for some j and thus a common extension +(w) exists. Otherwise i , , iz must belong to a set Y in B and thus il, iz is one of the Th and therefore no common extension of 4(wl), exists. Now consider the case when one of the nodes w l ,w 2is of C#J(w,)

337

MODELS OF THE GODELBERNAYS AXIOMS

the form ui,e.g. w I = q.If w z = i and w I non s w z then i f Si and therefore Si u {i} contains one of the sets Th.Thus no common extension of M i and M iexists because otherwise there would exist an extension of all the Mi,, i l € Th.Similarly if w z = ui, there cannot be a joint extension of Miand Mi, because sju sir contains one of the sets Th. EXAMPLE. Let N

=6

So = {(),I},

and A consist of the four sets

SI= {1,2,3}, Sz = {2,3,4}, S , = (5)

and of their non-void subsets. The family A has 14 sets. Let B consist of the remaining 49 non-void subsets of 0, 1 , 2 , 3 , 4 , 5 . The tree A,(A,B) has the following form: w

Theorems 4.1 and 4.2 show how complicated the family of transitive C-extensions M GBC of a given transitive denumerable model M of GBC is. From Theorem 4.2, we obtain

COROLLARY4.1. Given a transitive denumerable model M exists 2"' transitive C-extensions M bGBC of power wl.

+ GBC, there

PROOF. To prove this we consider the full binary tree of height w Iand a function C#J as described in Theorem 4.2 by Lemma 2.1 the union U C#J(w) taken over nodes w lying on a branch g gives us a C-extension M, of M of power w 1 such that M, F G B C and for different branches g , , g,, we obtain different models. Observing that models M,,, M, corresponding to two different branches do not have a common transitive C-extension, we obtain furthermore COROLLARY 4.2. There are at least 29 maximal-transitive models M +GBC with a given denumerable VM. Under the assumption of the continuum hypothesis, this estimate of the number of maximal-transitive models with a given denumerable VM is sharp. Without this assumption, no such sharp estimate exists but it seems probable that Martin's axiom implies that their number is 2*"1).

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ANDRZEJ MOSTOWSKI

5. Models for the predicative arithmetic of second order

A Godel-Bernays-type extension of a theory is possible not only for the theory ZF but for arbitrary first-order theories. Let us consider briefly the Godel-Bernays-type extension of Peano’s arithmetic 9.A detailed description of 9 can be found, e.g. in KLEENE[1952], p. 82. We extend 9 by adding to it a new sort of variable X , Y, Z,. .. called set variables and a binary predicate E. Formulas with no bound-set variables are called predicative. As axioms of the extended theory we take axioms of Peano’s system (Kleene’s axioms 14-21) as well as the following ones: + [(x E X) = (y E Y ) l , ( X = Y )= (x)[(x E X ) = (X E Y ) ] , ( E X ) ( x ) [ ( xE X ) = c p ] (set existence scheme), (0 E X ) & (x)[x E X + (x’ E X)] + (x EX).

(x = Y )

(All (A2) (A3) (A4)

In the set-existence scheme, cp can be any predicative formula in which X does not occur. The resulting system will be called predicative arithmetic of second order and denoted by Apr.It is possible to replace the predicative set-existence scheme by a finite number of axioms as it is done in the case of set theory. Models of A,, can be assumed to have the form (0,S, 0, I , +, C ) where (0,0, ’, +, *) is a model of Peano arithmetic and S C P ( 0 ) . Moreover, 0 can be assumed to have as its initial segment the set 0, of ordinary integers. The operations I , +, * restricted to 0, coincide with the usual arithmetical operations and 0 is the integer zero. If 0 = 0, then the model (0,S, 0, ’, +, *, C ) is an w-model. If Mi = (0,Si, O , ’ , +, *, C ) for i = l , 2 and S , C S2,then we say that M 2 is a C-extension of MI. Later, we shall abbreviate our notation for models and write (0,s) instead of (a,S, 0, *,

I ,

+,

-9

C).

LEMMA5.1 Lemma 2.1 (mutatis mutandis) holds for models of Apr. We define the family Def, and Def, (X) similarly as on p. 326. If Apr. Similarly, if M = (0,s)k A,,, then (n,Def,) Apr.Instead of Lemma 2.2, we have the following result: If (0,s) A,, and X C IR, then all the axioms of A,, with the possible exception of (A4) hold in (O,DefM(X)). The notion of a generic set is defined similarly as on p. 4 (cf. SIMPSON [ m ] ) .We can prove easily (cf. SIMPSON [a])that if (In,M) A,, u M is denumerable and P C Def, is a set partially ordered by a relation R C DefM then generic sets G C P exist. Moreover, (Cl,DefM(G)) ‘pA,. Our main results are as follows:

nk 9 , then (n,Def,)

+

+

+

+

MODELS O F THE GODEL-BERNAYS AXIOMS

339

THEOREM5.1. Given an integer N , a tree A=A,(A,B) and a model M = (R,S) A,, such that 0 u S = o and, R # Ro, then there exists a mapping C$ of the nodes o f A into denumerable C-extensions of M such that (1) the initial node of A is mapped onte M, ( 2 ) if w I , w z are nodes of A, then C $ ( w l ) > & ( w 2 )is equivalent to w1 c w,; ( 3 ) if w I , wz are nodes of A, then C$(w,),C$(wz)have a joint C-extension which is a model of A,, i f f there exists a w such that w =sw I and w S w2.

+

THEOREM5.2. tree.

Theorem 5.1 holds if one replaces A,(A,B) by a Am,-

These theorems are proved essentially as Theorems’4.1 and 4.2. The crucial lemma corresponding to Lemma 3.1 is proved by considering conditions which are finite two-valued functions with domains of the form {x : x =sn } x N , where n E R and s is the “less-than’’ relation of the model R. The word “finite” is meant here in the sense of R and the whole function has to be coded by a single element of 0. We fix an increasing sequence {a”}, n €Ro of elements of R which is cofinal with R. This sequence does not belong to the model. We arrange the construction of generic sets Gi,i < N , in such a way that the intersections ni,,,Dom(Gi) be equal to a set X consisting of infinitely many an’s. This is possible because for each condition p only finitely many an’scan belong to dom(p). We show that there is no model (R,S*) of A,, such that S * > U {Si: i E Th}.Otherwise, the set X would be an element of S * . Using the theorem on inductive definitions we could then find in S * a functional set f whose domain is a segment of R such that f(0) = ao, f ( x + 1) = the least element of X greater than f (x). Thus, the domain of f would be the set of standard integers which is a contradiction because this set belongs to no model (a$*) of Apr. It is rather remarkable that we have here a complete analogy between models of GBC and models of A,, with non-standard integers. For models of A,, with standard integers Theorems 5.1 and 5.2 are false because any two such models have a joint C-extension. References FELGNER,U. (1971) Comparison of the axioms of local and universal choice. Fundamenta Mathematicae 71, 43-62. GODEL,K. (1940) The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory. Annals of Math. Studies. Vol. 3 . Princeton University Press, Princeton, N.J.,66 pp. (7th printing (1966))

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KLEENE,S. C. (1952) Introduction to metamathematics. North-Holland, Amsterdam, 550 pp. (2nd printing (1957).) SIMPSON,S. G. (m) Forcing and models for arithmetic. Proceedings of the American Mathematical Society. (To appear.) VOPENKA,P. and HAJEK,P. (1972) The Theory of Semisets. North-Holland, Amsterdam, 332 PP.