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The effectivity of existential statements in axiomatic set theory
1~FORMA~ON SCIENCES
The Effectivity
of Existential
119
Statements
in Axiomatic
Set Theory
AZRIEL Ll%‘Yt Hebrew university, Jerusalem, and ~tarI~~r~Uni~ersiiy,Cai~fornia
ABSTRACT
Mathematical experience indicates a close connection between the axiom of choice and noneff~tive existence proofs. In order to make this connection a subject of a systematic study it seems appropriate to approach this probfem from the point of view of the hierarchy developed by the author in [8]. It turns out, as can be Seen from various related results and examples, that the connection between the axiom of choice and effectivity comes up when one considers sentences of the form (V7~)(3y)x(x,y), where, throughout this abstract, x(x,y) is a formula which does not contain quantifiers other than quantifiers restricted to sets (such as (Vt)(t E s + . I .) or (3t)(t E s.. .). Many statements of set theory are of this form; e.g. the ordering theorem, the Boolean prime ideal theorem, the generalized continuum hypothesis. If one can prove in the Zermelo-Fraenkel set theory without the axiom of choice (ZF) that (Vx)(~y)~(x,y) then there is a term T(z~)of set theory such that ZF t (Vx) (there is a finite subset u of the transitive closure of X) x(x, 7(u)). One cannot expect to get a “more effective” solution for y since no such solution exists even for y in (Vx)(sy)(x = Ov y E x). If one can prove(V~)(!ly)x(x,y) with the aid of the axiom of choice then there is a term 7 such that ZF I (for every well-ordering r of the transitive closure of x) X(X.4%.
When in set theory, or in some related field, one comes across an existential statement, i.e., a statement which asserts that there exists a set y which has a given property, one often asks also if there is an effective example of such a I’. We shall not give at this point a formal version of the intuitive notion of an effective example; as we shall see there are several different formal versions, depending on the circumstances. Let us only remark now that the meaning of the word “effective” here has nothing to do with its meaning in the theory of recursive functions. The present notion of effectivity is discussed in detail in Sierpixiski [I I]. The present paper consists of some general theorems concerning the existence of effective examples. t This work was written with the partial support of the U.S. National Science Foundation (Grant No. GP-3926) and the Information Systems Branch, Office of Naval Research, Washington, D.C. under Contract F-61052 67 C 0055. i~~r~ati~~ Sciences 1 ( 1969), I 19-I 30 Copyright @ 1969 by American Elsevier ~blishing
Company, fnc.
AZRIEL Lh’Y
Quite early in the development of set theory it was noticed that by means of the axiom of choice one can prove existential statements which do not seem to be effective. For example, one can prove by means of the axiom of choice that there is a well-ordering of the real numbers but one cannot produce an effective example of such a well-ordering (see Feferman [2, bottom of p. 3421). This seems to point to the axiom of choice as the culprit, but the axiom of choice has the following alibi. Let us say that a set y has the property P if it is an ordering of the set of the real numbers which is as good (or as well) as possible, i.e., y is a well-ordering of the real numbers if the real numbers can be well-ordered, and is an ordering of the real numbers otherwise. One can now prove, without using the axiom of choice, that there is a set y which has the property P and yet one cannot produce an effective example of such a y. It turns out however that, as we shall see later, there is a definite difference between the property of being a well-ordering of the real numbers and property P; the syntactical structure of the former is simpler than that of the latter. It will be pointed out that in the case of existential statements of relatively simple syntactical structure the lack of an effective example can indeed be blamed on the axiom of choice. To make our discussion more formal, we shall deal with the ZermeloFraenkel set theory ZF where all objects are sets and which is formulated in the first-order predicate calculus with equality and which consists of the axioms Ia-V, VII-IX of [3]. As our scale for measuring the syntactical complexity of formulas of the language of ZF we take the ordinary quantifier hierarchy, except that we disregard quantifiers of the type (Vx)(x E y + . . . and (3x)(x E y A . . ., which will be called restricted quantifiers. We add to the language of ZF the definite article, i.e., the L operator, which enables us to form descriptions. The terms of the language (i.e., the variables and the descriptions) will be denoted with T and u. We shall read (~y)@(y,x~,...,x,,) as “that y for which @(y,x ,, . . .,xJ if there is a unique such y and 0 otherwise”; the part “if there is a unique such y and 0 otherwise” will always be omitted. Throughout the present paper upper case Greek letters will denote arbitrary formulas of ZF and lower case Greek letters, other than T and u, will denote restricted formulas only, i.e., formulas of ZF in which the only quantifiers occurring are of the form (Vx E y) or (3x E y) and which do not contain descriptions. All the free variables of a formula or a term will always be written to the right of the symbol for the formula, or the term, surrounded by parentheses; thus @ and 4 are sentences, T is a constant term, Q(y), x(y), T(Y) are formulas_ and a term, respectively, with the only free variable y, etc. We denote with DC, AC, GCH and V = L the axiom of dependent choices, the axiom of choice, the generalized continuum hypotheses and the axiom of constructibility, respectively. ZF f AC + GCH I- dr stands for “0 is provable
EXISTENTIALSTATEMENTS IN AXIOMATIC SET THEORY
121
from the axioms of ZF by means of AC and GCH”, etc. ; k @ stands for “@ is a theorem of logic.” Not all the theorems presented here will be followed by a full proof. Whenever the proof is titled “Outline of proof” this means that enough details are supplied for the reconstruction of the full proof by the reader. On the other hand, whenever the title is “Method of proof” this means that in the proof one has to construct a particular model of set theory by Cohen’s method of forcing, and the construction is not carried out here. The author hopes to describe the construction of the required models in some later publication. We naturally try to obtain a positive solution for the effectivity problem in ZF, whenever possible, but it seems that in some cases we can get much better results in some stronger set theory like ZF + DC or ZF + AC. If we go as far as adding to ZF the axiom of constructibility or even the axiom that every set in ordinal-definable (see GGdel [5]) then the effectivity problem has trivial solution for every existential statement. In these theories there is a definable well-ordering W of the universe, thus we can prove in them (Zly)@(y,x,, . . .,x,) -+ @(T(x,, . . .,x,),x,, . . .,x,)) where 7(x,, . . .,x,,) is the term “the least set y in the well-ordering W such that @(y,x,, . . .,x,,).”
In order to examine the kind of assumptions that we have to make in order to obtain effective examples for an existential statement let us look at the existential statement Y, which for the sake of illustration we assume to be of the form @y)@(y). In order to obtain an effective example of a y such that Q(y) we must be sure that there is at all a y such that Q(y), so the least we have to assume is that Y, i.e., @y)@(y), is true. However, as we shall see, in many cases the assumption that !P holds is too weak to entail the existence of an effective example, and we may have to assume one of the following assumptions on Y, listed in the order of increasing strength: ZF+ V= L t Y, ZF + AC + GCH I- Y, ZF + AC k Y, ZF I- Y, I- Y. The theory mentioned in this assumption should not be confused with the theory in which we can prove the existence of an effective example. All the negative results in the present paper make use, of course, of the tacit assumption that ZF is consistent. Case A. @y)@(y)
In this case a positive solution to the effectivity problem consists in finding a term T such that O(T). The subcases of Case A deal, respectively, with the formulas (~Y)#Y), @Y)(~+$(Y,z), (~Y)(~z)(~~)~(Y,z,u), .... As soOn as we get a negative answer somewhere down the line there is no point in going further without making some stronger assumption. No formulas with two adjacent quantifiers of the same type (i.e., universal or existential) is listed here since any two adjacent quantifiers of the same type can be replaced by a single quantifier of that type by means of ordered pairs ([8], Lemma l(d)]); Information Sciences 1(1969), 119-130
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AZRIEL Lk’Y
the only axiom needed for this replacement is the axiom of pairing and therefore adjacent quantifiers of the same type need be considered only in the case of provability in logic. (Even in the latter case we shall regard in our discussion, though not in the statement of the theorems, two adjacent quantifiers of the same type as one quantifier.) Actually, it might have been more natural to replace logical provability by provability from the axiom of pairing throughout the paper. Case Al. (%+#(Y) THEOREM 1. ZF+
DC k(3y)&)
+ (3y)(y
is constructible A 4(y)),
and
hence
where T is “the constructible y of least order such that #J(Y)” where the notion of a constructible set and the order of a constructible set are as in Giidel[4]. Proof. This is, essentially, Theorem 43 in [8], which is based on the theorem of Shoenfield [lo]. Moreover, when one checks the proof in [8] one can easily verify that there is a set y as required which is a &-set in the sense of Scarpellini [9] (by means of [9, Lemma 4 and Theorem 21).
If one wants to avoid the use of DC in Theorem 1 it seems that one has to make the stronger assumption that ZF + V = L t @y)+(y) and then one easily obtains ZF k (3y)(y is constructible A 4(y)) (see Godel [4] and [8, Lemma 341) and hence ZF 1 d(7), where 7 is as in Theorem 1. Case A2. (~Y)(~z)~(YJ) THEOREM2. There is a formula +(y,z) such that ZF I- (3y)(Vz)#y,z) for no term T does ZF + AC + GCH I-(VZ)$(T,Z).
and
Outline of proof. We take for (3y)(Vz)+( y, z ) a sentence which asserts that there is a nonconstructible set of natural numbers or else the set of all sets of natural numbers is constructible. (We use the formula at the top of p. 61 in [S]). A model of ZF + AC + GCH, defined by the method of forcing of Cohen [l], is constructed in [7] in which there are nonconstructible subsets of w yet there is no such definable one; this establishes the present theorem. THEOREM3. Let ZF- be ZF without the power-set axiom. For every formula &y,z) if ZF- I-@y)(Vz)+(y,z) then ZF+ DC t @y)(y is constructible A (Vz)$(y,z)) and therefore there is a term r such that ZF+ DC !- (VZ)$(T,Z).
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EXISTENTIALSTATEMENTS IN AXIOMATIC SET THEORY
Outline of proof. Let us say that a set x is constructibly countable if there is
a constructible one-one mapping of x on w or on some finite ordinal. Given a property P we say that x is hereditarily of the property P is x has the property P as well as all its members, all the members of its members, etc. If ZF- I-(3y)(Vz)&y,z) then @y)(Vz)&y,z) is true in the model of ZF- which consists of all hereditarily constructibly countable sets (hcc-sets). Therefore (Ihcc-sety)(Vhcc-setz)&y,z). By means of a proof which is very similar to the proof of Theorem 1 one can show that for an hcc-set y if (32) - 4(y,z) then (3hcc-setz) - $(y,z), h ence the y of the statement above is indeed a constructible y such that (Vz) 4( y, z). Case A3.
(~Y)W~(~U)&Y,Z,U)
such that I-(3y)(Vz,,z,)(3u) Y(y,z,,z2,u) but for no term T does ZF+ AC + GCH I- (Vz,,z2)(3u) s+!J(T,z,,z~,u).(The author does not know whether this result holds also for some THEOREM
4. There is a formula $(Y,z,,z~,u)
(~Y)Wz)(WJ(Y>z~u).) Proof:
KY,z1) v -
Let +(y,z)
be as in Theorem
2. Take
$(y,z,,z2, u) to be
4(zz,u).
Case B. (Vz)@y)@(x, y)
In this case the best effectivity result one could hope for is to have, for some term T(X), (VX)~(X,T(X)). H owever, that is too good to be true, even under the assumption that l- (Vx)(!ly)&x,y), as one immediately sees from the example (Vx)@y)(y E xv - (32 E x)); the latter sentence is a theorem of logic but for no term T(X) can one prove, not even in ZF + AC + GCH, that (VX)(T(X) E xv - (32 E x)), since it is consistent with ZF + AC + GCH to assume that there is a definable set which has no definable member (see [7]). As we shall see, the natural positive results here are, in order of decreasing strength, (i)-(iii) below. Let us say that a set o is transitice if (Vx,y)(x E y E L’ -+ x E t;). Let Tc(x) (read: the transitive closure of x) denote the least transitive set which contains x, i.e., the set whose members are x, the members of x, the members of the members of x, etc. For every set v let P* denote the set of all finite sequences of members oft’. (i) For some term (ii) For some term (iii) For some term
(Vx)(3u E TC(X)*)@(X, T(U)) (V’x)(V well-ordering r of Tc(X))@(x,~(r)) T(X,S) (VX)(~SG Tc(x) x Tc(x))@(x,~(x,s)).
T(U) T(r)
It is immediate that (ii) is at least as strong as (iii). To see that (i) is at least as strong as (ii) we proceed as follows. Let us mean be a well ordering r
hformarion
Sciences 1(1969), 119-130
124
AZRIEL LlhY
of Tc(x) a one-one map of some ordinal doon Z%(X),and let ‘S(r) denote the range of r. Suppose that we have (i), then we define o(r) as “the value of+) fur the least u 6 S(r)* in the lexicographic we~i-ordering of S(r)* (which is obtained from the well-ordering r ofTc(x)) such that @(x, r(u)), where x is that member of 93(r) which is not contained in any other member of S(r)“; we have,
obviously (Vx) @(x, u(r)). At first sight even the strongest condition (i) seems disappointing, but after one reflects one sees that this is very much like what one usually calls an effective example in mathematics. For example, when we consider a Boolean algebra with atoms, in the general case we do not have an effective example of a nonprincipal prime ideal (see Feferman [2] and the remarks in [7, p. 1281); however one considers a principal prime ideal to be given effectively when it is described as the set of all members which include a given atom. If x is the Boolean algebra a principal prime ideal is thus given not as T(X) but as 7(y), where y is a member of that Boolean algebra.
CaseB1.(VX)(~Y)&X,Y) We shall first see that efhzctivity as in (i) above can indeed be obtained under the assumption that ZF I-(Vx)(~y)#(x, y). THEOREM5. If ZF 1 (Vx)(3y)#x,y) ZF t-(Vx)(3u E Tc(x)*)#(x, T(U)).
then for some term T(U) we have
Outline of proof. Given a set x let us look at the class L., of all sets constructjb~e relative to x. The best way to approach the notion of constru~tibility
relative to x in the present context is to consider a ramified language S-’ with constants for the members of l”c(x) (it is similar to the language 3 in [7] suggested by D. Scott for handling forcing). The sets constructible relative to x are those which are the values of the terms of 2’. For u E Tc(x)* the class of all the terms of 2 which contain no constants for members of Tc(x) other than those in u can be given a well-ordering which is definable with tl as a parameter. Since ZF I-(Vx)(!Iy)#(x,y) and L, is a model of ZF the relativization of (Vx)@y)+(x,y) to LX is also provable in ZF. #x,y) is absolute with respect to L, (see, e.g., [8, Lemma 341) hence we haveZF I- (Vx)(GIyE L&#(x,y), Remembering what we mentioned above we have ZF I-(Vx)(& 4 Tc(x)*)&c,T(u)), where T(U)is the term “the value y of the least term of 9 with no constants other than those for the sets which occur in u (in some canonical well-ordering as mentioned above) such that 4(x, y).”
We define So(x) = {x>, P+*(x) = S”(x) U U@“(x)) where U(y) is the union ofy (i.e., the set of all members of members ofy). Obviously Tc(x) = U,,, P(x). The question whether Theorem 5 can be improved by replacing Tefx) by some
125
EXISTENTIAL STATEMENTS IN AXIOMATIC SET THEORY
S”(x), or by restricting the length of u is answered negatively by Theorem 6. Thus Theorem 5 is, essentially, the best result one can get when one assumes only ZF l- (~#~Y)&v). THEOREM 6. There is a formula $(x,y) such rhat ZF k (Vx)(3y)$(x,y) but for no term T(U) does ZF + AC + GCH k (51 E w)(Vx)(3u E S”(X)*)+(X,T(U)) or ZF + AC + GCH k (31 E w)(Vx)(3 u E Tc(x)* of length < n)$(x,~(u)).
Method of proof. Consider the formula (Vx)(Vm)(Elf )(m E w --f Fnc( f) A ~(f)=nhf(O)=xh(Vk~m)(k+1#m~(3z~f(k))(z~w)-tf(k+1)~f(k)~ f(k + 1) $ w)) where Fnc( f) and Tr)(f) stand for “fis a function” and “the domain of f,” respectively. This formula is converted to the form (Vx)@y)$(x,y) by the standard technique (see [8]). Since the original formula is provable in ZF so is (Vx)@y)qS(x,y). Now, one uses P. Cohen’s method of forcing (Cohen [l]) to construct a model of ZF in which there is a sequence a = (a,,,~,, . . .) such that each ui is a set of subsets of w and no member of a, is definable in terms of the sequence a and of the other members of U,,, ui. Define now co = uo, ci+l = {ci U t (t E ai+l}a It is now easily seen that for m > n and for x = c, an f as required by the formula above is not definable in terms of members of S”(x) or of n members of Tc(x).
Under the weaker assumption ZF + AC t (Vx)@y)&x,y), do have (ii), as shown in Theorems 7 and 8.
(i) fails but we
THEOREM 7. There is a formula $(x,y) such that ZF + AC I- (Vx)@y)&x,y) but there is no term T(U) such that ZF + AC + GCH 1 (Vx)(% E Tc(x)*)
44x7 44). ProoJ This follows immediately from Theorem
10, as mentioned in its
proof. Theorems 5 and 7 establish that in Case Bl all the statements proved in ZF are effective but some statements proved in ZF+ AC are not. It is now natural to ask whether there are sentences (Vx)(ily)&x,y) which are shown to be effective in ZF+ AC but are not provable in ZF. The answer is trivially positive. Let &x,y) be y # 0 A (Vf E y) (f is a one-one function on x and its u&es are ordinals). It is easily seen that this &x,y) does not contain unrestricted quantifiers (see [S]). Let T(X) be the “the set of all one one mappings of x on the least ordinal equinumerous (equioulent) to x.” Obviously O’$(~Y)#~,Y) and @x)4( x p7 ( x N are equivalent to AC, hence unprovable in ZF (by Cohen [l]). Information Sciences 1(1969), 119-130
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AZRIELL&Y
THEOREM 8.ZfZF + AC k (Vx)(3y)+(x,y) rhen for some term T(r) we have ZF I- (Vx)(V well ordering r ofTc(x))&x, T(r)).
Outline of proof. Let x be a set and let r be any well-ordering of Tc(x). Let cm(r) = (@,y)Ifl E 3(r) A y E r(p)>>.Let L,,,,, be the class of sets which are constructible relative to cm(r). L,,,,, is a model of ZF+ AC and x E Lcmfrj(see the second part of the proof of [6, Theorem 1J); therefore, using the absoluteness of #x,y) with respect to L,,,,, we have (3y E ~=~~~~)#(x,y). Thus we have #(x,~r)), where r(r) stands for “the member y of&,,,, of feast cm(r)order such that #(x, y).‘” COROLLARY 9. ZfZF + AC 1 (Vx)(Jy)+(x, well-ordered --f (Jy)&x,y)).
y) then ZF 1 (Vx)(Tc(x)
can be
The following Theorem 10 rules out the possibility of improving Theorem 8 in the most obvious direction. THEOREM 10.There is ~forrn~~u Rx,y) such that ZF + AC k (Vx)(~y)~(x,y~ but there is no term u(r) such fhat ZF + AC + GCH I-(Vx)(3n E w)(V we& ordering r of S”(x))Ql(x, u(r)). (S”(x) is defined just before Theorem 6.) Method of proof. Let us remark first that Theorem 10 implies Theorem 7 rather directly. If for +(x, y) there were a term T(U)such that ZF -t AC + GCH I(Vx)(3u E Tc(x)*)rj(x,~(u)) then we could easily get a term u(r) such that ZF + AC + GCH I- (Vx)@n E w)(V we//-ordering r of S”(x))&x,o(r)), using the same idea as in the proof that (i) is at least as strong as (ii), which occurs at the beginning of our discussion of Case B. For the formula $(x,y) of Theorem IO we choose‘y is ~~~~~~~0~A ‘I)(y) is c~1ordinal A S(y) is a trunsitiue set A x E S(y).” +4(x,y) asserts that y is a wellordering of some transitive set which includes n(x). A certain model, constructed by Cohen’s method of forcinp,0 is used tolestablish for &x,y) what is claimed in the present theorem.
If we weaken our assumption to ZF+ AC f GCH !- (Vx)(3y)&x,y) also (ii) fails (Theorem 11) but (iii) holds even under the weakest assumption, namely that for a given x (Zly)&x,y) (Theorem 12). THEOREM
11. The
simple
con~in~~rn hypothesis can be
written as
(Vx)(3y)&x,y) it,r such a way that there is no term T(r) for which ZF + AC + GCH k (Vx) (V we/l ordering r of Tc(x))&x, T(r)). Method of proof. The simple continuum hypothesis can be written as W) v (%)(I c w A r # x) v @t,z,f) (f is an ordinals (VU E t)
(Vx)[(S E x)(s$
EXISTENTIALSTATEMENTS IN AXIOMATIC SET THEORY
127
(3g E z) (g is a one-one map of u into w) Af is a one-one map oft on x] which is
easily brought to the form (Vx)(3y)#(x,y) (see [S. pp. SO-611). In a suitable model of ZF + AC + GCH defined by means of forcing, namely in one where one adds to the constructible universe No generic subsets of w and a generic mapping of x2 on x1 (the latter by means of countable conditions), we have for no term 7(r) (V’x)(V dell-ordering r ~~~e(x})~(,~,7(r)). THEOREM
12. For every formula &x,y)
ZF -t AC k (Vx)[(3y)+(x,y)
there is a term T(X,S) such that -+ (3s~ 7-c(x) ;: Tc(x))&x,~(x,s))].
Outline of proof. We shall deal separately with the two cases where Tc(x) is finite and Tc(x) is infinite and obtain respective terms T,(X) and Q(S) which perform as required in the corresponding case. Then we take for T(X,S) the term “that y ~~?ichisequal to r,(x) ifTc(x) is$nite andjsequa~~o ~~(s)other~7ise.‘~ If Tc{x) is finite then, by 18, Theorem 431 one can prove that there is a constructible y such that #+,y), hence one can easily get a term T,(X) as required. If Tc(x) is infinite then, by [S, Theorem 361 (which is a simple use of the Skolem-LGwenheim theorem) there is a y such that ITc(y)[ G ITc(x)j and +(x,y) (where IuI stands for the cardinality of u). As a result of the axiom of foundation every nonvoid set y is uniquely determined by the isomorphism type of the E relation on 7’c(y). This isomorphism type can be given by a subset s of Tc(x) x Tc(x). Thus we can take T(X,S) to be “the set y such thaz the E relation on Tc(y) is isomorphic to the relation s.”
We started with the assumption ZF 1 (V,~)(~~)~~x,y) and we went on to weaker and weaker assumptions. Let us examine now what we can get from the stronger assumption t (Vx)(3y)$(x,y). We mentioned already earlier that even now we do not have (VX)$(X,T(X)); a stronger negative result is given in Theorem 13. A positive result is given in Theorem 14. THEOREM
13.For euery natural member II there is a,formrrIa $(x,y) such that
I-(Vx)@y)#(x,y)
but for
no
term
T(U)
does
ZF i- AC + GCH k (V’x)
@u E SyX)*)+(x, T(U)). Proof1 Let us write y E’ x for y E x. and for II > I we write y E” x for (32, E y)f3z, E z,)...@z,_, E z,_~)(x E z,-I). For the given n let $(x,y) be - (32 E”’ ’ x) v y E”*’ x. Obviously F-(Vx)@y)#~(x,y). Suppose that there is a term T(U) as above, take x to be the set (. . . {t). . .) (t flanked by n braces on each side), where t is the set of all nonconstructible subsets of o. Obviously x is a definable set and so are all the members of S”(X); therefore every finite subset of S”(x) is definable. By our assumption the definable set T(U) is a nonconstructible subset of w, but it is consistent with ZF i- AC + GCH that no nonconstructibie subset of LOis definable (see [7]).
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AZRlELLk?VY
THEOREM 14. For euery formula qb(x,y) such that I-(Vx)(3y)+(x,y) afinite number Msuch that I-(Vx)(3y)((y=xv
YE’XV YE2XV . ..v yPx)n
there is
#(x,y))
Proof.
If what is claimed does not hold then the set of sentences @ = I- #(MY N @Y E” 44(%Yll n E w> is consistent. If (A&z) is a model x 1x R “a}, wh ere xR”a stands for x - a and xR”+’ a of 0 thenletB=U,,,( for (3z)(xRz~ zR”a). As easily seen (B,R,a) is also a model of 0. On the other hand since I- (Vx)(3y)+(x,y) (3y)#(a,y) is true in (B,R,a}, i.e., for some bR”a we have #(a, b), contradicting 0.
THEOREM 15. There is a formula $(x, y,z) such that I- (Vx)(kly)(Vz)$(x, y,z) but for no term T(r) does ZF -t AC-t GCH I-(Vz)(V well ordering r of Tc(x))(Vz)#x,
T(r), 4.
Proof.
Let r&x,y) be as in Theorem 11. Take $(x,y,z) to be 4(x, y) v N $(y, z). Obviously f- (V~)(~y)(Vz)~(~, y, z) ; the rest follows easily from Theorem 11. THEOREM 16.For every formula $(x,y,z) ifZF- I-(Vx)(3y)(Vz)~(x, y,z), where ZF- is ZF without the power-set axiom, then there is a term r such that ZF + AC I- (VX)(~SS Tc(x) x Tc(~))(Vz)~(x,~(x,s),z). Outline of proof. As in the proof of Theorem 12 we get first two distinct terms TV and ~~(5) to deal with the respective cases of finite and infinite Tc(x). If R(x) is finite then we prove that there is a constructible set y such that (Vz)#x,y,z) and thus we take T,(X) to be “the constructible y of least order such that (Vz)#(x,y,z).” If Tc(x) is infinite, let u = {t ] ITc(t)j Q ITc(x)]), where IuI is the cardinality of the set a. x E u, and u is a model of ZF-, hence there is a y E u such that (Vz E ~)#(~,y,z). We claim that (Vz)~x~y,z~, since if (32) - cb(x,y,z) then by [8, Theorem 361 there is a z E u such that - +(x,y,z), contradiction. Since y E u we take s as in Theorem 12; the choice of +,s) is now obvious. THEOREM 17. There is a formula $(x,y,z) such that ZF k (Vx)(3y)(Vz) #(x,y,z) but for no term ~(x,s) does ZF-k AC-t GCH k (Vx) (3s~ Tc(x) x Tc(x~)(Vz)~~x, ~(x,s),.+
EXISTENTIAL STATEMENTS IN AXIOMATIC SET THEORY
129
Proof. Follows immediately from Theorem 2 when we take for #(x,y,z) #y, z) of Theorem 2. Case B3. (Qx)(~~}(Qz)(~~)~(x,y,z,u). Because of Theorem 15 we have to check only the case of provability in logic. A negative answer for the case t- (Qx)(~y)(Qz~,z~)(~~)~(x,y,z,,z~,~) follows immediately from Theorem 4. Case C. The quantifier (3~) is not one of the first two unrestricted quantifiers. The complete answer is given by Theorem 12 when (3~) is the innermost unrestricted q~nti~er, by Theorems 17 and 16 when (3~) is the innermost unrestricted quantifier but one, and by Case B3 in, essentially, all other cases. The results of the paper are presented in the Table 1. Under the heading “Assumption” we write what is assumed on the existential statement 0, e.g., “ZF I-” means that we assume ZF I- Cp, whereas “true” means that we only assume that Qi holds (as in Theorems 1 and 16). Under the heading “Effective solution for y” we write for Case A, namely the case of @y)@(y), “positive” or “negative”. For cases B, namely the case of (Qx)(3~~)~(~), and Case C, we write (i), (ii), or (iii) referring thereby to (i)-(iii) on page 123,
TABLE I Statement --..
(39 lb(Y)
True ZFtZF- k t i-
(~YXQ.@rbb, d (3~NQz1,zdW #Y, 2,~) (Q’x)(~Y)4(x, Y)
(QUERY)
$4.~Y, 2)
(Qx)(~Y)(‘~z,,z,)(~u)~(x,Y,
Assumption _ _~
21,
zz,u)
* f * (3Y)NY,. +.)
. . . (~Y)(QzM(Y,z, . . .) . . .(3y). . .$(y,. ..) (all other cases)
ZFt ZF+ AC t ZF+ AC+ CCH True lZF- t ZFt t True ZF- t ZF b I-
Effective solution for y Positive Negative Positive Negative Better than (i), but not completely positive (i) (ii) ((i) fails) (ii) fails (iii) (ii) fails (iii) (iii) fails (iii) fails Analog of (iii) Analog of (iii) Analog of (iii) fails Analog of (iii) fails
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130
AZRIEL LtiVY
REFERENCES
1 P. J. Cohen, The independence of the continuum hypothesis, Proc. Nat. Acad. Sci. 50 (1963), 1143-1148, and51 (1964), 105-110. 2 S. Feferman, Some applications of the notions of forcing and generic sets, Fundamenta Mathematicae 56 (1965), 325-345. 3 A. A. Fraenkel and Y. Bar-Hillel, Foundations of Set Theory, Amsterdam, 1958. 4 K. Godel, The Consistency of the Continuum Hypothesis, 3rd printing, Princeton, 1953. 5 K. Godel, Remarks before the Princeton Bicentennial Conference, in The Undecidable (M. Davis, ed.), New York, 1965, pp. 84-88. 6 A. Levy, A generalization of Godel’s notion of constructibility, Journal of Symbolic Logic 25 (1960), 147-155. 7 A. Levy, Definability in axiomatic set theory I, Proceedings of the 1964 Znternational Congress for Logic, Methodology and Philosophy of Science, Amsterdam, 1965, pp. 127151. 8 A. Levy, A hierarchy of formulas in set theory, Memoirs of the Amer. Math. Sot. 57 (1965). 9 B. Scarpellini, A characterization of &sets, Trans. Amer. Math. Sot. 117 (1965), 441450. 10 J. R. Shoenfield, The Problem ofpredicativity: Essays on the Foundations of Mathematics, Jerusalem, 1961, pp. 132-139. 11 W. Sierpitiski,~CardinaI and Ordinal Numbers, Warsaw, 1958.
Received June 16,1968
Revised manuscript received October 17,1968
The Effectivity
of Existential
119
Statements
in Axiomatic
Set Theory
AZRIEL Ll%‘Yt Hebrew university, Jerusalem, and ~tarI~~r~Uni~ersiiy,Cai~fornia
ABSTRACT
Mathematical experience indicates a close connection between the axiom of choice and noneff~tive existence proofs. In order to make this connection a subject of a systematic study it seems appropriate to approach this probfem from the point of view of the hierarchy developed by the author in [8]. It turns out, as can be Seen from various related results and examples, that the connection between the axiom of choice and effectivity comes up when one considers sentences of the form (V7~)(3y)x(x,y), where, throughout this abstract, x(x,y) is a formula which does not contain quantifiers other than quantifiers restricted to sets (such as (Vt)(t E s + . I .) or (3t)(t E s.. .). Many statements of set theory are of this form; e.g. the ordering theorem, the Boolean prime ideal theorem, the generalized continuum hypothesis. If one can prove in the Zermelo-Fraenkel set theory without the axiom of choice (ZF) that (Vx)(~y)~(x,y) then there is a term T(z~)of set theory such that ZF t (Vx) (there is a finite subset u of the transitive closure of X) x(x, 7(u)). One cannot expect to get a “more effective” solution for y since no such solution exists even for y in (Vx)(sy)(x = Ov y E x). If one can prove(V~)(!ly)x(x,y) with the aid of the axiom of choice then there is a term 7 such that ZF I (for every well-ordering r of the transitive closure of x) X(X.4%.
When in set theory, or in some related field, one comes across an existential statement, i.e., a statement which asserts that there exists a set y which has a given property, one often asks also if there is an effective example of such a I’. We shall not give at this point a formal version of the intuitive notion of an effective example; as we shall see there are several different formal versions, depending on the circumstances. Let us only remark now that the meaning of the word “effective” here has nothing to do with its meaning in the theory of recursive functions. The present notion of effectivity is discussed in detail in Sierpixiski [I I]. The present paper consists of some general theorems concerning the existence of effective examples. t This work was written with the partial support of the U.S. National Science Foundation (Grant No. GP-3926) and the Information Systems Branch, Office of Naval Research, Washington, D.C. under Contract F-61052 67 C 0055. i~~r~ati~~ Sciences 1 ( 1969), I 19-I 30 Copyright @ 1969 by American Elsevier ~blishing
Company, fnc.
AZRIEL Lh’Y
Quite early in the development of set theory it was noticed that by means of the axiom of choice one can prove existential statements which do not seem to be effective. For example, one can prove by means of the axiom of choice that there is a well-ordering of the real numbers but one cannot produce an effective example of such a well-ordering (see Feferman [2, bottom of p. 3421). This seems to point to the axiom of choice as the culprit, but the axiom of choice has the following alibi. Let us say that a set y has the property P if it is an ordering of the set of the real numbers which is as good (or as well) as possible, i.e., y is a well-ordering of the real numbers if the real numbers can be well-ordered, and is an ordering of the real numbers otherwise. One can now prove, without using the axiom of choice, that there is a set y which has the property P and yet one cannot produce an effective example of such a y. It turns out however that, as we shall see later, there is a definite difference between the property of being a well-ordering of the real numbers and property P; the syntactical structure of the former is simpler than that of the latter. It will be pointed out that in the case of existential statements of relatively simple syntactical structure the lack of an effective example can indeed be blamed on the axiom of choice. To make our discussion more formal, we shall deal with the ZermeloFraenkel set theory ZF where all objects are sets and which is formulated in the first-order predicate calculus with equality and which consists of the axioms Ia-V, VII-IX of [3]. As our scale for measuring the syntactical complexity of formulas of the language of ZF we take the ordinary quantifier hierarchy, except that we disregard quantifiers of the type (Vx)(x E y + . . . and (3x)(x E y A . . ., which will be called restricted quantifiers. We add to the language of ZF the definite article, i.e., the L operator, which enables us to form descriptions. The terms of the language (i.e., the variables and the descriptions) will be denoted with T and u. We shall read (~y)@(y,x~,...,x,,) as “that y for which @(y,x ,, . . .,xJ if there is a unique such y and 0 otherwise”; the part “if there is a unique such y and 0 otherwise” will always be omitted. Throughout the present paper upper case Greek letters will denote arbitrary formulas of ZF and lower case Greek letters, other than T and u, will denote restricted formulas only, i.e., formulas of ZF in which the only quantifiers occurring are of the form (Vx E y) or (3x E y) and which do not contain descriptions. All the free variables of a formula or a term will always be written to the right of the symbol for the formula, or the term, surrounded by parentheses; thus @ and 4 are sentences, T is a constant term, Q(y), x(y), T(Y) are formulas_ and a term, respectively, with the only free variable y, etc. We denote with DC, AC, GCH and V = L the axiom of dependent choices, the axiom of choice, the generalized continuum hypotheses and the axiom of constructibility, respectively. ZF f AC + GCH I- dr stands for “0 is provable
EXISTENTIALSTATEMENTS IN AXIOMATIC SET THEORY
121
from the axioms of ZF by means of AC and GCH”, etc. ; k @ stands for “@ is a theorem of logic.” Not all the theorems presented here will be followed by a full proof. Whenever the proof is titled “Outline of proof” this means that enough details are supplied for the reconstruction of the full proof by the reader. On the other hand, whenever the title is “Method of proof” this means that in the proof one has to construct a particular model of set theory by Cohen’s method of forcing, and the construction is not carried out here. The author hopes to describe the construction of the required models in some later publication. We naturally try to obtain a positive solution for the effectivity problem in ZF, whenever possible, but it seems that in some cases we can get much better results in some stronger set theory like ZF + DC or ZF + AC. If we go as far as adding to ZF the axiom of constructibility or even the axiom that every set in ordinal-definable (see GGdel [5]) then the effectivity problem has trivial solution for every existential statement. In these theories there is a definable well-ordering W of the universe, thus we can prove in them (Zly)@(y,x,, . . .,x,) -+ @(T(x,, . . .,x,),x,, . . .,x,)) where 7(x,, . . .,x,,) is the term “the least set y in the well-ordering W such that @(y,x,, . . .,x,,).”
In order to examine the kind of assumptions that we have to make in order to obtain effective examples for an existential statement let us look at the existential statement Y, which for the sake of illustration we assume to be of the form @y)@(y). In order to obtain an effective example of a y such that Q(y) we must be sure that there is at all a y such that Q(y), so the least we have to assume is that Y, i.e., @y)@(y), is true. However, as we shall see, in many cases the assumption that !P holds is too weak to entail the existence of an effective example, and we may have to assume one of the following assumptions on Y, listed in the order of increasing strength: ZF+ V= L t Y, ZF + AC + GCH I- Y, ZF + AC k Y, ZF I- Y, I- Y. The theory mentioned in this assumption should not be confused with the theory in which we can prove the existence of an effective example. All the negative results in the present paper make use, of course, of the tacit assumption that ZF is consistent. Case A. @y)@(y)
In this case a positive solution to the effectivity problem consists in finding a term T such that O(T). The subcases of Case A deal, respectively, with the formulas (~Y)#Y), @Y)(~+$(Y,z), (~Y)(~z)(~~)~(Y,z,u), .... As soOn as we get a negative answer somewhere down the line there is no point in going further without making some stronger assumption. No formulas with two adjacent quantifiers of the same type (i.e., universal or existential) is listed here since any two adjacent quantifiers of the same type can be replaced by a single quantifier of that type by means of ordered pairs ([8], Lemma l(d)]); Information Sciences 1(1969), 119-130
122
AZRIEL Lk’Y
the only axiom needed for this replacement is the axiom of pairing and therefore adjacent quantifiers of the same type need be considered only in the case of provability in logic. (Even in the latter case we shall regard in our discussion, though not in the statement of the theorems, two adjacent quantifiers of the same type as one quantifier.) Actually, it might have been more natural to replace logical provability by provability from the axiom of pairing throughout the paper. Case Al. (%+#(Y) THEOREM 1. ZF+
DC k(3y)&)
+ (3y)(y
is constructible A 4(y)),
and
hence
where T is “the constructible y of least order such that #J(Y)” where the notion of a constructible set and the order of a constructible set are as in Giidel[4]. Proof. This is, essentially, Theorem 43 in [8], which is based on the theorem of Shoenfield [lo]. Moreover, when one checks the proof in [8] one can easily verify that there is a set y as required which is a &-set in the sense of Scarpellini [9] (by means of [9, Lemma 4 and Theorem 21).
If one wants to avoid the use of DC in Theorem 1 it seems that one has to make the stronger assumption that ZF + V = L t @y)+(y) and then one easily obtains ZF k (3y)(y is constructible A 4(y)) (see Godel [4] and [8, Lemma 341) and hence ZF 1 d(7), where 7 is as in Theorem 1. Case A2. (~Y)(~z)~(YJ) THEOREM2. There is a formula +(y,z) such that ZF I- (3y)(Vz)#y,z) for no term T does ZF + AC + GCH I-(VZ)$(T,Z).
and
Outline of proof. We take for (3y)(Vz)+( y, z ) a sentence which asserts that there is a nonconstructible set of natural numbers or else the set of all sets of natural numbers is constructible. (We use the formula at the top of p. 61 in [S]). A model of ZF + AC + GCH, defined by the method of forcing of Cohen [l], is constructed in [7] in which there are nonconstructible subsets of w yet there is no such definable one; this establishes the present theorem. THEOREM3. Let ZF- be ZF without the power-set axiom. For every formula &y,z) if ZF- I-@y)(Vz)+(y,z) then ZF+ DC t @y)(y is constructible A (Vz)$(y,z)) and therefore there is a term r such that ZF+ DC !- (VZ)$(T,Z).
123
EXISTENTIALSTATEMENTS IN AXIOMATIC SET THEORY
Outline of proof. Let us say that a set x is constructibly countable if there is
a constructible one-one mapping of x on w or on some finite ordinal. Given a property P we say that x is hereditarily of the property P is x has the property P as well as all its members, all the members of its members, etc. If ZF- I-(3y)(Vz)&y,z) then @y)(Vz)&y,z) is true in the model of ZF- which consists of all hereditarily constructibly countable sets (hcc-sets). Therefore (Ihcc-sety)(Vhcc-setz)&y,z). By means of a proof which is very similar to the proof of Theorem 1 one can show that for an hcc-set y if (32) - 4(y,z) then (3hcc-setz) - $(y,z), h ence the y of the statement above is indeed a constructible y such that (Vz) 4( y, z). Case A3.
(~Y)W~(~U)&Y,Z,U)
such that I-(3y)(Vz,,z,)(3u) Y(y,z,,z2,u) but for no term T does ZF+ AC + GCH I- (Vz,,z2)(3u) s+!J(T,z,,z~,u).(The author does not know whether this result holds also for some THEOREM
4. There is a formula $(Y,z,,z~,u)
(~Y)Wz)(WJ(Y>z~u).) Proof:
KY,z1) v -
Let +(y,z)
be as in Theorem
2. Take
$(y,z,,z2, u) to be
4(zz,u).
Case B. (Vz)@y)@(x, y)
In this case the best effectivity result one could hope for is to have, for some term T(X), (VX)~(X,T(X)). H owever, that is too good to be true, even under the assumption that l- (Vx)(!ly)&x,y), as one immediately sees from the example (Vx)@y)(y E xv - (32 E x)); the latter sentence is a theorem of logic but for no term T(X) can one prove, not even in ZF + AC + GCH, that (VX)(T(X) E xv - (32 E x)), since it is consistent with ZF + AC + GCH to assume that there is a definable set which has no definable member (see [7]). As we shall see, the natural positive results here are, in order of decreasing strength, (i)-(iii) below. Let us say that a set o is transitice if (Vx,y)(x E y E L’ -+ x E t;). Let Tc(x) (read: the transitive closure of x) denote the least transitive set which contains x, i.e., the set whose members are x, the members of x, the members of the members of x, etc. For every set v let P* denote the set of all finite sequences of members oft’. (i) For some term (ii) For some term (iii) For some term
(Vx)(3u E TC(X)*)@(X, T(U)) (V’x)(V well-ordering r of Tc(X))@(x,~(r)) T(X,S) (VX)(~SG Tc(x) x Tc(x))@(x,~(x,s)).
T(U) T(r)
It is immediate that (ii) is at least as strong as (iii). To see that (i) is at least as strong as (ii) we proceed as follows. Let us mean be a well ordering r
hformarion
Sciences 1(1969), 119-130
124
AZRIEL LlhY
of Tc(x) a one-one map of some ordinal doon Z%(X),and let ‘S(r) denote the range of r. Suppose that we have (i), then we define o(r) as “the value of+) fur the least u 6 S(r)* in the lexicographic we~i-ordering of S(r)* (which is obtained from the well-ordering r ofTc(x)) such that @(x, r(u)), where x is that member of 93(r) which is not contained in any other member of S(r)“; we have,
obviously (Vx) @(x, u(r)). At first sight even the strongest condition (i) seems disappointing, but after one reflects one sees that this is very much like what one usually calls an effective example in mathematics. For example, when we consider a Boolean algebra with atoms, in the general case we do not have an effective example of a nonprincipal prime ideal (see Feferman [2] and the remarks in [7, p. 1281); however one considers a principal prime ideal to be given effectively when it is described as the set of all members which include a given atom. If x is the Boolean algebra a principal prime ideal is thus given not as T(X) but as 7(y), where y is a member of that Boolean algebra.
CaseB1.(VX)(~Y)&X,Y) We shall first see that efhzctivity as in (i) above can indeed be obtained under the assumption that ZF I-(Vx)(~y)#(x, y). THEOREM5. If ZF 1 (Vx)(3y)#x,y) ZF t-(Vx)(3u E Tc(x)*)#(x, T(U)).
then for some term T(U) we have
Outline of proof. Given a set x let us look at the class L., of all sets constructjb~e relative to x. The best way to approach the notion of constru~tibility
relative to x in the present context is to consider a ramified language S-’ with constants for the members of l”c(x) (it is similar to the language 3 in [7] suggested by D. Scott for handling forcing). The sets constructible relative to x are those which are the values of the terms of 2’. For u E Tc(x)* the class of all the terms of 2 which contain no constants for members of Tc(x) other than those in u can be given a well-ordering which is definable with tl as a parameter. Since ZF I-(Vx)(!Iy)#(x,y) and L, is a model of ZF the relativization of (Vx)@y)+(x,y) to LX is also provable in ZF. #x,y) is absolute with respect to L, (see, e.g., [8, Lemma 341) hence we haveZF I- (Vx)(GIyE L&#(x,y), Remembering what we mentioned above we have ZF I-(Vx)(& 4 Tc(x)*)&c,T(u)), where T(U)is the term “the value y of the least term of 9 with no constants other than those for the sets which occur in u (in some canonical well-ordering as mentioned above) such that 4(x, y).”
We define So(x) = {x>, P+*(x) = S”(x) U U@“(x)) where U(y) is the union ofy (i.e., the set of all members of members ofy). Obviously Tc(x) = U,,, P(x). The question whether Theorem 5 can be improved by replacing Tefx) by some
125
EXISTENTIAL STATEMENTS IN AXIOMATIC SET THEORY
S”(x), or by restricting the length of u is answered negatively by Theorem 6. Thus Theorem 5 is, essentially, the best result one can get when one assumes only ZF l- (~#~Y)&v). THEOREM 6. There is a formula $(x,y) such rhat ZF k (Vx)(3y)$(x,y) but for no term T(U) does ZF + AC + GCH k (51 E w)(Vx)(3u E S”(X)*)+(X,T(U)) or ZF + AC + GCH k (31 E w)(Vx)(3 u E Tc(x)* of length < n)$(x,~(u)).
Method of proof. Consider the formula (Vx)(Vm)(Elf )(m E w --f Fnc( f) A ~(f)=nhf(O)=xh(Vk~m)(k+1#m~(3z~f(k))(z~w)-tf(k+1)~f(k)~ f(k + 1) $ w)) where Fnc( f) and Tr)(f) stand for “fis a function” and “the domain of f,” respectively. This formula is converted to the form (Vx)@y)$(x,y) by the standard technique (see [8]). Since the original formula is provable in ZF so is (Vx)@y)qS(x,y). Now, one uses P. Cohen’s method of forcing (Cohen [l]) to construct a model of ZF in which there is a sequence a = (a,,,~,, . . .) such that each ui is a set of subsets of w and no member of a, is definable in terms of the sequence a and of the other members of U,,, ui. Define now co = uo, ci+l = {ci U t (t E ai+l}a It is now easily seen that for m > n and for x = c, an f as required by the formula above is not definable in terms of members of S”(x) or of n members of Tc(x).
Under the weaker assumption ZF + AC t (Vx)@y)&x,y), do have (ii), as shown in Theorems 7 and 8.
(i) fails but we
THEOREM 7. There is a formula $(x,y) such that ZF + AC I- (Vx)@y)&x,y) but there is no term T(U) such that ZF + AC + GCH 1 (Vx)(% E Tc(x)*)
44x7 44). ProoJ This follows immediately from Theorem
10, as mentioned in its
proof. Theorems 5 and 7 establish that in Case Bl all the statements proved in ZF are effective but some statements proved in ZF+ AC are not. It is now natural to ask whether there are sentences (Vx)(ily)&x,y) which are shown to be effective in ZF+ AC but are not provable in ZF. The answer is trivially positive. Let &x,y) be y # 0 A (Vf E y) (f is a one-one function on x and its u&es are ordinals). It is easily seen that this &x,y) does not contain unrestricted quantifiers (see [S]). Let T(X) be the “the set of all one one mappings of x on the least ordinal equinumerous (equioulent) to x.” Obviously O’$(~Y)#~,Y) and @x)4( x p7 ( x N are equivalent to AC, hence unprovable in ZF (by Cohen [l]). Information Sciences 1(1969), 119-130
126
AZRIELL&Y
THEOREM 8.ZfZF + AC k (Vx)(3y)+(x,y) rhen for some term T(r) we have ZF I- (Vx)(V well ordering r ofTc(x))&x, T(r)).
Outline of proof. Let x be a set and let r be any well-ordering of Tc(x). Let cm(r) = (@,y)Ifl E 3(r) A y E r(p)>>.Let L,,,,, be the class of sets which are constructible relative to cm(r). L,,,,, is a model of ZF+ AC and x E Lcmfrj(see the second part of the proof of [6, Theorem 1J); therefore, using the absoluteness of #x,y) with respect to L,,,,, we have (3y E ~=~~~~)#(x,y). Thus we have #(x,~r)), where r(r) stands for “the member y of&,,,, of feast cm(r)order such that #(x, y).‘” COROLLARY 9. ZfZF + AC 1 (Vx)(Jy)+(x, well-ordered --f (Jy)&x,y)).
y) then ZF 1 (Vx)(Tc(x)
can be
The following Theorem 10 rules out the possibility of improving Theorem 8 in the most obvious direction. THEOREM 10.There is ~forrn~~u Rx,y) such that ZF + AC k (Vx)(~y)~(x,y~ but there is no term u(r) such fhat ZF + AC + GCH I-(Vx)(3n E w)(V we& ordering r of S”(x))Ql(x, u(r)). (S”(x) is defined just before Theorem 6.) Method of proof. Let us remark first that Theorem 10 implies Theorem 7 rather directly. If for +(x, y) there were a term T(U)such that ZF -t AC + GCH I(Vx)(3u E Tc(x)*)rj(x,~(u)) then we could easily get a term u(r) such that ZF + AC + GCH I- (Vx)@n E w)(V we//-ordering r of S”(x))&x,o(r)), using the same idea as in the proof that (i) is at least as strong as (ii), which occurs at the beginning of our discussion of Case B. For the formula $(x,y) of Theorem IO we choose‘y is ~~~~~~~0~A ‘I)(y) is c~1ordinal A S(y) is a trunsitiue set A x E S(y).” +4(x,y) asserts that y is a wellordering of some transitive set which includes n(x). A certain model, constructed by Cohen’s method of forcinp,0 is used tolestablish for &x,y) what is claimed in the present theorem.
If we weaken our assumption to ZF+ AC f GCH !- (Vx)(3y)&x,y) also (ii) fails (Theorem 11) but (iii) holds even under the weakest assumption, namely that for a given x (Zly)&x,y) (Theorem 12). THEOREM
11. The
simple
con~in~~rn hypothesis can be
written as
(Vx)(3y)&x,y) it,r such a way that there is no term T(r) for which ZF + AC + GCH k (Vx) (V we/l ordering r of Tc(x))&x, T(r)). Method of proof. The simple continuum hypothesis can be written as W) v (%)(I c w A r # x) v @t,z,f) (f is an ordinals (VU E t)
(Vx)[(S E x)(s$
EXISTENTIALSTATEMENTS IN AXIOMATIC SET THEORY
127
(3g E z) (g is a one-one map of u into w) Af is a one-one map oft on x] which is
easily brought to the form (Vx)(3y)#(x,y) (see [S. pp. SO-611). In a suitable model of ZF + AC + GCH defined by means of forcing, namely in one where one adds to the constructible universe No generic subsets of w and a generic mapping of x2 on x1 (the latter by means of countable conditions), we have for no term 7(r) (V’x)(V dell-ordering r ~~~e(x})~(,~,7(r)). THEOREM
12. For every formula &x,y)
ZF -t AC k (Vx)[(3y)+(x,y)
there is a term T(X,S) such that -+ (3s~ 7-c(x) ;: Tc(x))&x,~(x,s))].
Outline of proof. We shall deal separately with the two cases where Tc(x) is finite and Tc(x) is infinite and obtain respective terms T,(X) and Q(S) which perform as required in the corresponding case. Then we take for T(X,S) the term “that y ~~?ichisequal to r,(x) ifTc(x) is$nite andjsequa~~o ~~(s)other~7ise.‘~ If Tc{x) is finite then, by 18, Theorem 431 one can prove that there is a constructible y such that #+,y), hence one can easily get a term T,(X) as required. If Tc(x) is infinite then, by [S, Theorem 361 (which is a simple use of the Skolem-LGwenheim theorem) there is a y such that ITc(y)[ G ITc(x)j and +(x,y) (where IuI stands for the cardinality of u). As a result of the axiom of foundation every nonvoid set y is uniquely determined by the isomorphism type of the E relation on 7’c(y). This isomorphism type can be given by a subset s of Tc(x) x Tc(x). Thus we can take T(X,S) to be “the set y such thaz the E relation on Tc(y) is isomorphic to the relation s.”
We started with the assumption ZF 1 (V,~)(~~)~~x,y) and we went on to weaker and weaker assumptions. Let us examine now what we can get from the stronger assumption t (Vx)(3y)$(x,y). We mentioned already earlier that even now we do not have (VX)$(X,T(X)); a stronger negative result is given in Theorem 13. A positive result is given in Theorem 14. THEOREM
13.For euery natural member II there is a,formrrIa $(x,y) such that
I-(Vx)@y)#(x,y)
but for
no
term
T(U)
does
ZF i- AC + GCH k (V’x)
@u E SyX)*)+(x, T(U)). Proof1 Let us write y E’ x for y E x. and for II > I we write y E” x for (32, E y)f3z, E z,)...@z,_, E z,_~)(x E z,-I). For the given n let $(x,y) be - (32 E”’ ’ x) v y E”*’ x. Obviously F-(Vx)@y)#~(x,y). Suppose that there is a term T(U) as above, take x to be the set (. . . {t). . .) (t flanked by n braces on each side), where t is the set of all nonconstructible subsets of o. Obviously x is a definable set and so are all the members of S”(X); therefore every finite subset of S”(x) is definable. By our assumption the definable set T(U) is a nonconstructible subset of w, but it is consistent with ZF i- AC + GCH that no nonconstructibie subset of LOis definable (see [7]).
128
AZRlELLk?VY
THEOREM 14. For euery formula qb(x,y) such that I-(Vx)(3y)+(x,y) afinite number Msuch that I-(Vx)(3y)((y=xv
YE’XV YE2XV . ..v yPx)n
there is
#(x,y))
Proof.
If what is claimed does not hold then the set of sentences @ = I- #(MY N @Y E” 44(%Yll n E w> is consistent. If (A&z) is a model x 1x R “a}, wh ere xR”a stands for x - a and xR”+’ a of 0 thenletB=U,,,( for (3z)(xRz~ zR”a). As easily seen (B,R,a) is also a model of 0. On the other hand since I- (Vx)(3y)+(x,y) (3y)#(a,y) is true in (B,R,a}, i.e., for some bR”a we have #(a, b), contradicting 0.
THEOREM 15. There is a formula $(x, y,z) such that I- (Vx)(kly)(Vz)$(x, y,z) but for no term T(r) does ZF -t AC-t GCH I-(Vz)(V well ordering r of Tc(x))(Vz)#x,
T(r), 4.
Proof.
Let r&x,y) be as in Theorem 11. Take $(x,y,z) to be 4(x, y) v N $(y, z). Obviously f- (V~)(~y)(Vz)~(~, y, z) ; the rest follows easily from Theorem 11. THEOREM 16.For every formula $(x,y,z) ifZF- I-(Vx)(3y)(Vz)~(x, y,z), where ZF- is ZF without the power-set axiom, then there is a term r such that ZF + AC I- (VX)(~SS Tc(x) x Tc(~))(Vz)~(x,~(x,s),z). Outline of proof. As in the proof of Theorem 12 we get first two distinct terms TV and ~~(5) to deal with the respective cases of finite and infinite Tc(x). If R(x) is finite then we prove that there is a constructible set y such that (Vz)#x,y,z) and thus we take T,(X) to be “the constructible y of least order such that (Vz)#(x,y,z).” If Tc(x) is infinite, let u = {t ] ITc(t)j Q ITc(x)]), where IuI is the cardinality of the set a. x E u, and u is a model of ZF-, hence there is a y E u such that (Vz E ~)#(~,y,z). We claim that (Vz)~x~y,z~, since if (32) - cb(x,y,z) then by [8, Theorem 361 there is a z E u such that - +(x,y,z), contradiction. Since y E u we take s as in Theorem 12; the choice of +,s) is now obvious. THEOREM 17. There is a formula $(x,y,z) such that ZF k (Vx)(3y)(Vz) #(x,y,z) but for no term ~(x,s) does ZF-k AC-t GCH k (Vx) (3s~ Tc(x) x Tc(x~)(Vz)~~x, ~(x,s),.+
EXISTENTIAL STATEMENTS IN AXIOMATIC SET THEORY
129
Proof. Follows immediately from Theorem 2 when we take for #(x,y,z) #y, z) of Theorem 2. Case B3. (Qx)(~~}(Qz)(~~)~(x,y,z,u). Because of Theorem 15 we have to check only the case of provability in logic. A negative answer for the case t- (Qx)(~y)(Qz~,z~)(~~)~(x,y,z,,z~,~) follows immediately from Theorem 4. Case C. The quantifier (3~) is not one of the first two unrestricted quantifiers. The complete answer is given by Theorem 12 when (3~) is the innermost unrestricted q~nti~er, by Theorems 17 and 16 when (3~) is the innermost unrestricted quantifier but one, and by Case B3 in, essentially, all other cases. The results of the paper are presented in the Table 1. Under the heading “Assumption” we write what is assumed on the existential statement 0, e.g., “ZF I-” means that we assume ZF I- Cp, whereas “true” means that we only assume that Qi holds (as in Theorems 1 and 16). Under the heading “Effective solution for y” we write for Case A, namely the case of @y)@(y), “positive” or “negative”. For cases B, namely the case of (Qx)(3~~)~(~), and Case C, we write (i), (ii), or (iii) referring thereby to (i)-(iii) on page 123,
TABLE I Statement --..
(39 lb(Y)
True ZFtZF- k t i-
(~YXQ.@rbb, d (3~NQz1,zdW #Y, 2,~) (Q’x)(~Y)4(x, Y)
(QUERY)
$4.~Y, 2)
(Qx)(~Y)(‘~z,,z,)(~u)~(x,Y,
Assumption _ _~
21,
zz,u)
* f * (3Y)NY,. +.)
. . . (~Y)(QzM(Y,z, . . .) . . .(3y). . .$(y,. ..) (all other cases)
ZFt ZF+ AC t ZF+ AC+ CCH True lZF- t ZFt t True ZF- t ZF b I-
Effective solution for y Positive Negative Positive Negative Better than (i), but not completely positive (i) (ii) ((i) fails) (ii) fails (iii) (ii) fails (iii) (iii) fails (iii) fails Analog of (iii) Analog of (iii) Analog of (iii) fails Analog of (iii) fails
Infor~arion .Sciences1 (I969), 119-130
130
AZRIEL LtiVY
REFERENCES
1 P. J. Cohen, The independence of the continuum hypothesis, Proc. Nat. Acad. Sci. 50 (1963), 1143-1148, and51 (1964), 105-110. 2 S. Feferman, Some applications of the notions of forcing and generic sets, Fundamenta Mathematicae 56 (1965), 325-345. 3 A. A. Fraenkel and Y. Bar-Hillel, Foundations of Set Theory, Amsterdam, 1958. 4 K. Godel, The Consistency of the Continuum Hypothesis, 3rd printing, Princeton, 1953. 5 K. Godel, Remarks before the Princeton Bicentennial Conference, in The Undecidable (M. Davis, ed.), New York, 1965, pp. 84-88. 6 A. Levy, A generalization of Godel’s notion of constructibility, Journal of Symbolic Logic 25 (1960), 147-155. 7 A. Levy, Definability in axiomatic set theory I, Proceedings of the 1964 Znternational Congress for Logic, Methodology and Philosophy of Science, Amsterdam, 1965, pp. 127151. 8 A. Levy, A hierarchy of formulas in set theory, Memoirs of the Amer. Math. Sot. 57 (1965). 9 B. Scarpellini, A characterization of &sets, Trans. Amer. Math. Sot. 117 (1965), 441450. 10 J. R. Shoenfield, The Problem ofpredicativity: Essays on the Foundations of Mathematics, Jerusalem, 1961, pp. 132-139. 11 W. Sierpitiski,~CardinaI and Ordinal Numbers, Warsaw, 1958.
Received June 16,1968
Revised manuscript received October 17,1968