Saturated Ideals

Saturated IdealsAuthor(s): Kenneth KunenSource: The Journal of Symbolic Logic, Vol. 43, No. 1 (Mar., 1978), pp. 65-76Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2271949 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 43, Number 1, March 1978

SATURATED IDEALS

KENNETH KUNEN'

?0. Introduction. In this paper, we give consistency proofs for the existence of a K-saturated ideal on an inaccessible K, and for the existence of an co2-saturated ideal on w,. We also include an historical survey outlining other known results on saturated ideals.

?1. Forcing. We assume that the reader is familiar with the usual tech- niques in forcing and Boolean-valued models (see Jech [3] or Rosser [11]), so we shall just specify here some of the less standard notation.

"cBa" abbreviates "complete Boolean algebra". If P ( = (P, <)) is a notion of forcing, 2(P) is the associated cBa. We write

VP for VAMP). Notions like K-closed, K-complete, etc. always mean < K. Thus, P is

K-closed iff every decreasing chain of length less than K has a lower bound, and a Boolean algebra O is K-complete iff sups of subsets of 2 of cardinality less than K always exist.

If 0 is a cBa, xv is the object in Vi representing x in V. In many cases, especially with ordinals, the v is dropped. V is the Boolean-valued class representing V -i.e.,

[uE V'= V{fu = :xE V}.

If X' is a cBa and 2 E V' is with value 1 a cBa, then X' * , is the cBa which corresponds to the extension V"'. Thus, the base set of -' * O consists of elements in V' which have value 1 to be in 02. We identify 4 as a complete subalgebra of -' * A by identifying a E X4 with the b E V' such that

Rb = 1, = a and fb = ?1= a' (the complement of a). If X' = T(P), Q E V' and J/3 = T(Q)J = 1, then a dense set for X * A is obtained by taking pairs (p, q) with p E P and fq E Q] = 1.

If W is a complete subalgebra of X, then V' may be considered to be a Boolean extension of V' by a cBa, q: W in V'. q: 1 is the 6 -valued Boolean algebra whose elements are those of X, but

?di = d21 = (A {c E 1C: diAd2?c})'.

Then _2 is isomorphic to '6 * (2: 16), and, if X', @ are as in the previous paragraph, X is isomorphic (in V') to (_1 * 0):.

Received January 14, 1974; revised November 12, 1976. 'Research supported by NSF Grant GP-27633. The referee is responsible for a number of

improvements in this paper over the version first submitted, including the replacing of a fake proof of the result in ?4 by a (hopefully) correct one.

65 ? 1978, Association for Symbolic Logic



66 KENNETH KUNEN

Finally, if 9, 9 are cBa's, W 3 9 is the completion of their direct sum, or, equivalently, the regular open algebra of the product of their Stone spaces.

?2. Historical survey. K, A will range over infinite regular cardinals, with K > w. Let S(K, A, 1') abbreviate the statement that 1 is a K-complete, A-saturated, nontrivial ideal in OP(K). Here, A-saturated means that 0P(K)I/0

has the A -chain condition (A -c.c.), and nontrivial means that ji contains all singletons (and hence all sets of cardinality less than K). S(K, A) abbreviates 3 JS(K, A, ).

We consider the possibility of S(K, A) holding for various K, A. The property S(K, A) gets weaker as A gets bigger. S(K, (2K)?) holds trivially, and S(K, W) is equivalent to K being measurable, so we are interested only in the cases where K is nonmeasurable and wc ? A ? 2K.

If S(K, K+), then K is measurable in an inner model [6, Theorem 11.12]; the proof uses methods of Solovay [13], plus the theorem of Smith and Tarski [12] that S(K, K+, X') implies that 9(K)1/0 is a complete Boolean algebra.

Not much is known about the consistency-wise strength of S(K, A) for K + < A < 2K; we comment briefly on this situation and then concentrate on the cases where A < K+. It is known (Baumgartner [1]), that statements like

S(W1, W3,J) & 2w1 = 2w2 = 04 are consistent, where J is the ideal of countable sets; one obtains this by adding W4 Cohen reals to any model of ZFC + GCH. However, for this J, P(wl)IJ can never be complete. In general, if ji is a K -complete nontrivial ideal on K and P (K)I1' is complete, then S (K, 2K, X') (by

a simple cardinality argument), but there are no known consistency proofs regarding such an 1' for which S(K, K +, 1) fails to hold. It is also unknown whether the completeness of PP(K)I/0 alone is sufficient to imply that K is measurable in an inner model.

We return now to A K+. Let S*(K,A) mean that there is an ideal of saturatedness exactly A -i.e.,

3J[S(K, A, 1') A - i3A' < AS(K, A', i')].

We consider when S *(K, A) is possible for various K and A. The results are summarized in the following table, which lists the cases A < K, A = K, A = K +

paired with various nonmeasurable K. We shall not, in this paper, analyze in more detail the possible values for A < K.

For certain combinations of K and A, S(K, A) can be refuted outright in ZFC. There are two main arguments for doing this, both due essentially to Ulam [16]. The first, Ulam matrices, gives the results in the upper left of the table. The second is a tree argument, used by Ulam to show that real-valued measurable cardinals are not strongly inaccessible, but later modified by Tarski and by Levy and Silver to give the results in the lower left of the table. The Levy-Silver argument in fact produces a K-Suslin tree from the assumption S(K, K) with K inaccessible.

Since S (K, K +) implies that K is measurable in an inner model, no entry in the table can be proved in ZFC, but one can give relative consistency results for those not refutable. All results except the consistency of S (K, K +) for successor



SATURATED IDEALS 67

TABLE

S*(K, A) for K not measurable

C < A < K A = K A = K

K a successor FALSE FALSE Consistent cardinal Ulam [161 Ulam [16] ?4

K weakly, but Consistent Consistent Consistent not strongly, Prikry [10] [8] inaccessible

K strongly FALSE Consistent Consistent inaccessible, Tarski [15] ?3 Boos [2] not weakly compact

K strongly FALSE FALSE Consistent inaccessible Tarski [15] Levy-Silver [8] and weakly compact

K (which we discuss further in ?4) show that whenever K is measurable, the desired property holds in some generic extension of the universe. There are two methods used here. The first, due to Prikry, uses

LEMMA 1 (PRIKRY [10]). If K is measurable, Vl a K-complete nonprincipal ultrafilter on K, and A a A -c.c. cBa, then in V@, define i so that with value 1,

i = {X EB P(K): 3Y E BP(K) n V[Y EY Ay n x = o]}. Then, if A < K, JS(K, A, J)D = 1.

Lemma 1 is used to establish various consistency results regarding S (K, A) for A < K. This lemma is false for A ? K (consider, e.g., the Levy algebra for collapsing K to to,). To construct models for S(K, K) and S(K, K'), one uses Lemma 2 instead.

Following [8, Definition 1.2], if M is a transitive model of ZFC and K is a regular cardinal in M, an M - K-complete ultrafilter on K is a nonprincipal ultrafilter U on 9fA(K) n M such that VU is closed under < K intersections in M-i.e.,

VaEKVhEtM nm[ n rangehE Ej.

Note that 6l need not be in M. Then, as in [8, Lemma 4.6], one has, for any A, LEMMA 2. If 2 is a A-c.c. cBa, Vl E V',

[jU is a V - K -complete ultrafilter on K Dj = 1,

and J ={X CK : X E qt] = =0}, then S(K, A, J). PROOF. 1 is clearly K -complete and nontrivial. If J were not A -saturated,

let A4 ({ < A) be such that A4 0 J but ? -q => A, n A, C J. Then the values eAS 6 V D would be non-0 but pairwise disjoint, contradicting the A -c.c. of 2.

In practice, Lemma 2 is often applied within some V' with respect to some 2 in V5, using Lemma 3. The following general formulation of the method is due to Paris. Assume that j is an elementary embedding from V into some



68 KENNETH KUNEN

transitive class M, with K the first ordinal moved. If O is a cBa and X E V@, then j(X) E Mi`), i.e., the extension of M by j(60), defined within M. Suppose that O is a complete subalgebra of some cBa XC, and that there is a K-complete homomorphism w: j(PJ) --> 1 with 7T oj [2 the identity. Define V1 E V' so that Sl C2 ?(K)fn VO]JI = 1 and, if X E V` with IX C KR = 1, ?X E z11]J' = 1-(iK E j(X)ff'"@). Then, as in [8, Lemma 1.4] (see also Jech [4]), we have

LEMMA 3. With the above notation, it is '6-valid that 1 is a Vi - K-

complete ultrafilter on K.

If 2 = 1: X, then V` may be regarded as an extension of Vi by 2. By suitably arranging chain conditions one may then apply Lemma 2 in Vi to get S(K, A) there. The consistency results in the table from [2], [8], and ?4 are all proved in this manner. The two results from [8] take O to be two different variants of Easton algebras, and it is trivial to modify the construction there using a third variant to get the consistency of S *(K, K+) for K weakly but not strongly inaccessible (add K Cohen reals to the model obtained there for S *(K, K+) and K weakly compact). In [2] and in ?4, non-Easton algebras are used.

?3 uses Lemma 2 directly, although it depends on a theorem of Silver which could be proved by applying Lemma 3 to reverse Easton forcing.

??3 and 4 may be read independently of each other.

?3. S (K, K) for K inaccessible. THEOREM. If K is measurable, then for some cBa X2, it is valid in V` that

S (K, K), K is inaccessible, and K is not measurable. We begin by outlining the plan of the proof. Let P be the usual partial order

for adding a Cohen subset to K; so P is the set of all functions p with domp C K, ran p C 2, and P < K. K may fail to remain measurable in VP; this happens, for example, if V = Lt', where 611 is a normal ultrafilter on K.

However, we have THEOREM (SILVER). If K is measurable, then there is a cBa - such that in VS,

it is valid that K is measurable and remains measurable in the extension by P. Note that by P we mean the Cohen forcing defined within V'. The theorem

follows easily by Silver's work on reverse Easton extensions. See, e.g., Menas [9] for an account of this subject. We indicate now more specifically the particular - we have in mind. First, if 2K = K', X simply adds one Cohen subset to every inaccessible ? K via a reverse Easton extension. Silver actually showed that K was measurable in the extension of V by some ultrapower of X, but in fact K is measurable in V' since Vi' = 2K = K + (see the remark on p. 371 of [8] or Lemma 4 of ?4). K also remains measurable upon extending V' by adding another Cohen subset to K, since this extension is obtained over V by a cBa isomorphic to - (since adding two Cohen sets is the same as adding one). Finally, if 2K # K', we may make a preliminary extension to collapse 2K to K+

without adding any subsets of K, and then do the procedure described above. Using this theorem, we may and shall assume, by making a preliminary

extension, that K is indeed measurable in VP.



SATURATED IDEALS 69

The ~ of our theorem will be Z(R), where R is a notion of forcing which adds a K-Suslin tree, T. This is not surprising in view of the fact that there must be a K-Suslin tree in Vi if the conclusion of the theorem is to hold. In V@, T itself is a partial order which adds a generic path thru T, and T has the K -C.C.

We shall check that the iterated extension VRT is equivalent to VP, so that K is

measurable in VRT. Then in VR, K is inaccessible, K is not measurable (since there is a K-Suslin tree), and S(K, K) (by Lemma 2 of ?2).

We now describe R. This is a minor modification of the method of Prikry and Silver for adding a K-Suslin tree, which in turn is a generalization of Jech's construction for )1. Our tree, unlike the Jech-Prikry-Silver tree, will be homogeneous, so we shall force with homogeneous trees of height less than K.

This modification will facilitate checking that VRT is equivalent to VP. To simplify notation, all trees will be subtrees of the full binary tree.

For any ordinal a,`''2 = U {f2: < a}. An a-tree is a nonempty subset t of '-'2suchthatVs E t V O<doms [s r E t] andV4 < a 3s E t [doms = ].We call a the height of t (ht(t)). A tree is an a-tree for some a.

For any tree t and s E t, let t, = {s': s's' E t}, where n denotes concatena- tion.

In the following, we always use My, 8 for limit ordinals. We call a y-tree homogeneous iff for all s E t, t, = t; this is equivalent to saying that for all si

and S2, sI's2 E t iff s, E t and s2 E t. The existence of a homogeneous y-tree implies that y is indecomposable (i.e. Va < y[a + y = y]), and if y is indecomposable D2 is homogeneous. Also, we call a (y + 1)-tree homogeneous iff Vs E t (doms < y --t = t); this implies that {s E t: doms < y} is a homogeneous y-tree.

Let R be the set containing the one-point tree, {O}, together with homogeneous trees t of successor height less than K such that (0) and (1) are in t. This last condition, together with homogeneity, insures that t branches at every point below the top. Partially order R by end extension: t1 < t2 iff ht(t1) > ht(t2) and t2 = {s E t1: dom(s) < ht(t2)}.

We make some elementary remarks on R. First, it has a largest element, 1 = {0}, which was thrown in artificially for this purpose. Second, any homogeneous y-tree t with a path thru it can be extended to a condition of height y + 1 simply by adding at level y all paths thru t. If s0 is any path thru t, there is a minimal extension of t to a condition t' of height y + 1 containing so; thus, let t'= t U{s (so\a): sE t&a < y}, where soc\a is the unique s1E 2 such that (s0 [ a)'s1 = s0. We shall call this t' m (t, so).

Also,-any condition t has a proper extension in R. To see this, let t have height y + 1, and let t' be the tree of all sequences of the form sns2n. * * 'sk, where k E w and si,. .., k E t. By homogeneity of t, t' is a homogeneous end extension of t of height y * w. Since t' has a path thru it, it can be extended to a condition of height y * w + 1.

Thus, if G is an R -generic filter and T= U G, then T is a homogeneous tree of height some limit ordinal ? K. We shall see that in fact, T is, in VR, a K-Suslin tree.

R is w1-closed but may fail to be A-closed for larger A. If 8 < K and t (4 < 8)



70 KENNETH KUNEN

is a decreasing chain of conditions of height -y + 1 respectively, then t =

U {to: ( < 8} is a homogeneous tree of height y = sup{y~: ( < 8}, but it may happen that there is no path thru t, in which case there would be no common extension of the to. Of course, this cannot happen if cf(8) = w. In the present context, the Prikry-Silver trick consists in noting that if we are also given a sequence sf (4 < 8) where dom(s~) = ye, sf E tf, and the sf extend each other (( < r <8 -> s Cs,), then U {so: ( < S} is a path thru t, so in this case, the tf have a common extension. One applies this trick to obtain many of the same results about R that are known for A -closed notions of forcing.

The first application is that T is indeed a K -tree, since any condition can be extended inductively to conditions of arbitrarily large height below K.

Next, if a < K, R adds no new functions from a into V, since if t 6F "f: t -> V", then we may, by induction on (, find a decreasing chain of conditions tf (( < a), where to = t and t~+ 11 f(f) for each (. If we also choose sf as above, this construction will pass thru the limit stages, and we also obtain a condition extending all the tf which decides all values of f.

Finally, to see that T is with value one a Suslin tree, suppose t 1- "A is a maximal antichain". By induction, define a decreasing chain of conditions tf (( < K) with to = t and height (ta) = -y + 1. At the same time, inductively pick sf E tf with dom(s~) = -y so that the sf extend each other. If ( is a limit ordinal, let oyf = sup{y,: -q < (}, sf = U {s: -q < (}, and tf = m( U {t,: <q <J}, s). We also arrange for the following:

(1) For any (, if s E tf then t~+1 11 "'' E A ".

(2) For each s E U {to: ( < K} and each a < K, there is an -q such that y, > a and for some ,3 <-y7,

trn F "(s (s, \at)) [ /3 E A

Note that (2) implies also that if ri'> m tryIF "(s (s, \a)) /,3 E A

(1) implies that if ( is a limit ordinal, s E tf, and dom(s) < yf, then to I1 "s E A ".

To show that (2) can be accomplished we need only see how to handle a particular s and a. Fix a limit ( such that -y > a and -y > dom(s). If

3,3 < yftf IF "sn(s, \a) ,X3 E A "],

take r1 = (. Otherwise, since to H "A is maximal", there is a t' extending tf and an s1 such that sn(s~\ap)ns E t' and

t' H "sn(s,\a )ns E A ".

Let t~+j extend t' and satisfy (1). Choose s~+j to be a path thru t~+j extending so so that s~+?\a extends (s\ap)js. Then (2) is satisfied with rq = ( + 1.

Now, there must be a limit ordinal ( such that for each a < -y and each s E U {to,: r < (}, there is a ,3 < y such that sn(Sg\a) f, is forced to be in A. Since tf = m ( U {to,: r < (}, so), every point in tf at level -y is forced to lie above some point in A, so t~f V"A 5 t9", so t~fl "A < K".

Thus, every condition which forces A to be a maximal antichain has an



SATURATED IDEALS 71

extension which forces A to have cardinality less than K, so T is, with value 1, a Suslin tree in VR.

To complete the proof, we need only show that the iterated extension, VRT, is equivalent to the Cohen extension VP. As usual in iterated forcing, VRT is equivalent to VQo, where

Qo={(t,s): tER&s Et2&tIF"sE T"}. If

Q1= {(ts)E Qo: ht(t)= dom(s)+ 1 &s E t},

then Q1 is dense in Q0, so VQo is equivalent to VQt. But now 23(P) and T(Q1) are both nonatomic complete Boolean algebras generated by K-closed sets of cardinality K; hence, they are isomorphic.

We remark that the same method can be used to show the independence of a combinatorial result which Jensen proved held in L, namely:

THEOREM (JENSEN [5]). If V = L and K is not weakly compact, then there is a stationary X C K such that X nf a is nonstationary in a for all limit a < K.

Our method of proof gives: THEOREM. It is consistent with ZFC that there is an inaccessible K such that K

is not weakly compact but (1) For all stationary X C K, there is an inaccessible a <K such that X n a is

stationary in a, and (2) V=L[A] for someACK. PROOF. Of course, by Solovay [13], S(K, K) plus K being inaccessible implies

(1), but it also implies that (2) is false. To obtain (2), we start with V = L and K weakly compact, and then proceed as before to obtain a situation in which there is a K-Suslin tree T such that K is weakly compact in VT. One easily checks that the preliminary reverse Easton extension described above (adding one Cohen subset to each inaccessible ? K) plus the one which adds T only extends L to L [A] for some A C K.

To check (1), if X 5 K is stationary, then X remains stationary in VT, since by K-c.c., any closed unbounded subset of K in VT contains one in V. Hence, by weak compactness of K in VT, X n na is stationary in a for some inaccessible a < K (note that V and VT have the same bounded subsets of K).

?4. S(01,co2). THEOREM. Suppose that there is an elementary embedding, j, from V into

some transitive class, M, with K the first ordinal moved. Let A = j(K) and assume that AM1 CM. Then for some cBa A, it is true in V' that K' = W1, A = W2, and S(K, A).

A K satisfying the hypothesis of the theorem is called huge. If one weakens AM CM tod'M CM, K is called almost huge. Neither of these properties has, to our knowledge, been shown to be inconsistent, although the somewhat stronger property that ?P (Sup {j n(K): n E w }) C M has been (see [7]). If K is almost huge, then K is a -supercompact for all a <A, so R(A)t= "K is supercompact". If LL = {X 5 K: K E j(X)}, then V is a normal ultrafilter on K

and



72 KENNETH KUNEN

{ a < K: R (K) I= "a is supercompact"} E JU.

We do not know whether the assumption S(w,, w2) is really consistency-wise very strong at all, since all one can get for sure from it are inner models with several measurable cardinals (using the method of [6]). However, the following heuristic argument suggests that S(wo, cc2) may in fact be stronger than had been suspected. If S(ol, W2, A6) and At = P(o)IJ, then Solovay [13] shows how to get an elementary embedding, i, from V into some transitive subclass N of Va. It is easy to see that i(w01) = w02 and for all a <w02, aN CN, i.e., any d4t-valued function from a into N is in N. It thus appears that wi is almost huge -except, of course, that N is a subclass of V', not of V.

We now outline briefly the strategy for proving our theorem before plunging into the gory details. The idea is to apply Lemmas 2 and 3 of ?2, together with a third principle, Lemma 4, below.

We shall describe a cBa 2 such that in V@, K, = o1, A = W2, 2W = wo, and 2`1 = &02. Lemma 2 will be used in V" to check S(Ki, AX) in V"; thus 6 will be a complete subalgebra of some cBa A, where W has the A-c.c., and where for some 61 E VW, I61I is a V - K-complete ultrafilter on K]J = 1, so, since VW is a A -c.c. extension of V" by W' X, IS (K, A)r0 = 1.

W will be obtained as in Lemma 3, but with an added complication. We shall get a o-, IC as in ?2, with 6 C W C IC, and Lemma 3 will yield immediately a Vl in V' with the required properties. IC will not have the A -c.c. However, V' will be an extension of VW by a A -closed notion of forcing. Furthermore, the statement that there is a V" - K -complete ultrafilter on K is, in any extension of V@, a ;1 property of A -i.e., it is expressible by a E1 sentence of second-order logic using a subset of A as a parameter to code up PP(K) n vi (since V" 1= "2K = A "). One may thus, to obtain a OU E VW, apply Lemma 4 within Ve.

LEMMA 4. Let a be regular and > w, P an a-closed partial order, A a relation on a, and up a E1 sentence. If in V', ((a, A) I= (p D > O, then (, AA) I= up.

This lemma is well known and is part of the folklore of the subject. For completeness, we sketch a proof. Say p is 3Xq(A, X) where tf is first order and X ranges over subsets of a. Since 'p has positive value to hold in V', there is a term r and a p0&E P such that p0 I'(aA, r)1=qf". By a Ldwenheim-Skolem argument plus A -closedness, we find an increasing sequence of ordinals -y (( < a), sets be 9 lye, and a decreasing sequence of conditions p (( < a) such that for each (, pa v- "r n Ay = (be)" and

pi 1F "(yf, A r y, r n ye ) < (a, A, r) ".

Since the p6 are decreasing, there is a B C a such that each be = B n By. But then the structures (-y, A [ ye, B n Ayd) form an elementary chain and they all satisfy qf, so (a, A,B)I= q, so (a,A)I= 'p. Note that there need not be any condition which forces "r = B".

Aside. A well-known application of this result is that no a-closed notion of forcing can destroy an a-Suslin tree.

It remains now to describe 6 specifically and to check that the situation of Lemmas 2-4 obtains.



SATURATED IDEALS 73

Let - be a cBa with the following properties: (I) alphas the K-C.C.

(II) A4= K. (III) In V', Rk = w1J= 1. Certainly the ordinary Levy collapse of K has these properties. It will

become necessary later for X to have a fourth property not shared by the Levy algebra. By condition (II), we may and shall assume, to simplify notation, that s4 CR(K), so that j [ X is the identity. Also, note that since AM CM, j('4) is A-complete and A-c.c., and hence complete.

2 will correspond to following the extension s by Silver's modification of the Levy collapse to make A = w2 without adding any reals. Thus, within Vffl, let P be the set of all p such that:

(1) p is a function, (2) dom(p) C {(ac 4): a < A & < K}, ran(pa) C a, (3) pI -K, and (4) I U a dom (Pa)I < K.

Here, Pa = {((,p(a, 4)): (a, 4) E dom(p)} is a partial function from K to a. So p is thought of as coding a A -sequence of such partial functions, all but K of which are the empty function (3). As a special case of (4), | dom (pa ) I < K, so that the generic extension defined by P contains a function from K onto a for all a < A. By (3), P has the A -c.c., so that A remains a cardinal in the extension by P. Let OX = - * 23(P), where 23(P) is defined within Va. Note that, via the usual theory of iterated forcing, a dense set D for OX is defined as the set of pairs (a, p) such that a E s4 - {O} and fp E Pf = 1. The map which sends a to (a, 1) is the canonical complete embedding of 4 into 06.

The map j [3: 3 ->j(P/) is an invective homomorphism which is K-

complete but not complete. However, j [ s ( = the identity on A) is complete since s has the K-C.C. We may now state the fourth property that we assume for 4:

(IV) There is a complete invective homomorphism, h: 0 -> j(s4) extending jr&,'.

We defer until later the proof that an 4 satisfying (I)-(IV) exists. Note that (IV) is a property of X, since 0 has been explicitly defined from 4. The W in the above discussion will be j(sd), identified via h as a complete superalgebra of 06.

To set up for an application of Lemma 3, we now try to define a cBa W and a K-complete homomorphism iT: j(P/)-> W such that or o0j [ 0 is 1-1 and complete (so that we may then identify or oj [ 0 as the identity). W will be the completion of j(P/) modulo a principal ideal J. Thus, J will be of the form {b E j(0): b A N = O} for some b0 E j(P/); W may also be viewed as the algebra below bo. or will be the canonical embedding. bo will be defined so that:

(a) J Uj(s4)=O, and (b) or o h = X L ? / 3; equivalently, Vb E O (h(b) A bo = j(b) A bo). Note that (a) and (b) imply that or oj [ 3 is 1-1 and complete. Also, since

j(&i) has the A-c.c., or embeds j(sd) as a complete subalgebra of W. The reader may note that or itself is in fact complete -i.e., it preserves all sups which exist



74 KENNETH KUNEN

in j(O); but since j fOA is not complete, we needed something like (b) to guarantee that Xr o1j 91 is complete.

To define b0, it will be helpful to look more carefully at j(,d) and j(s). Since D is dense in 64, j(D) is dense in jQ(90). Since MA CM, j(D) may be defined in V, as the set of pairs (a, p) such that a E j (sd) - {O} and p in Vi(s) with value 1 satisfies the conditions (1)-(4) above with A replacing K and j(A) replacing A. Call these (1*)-(4*). b0 will be in j(D) and of the form (1, q), so that (a) will automatically be true.

We define q as follows: For each p in V' such that lp E PI= 1, let qp E V'() be such that ?qp = j (p) = h ((1, p)) and Rqp = 0] = h ((1, p))' (O here is the empty function). Since h and j are the identity on X, the map which assigns qp to p may be thought of as existing in V"). In Vi(,), let q = U {qp: p E P}; then q, with value 1, satisfies (1*)-(4*). Equivalently, in V we may define, in the Boolean algebra j(/), (1, qp) = (h(1, p), j(p)) v (h(1, p)', 1), and bo = (1, q) = Ap(l, qp ).

We now check that bo satisfies (b). Since h((1, p)) is identified with (h((1, p)), 1) E j (si) Cj($0/3), whereas j((,p)) = (1,j(p)), we have h((1,p)) A(1,qp)? j((1,p)) A(1,qp). Thus, for all p, h((1,p))Abo? j((1,p))Abo. Thus, for all d =(a,p)=a A(1,p) ED, h(d)Abo'j(d)Abo. If b E 13 is arbitrary, write b = V {d,: i E I} for suitable d, E D. Since h is complete,

h (b) A b = V (h (d, ) A bo) ' V (j(dl ) A bo) ? j(b) A bo.

Likewise, h(b') A bo ' j(b') A bo, so h(b) A bo = j(b) A bo. Note also that one may view VW as the extension of Vi'i) obtained by forcing

with those conditions in j(P) which agree with q. Thus, VW is a A-closed extension of VI'(). Thus, we may apply the scheme outlined earlier (with W = j(sd)), so we are done, modulo the construction of an s4 satisfying (I)-(IV).

We obtain such an 4 by a minor modification of the usual procedure for constructing models for Martin's Axiom (see [3], [14]). In general, for an inaccessible p, call a cBa 4 a p-collapse iff s satisfies (I)-(III) (with p replacing K). If cu is a larger inaccessible, let Oh (A', cr) be the algebra obtained from s as $9/ above was obtained from a K -collapse (so that our $1 above was O1 (,4, A)). A o--collapse 4 is called universal iff for every inaccessible p less than a, every p-collapse 4', and every complete injective homomorphism f: s ', f has a complete injective extension to a g: O (,d', u) -- A.

LEMMA. For any inaccessible co, there is a universal o-collapse. PROOF. We shall construct an increasing chain, s4 (5 c cr), of cr-collapses,

with sio, say, the standard Levy algebra. 4, will be a universal cu-collapse. If f <ii, 46 will be a complete subalgebra of 4, At limit stages we just take the completion of the union; that this preserves o--c.c. is trivial at stages below o-, and follows at stage a- by the proof of the corresponding fact for c.c.c. in Solovay and Tennebaum [14] (see also Theorem 49 of Jech [3]).

There are only 0a possible embeddings f: .A" for p-collapses 4' with p < , and the range of such an f is contained in some 46 for f < , so, by the usual diagonal argument, we need only show how to handle one such f at a



SATURATED IDEALS 75

successor stage. Thus, we are confronted with a complete injective f: d' > de

and we wish to define 4+, so that f extends to a complete injective h: 3 (d', o-) -> 1.To simplify notation we may assume that f is the identity, so d' C Hid. Now, we may take 46, to be the amalgamated product of s4 and 0 (d', a) over d'. Thus, within V", we have the cBa's h4: sd' and 0 (d', o): d'. In Vat, let W = (a4h :,d') (D ($0, (d', o-): s'), and let , =

It is easy to see that 4+, has the required properties if we can check that W has the o-c.c. in Vi. This is trivial when oa is weakly compact (since then the sum of any two o-c.c. Boolean algebras is o-c.c.) but follows anyway since 0 (d', o): s' has the property (in V") that given nonzero b,,. (g <ov),

3X C [JIXI = a& Vg, v E X[b,, A bk 0]],

and the sum of any o-c.c. Boolean algebra with one which has this property also has o-c.c.

This completes the proof of the lemma. If X is a universal K -collapse, then clearly s will satisfy conditions (I)-(III). Since AM CM, j(sd) is a universal A -collapse. Hence, s satisfies (IV) also.

This completes the proof that S(w,, w2) is consistent. The question of the consistency of S (w,, w2,J ), where J is the ideal of nonstationary sets, remains open.

In conclusion, we remark that it is no accident that our construction resembles Silver's proof of the consistency of Chang's Conjecture (CC), although Silver just needed a Ramsey cardinal (or even just A ->(1)<@). We do not know whether S(w,, w2) implies CC, but CC holds in our model, for, following Jech [4], the fact that or is complete means that the elementary embedding j: V -> M extends to an elementary embedding j': Vi -> M' C V`,

where M' is closed under W -valued A-sequences. j(K) = A, so A is the W, of M', and K is countable in M'. In V@, let ? = (A; K,... ) be a structure of type (A, K)

(i.e., (W2, w1)). Then, since the range of j on A (j"A) is in M', M' thinks that j'(9R) has an elementary submodel of type (w1, wco) (namely (j"A; j"K,.. .)). By elementarity of j', U? has an elementary submodel of type (w1, wo) in V@.

REFERENCES

[1] J. E. BAUMGARTNER, Results and independence proofs in combinatorial set theory, Doctoral Dissertation, University of California, Berkeley, 1970.

[2] W. Boos, Boolean extensions which efface the Mahlo property, this JOURNAL, vol. 39 (1974), pp. 254-268.

[3] T. J. JECH, Lectures in set theory, Lecture Notes in Mathematics, no. 217, Springer-Verlag, Berlin,,1971.

[4] , Two remarks on elementary embeddings of the universe, Pacific Journal of Mathema- tics, vol. 39 (1971), pp. 395-400.

[5] R. B. JENSEN, The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229-308.

[6] K. KUNEN, Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179-227.

[7] , Elementary embeddings and infinitary combinations, this JOURNAL, vol. 36 (1971), pp. 407-413.



76 KENNETH KUNEN

[8] K. KUNEN and J. B. PARIS, Boolean extensions and measurable cardinals, Annals of Mathematical Logic, vol. 2 (1971), pp. 359-378.

[9] T. K. MENAS, On strong compactness and supercompactness, Doctoral Dissertation, Univer- sity of California, Berkeley, 1973.

[10] K. L. PRIKRY, Changing measurable into accessible cardinals, Dissertationes Mathematicae, Rozprawy Matematyczne, vol. 68 (1970).

[11] J. B. ROSSER, Simplified independence proofs, Pure and Applied Mathematics, vol. 31, Academic Press, New York, 1969.

[12] E. C. SMITH, JR. and A. TARSKI, Higher degrees of distributivity and completeness in Boolean algebras, Transactions of the American Mathematical Society, vol. 84 (1957), pp. 230-257.

[13] R. M. SOLOVAY, Real-valued measurable cardinals, Proceedings of Symposia in Pure Mathematics, vol. 13, Part I, 1967 UCLA Summer Institute, American Mathematical Society, Providence, Rhode Island, 1971, pp. 397-428.

[14] R. M. SOLOVAY and S. TENNENBAUM, Iterated Cohen extensions and Souslin 's problem, Annals of Mathematics, vol. 94 (1971), pp. 201-245.

[15] A. TARSKI, Ideale in vollstandigen Mengenkorpen II, Fundamenta Mathematicae, vol. 33

(1945), pp. 51-65. [16] S. ULAM, Zur Masstheorie in der allgemeinen Mengenlehre, Fundamenta Mathematicae, vol.

16 (1930), pp. 140-150.

UNIVERSITY OF WISCONSIN-MADISON

MADISON, WISCONSIN 53706



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