Elementary Embeddings and Infinitary Combinatorics

Elementary Embeddings and Infinitary CombinatoricsAuthor(s): Kenneth KunenSource: The Journal of Symbolic Logic, Vol. 36, No. 3 (Sep., 1971), pp. 407-413Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2269948 .

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THE JOURNAL OF SYMBOLIC Looic Volume 36, Number 3, Sept. 1971

ELEMENTARY EMBEDDINGS AND INFINITARY COMBINATORICS

KENNETH KUNEN1

?0. Introduction. One of the standard ways of postulating large cardinal axioms is to consider elementary embeddings, j, from the universe, V, into some transitive submodel, M. See Reinhardt-Solovay [7] for more details. If j is not the identity, and K is the first ordinal moved byj, then K is a measurable cardinal. Conversely, Scott [8] showed that whenever K is measurable, there is such a j and M. If we had assumed, in addition, that A(2K) c M, then ic would be the Kth measurable cardinal; in general, the wider we assume M to be, the larger K must be.

A natural question, posed by Reinhardt, is whether one can actually assume that M = V. The answer is no, however, as we shall show in ?1 by a combinatorial argument. In ?3, we introduce a combinatorial partition property, P. We show, using the existence of a super-compact cardinal, that there are cardinals possessing property P. By ?5, this fact could not have been proved by using merely the existence of a measurable cardinal.

We emphasize that when we say that a map k: M1 -- M2 is elementary, we mean that M1 and M2 are considered to be first-order structures with the e-relation understood; i.e., for every first-order formula (vl ... vm) in a symbol for e, and every xi ... x. e M1,

9(M,)(X 1 ... Xm) 9(M2)(k(xj) ... k(xm)),

where 9(ML) is the relativization of q to Mi. It is intended that our results be formalized within the second-order Morse-Kelley set theory (as in the appendix to Kelley [4]), so that statements involving the satisfaction predicate for class models can be expressed.

?1. Embeddings from V into V. Our main tool here is the following: THEOREM (ERDos-HAJNAL [1]). Let A be any infinite ordinal. There is function

F: OA -* A such that whenever A c A and A = A, F(A) = A. Here, as in Gbdel [3], F"X = range (F r X). The proof of the theorem for an arbitrary A involves a rather ingenious construc-

tion, but in the one special case which we shall need, namely, when A is a strong limit cardinal of cofinality w, the proof is trivial. In this case, AO = 2A, so, if <<?a, ea>: a < 2A> enumerates {X a A: X = A} x A, we may simply pick, by induction on cc, so e OAa {s8: P < oc}, and set F(s,) = {a

Now, as in ?0, assume that j is a nontrivial elementary embedding from V into some transitive class M, with K the first ordinal moved byj. Letj'I = j - jn j O in.

Received May 10, 1970. 1 The author is a fellow of the Alfred P. Sloan Foundation. Financial support received also

from NSF Grant GP 14569.

407



408 KENNETH KUNEN

Let A = sup {ji(K): n < w}. Note that j(A) = A, since j({jn(i): n < w}) = {jR(K):

1 < n < )}. THEOREM. 91(A) # M. PROOF. Let F: "A -- A be as in the statement of the Erdbs-Hajnal theorem.

Let A = {j(o): e < A). If A X M, then, by elementarity ofj, there is an s e "0A C M such that (j(F))(s) = K. But s = j(t), where j(t(n)) = s(n) (n < i), so

K = (j(F))(j(t)) = j(F(t)) e range (j), which is impossible.

In the proof, we could have first assumed, by way of obtaining a contradiction, that 9(A) c M. Then, by induction on n, each jR(K) is measurable, so A is a strong limit cardinal of cofinality w. Thus, we really needed only this special case of the Erdbs-Hajnal theorem.

Of course, we have immediately, COROLLARY. Ifj is an elementary embedding from V into V, j is the identity. Remark. At the suggestion of the referee, we point out that the Axiom of

Choice (AC) is used in an essential way in the proof of the Erdis-Hajnal theorem. For example, one may ask whether the Axiom of Determinateness (AD) plus the Axiom of Dependent Choices (DC) implies that the Erdos-Hajnal theorem fails for some A. Furthermore, it is unknown whether the Corollary can be proved in ZF, or, for that matter, whether it can be refuted in ZF + AD + DC.

?2. Iterating elementary embeddings. We continue to assume that J, M, K are as in ?1. In this section, we show that each JR is an elementary embedding of V into some transitive submodel. There is really nothing essentially new here; it is merely a matter of checking that Gaifman's construction [2] for the case that is definable from a set goes through in general.

DENm ON. If K is any class and k is a function on V, k(K) = U {k(K r R(a)): a e ORD}.

Note in particular that j(V) = M. By an application of the reflection theorem, LEMMA. If K1 .. * *, Km are any classes, j is an elementary embedding from

(V; e, K1 ... * Kj) into (M; E,j(K1), * *, j(.m)) TEOEM. Each j" is an elementary embedding from V into some transitive class

Mn. If OM c M', then aM, c Ms. PROOF. We proceed by induction on n. Assuming the theorem for n, let

Mn+l - j(M. By the lemma applied to (V; jR, M), M,+l is transitive and j(jR): Ma- M,+l is elementary, so that j(J") o j: V-* M. +1 is elementary. But j(UR) oj=j" + since

}(Jn) 2) {<i(3 ill+(X)> XE ev}. If OM c M and we assume inductively that 'M, c M., then by applying the lemma to(V; Mj)wehavethatl(a)MR+l ( r M c M,+1,soOM,+l = Mn +1 M C Ms+1.

The M, of the theorem are actually determined uniquely, since M,, = jR(V). We could, following Gaifman, define M? for e E ORD, and elementary embed-

dings J4,: Mc .- M, for f s 71. Then JR would be jok. Gaifman's proof that all the M? are wellfounded goes over without change here.



ELEMENTARY EMBEDDINGS 409

?3. Super-compact cardinals. DEFINITION. K is super-compact iff for all a, there is a transitive M and an

elementary embedding j: V-+ M such that: (i) aM cM; (ii) j(f) = for all < K;

(iii) j(K) > a.

This is the usual definition which, in [7], is shown to be equivalent to the existence of certain types of ultrafilters on S,(a) for arbitrarily large a. As a technical point, we remark that

THEOREM. In the above definition, clause (iii) may be weakened to j(K) > K.

PROOF. Suppose Mj satisfy (i) and (ii) for a particular a. By (i) and ?1,jn(K) > a for some n. By ?2, jl( V), jn satisfy (i)-(iii) for a.

Now consider the following combinatorial property: DEFINITION. P(A) is the assertion: Whenever F: 'A - A, there is an infinite

A c A such that A $F(A). Of course, A' > A A P(A) -* P(A'). THEOREM. If K is super-compact then 3AP(A). PROOF. We show P(K+). Let F: O(K+) -K+. Let M, j satisfy tie definition of

super-compactness for a = K+. Let A = {j(e): 6 < K+}. Then A e M and, as in ?1, K ? U(F))#(')A), so

M P 3X c j(K +)[Y $ (j(F))'"(OX)].

By elementarity ofj, 3 XC K+ [TF"(10X)].

Of course, by super-compactness, 3AP(A) - A < KP(A). The Erdos-Hajnal theorem is a refutation of a stronger version of P in which we

require that A=A. Many similar partition properties involving infinite sequences are refutable by similar arguments, although we are unable to refute P itself. How- ever, by the results of ?5, 3AP(A) is not provable in ZFC, even under the assumption of the existence of several measurable cardinals.

?4. Remarks on partition properties. Examining the proof Of P(K+) for K super- compact yields a seemingly stronger property, in which F'(01A) is replaced by the closure of A under F. However, by part (ii) of the following lemma, these properties are always equivalent.

LEmmA. Suppose B is any class and, for a < 20, Fa: 0B -- B. Then there are functions G1, G2: GB -. B such that whenever A c B and I > 2,

(i) G;(1A) = U{Fa(0A): a < 2X1}; (ii) G2(0A) is the closure of A under the Fa; i.e.,

Gf(*A)= n{X c B: A a X A Vs e '0XVa < 2w[Fa(s) e X]}. PROOF. For (i), let fl: '0B -* 2c0, f2: 0B -* "B be such that

Vx a BVa < 20'Vt e WB[[2 < X < to A range (t) c x] - s E cOB[range (s) = x A f1(s) = a A f2(s) = t]

Now, let G(s) = Ff1(,)f2(s)). For (ii), let i"t(s) = s(n) for s e 10B, n < w. We may assume that the In are among



410 KENNErH KUNEN

the FOR since this will not change closures. Let H8 (8 < 2'0) enumerate the collection of all functions obtainable from the F,, by composition-i.e. the least collection of functions, A', containing the FOO such that whenever H, E E*F (p < cv),

{<s, H,(<HH(s): n < w>)>: s E W0B} e i. Then the closure of any A under the H" is the same as the closure under the F,,. Furthermore, this closure is just U{H '(,WA): , < 2W}, since any H(<H(K (sn): n < a)>) is equal to

HY(<H,0nW<2n.2M(*) m < co>): n < co>), where t(2n. 3n) = sn(m)(n, m < Co). (ii) now follows by applying (i) to the Hi.

Actually, the statement of the Erdos-Hajnal theorem in [1] is with the closure of A under F, rather than with F"(W0A), and was intended as a proof of the existence of Jonsson algebras in every cardinality if one admits infinitary functions. Of course, by our lemma, the two forms of the theorem are equivalent. Also, a trivial modification of the proof in [1] yields directly the form quoted here.

?5. Further discussion of property P. It follows from the Erdos-Hajnal theorem and the lemma of ?4 that P(A) implies A > (2W)+. We do not know whether it is possible that P((2W) + +). However, P is a large cardinal property in the sense that if 3AP(A), then there are inner models with several measurable cardinals. More precisely,

THEOREM. Suppose 3AP(A). Then for all ordinals 0 < w1, there is a transitive proper class, N, such that

N F [ZFC + there are 0 measurable cardinals]. [5, ?10] and [6] present two somewhat different methods for deriving the same

conclusion (but for all 0) from the assumption of the existence of a strongly compact cardinal. Many parts of the present proof can be transcribed directly from corresponding parts of one or the other of these earlier proofs. We shall concen- trate here on showing how the parts fit together, referring the reader to [5] and [6] for those details which are not essentially new. The general outline will follow [6]; in particular, Definitions 1-4 will be taken verbatim from there.

The proof of the theorem occupies the rest of this section, and we assume 3AP(A) from now on. We first derive, from the combinatorial property P, a statement involving elementary embeddings. This type of argument is due to Rowbottom and Silver.

LEMMA 1. There at least (2W) + different ordinals, 8, such that there exist M, i with the following properties:

(i) s C M. (ii) M is a transitive model of ZFC. (iii) i is an elementary embedding from M into V. (iv) i(8) > S. (v) i r 8 is the identity. (vi) WM c M. PROOF. We first show that there is at least one such S. Let F,: WV-E V be

Skolem functions for < V; e>. More precisely, we assume that whenever .v0 ?... Vk -)




is a first order formula of set theory of the form 3vfA.(V. * k - lsk), then for some n and all s e 0V,

p(s(O) e s(k - 1)) -> 1(s(O) ... s(k - l)F.(s)). Also, let F.: 0 V- V be the identity (i.e. F0(s) = s for s E V). By (ii) of the lemma of ?4, let G: 0 V- V be such that G "(OA) is the closure of A under the Fa(a < w) whenever A 2 2. Assume P(A), and define C: 0A -*,A so that G(s) = G(s) if G(s) E A; C(s) = 0 otherwise. Let A be an infinite subset of A such that A $ Cf(OA). Let C = G'(0A), and let S be the first ordinal not in C. Then <C; e> < <V; e>, XC c C, and card (C n ORD) 2 a +. Thus, if M is the transitive model isomorphic to C and i is the isomorphism from M onto C, it follows that 8, M, i satisfy (i)-{vi).

Suppose now that 8. (p < 2c) satisfy the conditions of Lemma 2. Define F. +1 + : 0V -- V so that F,, + +8(s) = Sfl for all s. Go through the above proof, but now using all the F, (a < 20). Then the C we obtain will contain all the 8a, so the 8 we obtain will be different from all the 8Sf. Thus, there are at least (20) + such 8, proving Lemma 1.

We remark on some technical points. In carrying out the proof of Lemma 1 in Morse-Kelley set theory, the global form of the axiom of choice (needed to define the F,) could be replaced by the local form by using Skolem functions just for some R(a), where a > A, cf(a) > w, and <R(a), e> < < V, e>. Actually, the whole proof of the theorem could be formalized within ZFC by considering elementary submodels of a suitable R(a) rather than of V.

We really need only co, ordinals satisfying Lemma 1. Let 8 (I < wl) be an increasing sequence of such ordinals, with Mc and ic the corresponding models and embeddings. Also, fix K to be some regular cardinal greater than all the 8.

We shall now forget about property P and argue directly with the 8c, Mc, ic. The power of Lemma 1 lies in condition (vi). If we had omitted this condition, we could have dealt just with the Skolem functions, F,, which are essentially finitary. The lemma would then follow from the existence of a Ramsey cardinal, which is clearly insufficient to imply the theorem.

DEFINITION 1. For any ordinal 0, a 6-set is a set b c ORD such that b has order type w 0 and such that for each 4 E b, 4 > sup (b ( 0).

DEFINITION 2. If b is a 0-set, e < 0, and n < w, (a) y(b, f, n) is the wA g + nth ordinal in b and A(b, e) = sup ({y(b, e, m): m < a}D. (b) JF(b, e) = {x c A(b, 6): 3mVk > in[y(b, A, k) E x]}; S#F(b) = <F(b, -):

77 < 0>.

(c) 1(b, e) is the statement that .F(b, e) n L[.F(b)] is, in L[,F(b)], a normal ultrafilter on A(b, e).

DEFINITION 3. (a) K0 = {a:a is a cardinal A cf(a) > K A VT < a[Tc' < a]}. (b) K. + 1 = {a e K: card (K, n a) = a}. (c) K8 = nfle< aKe for limit ordinals P. DEFINITION 4. If a c ORD and X a L[a], then H(a, X) = {y e L[a]: y is

definable in L[a] from elements of a U X U {a}}. Similarly to [6], we want to show that if 0 < co,, b is a 6-set, and b c KR.6 + 1,

then Ve < 60(b, e). There, we obtained our results by iterating ultrapowers by



412 KENNETH KUNEN

an ultrafilter on K. Now, K need not even be measurable. Instead, we shall use the ultrafilters on the 8 which arise from the embeddings ic.

DEFINITION 5. For X < co,

VC = {x e Y(S,) ri Ma: 8S e ic(x)}. LEMMA 2. For ; < w1, (i) Otc is an ultrafilter in the Boolean algebra Y(8f) n Mc. (ii) q&c is closed under countable intersections. LEMMA 3. If b is a countable set of ordinals and ; < cu1, then 6Y(8c) n L[b] c Mc. PROOF. If x e- t(8) rn L[b], then, by a L6wenheim-Skolem and collapsing

argument, x e L,,[b'] for some z < 8 and b' a countable subset of ,u. Hence, x e Mc by (i) and (vi) of Lemma 1.

LEMMA 4. Suppose b is a countable set of ordinals, C < 1, and b - S' c K0. Then (i) '?c n L[b] is an L[b]-ultrafilter on 8S (see [5, Definition 1.1]), (ii) Ultv(L[b], Vc 'r L[bD is wellfoundedfor all v. PROOF. By Lemma 2 (ii), part (ii) will follow once we establish (i) (see [5,

Theorem 3.6]). Let Y1 = '14 rf L[b]. By Lemmas 2 (i) and 3, / satisfies all the conditions for being an L[b]-ultrafilter except possibly (v) of [5, Definition 1.1]: namely, that whenever <x;,: p < 8C> e Mc, {,u: x,, e r } e Mc. This condition is necessary for the iterated ultrapowers to be defined, but in any case we may define Ult1(L[b], 1'), which will be well founded by Lemma 2(ii). The assumption that b 8 c K0 insures that b is fixed by the embedding i<: L[b] Ult1(L[b], 1'). Hence, Ult1(L[b], Y) = L[b], so r (which equals {x E 91(C) n L[b]: 8; e iol(x)}) is an L[b]-ultrafilter on 8; (see [5, Lemma 4.6]).

LEMMA 5. If b isa 0-set, C < w1, 0 < w1, and f < 0 is such that V77 < f[A(b, 7q) < AC] and VnV[e < < 0 -< y(b, -q, n) E KO], then F(b, e).

PROOF. Use iterated ultrapowers of L[b] by '14 r) L[b]. The details are exactly the same as in the proof of [5, Lemma 10.10], so we omit them.

Lemma 5 is analogous to Lemma 6 of [6]. The easier proof given there will not work, however, since the S may not really be measurable.

We now state a more complicated version of [6, Lemma 7], which will enable us to collapse down a 0-set for an application of Lemma 5. The proof is almost identi- cal to the one in [6]. If x c ORD, we use ord (x) to denote the order type of x.

LEmmA 6. Let a c K. be of order type a, where a < w1. Say a = {a:s < cc} in increasing enumeration. Then for each ; < a, ord (ac r) H(a, KC+1 ) a < SC.

Now, to prove the theorem, let b be a 0-set, 0 < co,, and b C K0..+,. Let , < 0. Let A = H(b, K. - sup ({A(b, 77): < e})), and j the isomorphism from A onto the transitive collapse of A. For e < q < 0 and n < co, j(y(b, -q, n)) = y(b, rw, n) e K0, and for t < e and n < c, j(y(b, -q, n)) < S,, (by Lemma 6). By Lemma 5 (applied with some C between wc t and wl), OD(j(b), J0, and hence b(b, 0). Thus, Ve < 00(bg e).

?6. Conclusion. There are many other properties related to P that one may obtain by suitable large cardinal axioms. For example, let Q(K, X, A) be the statement that K, 9r, A are cardinals, K < 1T< A, and whenever F: 0(A) -* r, there is an A a A such that A = ir and card (F"(WA)) < K. Clearly Q(K, X, A) implies P(A).




To obtain a triple satisfying Q, assume in the notation of ?1, that j: Va- M is such that 12(K)M c M. Then Q(K, j(K), j2(K)).

Similarly to Lemma 1 in ?5, we can, via Skolem functions, translate Q(K, 7r, A) into a statement about models, obtaining a form of Chang's two-cardinal con- jecture in the language i' Namely, whenever we have a two-cardinal model, <B, U,. * * > for 2 of type <A, 7r>, it has an elementary submodel (in the sense of 2 of type <(O, K>. Conceivably, these ideas may be used to prove theorems about ,, We have been unable to discover any such theorems, however, that do not follow, as above, by trivially tracing through definitions.

REFERENCES

[1] P. ERDws, and A. HAJNAL, On a problem of B. Jdnsson, Bulletin de l'Acaddmie Polonaise des Sciences. Sirie des Sciences Mathimatiques, Astronomiques et Physiques, vol. 14 (1966), pp. 19-23.

[2] H. GAIFMAN, Pushing up the measurable cardinal, Lecture notes 1967 Institute on Axiomatic Set Theory (University of California, Los Angeles, Calif., 1967), American Mathematical Society, Providence, R.I., 1967, pp. IV R 1-16. (Mimeographed).

[3] K. GODEL, The consistency of the continuum hypothesis, Princeton Univ. Press, Princeton, 1940.

[4] J. L. KELLEY, General topology, D. Van Nostrand Co., Inc., Princeton, 1965. [5] K. KUNEN,, Some applications of iterated ultrapowers in set theory, Annals of Mathematical

Logic, vol. 1 (1970), pp. 179-227. (61 , On the GCH at measurable cardinals, Proceedings of 1969 Logic Summer School

at Manchester, pp. 107-110. [7] W. RENHARDT and R. SOLovAY, Strong axioms of infinity and elementary embeddings,

to appear. [81 D. Scorr, Measurable cardinals and constructible sets, Bulletin de l'Acadimie Polonaise

des Sciences. Serie des Sciences Mathimatiques, Astronomiques et Physiques, vol. 9 (1961) pp. 521-524.

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Elementary Embeddings and Infinitary Combinatorics