Logic, Foundations of Mathematics, and Computability Theory || Ineffability Properties of Cardinals II

JAMES E. BAUMGARTNER'

INEFFABILITY PROPERTIES OF CARDINALS II

O. INTRODUCTION

This paper applies the methods of [1] to several classes of cardinals other than ineffables. The central point is the same as [1], namely that many 'large cardinal' properties are better viewed as properties of normal ideals than as properties of cardinals alone, and that in order to understand these properties fully it is necessary to consider the associated normal ideals.

We begin in Section 2 with a treatment of the weakly compact ideal on a weakly compact cardinal. This ideal is easily defined in terms of lI~-indescribability (see [8] and [1]), but not in terms of the usual combinatorial definitions of weak compactness. We take three such common definitions (in terms of partitions, trees, and ultrafilters in fields of sets) and show how to strengthen them to obtain combinatorial properties which easily yield the existence of the weakly compact ideal. The existence of the weakly compact ideal easily implies strong versions of the usual theorems that weakly compact cardinals are 'large' (i.e. Mahlo, hyperMahlo, etc.), so one by-product of this approach is a purely combinatorial proof that weakly compact cardinals are large. Other such proofs are known, and some of them are much shorter (see Kunen [6], for example), but none of them yield the weakly compact ideal.

The weakly compact ideal has also been exploited in [3], where it is shown that the non-stationary ideal on a weakly compact cardinal K is not K + -saturated.

In Section 3 we consider the result of iterating into the transfinite the 'operation' which originally produced ineffable sets from stationary sets. The eventual results about this iteration, however, do not seem as interesting as the method which is developed to treat stages a of the

1 Preparation of this paper was partially supported by National Science Foundation grant number GP-38026.

Buns and Hintikka (eds.), Logic, Foundations of Mathematics and Computability Theory, 87-106. Copyright © 1977 by D. Reidel Publishing Company, Dordrecht-Holland. All Rights Reseroed.

88 J. E. BAUMGARTNER

iteration on a cardinal K such that K ~ a < K +. The method involves 'canonical' sequences of functions ordered by eventual dominance. Several results about these sequences are obtained, including a new characterization of ineffablos in L. It seems quite likely that this eventual dominance method may be used to handle other unrelated iterations as well.

Section 4 is devoted to extending the results of [1] to Erdos and Ramsey cardinals. We begin by associating normal ideals with Erdos cardinals and with the a-partition cardinals of Drake [4]. This leads naturally to the definition of new classes of cardinals, the a-Erdos cardinals, which generalize the properties of Erdos and a-partition cardinals. Moreover, this definition yields an interesting class of cardinals even when a is a successor ordinal.

Jensen and Kunen [5] showed that ineffable cardinals are n~indescribable and subtle, but that the converse does not hold. Perhaps the most striking illustration that properties like ineffability, indescribability, and subtlety are really properties of normal ideals is the result in [1] that a cardinal K is ineffable iff the ideal of non-subtle sets and the ideal of non-n~-indescribable sets together generate a nontrivial K-complete normal ideal, which then coincides with the ideal of non-ineffable sets. This result is extended in Section 4 to Ramsey and ineffably Ramsey cardinals. An analogue of subtle and inaccessible cardinals, the pre-Ramsey cardinals, is defined and a version of the theorem above is obtained in which ineffable, subtle, and n~indescribable are replaced by Ramsey, pre-Ramsey, and n~indescribable, respectively.

1. NOTATION AND TERMINOLOGY

Our set-theoretical usage is standard. If x is a set, then ~(x) is the power set of x and Ixl is the cardinality of x. If x and yare sets then Xy is the set of all functions from x into y. If x is a set and A is a cardinal, then [x]" = {y s; x:lyl = A} and [xt A = {y s; x: Iyl < A}.

We assume the axiom of choice throughout, so cardinals are identified with intial ordinals.

A tree is a partially ordered set (T, ~T) such that for all t E T, {s E T: S~Tt} is well-ordered by ~T' and such that T has a unique

INEFFABILITY PROPERTIES OF CARDINALS II 89

minimal element, the root of the tree. A branch through a tree (T, :O::;;;T)

is a linearly ordered subset; its length is its order type. The ath level of the tree (T, :O::;;;T) is {t E T: {s E T: S~Tt} has order type a + 1}.

The set of all sets of rank less than a is denoted by Va. A second order formula cp is II~ if cp = (Q1X1) ... (QnXn)t/I, where t/I is firstorder, each Xi is a second order variable, and Qi is V if i is odd and 3 if i is even. A cardinal K is' II~-indescribable iff for any II~-sentence cp and relations R b ... , Rm on VK , if (VK , E, R b ... ,Rm)Fcp, then there is a < K such that (Va, E, R1iVa, ... , Rml Va)Fcp (here Ri I Va denotes the restriction of Ri to Va).

A function f defined on a set of ordinals A is regressive on A iff f(a) < a for all a E A such that a> O. An ideal I on a cardinal K is K - complete if I is closed under unions of size < K; I is normal if for any A£; K, if Ai I and f is regressive on A, then there is a < K such that rl({a}) i 1. Fodor's Theorem asserts that the non-stationary ideal on a regular uncountable cardinal is normal.

If f is a function on [Xr for some n < w, then we say a set Y £; X is homogeneous for f if f is constant on [yr. The notation K~(A)~ means that for any f: [Kr ~ IL, there is X£; K such that IXI = A and X is homogeneous for f. The notation K~(a)~W means that for any f: [KrW ~ A there is X£; K such that X has order type a and for each n < w, X is homogeneous for f I [Kr.

If X is a set of ordinals, then a function f defined on [X]n (or on [X]<W) is regressive if f(x) < min (x) for all x E [x]n (resp.: all x E [Xr W

). A cardinal K is subtle if for any regressive f on [K]2 and any closed unbounded set C £; K, there is A £; C such that IA I ;;l!: 3 and A is homogeneous for f. The definitions of almost ineffable and ineffable are obtained from the definition above by replacing 'IA I ;;l!: 3' by 'IAI = K' and 'A is stationary', respectively. All three notions have reformulations in the following style: e.g. K is ineffable iff for any (Sa: a> K), if Sa £; a for all a then there is A£; K such that A is stationary and for all a, f3 E A, if a < f3 then Sa = S~ n a. In the latter situation we say A is homogeneous for (Sa: a < K). The fact that both types of formulations exist is used in Section 3, where we state without proof the analogous fact for a generalization of ineffability. The proof is the same (see [5]).


2. THE WEAKLY COMPACT IDEAL

The purpose of this section is to show that every weakly compact cardinal bears a natural norI,!1al ideal, the weakly compact ideal.

There are a great number of equivalent definitions of weak compactness (see [10]) but the following four are probably the most frequently used:

Assume that K is strongly inaccessible and uncountable. Then the following are equivalent.

(a) K is II~ -indescribable. (b) K~(K)~. (c) K has the tree property, i.e. if (T, ~) is a tree of cardinality K

and every element of T has < K immediate successors, then T has a branch of length K.

(d) If S s; ~(K) is a K-complete field of sets and lSI = K, then there is a non-principal K-complete ultrafilter on S.

If K is weakly compact then K is not the first strongly inaccessible cardinal but this result is obvious only from definition (a) above. For to say that K is strongly inaccessible is a II~ statement about (VK , E); hence there is a < K such that a is inaccessible. Apparently, then, the combinatorial definitions (b), (c), and (d) are not as powerful as (a). We would like to argue that the reason is that there is a normal ideal on K naturally definable from (a), but not from (b), (c), or (d).

Let us say that A S; K is II ~ - indescribable iff for any II ~ -sentence lP and any relations R b ..• , Rn on VK , if lP is true in (VK , E, Rb ... ,Rn) then for some a E A, lP is true in (V"" E, R1jV"" ... , RnIV",). Let 11 ={A S; K: A is not II~-indescribable}. It is easy to see that 11 is an ideal. Moreover, 11 contains all nonstationary subsets of K. For if C is closed and unbounded in K then (VK' E, C)FlP, where lP says C is closed and unbounded in K. If a < K

and (V"" E, cn V",)FlP, then clearly a E C. Hence K - CE 11. In fact, 11 is normal, as will follow from Theorem 2.7 below. This

result is essentially due to Levy [8]. It is also proved in [1]. It turns out, however, that if we replace definitions (b), (c), and (d)

with (b'), (c'), and (d') below, then the ideal is readily obtainable from each of them, and so are the results about the size of K.


(b') K~(K, stationary set)2. i.e. if f: [KY~2, then there is As; K such that either IAI = K and f is constantly 0 on [AY, or else A is stationary and f is constantly 1 on [A]2.

(c') Suppose (T, ~ T) is a tree and T = K. If a ~ T f3 implies a ~ f3 and the set of immediate successors of each a E T is nonstationary, then T has a branch of length K.

(d') If S is a K -complete field of sets, F is a collection of regressive functions on K,rl({a})ES for each fEF and aEK, and ISI= IFI = K, then there is a non-principal K-complete ultrafilter U on S such that every f E F is constant on a set in U.

Note that in each case we have obtained the primed version from the unprimed version by substituting the ideal of non-stationary sets for the ideal of sets of cardinality less than K at an appropriate point.

Let PI = {X S; K: X is II~-indescribable}. Let Pz be the set of all X S; K such that X ~ (K, stationary set)2, i.e.

such that in (b') we can always find A so that A S; X. Let P3 be the set of all X S; K such that every tree (T, $T) as in (c')

with universe T = X has a branch of length K. Let P4 be the set of all X S; K such that, given F and S as in (d') with

XES, there is U as in (d') with X E U.

THEOREM 2.1. Assume K is strongly inaccessible. Then PI = P2 = P3 = P4 •

The proof follows from Lemmas 2.2-2.5. Assume K is strongly inaccessible.

LEMMA 2.2. PI S; P2 •

Proof. Suppose X E Pl' Let f:[XY~2. For each a E X define a sequence (s~, ~ < ()",) inductively as follows. Given s~ for all ~ < TJ, let s~ be the least SEX such that s~< s < a and f({s~, s}) = f({s, a}) = 0 for all ~ < TJ, provided such s exists; otherwise let ()a = TJ. Now let h(a) = sup {s~: ~ < ()a}.

Case 1. {a: h(a)<a} is stationary in K. Then by Fodor's Theorem there is stationary S such that h is constant on S. But now, using the fact. that K is strongly inaccessible, there is stationary S's; S such that (s~: ~ < ()a) = <s~: g < ()(3) for all a, f3 E S'. But then if {a, f3} E [S,]2 it is clear that f({a, f3}) = 1.

92 1. E. BAUMGARTNER

Case 2. {a: h(a)<a} is non-stationary, hence belongs to II. Thus we may as well assume that h(a) = a for all a e X. Let cp be the assertion that there is no cofinal subset A of X such that I is constantly o on [At Then cp is a n~ assertion about (VK , E, I, X) so, since X e Ph if cp is true in (VK , E, I, X) then there is a e X such that cp is true in (Va, E, I IVa, XI Va), and the latter is false since {s;: ~ < 8a} is a counterexample. •

LEMMA 2.3. P2 !; P3•

Proof. Suppose X e P2. Let (T, ::;;;T) be a tree such that X = T, a ::;;;TP implies a::$; p, and the set of immediate successors of each a e T is non-stationary. For each a e T, let pa be the sequence obtained by writing {p:p::;;; ra} in ::;;;T-increasing order with the minimal element omitted. If a, peT, a < p, let I({a, P}) = 0 if plJ lexicographically (with respect to the usual ordering on ordinals) precedes pa; I({a, P}) = 1 otherwise.

Case 1. There is A!; X such that IAI = K and I is 0 on [At If a, peA, a < p, then plJ lexicographically precedes pa so there must be some ao such that if p > ao then the least element of plJ is always the same, say Xo. Repeating the argument, we can find al > ao so that the next element of each plJ for (3 > a 1 is always the same, say Xl.

Proceeding inductively, and using the fact that K is regular, we can find XE and aE for each ~ < K. But then {XE: ~ < K} is a branch of length K.

Case 2. There is stationary A!; X such that I is 1 on [A]2. If a, (3 e A, a < (3, then pa lexicographically precedes plJ. For each a e A let g(a) be the least element of pa. Then g is regressive on A so g is constant on a stationary set. Hence there is ao < K so that if (3 > ao then the least element of plJ is always the same, say Xo. Now proceed inductively as in Case 1. •

LEMMA 2.4. P3 !;P4 •

Prool. Let Sand F be as in (d'). Let h: K -+ S U F be one-to-one and onto. For each a, we define a partition Da of K inductively as follows: Let Do = {K}. If h(a) = A e S then Da+l = {Any: YeDa}U{Y-A: YeDa}. If h(a)=/eF then Da+1 = lfl({(3}) n Y: Y E Da , (3 E K}. If a is a limit ordinal then Da is the collection of all sets of the form n {Xli: ~ < a} where XIJ e DIJ. Note that if a < (3 then DIJ refines Da.


Now suppose X E P3. We define a tree (T, ~T) with T= X as follows. Let Ta denote the ath level of T. We determine Ta by induction on a. Given T{3 for all f3 < a, let Ta = {g EX: (3 Y E Da)g is the least element of Y - U {3<aT{3}'

If g ETa, 11 E T{3, f3 < a, g EYE Da , 11 E Z E D{3, put 11 <yg iff Z £ Y. It is easy to see that if h(a) E S then every member of Ta has at most two immediate successors, and if h(a) = IE F then I is one-to-one and regressive on the immediate successors of each element of Ta. Hence the set of immediate successors of each element of T is non-stationary. Since X E P3 there is a branch B £ T of length K. Let U = {A E S: IB - AI < K}. It is easy to see that U works. •

LEMMA 2.5. P4 £ Pl'

Proof. Let X E P4. Let q:J be 1I~ and assume (VK' E, Ilh ... ,R,,)Fq:J. Let At= (VK, E, <10 R h • •• , R,,) where <1 is a well-ordering of VK• Let G be the set of all functions from K into VK which are definable from parameters in At, and let F be the set of all regressive functions in G. Let S be the K-complete field of sets generated by X together with all the subsets of K which are definable from parameters in At. Let U be a non-principal K-complete ultrafilter over S such that X E U and every IE F is constant on a set in U.

Now form the 'definable ultrapower' of At modulo U. That is, if I, g E G, let 1= g iff {a: f(a) = g(a)}E U and let [f] be the equivalence class of f. Define relations R: on V' = {[f]: lEG} by R:([/d, ... , [fk]) iff {a: .AtFR j (ll(a), . .. , A(a))}E U. <~ and E' may be defined similarly. Let .At' = (V', E' <~, ... ,R~). Since <1 is a well-ordering, Skolem functions for .M, are definable, and so we have a version of ~s's Theorem:

where ." is any first-order formula appropriate for At. It follows that the embedding i: .At-.At' given by i(x)=[fx] is elementary, where Ix is the constant function with value x. Since U is K-complete, .M,' is well-founded, hence is isomorphic to a structure N = (N, e, <2, Sh ... , SrI) where N is a transitive set. We identify At' with N, By standard arguments the K-completeness of U implies that i is the identity on VK so VK £; N. Moreover, since each regressive function


in G is constant on a set in U, it follows that [id] is identified with K, where id is the identity function on K.

If cp = "I XI{I(X) , let cp(K) be "Ix(x £; VK ~ ~(x», where ~ is the same as I{I except that all quantifierS- are restricted to VK • Then cp(K) is first order, and since .MFcp and VK £; N (so VK EN), we have J{FCP(K). Since K=[id], we have Y={a:,AtFcp(a)}EU by tos's Theorem. Now ,AtFcp(a) iff (Va, E, Rl I Va,' .. , Rnl Va)Fcp, and since X E U we have xn Y;i'O. Thus XEP1 ••

COROLLARY 2.6. If K is strongly inaccessible then (b '), (c '), and (d ') are equivalent formulations of weak compactness.

Proof To say that K is weakly compact is the same as saying K E P b

so we are done by Theorem 2.1. •

In view of Theorem 2.1, it seems reasonable to say that weak compactness is really a property of subsets of a cardinal K. We refer to the set P = Pi (1 :s;; i :s;; 4) as the set of weakly compact subsets of K. The ideal 1= {X £; K: X $ P} is the weakly compact ideal, and the filter {X £; K: K - X E I} is the weakly compact Jilter.

THEOREM 2.7. The weakly compact ideal is normal.

Proof. The proof can be given equally well from any of the definitions (a), (b'), (c'), or (d'). We illustrate with (b'). It will suffice to show that if f is regressive on X E P2 then for some a, rl({a}) E P2. Suppose not. For each a, let ga: [rl({a})]2~2 be a counterexample to r 1({a})EP2. Define g:[X]2~2 as follows. If a,{3EX, a=f{3, let g({a, (3}) = 0 iff 3 . .J(a) = f({3) = y and g.y({a, (3}) = O.

Case 1. There is A £; X such that IAI = K and g is 0 on [A]2. Then clearly 3yg(a) = y for all a E A and g.y is 0 on [A]2, contradiction.

Case 2. There is stationary A £; X such that g is 1 on [Af Since f is regressive on A, by Fodor's Theorem there is y and stationary B £; A such that f is constantly y on B. But then gy is 1 on [B]2, contradiction. •

It follows from Theorem 2.7 and the proof of Lemma 2.2 that a much stronger version of (b') holds for weakly compact cardinals, namely, K~(K, weakly compact set)2. The proof is left to the reader. In fact, for this partition property one may even begin with a partition


of [AY, where A is weakly compact, rather than with a partition of [Kf

The fact that weakly compact cardinals are large is perhaps best illustrated by the following two theorems.

THEOREM 2.8. If K is weakly compact, then {a < K: a is strongly inaccessible} belongs to the weakly compact filter.

Proof. The easiest proof uses II~ -indescribability. Since we have asserted that the combinatorial definitions (b'), (c'), and (d') readily yield information about the size of K, however, we illustrate with (b'). First note that since K is strongly inaccessible, {a < K: a is a strong limit cardinal} is closed and unbounded in K, hence must belong to the weakly compact filter. It will suffice, therefore, to show that {a < K: a is regular} belongs to the weakly compact filter. Suppose not. Since the cofinality function is regressive on {a < K: a is singular}, we may find >.. and A E P2 such that efa = >.. for all a E A.

For a E A, let <s~: ~ < A) be an increasing sequence cofinal in a. If a, (3 E A, a < (3, let f({a, (3}) = 0 if (s~: ~ < >..) lexicographically precedes (s~: ~ < >..), and f({a, (3}) = 1 otherwise. Now arguing as in the proof of Lemma 2.3, we can find a homogeneous set X £; A, and SI; and al; for each ~ < >.. such that if a > al; and a E X then s~ = sl;' But this is a contradiction, since then (s~: ~<>")=(Sl;: ~<>..) whenever aEX, a> sup {al;: g < A}. •

Similarly, one can prove

THEOREM 2.9. If K is weakly compact and S £; K is stationary, then {a < K: S n a is stationary in a} belongs to the weakly compact filter.

It follows immediately that {a < K: a is Mahlo} belongs to the weakly compact filter, and so forth. Theorem 2.9 has also been utilized in [3] to show that if K is weakly compact, then the ideal of non-stationary subsets of K is not K + -saturated. In fact, the result yields that if X£; K is weakly compact, then the non-stationary ideal restricted to X is not K + -saturated.

It is natural to ask whether a purely combinatorial treatment of the theory of weakly compact cardinals can be given. To be more specific, can one eliminate the role of II~-indescribability and exhibit a direct combinatorial path from definitions (b), (c), and (d) to (b'), (c'), and


(d')? The answer is affirmative, and one approach is in Theorem 2.10 below. In this connection, it should be remarked that in Theorem 2.1 a direct proof that P4 S;;; P2 can be given.

Note that the proof of Theorem 2.10 is really the same as the proof that a measurable cardinal bears a normal ultrafilter.

THEOREM 2.10. (d) implies (d').

Proof Let Sand F be as in (d'). By enlarging Sand F, if necessary, we may assume that

(i) F is closed under composition of functions (ii) if XeS and fe F then rl(X)e S

(iii) if f, ge F then {a: f(a)< g(a)}e S.

By (d), there is a non-principal K-complete ultrafilter U on S. If every f e F is constant on a set in U then we are done, so suppose not. For f,geF, let f<g iff {a:f(a)<g(a)}eU. Since U is K-complete, it follows that < well-orders F (provided we identify functions f and g such that {a: f(a) = g(a)}e U. Let f be the <-least member of F such that f is not constant on a set in U. For XeS, let XeD iff rl(X) e U. It is easy to check that D is a K-complete non-principal ultrafilter on S. Suppose g e F. Then clearly gf < f so gf is constant on a set in U. Say {a: gf(a) = 'Y}e U. But then {a: g(a) = 'Y}e D so we are done. •

If one is only interested in a purely combinatorial proof that weakly compact cardinals are large, then considerably shorter proofs are available. See for example Kunen's proof in [6]. However, the shorter arguments do not seem to yield the weakly compact ideal, and therefore results such as Theorems 2.8 and 2.9 are unobtainable.

3. EVENTUAL DOMINANCE AND A HIERARCHY OF INEFFABLES

The ineffable sets may be regarded as having been produced from the stationary sets by an operation involving a combinatorial propery. In this section we consider the result of iterating this operation transfinitely. First, however, it is necessary to investigate briefly the notion of eventual dominance of functions.

If K is a cardinal, D is a filter of subsets of K, and f, g e KK, then we write f < g (mod D) iff {a: f(a) < g(a)}e D. We say g eventually


dominates f mod D. f ~g is defined similarly. Note that if D is countably complete, then < (mod D) is well-founded. We will usually be interested in the case when D is the filter generated by the closed unbounded subsets of a regular uncountable cardinal K. In this case we write simply f < g.

Now let K be regular and uncountable. Let us call a sequence (fa: a < K +) of elements of K K a canonical sequence iff a < {3 implies fa < f~ and for any other sequence (g.,: a < K +) well-ordered by <, we have fa ~ &ex for all a. Note that if (fa: a < K +) and (g.,: a < K +) are

both canonical, then for each a, {g: fa (g) == g.,(g)} contains a closed unbounded set.

The following theorem is part of the folklore.

THEOREM 3.1. If K is regular and uncountable, then there is a canonical sequence of elements of K K.

Proof. The fa can be determined inductively, but it will be convenient to have a direct construction. If a < K let fa (g) == a for all g < K.

Suppose K s; a < K +. Let ha: K ~ a be an isomorphism, and let <a be the relation on K induced by the usual ordering on a. For each g < K,

let fa(g) be the order-type of (~, <a I g). Now let a < (3. We must show fa < f~. Fix Tlo so that ha( Tlo) ~ a. Let

Ca~=={~<K: Tlo<~ and if TI<~ then h~lh",(TI)<~}. Ca~ is closed and unbounded, and if g E Ca~ then (g, <a I g) is carried into a proper initial segment of (~, <~ I~) by h~lh",. Thus fa(~)<ff3(~) and fa <ff3.

To complete the proof, it will suffice to show that if a < K+ and U: g(g) < fa(~)} is stationary, then there is (3 < a such that U: g(~) s; ff3 (~)} is stationary. We leave to the reader the case when cfa<K.

Suppose cfa == K. Let (ae: ~ < K) be an increasing sequence with limit a, and let Ce = Ca,a, defined as above. Also, if TI < K, let j(TI) be the least ~ such that h", ( TI) < a~. Let C be the diagonal intersection of the q, i.e. C == {~: \iTl < ~)~ E C.J Let C = C n {~: if 1) < ~ then j( TI) < ~}. Then C is closed unbounded, and it is easy to check that if ~ E C then f",(~) == sUP"l<da.(~). By Fodor's Theorem, there is '11 so that {~: g(~) s;

fa.(~)} is stationary. • Kunen [7] has shown that if D is a countably complete filter on a

cardinal K, then there is an ordinal a < (rt such that all subsets of KK


well-ordered by < (mod D) have order type less than a. The identity of the least such a is independent of the usual axioms for set theory, as Kunen also shows. Next we observe that two familiar combinatorial 'axioms' settle the question in different ways.

Let K be a regular uncountable cardinal. Then <>: is the following assertion: There is (Sa: a < K) such that Sa S; PJ(a) for each a, ISal::: la I, and for any X S; K, {a: X n a E Sa} contains a closed unbounded set.

Jensen [5] has shown that in L, K is ineffable iff K is regular and <>: fails.

THEOREM 3.2. <>: implies that there is a sequence (fa: a ~ K +) well-ordered by <, and such that fa (g) < Igl+ for all a and all g.

Proof. Let (fa: a<K+) be canonical. By the proof of Theorem 3.1, we may take fa(g)<lgl+ for all a<K+. By <>'~, there is (S~: g<K) such that S~s;PJ(gxg), IS~I=lgl and for any XS;KXK, {g:xn(gxg)ES~} contains a closed unbounded set. For each g, let fK+(g) = sup {y + 1: some Z E S~ is a well-ordering of type y}. Then fK+(g) < Igl+. If a < K + then there is a closed unbounded set Ca such that for all g E Ca ,

<a I gE Sa. Hence C. s;{g: fa (g) <fK+(g)}. •

Chang's Conjecture is equivalent to the partition relation

which is translated as follows: for any f: [W2r W ~ WI. there is A E

[W2r i such that the range of f on [Ar W is countable. In unpublished work, Silver has shown Chang's Conjecture to be consistent relative to the existence of certain large cardinals.

THEOREM 3.3. Chang'S Conjecture implies that there is no sequence (fa: a ~ W2) of elements of Wi WI well-ordered by <.

Proof. Suppose not. We will construct f: [W2r W ~ WI inductively. At the nth stage we determine f I [W2r· Let f I [W2r be arbitrary for n = 0, 1. For a < (3 < W2, let Ca (3 be a closed unbounded set such that for all ~ E Ca(3' fa(~) < f(3(~) < fw,(~)' Now if n ~ 2 and x E [W2r, let f(x) be the least element of n {Ca /3: a, (3 E x, a < {3} which is greater than f(y) for ail YE[Xr- i .


By Chang's Conjecture, there is A E [W2r' such that f has countable range on [Atw • Let 'Y be the least element of WI not in the range of f. Suppose l) < 'Y, l) E range f. Say l) = f(x). Let 0:, {3 E A, 0: < {3. Choose

y ;2 x, y;i x such that 0:, (3 E 'Y. Then f(y) E COt~, f(y) > l). Hence COt~ is· cofinal in 'Y, so 'Y E COt~. But then fOt ('Y) < f(3 ('Y) < f"'2( 'Y) whenever 0:, {3 E

A and 0: < {3, contradicting the fact that A is uncountable. •

The proof of Theorem 3.3 can easily be generalized to other cardinals.

THEO REM 3.4. If K is ineffable, then there is no sequence (fer.: 0: ::5 K +) of elements of I< K such that 0: < {3 implies fer. < f(3, and fer. (~) < I ~I+ for all g.

Proof. Suppose not. We may assume that (f,,: O:<K+) is as constructed in the proof of Theorem 3.1. For each ~, let S~ £ ~ x ~ be a well-ordering of ~ in order type fl<+(~). Since K is ineffable, there is S ~ K X K such that {~: .s n (~x~) = S~} is stationary. But if S has order type 0:, then {~: f,,(~) = fl<+(~)} is stationary, contradiction. •

Combining Theorems 3.2 and 3.4 with the result of Jensen mentioned earlier, we obtain:

COROLLARY 3.5. Assume V = Land K is regular. Then K is ineffable iff there is no sequence (f,,: 0: ::5 K +) such that 0: < (3 implies f" < f~, and fOt(~) < I~I+ for all ~ < K.

Now we turn to the process of iterating the operation that produced ineffables. We define In~ by induction on K. Let In~ = {X £ K: X is stationary in K}. If In~ ~ 0, let In~+1 be the set of all X £ K such that if S~ £ ~ for all ~ E X, then there exists S £ K such that {~E X: S n ~ = S~} E In~. If In~ = 0, let In~+1 = 0. If 0: is a limit ordinal, let In~ = n {In~: {3 < o:} .. Note that In~ is the set of all ineffable subsets of K. Also, 0: < (3 implies In~ £ In~.

It is not difficult to see that if 0: is a successor ordinal, or if cfo: > K, then g>( K) - In~ is a K -complete normal ideal. Also, as in the case of ineffables, there are other characterizations of In~+1 in terms of In~. For example, X Eln~ +-1 iff for any regressive f: [X]2 ~ K there is y E In~ which is homogeneous for f.

The In~ for 0: finite were studied in [1]. Here we are interested in 0:

infinite, particularly when K::5 0: < K +. Our goal is to show (Theorem


3.7 below) that if 13 < K + and In~ ~ 0 then for all a < 13, In~+l - In~ ~ O. Our basic tool is the notion of a canonical sequence.

For each a, let D: = {X £: K: K - Xi In:}.

THEOREM 3.6. Let (f",: a < II) be a sequence of members of KK such that if a < 13 then f", < f~ (mod D~). Then for all a < II and all X£: K, if {~: Xn~EInL~)}EIn: then XEIn: .•

Proof. By induction on a. If a = 0 this follows from the well-known fact that if {~: X n ~ E In~} E In~, then X E In~.

Suppose a = 13 + 1. Then Y = {f X n ~ E Intw}n{f f~(~) <f",(~)}E In~. Now let (S,,: 1/ E X) be such that S" £: 1/. If ~ E Y, then there exists T€ such that {1/ <~: T€ n 1/ = S,,} E In1~w. Also, since Y E In~, there is T such that {~E Y: Tn g = T€}E In~. But now if Z = {1/: Tn 1/ = S,,}, we have {g: Z n ~ E In1~m} E In~, so by inductive hypothesis Z E In~. Hence XEIn~.

Suppose a is limit and f3<a. Then since {g:f~(g)<f",({)}ED~, we have {g: Xn{EIn1~m}EIn~£:In~, so by inductive hypothesis XEIn~. Hence X E n {In~: 13 < a} = In:. •

THEOREM 3.7. If f3<K+ and In~~O, then for all a<{3, In:+l - In: ~ O.

Proof. Let (fa: a < K +) be a canonical sequence as constructed in the proof of Theorem 3.1. Let X", ={g: giIntw}. The theorem follows from Lemmas 3.8 and 3.9. •

LEMMA 3.8. For all a.s; {3, Xa E In:.

Proof. The proof is by induction on K. If {g: g E Intm}i In: we are done, so assume otherwise. But now if g E InL~) then by inductive hypothesis Xa n g E Intw. This uses the particular definition of fa (g) as the order-type of <a I g (the case a < K is left to the reader), since then it is clear that there is a canonical sequence (g,,: 1/ < g+) of elements of ~g such that gf~w=falg. Hence {g:XangEInt(~)}EIn:, so by Theorem 3.6, Xa E In:. •

LEMMA 3.9. For all a, XaiIn:+l'

Proof. Suppose Xa E In~+l' For each g E Xa there is an ordinal «fa(g) and a counterexample (S~: 1/<g) to eEIn~+l' Let R~ be a


well-ordering of € of order-type , (we leave the argument to the reader in case a < K or ,< €). Since X" E In:+1> there are (S",: 1} < K) and R such that E = {€: (S",: 1} <~) = (S~: 1} < €) and R n (~x~) = RdE In:. Then R is a well-ordering of K in type 13, say, where 13 < a.

Thus it will suffice to show by induction on a that there are no E, (S",: 1} E E), and 13 < a such that E E In: and

(1) for all ~ E E, (S",: 1} E E n~) is a counterexample to E n ~ E

InL~» and (2) for all € E E no subset of En 13 homogeneous for

(S",: 1} E E n 13) lies in In1~w.

First suppose a = 1. For € E E, let C~ ~ € be closed unbounded such that C~ n{1}: S~ n 1} = S",}= O. Since E E Ini, there are C and S such that E' = {€ E E: C n € = C~, S n ~ = S€} is stationary. Choose 1}, € E

E' n C, 1} < f Then 1} E C~ since C'" ~ q and C~ is closed. But also S", = S~ n 1}, contradiction.

Now suppose a>1. For ~EE, let T~={1}: s€n1}=s",}, and let (T~: 1} E T~) be a counterexample to TE E In1~w. Choose, < f/3(~) such that no homogeneous set for (~: 1} E T~) lies in In~, and let R~ be a well-ordering of € in order type , (again we leave to the reader the argument when, < €). Since E E In:, there are S, T, (T~:1} E T), and R such that if E' = {€ E E: S n € = S~, Tn € = T~, (T~: 1} E T~) = (T~: 1} E T~), and R n (€ x g) = R~}, then E' E In~. Let 'Y be the order type of R. Then 'Y < 13· If we can show that E' ~ T then (1) and (2) will be satisfied by E', (T€: gEE'), and 'Y < 13, so we can apply the inductive hypothesis. Let 1} E E'. If gEE' and 1} < g, then S~ n 1} = S"'. Hence 1} E T~ ~ T .•

It is clear that for some a, In: = In:+1. If when this happens In: ':I 0, we call K completely ineffable. If K is measurable and U is a normal measure ultrafilter on K, then it is easy to see by induction that U ~ In: for all a. Hence K is completely ineffable. The least completely ineffable, however, is much smaller than the least measurable.

One may also ask which is the least a such that In: = In:+1 when K

is completely ineffable. By Theorem 3.7, a ~ K+. In recent unpublished work, Kleinberg has determined the least such a.

Finally, note that if K is completely ineffable, In: = In:+1> X E

In:, n < w, and f: [Xr ~ 2, then there is Y E In: which is homogeneous for f. The proof is easy, and is left to the reader.


4. ERDOS AND RAMSEY CARDINALS

At the end of [1] it is asserted that the methods of that paper can be applied to Ramsey cardinals as well as to ineffable cardinals. In particular, there is an inaccessibility-like property which, when combined appropriately with m -indescribability, yields the Ramsey property. This section is devoted to a brief treatment of Erdos and Ramsey cardinals using the techniques of normal ideals and regressive partition functions.

Let us denote by K(a) the least cardinal K such that K~(a)~W, where a is an infinite ordinal. The cardinals K(a) are usually called Erdos cardinals, particularly when a is a limit ordinal. Drake [4] has extended this notion to what he calls a-partition cardinals, where a ~ cu and a is a limit ordinal. K is an a-partition cardinal iff for some ,\ < K, K is the least cardinal such that K ~ (a) ~w. It is shown in [4] that this notion successfully generalizes the notion of an Erdos cardinal.

We propose a further generalization. If a ~ cu, let us call K an a-Erdos cardinal iff for every regressive function f on [KrW and every closed unbounded subset C of K, there is A £; C such that A has order type a and for all n < w, A is homogeneous for f I [K]n. The latter condition is abbreviated by saying that A is homogeneous for f.

Some of the results below on a-Erdos cardinals were obtained independently by J. Henle.

In [9] Schmerl shows that if a is a limit ordinal then K(a) is an a-Erdos cardinal. Virtually the same proof extends to show that every a-partition cardinal is an a-Erdos cardinal.

The a-Erdos cardinals form a natural class for several reasons. It is clear from the characterization of subtle cardinals via regressive functions that every a-Erdos cardinal is subtle, and hence is strongly inaccessible, Mahlo, hyperMahlo, etc. Also, the a-Erdos cardinals are closed under 'stationary limits':

THEOREM 4.1. If K is a cardinal and {g< K: g is an a-Erdos cardinal} is stationary in K, then K is an a -Erdos cardinal.

Proof. Let f: [K r w ~ K be regressive and let C £; K be closed unbounded. Choose g < K so that g is a-Erdos and g is a limit point of C. Now apply the fact that g is a-Erdos to f I [gr w and en f


This notion naturally leads to normal ideals. Let us say that X£; K is a-Erdos iff for any regressive I on [X]<'" and any closed unbounded C £; K there is A £; X n C such that A has type a and is homogeneous for I. The a-Erdos ideal on K is {X£; K: X is not a-Erdos }. The a-Erdos filter on K is {X £; K: K - X is not a-Erdos}. •

THEOREM 4.2. The a-Erdos ideal on K is a normal K-complete ideal.

Prool· We check only normality. Suppose X£; K, h: X-+K is regressive, and for all a, h-l({a}) is not a-Erdos. Let la and Ca witness that h-l({a}) is not a-Erdos. Let p: K X K-+ K be one-one and onto. Let C = {f3: (Va < f3)f3 e Ca }n{f3: 'Y, ~ < f3 implies p( 'Y, ~) < f3}. Define regressive I on [X nCr'" by I({x}) = p(h(x), 'a({x})), where a = h(x), and I({Xb"" xn}) = 'a({Xb"" xn}), where n> 1 and a = h(Xl). Clearly there is no homogeneous A £; X n C of type a. Hence X is not a-Erdos .•

THEOREM 4.3. Suppose w ~ a < f3 and K is f3-Erdos. Then lor any K, U<K: ~ is a-Erdos} lies in the f3-Erdos filter on K.

Proof. Let X = {~< K: ~ is not a-Erdos} and suppose X is f3-Erdos. For each ~ e X, let I~ and C~ witness that ~ is not a-Erdos. Define I on [Xr'" by 1({Xb' .. , Xn}) = Ix. ({Xb ... , Xn-l}), where Xl < ... < Xn. Let A be a subset of X of type f3 which is homogeneous for f. Moreover, by using a pairing function p as in the proof of Theorem 4.2, we may assume that A is also homogeneous for a function g, where g is constructed so that A is homogeneous for g iff A is homogeneous for (q: ~ e X). It is easy to see that if X is the a th member of A, then the first a members of A are homogeneous for Ix. But since A is also homogeneous for (C~: ~ e X), we have C~ n 11 = C.,., whenever 11, ~ e A and 11 <~, and therefore 11 e CEo Hence the first a members of A lie in Cx, and this contradicts the choice of Ix and Cx. •

It is worth emphasizing that Theorem 4.3 holds for successor ordinals as well as limit ordinals. The usual definition of Erdos cardinals does not yield essentially new cardinals for successor ordinals for, as Galvin [2] has shown, if a is limit and 1 ~ n < w then K(a + n) = (2~~{t, where 20 = K and 2;:'+1 = 2'\ where ,\ = 2;:'. Thus K(a + 1) is never an (a + 1)-Erdos cardinal.


It is clear that if K is a-Erdos then K-+(a)~'" for all A < K. In general there are a-Erdos cardinals which are not a-partition cardinals. If a is limit and K is (a + I)-Erdos, for example, then by Theorem 4.3 K is a limit of a-Erdos cardinals, and no limit of a-Erdos cardinals can be an a -partition cardinal.

Now we consider Ramsey cardinals. A cardinal K is Ramsey iff K -+ (K );"', i.e. iff K is K - Erdos. If K is Ramsey then we replace the term 'K-Erdos' by 'Ramsey', and speak of the Ramsey ideal, filter, etc.

In [1] it was shown that a cardinal K is ineffable iff the subtle ideal and the rr~ ideal (defined using rr~-indescribability in the same way the weakly compact ideal is defined using rrl-indescribability) together generate a non-trivial ideal, which then coincides with the ineffable ideal. Moreover, one must speak of ideals; the least cardinal which is simultaneously subtle and rr~-indescribable is not ineffable.

Similar results hold for Ramsey cardinals. The analogue of subtlety is as follows. A set X£; K is pre-Ramsey iff for any regressive f: [Xr'" -+ K and any closed unbounded C £; K, there is a EX n C and A £; X n en a such that A is cofinal in a, A is homogeneous for f, and f is constant on [A U{a}r. The pre-Ramsey ideal on K is {X £; K: X is not pre-Ramsey}. Of course K is a pre-Ramsey cardinal iff K is pre-Ramsey as a subset of itself.

As in Theorems 4.1 and 4.2, we see that the pre-Ramsey ideal is normal, and that the pre-Ramsey cardinals are closed under stationary limits. Moreover, pre-Ramsey-ness and a-Erdos-ness, like inaccessibility and subtlety, can be described by m-formulas. Since Ramsey cardinals are weakly compact and hence rrl-indescribable, the least pre-Ramsey cardinal is smaller than the least Ramsey cardinal.

Note that if K is pre-Ramsey then K is a-Erdos for all a < K. In fact, the a-Erdos filter is a subset of the pre-Ramsey filter for each a < K, so by normality of the pre-Ramsey ideal {~ < K: ~ is a-Erdos for all a < ~} belongs to the pre-Ramsey filter. Thus the pre-Ramsey cardinals do not coincide with the cardinals K which are a-Erdos for all a < K.

THEOREM 4.4. A cardinal K is Ramsey iff the pre-Ramsey ideal and the weakly compact ideal generate a non-trivial ideal. In that case, the ideal generated is the Ramsey ideal.

INEFFABILliTY PROPERTIES OF CARDINALS II 105

Proof. Let I be the ideal on K generated by the pre-Ramsey ideal and the weakly compact ideal. It will suffice to show X e I iff X is Ramsey.

First suppose Xe 1. Let f: [xtw ~ K be regressive, and let C be closed unbounded. Since I is normal, C n X e 1. Suppose there is no cofinal A ~ C n X which is homogeneous for f. Then this fact can be expressed by a m -sentence cp over (VK , e, C, X, f) so Y = {a < K: (Va, e, C n a, X n a, f 1 [a t W

) 1= cp} belongs to the weakly compact filter on K. Since X n Ce I, X n C is not the union of a set in the pre-Ramsey ideal and a set in the weakly compact ideal. Hence YnXnC is pre-Ramsey. But if aEYnXnC and B~

Y n X n C n a is homogeneous for f and cofinal in a, this contradicts a E Y. Hence X is Ramsey.

For the other direction, it will suffice to show that if X is Ramsey then X is weakly compact and X is pre-Ramsey. If X is Ramsey then X is almost ineffable, so (see [1, Section 7]) X is weakly compact. For the other part, suppose f: [xt'" ~ K is regressive and C is closed unbounded. Since the Ramsey ideal is normal, there is Y ~ X such that f is constant on [y]l and Y n C is Ramsey, so there is Z ~ Y n C such that IZI = K and Z is homogeneous for f. Since Y n C is stationary, there is a E Y n C which is a limit point of Z. But then Z n a and a satisfy the definition of pre-Ramsey. •

As in the case of ineffable cardinals, we have:

THEOREM 4.5. If K is Ramsey, then {a < K: a is pre-Ramsey and weakly compact} belongs to the Ramsey filter. Hence the least cardinal which is pre-Ramsey and weakly compact is not Ramsey.

Proof. It will suffice to show that {a < K: a is weakly compact} and {a < K: a is pre-Ramsey} belong to the Ramsey filter. The first set belongs to the subtle filter on K (see [1]) and the subtle filter is a subset of the Ramsey filter. The second set belongs to the weakly compact filter since the pre-Ramsey property can be expressed with a 1I~formula, and by Theorem 4.4 the weakly compact filter is a subset of the Ramsey filter. •

One may ask whether the pre-Ramsey ideal must be considered in Theorem 4.4. Would the a-Erdos ideal for some a < K do as well? The


answer is negative. In fact, if K is the least (a + I)-Erdos cardinal then for some ~ < K the a-Erdos ideal on ~ and the weakly compact ideal generate a non-trivial ideal. The proof is left to the reader. It would be interesting to have a combinatorial, ideal-free characterization of such cardinals ~, but we do not know of one.

Finally we mention briefly the notion of ineffably Ramsey, which is defined like Ramsey except that the homogeneous set is required to be stationary.

Let us say that K is pre-ineffably Ramsey iff for any regressive f: [Kr'" ~ K and any closed unbounded C ~ K, there is a E C and A ~ C n a such that A is homogeneous for f, f is constant on [A U {ant, and A is stationary in a. We leave to the reader the theory of the ineffably Ramsey and pre-ineffably Ramsey ideals, etc.

It is not difficult to show that if K is pre-ineffably Ramsey, then {a < K: a is Ramsey} belongs to the pre-ineffably Ramsey filter. Also, K is ineffably Ramsey iff the pre-ineffably Ramsey and n~ ideals generate a non-trivial ideal, which is then the ineffably Ramsey ideal.

Dartmouth College

BIBLIOGRAPHY

[1] J. Baumgartner, 'Ineffability Properties of Cardinals 1', in Colloq. Math. Soc. Janos Bolyai 10, 'Infinite.and Finite Sets', Keszthely, Hungary, 1973, pp. 109-130.

[2] J. Baumgartner and F. Galvin, to appear. [3] J. Baumgartner, A. Taylor, and S. Wagon, 'On Splitting Stationary Subsets of

Large Cardinals', J. Symbolic Logic, to appear. [4] F. R. Drake, 'A Fine Hierarchy of Partition Cardinals', Fund. Math. 81, (1974),

271-277. [5] R. Jensen and K. Kunen, 'Some Combinatorial Properties of L and V', mimeog

raphed. [6] K. Kunen, 'Combinatorics', in J. Barwise (ed.), A Handbook of Mathematical

Logic, to appear. [7] K. Kunen, 'Inaccessibility Properties of Cardinals', Ph.D. dissertation, Stanford

University, 1968. [8] A. Levy, 'The Sizes of Indescribable Cardinals', in Axiomatic Set Theory, Proc. of

Symposia in Pure Math., Vol XIII. Part 1, Amer. Math. Soc., pp. 205-218. [9] J. Schmerl, 'On K-like Structures which embed Stationary and Closed Unbounded

Subsets', Annals Math. Logic, to appear. [10] J. Silver, 'Some Applications of Model Theory in Set Theory', Annals Math. Logic

3, (1971), 45-110.

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