filters and large cardinals

ANNALS OF PURE AND APPLIED LOGIC

Annals of Pure and Applied Logic 72 (1995) 177-212

Filters and large cardinals

Jean-Pierre Levinski

Received 19 August 1982; revised 15 May 1992; communicated by T. Jech

Abstract

Assuming the consistency of the theory “ZF’C + there exists a measurable cardinal”, we construct (1) a model in which the first cardinal ti, such that 2” > tit, bears a normal filter F whose associated boolean algebra is K+ -distributive (and indeed strongly K+-distributive as defined in Section 5), (2) a model where there is a measurable cardinal K such that, for every regular cardinal p -=z ti, 2p = p ’ + holds, (3) a model of “ZFC + GCH” where there exists a non-measurable cardinal K bearing a normal filter F whose associated boolean algebra is K’-distributive {and K+-saturated as well).

1. Notations

1.1. Let P be a set of conditions. If G is P-generic over V, k’[G] denotes the generic

extension of I/ by G and, if a E V, aG denotes its interpretation in V[G]. B(P) will be

the boolean completion of P and, if 4 is any statement, 11 Cp 11 W) will denote the

boolean value of 4 in B(F).

1.2. If S is a set, U an ultrafilter on P(S)n I/ and f~ k’“n V, [f Jv denotes the class of f in ( VSn V)/U.

1.3. Let S be a set and F a filter over S. B(F), the boolean algebra of F, is the quotient P(S)/I, where I is the ideal dual to F. For every X c S, [X& denotes the class of X in B(F), and we set F + = {X c SlX#I). If A E F+, F[A] denotes the filter generated over S by F u {A >. Since B(F) - {Oj is the separable partially ordered set associated with the (usually non-separable) set of conditions (F+, c ), we shall not distinguish

between B( F)-generic sets over I/ and (F +, c )-generic sets over V, which, themselves, are ultrafilters on P(S)n V.

1.4. 1x1 is the cardinality of the set X. cof(a) is the cofinality of the ordinal ti.

0168-0072/95/%09.50 0 1995- Elsevier Science B.V. All rights reserved

SSDI 0168-0072(94)00015-U

178 J.-P. Let:inski/Annals @Pure and Applied Logic 72 (I 995) 177-212

1.5. ‘%-cc” means “ < K-CC”, and so on.

1.6. “<” means “elementary substructure of”.

1.7. “ZFCM” is the theory “ZFC + there exists a measurable cardinal”.

1.8. Assume that B,C are complete boolean algebras, that i:B + C is a complete monomorphism, and let G be B-generic over V. In V[G], we denote the Solovay-Tennenbaum quotient by any of the expressions CjB, Cji” B, C/G, C/i” G. Recall that it is the quotient of C by the ideal I = {q E C j3p E G, q A i(p) = 0). If q E C, its class in C/B will be denoted by any of the expressions q/B, q/i”B, q/G, q/i”G.

2. Introduction

2.1. The critical cardinal. Let us say that IC is the critical cardinal if it is the first infinite cardinal y such that 2y > y+. Obviously, the critical cardinal does not need to exist [Godel]. On the other hand, it may exist and be equal to w [Cohen] or to some other regular cardinal p > o [Cohen, Solovay].

2.2. It is known [Silver] that the critical cardinal cannot be singular of cofinality 2 wi. On the other hand, Magidor showed, starting with some large cardinal

assumptions~ that it may equal K,.

2.3. Concerning the case where the critical cardinal is regular, Scott showed that it cannot be measurable, and Silver showed, assuming some large cardinal axioms, that there may exist a measurable cardinal IC such that 2” > K+, while Kunen proved that such an assumption is strictly stronger than the theory “ZFCM” itself (these results were later improved by Woodin, Mitchell and Gitik).

2.4. We concentrate here on the following question. How much of measurability can the critical cardinal retain, and at which consistency price?

2.5. Precipitousness is already a possibility, and we gave in [ll] a proof of “cons(ZFCM) * cons(ZFC + the critical cardinal bears a normal, precipitous filter)“. However, the filters in these models are not even ineffably closed (see 2.9 for a definition).

2.6. So we try here to consider properties of K being as near to measurability as possible. Let U be a normal measure on K, and set M = VK,KJ. One can go in two directions. (1) Saturatedness. B(U), being trivial, is x+-saturated. (2) Closure. M” c M and, in particular [P(K)]~ = [P(K)]‘.

J.-P. LevinskilAnnals of Pure and Applied Logic 72 (1995) 177-212 179

2.7. (1) is not the right direction, for it is known that, if there exists a normal, K’-saturated filter over K such that (z < ~12” = ct’} EF+, then 2” = K+.

2.8. Hence, we treat closure. Let F be a normal, precipitous filter over K, let D be B(F)-generic over I/ and form, as usual, ND = (I/” n V)/D in V[D]. We can demand the following conditions on N,:

(a) (Na)” n VCDI c ND, 03 Cp(~)l”” = Cp(~)l”.

Note that (a) does not imply (b), for if K is a successor cardinal and F is K+-saturated, then (a) holds, but (b) does not.

2.9. Concerning (b), let us recall the following definition.

Definition 1. Let K be an uncountable regular cardinal. (1) A filter F on K is inefably closed if

(i) F is uniform (i.e. for all l < K, ] t, K[ E F ),

(ii) whenever A E F+ and (S,),,, is such that, for all LY < K, S, c LX, there exists someBEF+suchthatBcAand,foralla,flEB,x</I=S,=SOnC(.

(2) K is completely in@able iff there exists over K an ineffably closed filter.

Remark. Details can be found in [l, 21, where it is shown that an ineffably closed filter is normal and that, if K is completely ineffable, there exists a least ineffably closed filter over K.

This being said, the reader will immediately check that a normal filter F on K is ineffably closed if Ik,,,, “[P(K)]~” = [P(K)]‘“, which is (b).

2.10. It turns out that the conjunction of the two conditions of generic closure we have just discussed is equivalent to a natural condition concerning the boolean algebra

B(F) of F, in the ground model I/.

Lemma 1. Let F be a normal jilter on K. The following are then equivalent:

(1) ItBcF) “[P(K)]~” = [P(K)]” and (Na)Kn V[D] c ND”, (2) B(F) is K+-distributioe.

Proof. Let us first observe that, since IB(F)I < 2”, a standard argument shows that B(F) is rc + -distributive iff

(*) II&F) "[P(K)]'@' = [P(K)]““.

For, assuming that (*) holds, we see, due to the existence of an absolute bijection from (2”)” onto 2(KXK), that [B(F)“]” = [B(F)“]“tD1and hence, if@,),,. is, in V, a family of

180 J.-P. LevinskilAnnals of Pure and Applied Logic 72 (1995) 177-212

open dense sets in B(F), we can find in V[D] a family (pa),<, E IZ((D,),,.)r\GK and, since this family is in V, find some q E G, such that q < inf { pal ) a < K}, which implies that q E n {D, 1 SC < K}. Hence, the only non-trivial thing to prove is as follows. We assume that B(F) is rcf-distributive and prove that ES(r) “(N)rn V[D] c ND”. To do this, let (rJacK be a family of terms in I’ such that, for all a < K, ItBC,, “z~ E ND” and, for z < K, set D, = {p E B(F) 1 for some h E VK, p Ik,(,, “z, = [h],“). Since each D, is open dense in B(F), we can find some p E fi {D, 1 a < K } n D and (h,), < K such that, for all a < K, p ItBCFJ “t, = [I&“. Defining H E Vn V by H(a) = {he;(a) I( < a} we see, due to the normality of F, that p ltBtFj “[HID = {t, I a < K}". 0

2.11. We shall hence consider the following as relevant.

Definition 2. Let K be a regular uncountable cardinal. (1) A premeasure on K is a normal filter F on K such that B(F) is K ‘-distributive. (2) K is premeasurable iff there exists a premeasure on K.

Remark. If a filter F is < w-complete and B(F) is < w-distributive, then F is already precipitous, as the reader will easily show.

2.12. Some properties of premeasurability. (1) A premeasure is ineflably Ramsey closed, as defined in [2]. This means that,

wheneverf:[K]<O~Kisregressive,then{AEF’Iforalln<o,If”[A]“I= l}is dense in F + . In particular, K is Ramsey, and much more.

(2) If $ is a second-order statement such that, whenever K is measurable, (K,P(K))+ c$, then, whenever K is premeasurable, (K,P(K)) k 4. For, if F is a premeasure on K and D is B(F)-generic over V, then K is measurable in V[O].

2.13. We shall give another definition.

Definition 3. Let K be an uncountable regular cardinal. (1) A quasi-measure on K is a normal filter F on K such that B(F) admits a dense,

< k--closed subset. (2) K is quasi-measurable iff there exists a quasi-measure on K.

The interest of this definition lies in the fact that, if K is quasi-measurable and 2” = K +, then K is measurable, as will be recalled in Section 5. We shall also define in Sections 5 and 6 an intermediary notion, strong premeasurability, which, if satisfied by K, implies, together with 2” = K+, that K is measurable.

2.14. This being said, we shall, starting with a model of “ZFCM”, construct (1) a model where the critical cardinal is strongly premeasurable (see Section 6), (2) a model where GCH holds and there is a premeasurable cardinal which is not

measurable (see Section S),

J.-P. Levinski /Annals qf Pure and Applied Logic 72 (199s) 177-212 181

(3) a model where there is a measurable cardinal K such that, for all regular cardinals p < K, 2p = p + + (see Section 5).

We shall also find a model where the critical cardinal is quasi-measurable, starting however from an assumption strictly stronger than “ZFCM” (see Section 7).

3. A word on the method used

3.1. Precipitousjilters. Assume that S is a set and that U is an o,-complete ultrafilter over S. Let us, as usual, define j: V + M = I/‘/U, and let K be the critical point ofj.

Let P be a set of conditions satisfying the K-CC and let B be its boolean completion. Let G be P-generic over I/. In V[G], let F be the filter generated over S by U, F = (X c S \3E E U, E c X}. It is a classical fact that F is precipitous, and we shall shortly review the main points of the proof of this fact, for generalization to some non-K-cc sets of conditions.

(a) Since P has the K-cc, jlp: P --f j(P) is complete. Let hence G* be anyj(P)-generic set over I/, such that j”(G) c G*.

(b) Let j*: V[G] -+ M[G*] be defined, for a E I/, by j*(a,) = (j(a)),*. It is known that j* is an elementary embedding, extending j.

(c) Let d denote the identity function restricted to S, set 9 = [d]” and D = {XEP(S)~V[G]~SE~*(X)}. D is a V[G]-ultrafilter over S and, if we set as usual i : V[G] + N = (V [G]‘n V[G])/D, there exists an elementary embedding k:N + M[G*] such that koi =j*: for ,f~ VIGlsn V[G] we set

k(C.fl~) =j*(f)(@. (d) Hence, N is well-founded and, if we prove that D is B( F)-generic over I/ [G], it will

follow that, for some A E F +, F [A] is precipitous. (e) Indeed, if we prove that every B( F)-generic set over I/ [ G] can be obtained in this way

from some j(P)-generic set G* over V extending G, then F itself will be precipitous.

3.2. Structure ofB(F). We are however interested in the structure of B(F) (degree of distributivity, of saturatedness, of closure of some dense sets). In order to characterize B(F) in V[G], we observe that (d) and (e) of 3.1. imply that B(F) has the same generic models as j(B)/j”G = j(B)/G (the Solovay-Tennenbaum quotient of j(B) by the complete map j). This means that, modulo trivial elements, these two algebras have isomorphic boolean completions. In order to get an isomorphism between them, we first observe that F can be redefined as follows. For a term a E VP such that It, a c S, we have a,; E F 0 3p E G, [j(p) Itj(p) QE j(a)l”. Hence, we define a map cp: B(F) + j(B)/G setting cp( [ac&) = ( 11 QE j(a) 11 j@))/G. The fact that cp is an isomorphism is easy to check.

3.3. Uniformly normed jilters. We can also observe that the map k of (b) is actually onto M[G*], since if b E M and if we set b = [.f]“, where ,f~ VSn V, then


b = k([((f(a))GIcr~~)hd~ I-I ence, both models being transitive, k is the identity and i = j*. In particular, forfe VSn I/, [flu = j(f)(S) = j*(f)(S) = [fJD. This was used in [9] in order to prove that the filter F is actually uniformly normed, answering a question of Jech. For the reader’s ease, we might recall a few words about uniformly normed filters. Let X be a set, and let F be an ol-complete filter over X. For h E OnX, we denote by II h IIF the Galvin-Hajnal rank or norm of the function h over the filter F.

Now, let 01 E On, and h E OnX. We say that h is of uniform norm u over F if, for all A E Ff, IlhllFIal = llhll, (note that, in general, IlhllF[al > lihll,). We say that F is uniformly normed if, for every CI E On, there exists a function h E OnX, of uniform norm u over F.

The interest of a uniformly normed filter is that it approximates an ultrafilter, at least from the point of view of ordinal-valued functions. These filters have been studied independently by Jech and the author long time ago. Every uniformly normed filter is precipitous, and, as just alluded to, the author showed in [9] that, from a measurable cardinal, one can get a uniformly normed filter over, for instance, ol. Other kinds of filters have been defined in [9], like normed filters, which were later studied further by Jech and Mitchell.

3.4. Regular sets of conditions. We keep our setting and try to extend the proof to some sets of conditions P such that jlp is not complete any more. Extending the elementary embedding j will be possible if the following holds: for every P-generic set G over I/ there exists a j(P)-generic set G* over V such that j”G c G*. Let us say that P is regular (with respect to j) if this condition is satisfied. Regularity is actually the condition used by Silver in order to extend elementary embeddings and, for example, produce a model where there is a‘ measurable cardinal K such that 2” > K+ (see, for instance, [6, Theorem 88, p. 4501).

Hence, let us assume for this section that P is regular. Our aim is to associate with P a filter F in VP (using, for simplicity, the same letter to represent the filter in V[G] and its name in VP), from which we show that it is precipitous. F would not be the filter generated by U, but we shall define it using the idea of 3.2. As usual, we let GP be a term denoting the P-generic set. For q E j(B) we say that q is a regular condition if q Ilj(p) “j - 1 (Gj(p)) is P-generic over I/ “, and we say that h E BS n I/ is a regularfunction if [h]” is a regular condition. Now, let G be P-generic over V. If a E VP is such that It-, a c S, we set: aG E F o there exists some p E G such that, for every regular condition q, if j(p) A q # 0, then j(p) A q Itj(p) 9 E j(a). We shall not go through the details of checking that the construction works, but only indicate the main points, since in the next section we shall do a more general one. Let us hence assume that the previous formula defines indeed a filter F in I/ [ G] and concentrate on the characterization of the boolean algebra B(F). Let fi denote the boolean completion of j(B) in V (j(B) might only be complete in M) and set b = sup { q 1 q is a regular condition}. Hence, b E B and aG E F if and only if, for some p E G, j(p) A b < /I 9 E j(u) I/j@). Let us now set C = {qefllq < b} and define f:B + C by f(p) = j(p) A b. Then, f is complete by definition of b and is one-one precisely because of the regularity property of B,

J.-P. LevinskilAnnals of Pure and Applied Logic 72 (1995) 177.-212 183

which indeed means that, for all p E P,j(p) A b > 0. Hence, in V[G], we can form the Solovay-Tennenbaum quotient C/G = C/f “G, and we shall prove that B(F) is dense in this algebra. The isomorphism cp : B(F) + C/G is constructed essentially as in 3.2. If

a E V is such that IFPa c S, then we set cp([aclF) = ( /I s~j(a) lIj(B’ A b)/G. It is easily seen that this works, and that the proof of the precipitousness of F goes through. Let us finally observe that F can also be defined in the following way: F={X~P(S)nI’[G]IVrregular function h, ~EEU, EnK’(G)cX}. Hence, if h happens to belong to j(B), F is a principal extension of U.

3.5. Weakly regular sets of conditions. Some of the sets we shall use will, however, not be regular. We hence have to seek ior a weaker condition, which still allows the proof to work. Hence, let us keep the previous setting and notations. What we needed was not G* to be j(B)-generic over I’, but only over M, in order to be able to form the extension M [G*]. On the other hand, G* must be found in some generic extension of

V, say by some complete boolean algebra C. This actually means that there must be a term of I/‘, still denoted by G*, such that Itc “G* is j(B)-generic over M and j- ‘(G*) is B-generic over I/“. Hence, using standard ways, we define a map x : j(B) + C setting, for 4 E j(B), n(q) = I/ 9 E G* 11’. 7~ must be M-complete, since, if G is C-generic over I/, then n-l (G) = G* is j(B)-generic over M. On the other hand, set f= 7c oj. By the same sort of argument, f is V-complete. f might not be one-one, in which case we set a = inf {p E B If(p) = l} and replace B by B, = { p E B I p d a>, which is enough for consistency purposes.

Definition 1. We shall say that B is weakly regular (with respect to j) if there exists a boolean algebra C and an M-complete boolean morphism rc: j(B) -+ C such that f = 7c cj is one-one and complete.

Remark. If we wish, we may assume that x is onto C, by replacing C with Im(n), or else that Im(z) is dense in C, and C is complete.

3.6. Kunen-Parisfamilies. We now have to associate, with a given weakly regular set B, a filter F in VB. In fact, the construction of F will have nothing to do with weak regularity, but only weak regularity will allow to prove that F is precipitous. Hence, let still j : I/ --+ M = I/‘/U be fixed. We still let d denote the identity function restricted to S and set 9 = Cd]“.

Definition 2. A Kunen-Paris family (for j) is a triple (B, rc, C) where B is a complete boolean algebra, C is a boolean algebra, n is a boolean morphism from j(B) into C, Im( x) is dense in C and f = n 0 j is one-one and complete.

The family (B, rr, C) will be called regular if 7c is complete and weakly regular if 7c is M-complete.

184 J.-P. LevinskilAnnab of Pure and Applied Logic 72 (1995) 177-212

Let now (B, A, C) be a fixed Kunen-Paris family. We shall associate with it a filter F E VB, following the lines of 3.2 and 3.4. Let G be B-generic over V. For a E I/ such that its a c S, we set aG E F o for some p E G, f(p) Q z( // S~j(a) /j@‘). We shall use the same letter for F E V[G] and the term of VB representing it.

Lemma 1. Let a E V. The following are equivalent: (1) p It,a E F, and (2) f(p) <

4 II sEj(4 I/ 1.

Proof. (2) * (1) is obvious. Assume hence that plta E F. Then, trp’ < p, 3~” < p’, f(p”) z+Z TC( 119 E j(u) 11). Since f is complete, this implies that f(p) Q TT( 119 e](a) II). 0

Lemma 2. F is a ~o~-t~iv~ui~Zter on S in V [G] and F n V = U.

Proof. (1) F is well-defined and closed for supersets. Assume that a, b E VB and p E G

are such that pk,a c b and f(p) Q x( 119 o j(a) 11). Then, j(p) ItjcB,j(a) c j(6) and hence j(p) A /I Qe j(u) // d I/ QE j(b) I/, which implies that f(p) < n( /I 9 E j(6) 11).

(2) 8ef.F. For, if it does, then, for some p E G, f(p) = 0, a contradiction to f being one-one.

(3) F is closed under intersections. Assume that a, 6, c E VB and p E G are such that

PM nb = c, f(p) < n( II 9 Ej(4 II) and f(p) d n( II 9 E j(b) II ). Since j(p)lkj&(a) n j(b) c j(c), we see that j(p) A II 9 E j(a) II A I/ 9 E j(b) II G II Qc j(c) II; hence f(p) <

a( // 9 Ej(C) // ).

(4) Fn V = U. Take X E V such that X CT S. Then // 9 &j(X) // equals 1 if X E U and 0 if X$U. We conclude using the fact that f is one-one. 0

Lemma 3. Let K be the critical point of j, and assume that n is < K-complete.

(I) Jf p < K and U is E$ ~-complete, then so is F.

(2) Zf S = K and U is normal, so is F.

Proof. We shall prove (2), the proof of (1) being similar and easier. Assume then that

a = (ai)i<n and b are given, as well as p E G such that p forces over B “(Vi < K) (ui E F)

& b = d (Ui / i < K)“. Then, for aI i < K, f(p) d rc( // 8~j(u,)l/ f (using Lemma 1). On the other hand, ~l~(V~)[(Vi < a)(~ E ai) =j E E b]. Hence, j(p) It [(Vi < K)(K f j(ai))

=B K E j(b)], which yields that j(p) A inf{ /I K E j(ai)ll 1 i < K} < II K E j(b)]]. Since II is d K-complete, this implies that f(p) < n( 11 QE j(b) 11). [II

Lemma 4. In V [ G], there exists a dense inject~ve boolean morphism from B(F) into C/G = C/f”G.

Proof. Take a E V such that I1-,a c S. Set IC/(uc) = n( II 9Ej(a)/Ij’B’)/G. II/ defines clearly a dense boolean morphism from P(S)n V [G] into C/G. Now,

J.-P. LevinskilAnnals of Pure and Applied Logic 72 (1995) 177~212 185

aG E F - 3P E G f(P) d “d I/ sEj(a) 11) o n( (/ 9 ~j(a) I/ ) = 1 in C/G. Hence, Ker($) is exactly the ideal dual of F, and $ yields the required map B(F) + C/G. q

3.7. Remarks. (1) It is now easy to show, following 3.1, that, if the family (B, n, C) is weakly regular, then F is forced to be precipitous. We omit the proof, since the filters we will construct in this paper will all be 6 w-complete and < w-distributive; hence

necessarily precipitous. (2) The construction we just gave is the one in [9]. Although led by

different considerations, it amounts basically to the same as the one in [S]. The difference is that [S] only considers ultrafilters in generic extensions. However, the filter F we just constructed can also be defined as the intersection of these ultrafilters and, in turn, they are exactly the B(F)-generic sets over V[G], that is what is expressed by Lemma 4 above. This is why we thought it might be appropriate to use the expression “Kunen-Paris families”. Observe also that [S] never uses non-regular families, as quoted by the authors.

(3) In [lo] we propose a more general construction, which works starting with an arbitrary filter, and not only an ultrafilter. Families are defined, as well as regularity and weak regularity, and are used to prove preservation results (for example of precipitousness) in generic extensions. The construction also works for non-well- founded models (i.e. for non-precipitous filters), and allows the proof of various equiconsistency results, like, for example, the following: if“ZFC + there exists a com-

pletely ineffable cardinal” is consistent, then so is “ZFC + the critical cardinal is

completely ineflable”. We obtain similar results with “completely ineffable” replaced by “weakly ineffable”, or “ineffable”, or “FZ 1 ,n -indescribable”, for any fixed n. Since we do not give these proofs here, we shall not bother the reader with a general construction.

(4) Why, would the reader ask, talk about regularity and weak regularity if we do not use them? The answer is twofold. Regularity will be used, and weak regularity will be implicit, for the following reason: it can be shown that, if B is a boolean algebra and F E VB a term such that F is forced over B to be a precipitous filter such that it is forced over B(F), the canonical elementary embeddings iD: V[G] -+

(V[G]” n V[G])/D associated with F extend j: V -+ Y’/U, then B is weakly regular and in fact F can be constructed from some weakly regular family (B, rr, C) E V. Observe also that there may exist precipitous filters in VB whose associated elementary embeddings do not extend j.

(5) The sets of conditions that we shall consider in the sequel of this paper will be of two kinds. Either they will outright have the K-CC (where IC is the critical point ofj), or they will be weakly regular but non-regular. In the latter case, in order to construct n:j(B) -+ C, we will proceed as follows. We will write B as a directed union of complete subalgebras which are themselves regular, say B = IJ { Bi 1 i E I}, and will, using regular conditions, construct maps ni : j(Bi) -+ Ci, then show that these maps cohere, in order to yield rc.

186 J.-P. LevinskilAnnals of Pure and Applied Logic 72 (199s) 1777212

4. The set of conditions

4.1. It is known that, in general, one cannot introduce generic subsets of a regular cardinal K and preserve some large cardinal properties of K without having previously introduced generic subsets of almost all M. < K, because of the reflexion properties of K entailed by these large cardinal properties. Hence, if we intend to get a premeasure on the critical cardinal ic by adding, say K ’ + Cohen subsets of K, we have to add either one, or OL+, Cohen subsets of almost all ct < K. it can be shown that one would not work, but GI+ will, for K will not be able to tell the difference between cl’ and ct+ +. Since we have several applications in mind, also with non-GCH beneath K, we will do a slightly more general construction.

From now on we assume that GCH holds (which is not necessary, but simplifies the computations)~ and that ic is a measurable cardinal. We fix a normal measure U over K, and construct j: V --+ F’/U.

Let ( ida G K denote the strictly increasing enumeration of all inaccessible cardinals

smaller than or equal to K, and let (e,),,, be a fixed fWICtiOn from K into K satisfying the following (Easton-type) properties: for c1 < K, 8, is a cardinal and Pa < cof(8,) 6 8, < pn + , (hence, if oL d b, then @= d 0,). For a set X and an infinite cardinal p, let Q( p, X) denote the set of all partial functions 4 : p x X -+ 2 such that [q( < p (the Cohen set for adding 1x1 subsets of p).

Let us define an iteration (P& G K as follows: PO = {S> . P, + 1 = P, * Q( pa, 0,) (where the second term is computed in VP*). If a is limit and c1< pa, then P, is the inverse limit

of (P<)r<*1 while if CL = pat then P, is the direct limit of (PS)y<n. Also set P = P,.

The following are standard facts of forcing theory. (1) forcing with each P,

preserves cofinalities, (2) for inaccessible CL [PaI = ct, (3) for u Mahlo P, has the < a-cc, (4) for tl < fi Pa/P8 is < p,-closed in I/‘=, and (5) VP satisfies “Vx < K, 2p= = 0,“.

4.2. For any set X, set Qx = Q(K, X). In order to get a normal premeasure on K, we shall force over V with the set P * Q1, where I is a cardinal such that i 2 K+ ‘, and Ql is computed in VP. This will be done in Section 6. The construction will be done in two steps. The first part, interesting in itself, will be used in Section 5 in order to negate GCH at every regular cardinal p < K, while keeping K measurable. We are now going to study the “master condition” properties of the sets P * Qx, which will allow to construct preciptitous filters in Tip*QA.

For every set X, set Px = P * Qx, and let Bx denote its boolean completion. A P,-generic set Gx over V will be decomposed canonically as Gx = G * Hx, where G is P-generic over V and Hx is (QX)“tG1 -generic over FCC]. We also set r = [(@, Ix < K)]“. in M, j(P,) has the following decomposition: j(P,) = P * Qr * Plr,j<r>[ * Qj<x,. Observe that, since M” c M and P * Qz has the f ~-antichain condition in V, it follows that (P * QT)M = (P * Qr)‘. We shall set Qz = Q’ and

p1K. j(K)1 = P’. If Gj(x, is j(P,)-generic, we will decompose it canonically as Gj(x, = G * H’ * G’ * Hj(x), with the obvious meanings. We will also set G* = G*H’*G’.

J.-P. Levinskil Annals of Pure and Applied Logic 72 (1995) 177-212 187

4.3. We shall first consider the problem of forcing over I/ with P * Qr and finding in the generic extension a quasi-measure on K. How could this work? Asume that F is such a quasi-mesure on K in VB.. Due to the remarks in Section 3, there must exist in V a boolean algebra C and a map 7c : j( B,) --t C, such that rc is onto C, rt is M-complete,

,f= n oj is one-one and I/-complete, and B(F) is isomorphic to C/f “G,, where G, denotes the &-generic set over I/. How could we get, in such a quotient of j(B,), a dense, < K-closed subset? We observe that there is a canonical complete embedding i : P * Qr -+ j(P * Qr), given by i(p, q) = (p, q, 1,l) (which could also be seen as the inclusion with an appropriate choice of supports). Since j( B,)/i”G, is < K-closed in M[G,], hence in V [CT], we would try to “coequalize” i and the restriction of j to B,,

i.e. to find some b E: j( B,) such that, for all I E B,, i(r) A b = j(r) A b. This is impossible,

since one can show that there are no j(B,)-generic sets Gjc,, over I/ such that j- ’ ( Gj(,,)

is &-generic over I/. The proof of this claim is given in Section 5, 5.5, Remark (2). (Observe also that, if we had j”r E M, then we could construct such generic sets. But, this would be an assumption with a strength not allowing the proof of equiconsistency results.) However, there will be such sets generic over M, and we are now going to proceed in order to construct them, defining, for each X c r such that 1x1 < K,

a condition bx E j(B,) coequalizing j and i,, where ix : Bx --) j(B,) is the restriction of i to Bx.

4.4. For technical reasons to be seen in Section 6, we shall do a slightly more general construction. Hence, let the iteration (P,),<, be as before and let I be a cardinal, such that i. 2 K+. Set J= {X c IllXl d K}. Say that a map g is good if Dom(g)EJ, Im(g) c r, and g is oneeone. Let now g: X + A c T be a fixed good map. To simplify the notation, let us still denote by g the extension K xX -+ K x A

mapping (a, x) on (a, g(x)). If G is P-generic over I/, g yields in I/ [ G] an isomorphism Z9 : Qx + QA defined by Z,(q) = q 0 g _ ‘. Let us denote by i, : P * Qx -+ j( P * Qx) the complete embedding it yields in I/. Hence,

i&q) = (p,40K1, 1,l).

Lemma 1. Assume that Gx is Px-generic over V. Then, there exists a j(Px)-generic set

Gjcx, over V such that (i,)- ‘(G,(x)) = j- ‘(G,(x)) = Gx.

Proof. Let, with the usual notations, Gx = G * Hx. Set HA = (Z,)“(H,), which is QA-generic over V [G]. Let H’ be any Q’-generic set over V [G] such that HA c H’,

and let G’ be any P’-generic set over V[G, H’]. Set G* = G* H’* G’ and let j*.: V [G] --) M [G*] be the canonical extension of j (recall that P has the K-CC in V).

We have: (j*)“(H,) = {j*(q)lq E H,} = {qoj-1 IKE H,} (since q E Hx =. \ql < K) =

{(q~g-l)~(g~j~l)lqE H,} = {q’o(goj-‘)lq’E HA). But gE M andjlxE M, since M is d K-closed in V. On the other hand, HA E M [G, H’]. Hence, finally, if we set d, = (j*)“(H,), we see that dge M[G,H’], A, c Qj(x) and M[G,H’]i= I A,[ d K <j(K). Hence, setting sg = U A,, we see that sg E Qj(x) and that, for all q E Hx, s, <j*(q). Now, let Hjcx, be any Qjtx,-generic set over V[G*] containing sg, and set

188 J.-P. Levinskil Annals of Pure and Applied Logic 72 (1995) I77-212

Gjcx, = G* * Hj(x,. By the choice of sg, it is clear that j-‘(Gjcx,) = Gx. On the other hand, (i,))‘(Gj(x,) = (i,)-‘(G * HA) = G * HX, by construction of HA. I-J

Lemma 1 can be rephrased as follows.

Lemma 2. Assume that r E BX - (0). Then, there exists a j(BX)-generic set Gj,,, over V such that j(r) E Gj(x, and j- ‘(Gj(x,) = ii ’ (Gj(x,).

4.5. Another way of setting things is as follows. Let still g : X + A be a fixed good map. Let Gj(x, denote the j(Px)-generic set over I’, say Gj(x, = G * H’*G’* Hj(x,. Set A,=(q~g~j-‘Iq~H’nQ,}ands,=UA,.Thens,isforcedinMoverj(P)tobean element of Qj(x) and we set b, = (1, 1, 1, sg) E j( Px). Observe that i, E M and jlB, E M. Hence, the following lemma.

Lemma 3. (1) b, = II(‘(Gj(x,) = j-‘(GjcxJ IIj’Bx)m (2) For all r E BX, j(r) A b, = iJr) A b,.

Proof. (1) The left inequality 6 follows from the definition of b, and the proof of Lemma 1. Conversely, assume that Gj(x, is j(Bx)-generic over V and that

(i,)- ‘(Gj(x,) = j- ‘(Gjcx,). In order to prove that b, E Gjcx), we have to show that, for all qEHA=HnQA, qOgOjmlEHj(x). But, if qEHA, (l,qog)E (i,)-‘(Gj(x,) = j-’ G ( j(x))* Then (1, 1, Lj(q og)) E Gj(x), and hence j(q”g) = qogoj-1 E Hj(x).

(2) By complementarity, it is enough to show that j(r) A b, < i&r). Assume not. Then, for some q E j(P,), q <j(r) A b,, but q A i,(r) = 0. If Gj(x) is any j(P,)-generic set containing q, we have b, E Gjcx), j(r) E Gjcx,, i,(r)$Gjcx), a contradiction. 0

4.6. Still assuming that g : X -+ t is a good map we set C, = {q E j( B,) 1 q < b,} and define n,:j(B,) + C, by Kg(q) = q A b,.

Now, assume that X, Y E J are such that X c Y, that h: Y -+ z is good and set g = hlx. It is clear that BX c By, j(B,) c j(B,) and that i,, restricted to j(B,) is equal to i,. We define hence kg,, : C, + Ch by k,,,(q) = q A b,,. We claim that kgh is a one-one boolean morphism from C, into C,,. This is clearly equivalent to the following.

Lemma 4. (1) bh < b, in j( BY) (2) IfqEj(Bx)issuchthatq<b,,thenqAb,>O.

Proof. (1) Assume that 1 (bh < b,), let q E j(B,) be such that q < bh but q A b, = 0, let Gj(u) be a j(B,)-generic over V containing q and set Gj,,, = Gj(y,nj(B,). Since

b,#Gj(x,, we have j-‘(Gj,,,) # (ig)-l(Gj(x)). On the other hand, since bh E Gj(y), we get (i,)-‘(Gj(X,) = (ih)-l(Gi(y))nj(B,) = j-‘(Gjcy,)nj(B,) = j-‘(Gjcx,), a contradiction.

J.-P. Levinskil Annals of Pure und Applied Logic 72 (I 995) 177-212 189

(2) Let Gjcx, be j(Px)-generic over V, such that q E Gjcx>. We shall find a j(Br)- generic set Gj(r) over I/ such that Gj,,, c Gj(r) and b,, E Gj(yh. TO this end, set 2 = Y - X, f= hlz, A = Im(g) and E = Im(f). Also set Gj<xj = G* H’*Hjcx,, HA = H’nQA, HE = H’nQ, and (i,))‘(Gj(x,) = G*Hx. Let US now set

HZ = {qoflq E HE}. Since HE is Q,-generic over V[G,N,], it follows that HZ is Q,-generic over V[G, HA] = I/ [G, Hx]. Hence, HY = Hx x HZ is Qr-generic over k’[G] and clearly (ih)-l(G*H’)=G*HZ. Now, set d,= {q~j_~lq~H~S =

fqofoj-l /q E He). We get: d, belongs to MEG, H’], is directed, of size 8 ic <j(K) and is a subset of Qj(z), Let hence Hjcz, be any Qjcz,-generic set over V[Gj,x,]

containing sf = u A, and set Gjcr, = Gj(x, x Hi(z). It is clear that Gj(r, works, since

sh = (s,,sf). ,,

The last lemma we need, to be used in Section 6, is the following.

Lemma 5. Assume that X E J and that g : X + 5 and h: X -+ z are two good maps such

that g # h. Then b, A bh = 0 in j(B,).

Proof. Assume the contrary, and let G](x) be a I-generic set over V such that

b, A bh E Gjcxj = G * H’ * G’ * Hj<x,. Hence, (ig)-‘(Gjtx,) =j-‘(Gj,,,) I=: f&)-‘(Gj~X~f = G * Hx, say. Now, let VEX be such that 5 = g(u) #h(q) = [ and set

a = {a < ~(H’(a,t) = l} and b = {z < ~JH’(cc,i) = 1). Hence, a # b, since b isgen- eric over V[G,a]. But, on the other hand, for all c( < K, H’(a,t) =

H’(~,g(~)) = Hx(n,q) = H’(~,h(~)) = H’(sc,if, a contradiction. fJ

4.7. Notations. In the next paragraph, we will apply this construction with 1 = r and will, for each X E J, consider only one good map, namely Id Ix. In this case, we will simplify the notation, writing bx, sx, Cx and nx_

Why, would the reader ask, talk about an arbitrary good map g : X -+ z, if we use only one? This will become clearer at the end of Section 6.

5. A problem of Kunen and Paris

5.1. Kunen and Paris asked, a long time ago in [IS], what the consistency strength of the theory “ZFC + there exists a measurable cardinal IC such that, for all regular cardinals p < K, l?' > p + ” is. We shall give the following answer.

Theorem 1. Assume that “ZFC + there exists a measurable cardinal” is consistent.

Then, so is “ZFC + there exists a measarab~e cardinal IC such that,~or every regular cardinaf p < K, 2' = p c + “.

5.2. We shall arrange that IC stays measurable, while, for every inaccessible cardinal p < K, 2p = p+ +. As quoted in [IS], in order to obtain 2p = pf + for all regular


cardinals p < K, it is sufficient to perform a preliminary extension through direct Easton forcing, making 2P = p ’ + for all successor cardinals < IC (and for w).

Let us start with a model V+ “ZFC + GCH + K is measurable + U is a normal measure on K”. We shall end with a generic extension of I’, say W, in which there is a normal filter F on K, such that B(F) admits a dense set E, which is < rc-closed. This, together with 2” = K+, implies that K is measurable. Let us shortly recall a proof of this fact. Let D be a B(F)-generic set over W. Since forcing with B(F) adds no subsets of K, L) is, in W CD], a normal ultrafilter over K. Let still D E WE be a term denoting such a generic ultrafilter, and let (A& < K + be an enumeration of all the subsets of K in W, each of them being repeated K + times. Now, define a decreasing sequence (p,), < Ki in E such that, for all c1 < K+, pa decides over E whether A, belongs to D. Finally, setting D = { S c K / 3a < k', pm ibE S E I) 1, we see that D is a normal ultrafilter over K. In order to prove this fact, we first observe that D is a ~~~o~~ ultrafilter over K. Hence, in order to prove that D is normal, it is enough to prove that it is closed under diagonal intersection, for this, together with uniformity, will imply that D is K-

complete, as well. So, let {S, 1 a < K> be, in V, a family of elements of D, and set S = A{ S, / OL < K). For each CI < K, there is an index p(a) K K+, such that S, = Aptal.

Now, let R be an index such that S = An and 2 2 sup(p(ol) I GI < K>. For OL < K, we

have PJ. % P~(+ and pPcti) KE S, E D; hence p1 It, S, E D. Since it is forced over E that D is W-normal (since F is a normal filter on K in W), it follows that pn ltES E D, showing hence that S ED.

5.3. Hence, let (p,), s K and (O,),,, be as in 4.1 (here, we set 8, = (p,)’ ‘, but the proof will obviously work for any function satisfying the conditions of 4.1). We still let j: I/ -+ M = VK/U and let r = (j(0),)“. Let us observe, at this point, that lr]” = K+. To see this, we note that, for c1 < K, 8, < K, which implies that T = (j(O),) <j(K). But we know that j(k) < [2”‘]“. Since 2” = K+, the latter gives r < K++.

We shall apply the constructions of Section 4 with ;i = t, i.e. force over V with the set P*QI. For each XrzJ= (X c 7 11 X / G K >, we will consider only one good map from X into r, namely the identity restricted to X. Hence, we adopt the notations of 4.7 and we have, for each X E J, the canonical complete inclusion ix : P * Qx -+ j(P * Qx), an element bx gj(P r(l &) which says “ix = j”, a map nx:j(Bx) + Cx = (4 ~j(&) I q Q b,f, defined by xx(q) = 4 A bx, and, for X c Y, both in J, a one-one boolean morphism kxu : Cx + Cy, defined by k,,(q) = q A by. We now observe the following lemma.

Lemma 1. (1) B,= U(&(XcJ),

(2) jU%) = U CA&) IX E Jf-

Proof. (1) Assume that q E B,. Since B, has the < K-CC, q is the supremum of K elements Yi of P,, each of them lying in P,,i,, for some X(i) E J. If we set X = IJ X(i),

then q E B,.

J.-P. Lecinski/Annals of Pure and Applied Logic 72 (1995) 177-212 191

(2) Assume that q E j(S,) and let h E (B,)” be such that q = [h]“. Applying (l), we can find some X E J such that h E (B,)“. Hence, q Ed. 0

Let C be the direct limit of the system (Cx,kXY)X,YEJ.X c r, with limit maps kx: Cx + C, which are clearly injective and complete, since each kxr is. Let us now define a map n : j(B,) + C as follows. For q E j(B,), take X E J such that q E j(Bx) and set n(q) = kx(nx(q)) = kx(q A b,). It is clear that rc is well-defined and is a boolean morphism onto C. Moreover, by Lemma 1, Dom( n) = j(B,). Set f = n oj, and let i: B, + j(B,) be the canonical complete inclusion, given on P * QI by i( p, q) =

(P34,1,1).

Lemma 2. (1) f is one-one and complete.

(2) f= 713 i.

Proof. (a) fis one-one. Let r E B, be such that r # 0 and let X E J be such that r E Bx. Then f(r) = kx(nx( j(r))) = k,(j(r) A b,) # 0, by Lemma 2 of Section 4 and the fact that kx is one-one.

(b) f is complete. Assume that G is C-generic over I/ and let us show that G, =f-l(G) is B,- generic over I/. Let X be a maximal antichain in B,. Then 1x1 d K and hence, for some X E J, X c Bx. On the other hand, G,nB, = j- *(n; ‘(kg l(G))) is &-generic over I/, since kx is complete, rcx oj = rtx 0 ix (see Section 4, Lemma 3) and rcx, ix are complete. Hence, GxnBxnX # 8.

(c) f = n 0 i. Let r belong to B, and let X E J be such that r E Bx. Then,

f(r) = Uzx(j(r))) = kx(nx(ix(r))) = n(i(r)). 0

5.4. Now we define, as in Section 3, 3.6, a filter F in T/Bc, with the help of the family (B, rr, C). Let G(z) be &-generic over I/. For a E VE’ we set +(r) E F o 3p E G,

f(P) < /I K Eda) II j@) F is then normal on K, and the algebra B(F) is isomorphic to . C/G(r) = CIf”G(r).

Remark. We could of course have defined F “directly” as follows. For a E I/B, we set UC(,) E F o for some X in J and some p E G(T)nBx, a E VEX and j(p)ltjtBx,

“K E j(a) v(ix)-‘(Gjcx,) # j-‘(Gjcx,)“, but this would not help at all to study B(F).

We are now going to construct a d K-Closed set E, dense in C/G,. E will be the union of a 6 K-directed family ( EX)XE.,, each E, being in some way defined from Cx.

For X E J set fx = nx 0 i. fx is clearly complete. Moreover, for r E Bx, (kx(fx(r)) = kx(zx(i(r))) = n(i(r)) =f(r), by Lemma 2. Hence, kxofx =f and, in particular,f, is one-one, since f is. Hence, kx yields, in V[G,], a complete injective morphism rx: C,/G, = C,/( fx)“GT + C/G,. We now seek for a < K-closed set Dx, dense in C,/G,. To do this, we observe that the standard forcing theory shows that Cx admits as dense subset the set P * Q’ * P’ * Sx, where Sx = {q E Qj(x) I q < b,) and

192 J.-P. Levinskil Annals of Pure and Applied Logic 72 (1995) 177-212

that, with this notation, for (p, q) E P * Q’, &(p, q) = (p, q, 1,l). Hence, C,/G, admits the set & = P’* Sx as dense subset, and this set is clearly < tc-closed in V[GJ.

Hence, set Ex = (tx)“(&), and let us check that these sets cohere.

Claim 1. Let X, Y E J be such that X t Y. Then Ex c: Ey .

Proof. Let c E Ex. Then, there is some rl = (p, q, pl, ql) tz P * Q’ * P’ * Sx such that

(p,q) o G, and c = r&t/G,) = (~~(~~))/G~. But then, c = (~~(k~~(r~)))/G~ = tY[(kXY(rl))/Gr]. On the other hand, kxy(rl) = r1 A& = rl /\(l,l, 1,~~) = (p,q,pl,ql r\sy). Since q1 ASy < sy and q1 ASy = q1 USy E Qj(y), we see that kXY(r,) E Dy, and hence that c E Ey. q

Now, set E = l.J (Ex / X E Jf . Claim 1 implies that E is d ic-closed, since each Ex is and J is < K-directed in V [Gr], On the other hand, E is dense in C/G,, because each Dx is dense in C,/G, and C/G, is the union of all ( tx)“( C,/G,). Hence, the properties of F are established and Theorem 1 is proven, as quoted in 5.2.

5.5. Remarks. (1) For X E J, C,/G, will be different from Cx/Gx (that is the point of the proof), since forcing over V[G,] with C,/G, adds no subsets of K, while forcing with CxlGx does.

(2) There is no j(P,)-generic set over V whose inverse image by j would be P,-generic over V. The proof we shall give of this fact is due to the referee. Assume, to the contrary, that such a generic Gjf,, = G* * Hj(+ exists, setj- ‘(G& = G *H, and let as usual j* : V[G] + M[G*] be the canonical elementary embedding extending j. Observe that, for q E Qj(c) there is some 5 < T such that Dom(q)n[j(@ x {j(t)}] = 8. For, there is some X E Jsuch that q E Bj(x, (for q is the class of a function h from K into

Q) and then Dom(q)n [ j(lc) x j”,] c j(K) x j”X, and any 4 E t - X will do. This implies that the set D of all q E Qjfr> such that, for some 5 < T, [K x (j(r)>] x {O> c q is dense in Qj(rI. Let hence q be any element of L)n Hi(,) and let 5 < ic be such that

[TK x (j(t))] X (0) = 4. Since ff, = (j*)-‘(Hjcr,), we see that K is the 5th Cohen generic subset of K determined by H,; obviously a contradiction to the fact that K cannot be a Cohen generic subset of itself.

5.6. Strong ~is~~ib~~ivity. The remarks in 5.2 can be done slightly sharper. To this end, let us recall the following.

Definition 1. Let P be a set of conditions and let A be a cardinal. P is said to be strongly I-distributive iff the following holds: for every r E P and every family (Dol)aci of open dense sets in P, there exists a decreasing sequence (p,), < 1 in P beneath r such that, for all tl < rZ, pa E&

Obviously, strong ;l-distributivity implies A-distributivity, and if P admits a dense, h-closed subset, then it is strongly I-distributive. On the other hand, strong ,I-

J.-P. Levinski 1 Annals of Pure and Applied Logic 72 (I 995) I77- 2 12 193

distributivity is usually stronger that A-distributivity: take, for example, a I-Suslin tree. One interest of strong distributivity lies in the following.

Lemma 3. Assume that there exists a strongly u+-distributive set of conditions P such

that Ikp “K is measurable”. If 2” = K+, then K is measurable in V.

Proof. Let D E VP be such that Itp “D is a normal ultrafilter on K”. Let (A,),, K t be an enumeration of P(K) in V, such that each subset of K is repeated K+ times. For each c( -C K +, set D, = { p E P ( p decides if A, E D} and letting (pa),<,+ be a decreasing sequence in P such that, for all a < K+, pd ED,, set U = (X c K ( 3a < K+, pm It X E D}. It is clear that U is a normal ultrafilter on K: the proof of this claim goes exactly as in 5.2. 0

Remarks. Assume that 2” = K+ and that E is a normal filter on K such that B(F) is strongly K+-distributive. If the term D of the preceding proof is taken to be the B(F)-generic set over V, then obviously F c U.

5.7. Games. There is a game-theoretic equivalent of strong distributivity, as well as

a closure-type one. Let P be a set of conditions, 1 an infinite cardinal and r an element of P. We define

the following game C( P, 1, r) of potential length 1. Players 1 and 2 both play elements pa of P, for 0 < a < 1. Player 1 plays for a even (including the limit stages) and player 2 for a odd, the sequence (p,) has to be decreasing and p. must be smaller than or equal to r. Player 2 wins the game iff, at some (necessarily) limit stage a < II, player 1 is unable to move, i.e. if inf(pc 15 < a} = 0. If not, then player 1 wins.

Lemma 4. The following conditions are equivalent. (1) For no r E P has player 2 a winning strategy in G( P, ,I, r).

(2) P is strongly d-distributive. (3) For every term a E VP and every r E P such that r It,, “a is a function from I into V”

there exists a function f E V’ n V such that, for all a < A, there is some p 6 r with

plFp ‘tfla = al,“.

Proof. (1) * (2): Assume that (Da)a<l is a family of dense open sets in P and that r E P. We consider the following strategy for player 2 in G(P,I, r). For all limit ordinals a < 1 and all integers n, pa+ *“+ 1 will be an element of D,,, smaller than or equal to p.+ 2n. Obviously, any sequence (p,), < 1 showing that this strategy is not winning will yield a decreasing sequence ( qa)a < I beneath r such that, for all a < I, qaEDa and go dr.

(2) =S (3): Assume that r It- “a is a function from ;1 into V”. For a < 1, let D, be the set of all p < r such that, for some s E V”, p forces that s equals al,. Letting (P,&<~ be a decreasing sequence in P such that, for all a < 1, pal ED,, we let f be the union of all s such that, for some a < I, pa forces that s equals al,. It is clear that this f works.

194 J.-P. LevinskilAnnals osPure and Applied Logic 72 (1995) 177?2/2

(3) =s (1): Assume that r E P and that cr is a winning strategy for player 2 in G( P, 2, r). Let G be any fixed P-generic set over I/, such that r E G. In V [G], we define a function a : i + V, defining a(a) by induction on oz. Assume the following. (1) u is an even ordinal, (2) a la has been defined and is a decreasing sequence of conditions in P,

(3) for all iJ < c(, a(t) E G and (4) if 5 is odd, then a(<) = o(aI<). We now define a(a) andu(a+1).Setq=inf{u(~)~~~cc)r\r.Sinceu~,~I/,weseethatq~G.Onthe other hand, the set D = { ~(a[, u {(a, p)} ) I p < q} is dense in P beneath q. Hence, there exists some p < q such that a(u(,u { (a,~)}) E G. We let u(a) be such a p and set

a(a + 1) = 4uJ{w-$9 a is then defined, and is in V [G] a play against 0. We still let a E VP be a term which is forced by r to denote such a function, and let f~ I/’ be a function such that, for all M < A, there is some p d r with p Itp “j-1, = ~1~“. It is clear that, in V, f is a play against cr, a contradiction. IJ

Remark. It is clear from Lemma 4 (3) that P is strongly l-distributive iff its boolean completion is.

5.8. Standard properties. We now need to state some properties of strong distributivity. The first is a slight strengthening of Easton’s lemma.

Lemma 5. Assume that 1 is an injinite cardinal, and that P and Q are two sets of

conditions such that P has the l-cc and Q is strongly l-distributive. Q is then strongly

i-distributive in VP.

Proof. We first observe the following.

Claim 1. Assume that DE VP is such that Itp ‘D is open dense in Q”. The set E = {q E Q 1 Itp “q ED”) is then open dense in Q.

Proof. E is obviously open in Q. To see that it is dense, let r E Q and let us define, in V,

a strategy 0 for player 2 in G(Q, 1, r). q=, ~1 even, denoting the moves of player 1, we define the answers qa+ 1 of player 2, together with elements pa of P, forming an antichain in P. Hence, assume that c1 is even and that (q<)< s a and { pc I 5 even and r < g} have been defined and let us define q=+ 1 and pd. Set s, = 1 - sup{ ps I 5 even and 5 < u}. If s, = 0, we set q b+l=qaandp,=O.Ifs,>OweletO<p,<s,and qa+ 1 < q. be such that pb kp “qa+ 1 ED”. Now, let (qo1)61< A be a play against 0 in G(Q, 1, r). Since P has the I-cc, there must exist some even CI < k such that s, = 0. If we set q = qol, we see that q d r and kp “q ED”. 0

Proof of Lemma 5 (Conclusion). Let now (Da)a< 1 be a family of terms in VP such that, for all a < 1, Ikp "D, is open dense in Q” and define E, E V, E, c Q, setting E., = {q E Q I Ikp “q E D."}. By Claim 1, each Em is open dense in Q, and then, given any r E Q, we can find a decreasing sequence ( qE)= < A in Q beneath r such that, for all a < 2, q. E Ea. It is clear that (qol)acl is the desired sequence in VP. 0

J.-P. Levinskil Annals qf Pure and Applied Logic 72 (1995) 177-212 195

Lemma 6. Strong ,I-distributivity passes to complete subalgebras.

Proof. To see this, let i: B -+ C be a complete injective boolean morphism between complete boolean algebras and let n: C --) B be its associated normal projection.

Assume that (DJacl is a family of dense open sets in B and that r E B, and set E, = {q E C IZIp ED,, q 6 i(p)}. E, is then open dense in C, and if we let (qa)a<2 be a decreasing sequence beneath i(r) such that, for all ~1, 4% E E,, then (rc(q,)),, i. is a decreasing sequence beneath r such that, for all a, x(9,) E D,. 0

Lemma 7. Zf P is strongly i.-distributive and if Q is strongly i-distributive in VP, then

P * Q is strongly /l-distributive.

Proof. To see this, assume that a E VP*Q and (r, s) E P * Q are such that V + [(r, s) II,,,

“a: A + On”]. Let us identify Vp*Q and ( VP)Q, and let G be a P-generic set over V containing r. Then, V [G] /= [so IF, “ao: A-+ On”]. Hence, there exists a function g E 0n”n V[G] such that V [G] + [(Ya <2)(3q d s,)(q It, “gld = (ac)lb)“]. Let b E VP be a term forced by r to denote such a function g, and let f~ V’ n V be such that, for all a < E,, there is some p < r with p Ik, “jl, = bl,“. It is clear that this f works for P*Q. 0

Lemma 8. If P is E.-closed and Q is strongly ,I-distributive, then P x Q is strongly A-

distributive.

Proof. P is obviously strongly /l-distributive in VQ, which implies, by Lemma 7, that Q * P is strongly i-distributive. But, P x Q can be densely embedded in Q * P. 0

Lemma 9. Assume that i is regular. Then, every strongly L-distributive set of conditions

P preserves stationary subsets of i.

Proof. Assume that S c i is stationary and that C E VP is such that Ikp “C is club in i and SnC=@“. For CC<~, set D,={p~Plfor some [>a, pk<~Cj, and let

(Pa)ol<A be decreasing in P, such that, for all c( < 2, pd ED,. We now define a strictly increasing sequence (t,), < 2 in 2 by induction on a. Assuming that a < i and (t,), <31

has been defined, we set a’ = sup{ <, 1 v < a> and let 5, be such that 5, > ci and pa, II- 5, E C. Finally, letting D be the set of limit points of { 5, I a < I,}, we see that D is club in 2 and D n S = 8, a contradiction. 0

Remark. It follows from Lemma 9 and [S] that the converse to Lemma 7 is false. For [S] gives an example where B, C are two complete boolean algebras, both admitting a dense, d o-closed subset, and i : B + C is a complete monomorphism, such that, in VB, forcing with the quotient C/i”B does not preserve stationary subsets of or.

196 J.-P. Levinski JAnnals of Pure and Applied Logic 72 (1995) 1777212

6. A premeasure on the critical cardinal

6.1. Recall that, in Section 2, we have defined a premeasure on K as a normal filter Pi such that B(F,) is K+-distributive, and called K premeasurable iff there exists a premeasure over K. Recall also that K is the critical cardinal iff it is the first infinite cardinal t such that 2’ > z+. We intend now to prove the following.

Theorem 1. Assume that Vk “ZFC + GCH + U is a normal measure on K”. Let ;I be

a cardinal such that 1 2 IC+ +. Then, there exists a generic extension W of V satisfying “K is the critical cardinal and K is premeasurable and 2” 2 II”. Moreover, if V k “cof(1) > K", then W + “2” = 1”.

6.2. The set of conditions. We will keep the notations of Section 3. For an infinite cardinal p and a set X, we denote by Q(p, X) the set of all partial functions q:pxX+2suchthatlql <P.Welet(p,),,. denote the increasing enumeration of all inaccessible cardinals less than or equal to K and define an iteration (P,), s K as follows.

PO = (8); Pa+, = P, * Q(pa,(pa)+) (where the second term is computed in VP,); for CI = Pa, P, is the direct limit of the family (I’,), <a and for o! limit such that CI < pa, P, is the inverse limit of (P&+ We also set P = P,. It is clear that forcing with P preserves

cofinalities and the GCH, that P has the K-CC, and is of size K. We shall construct our premeasure over K in the generic extension of V by the set Pi = P * Q(K, 2). In order to proceed, we fix some more notations. For each set X, we set Qx = Q( K, X) (wherever it is computed), Px = P * Qx, and let Bx denote the boolean completion of Px. Any P,-generic set Gx over V will be decomposed canonically as Gx = G * Hx, where G is P-generic over V and Hx is ( Q,)“tG1-generic over V [ G].

We also let j: V + VK/U = M denote the canonical elementary embedding, set r = (K + )M = (K + )“, Qr = Q’, and decompose j( Px) as j( Px) = P * Q’ * P’ *j( Qx), where P’ = P,,,j(K)f+ Any j(P,)-generic set Gj(x) over M will be decomposed canoni-

cally as Gj(x) = G * H’ * G’ * Hj(x), with the obvious assignments. We also set

I = Qjtx).

6.3. We shall also use the constructions of Section 5, concerning a filter F on K in VP*Q’, which we recall now. We set J = (A ctlIAl<~}and,foreachA~J,define an element bA EJ’(P~), which is the largest b forcing over j(P,) that j- ’ (Gj(A)) = ( iA)- ’ (Gj(A)), where iA is the canonical complete inclusion from P * QA in j(P,), given by iA(p, q) = (p, q, 1, l), and Gj(a) denotes the j(P,)-generic set. We also set CA = {q E j(B,)) q d bA), and, for S, T E J such that S c T, we define a complete one-one boolean morphism kST : Cs + CT setting k,,(q) = q A bT. Finally, we let (C, ( kA)AEJ) be the direct limit of the < K-directed family (( CA)AE.,, (ksT)s C T,sEJ, TEJ). Each kA: CA + C is a complete one-one boolean morphism. We define an M-complete boolean morphism TT from j(B,) onto C setting, for A E J and q E j(B,), n(q) = k,(j(q) A bA) = k,(i(q) A bA), where i is the canonical complete inclusion from P*Q’ into j(P* Q’) given by 0, q) = (P, q, 1, 1) and observe that

J.-P. Levinski / Annals of Pure and Applied Logic 72 (I 995) I777212 197

rt 0 j = rc 0 i = f‘: B, + C is one-one and complete. Hence, if G * H’ is P * Q’ generic over V, we define, as in Section 3, a filter F on K in I/ [G, H’], such that the boolean algebra B(F) is isomorphic to C/( G * H’) = C/f”(G * H’). We have also proven that, in I/ [ G, H’], B(F) contains a 6 K-closed, dense subset. The letter F will be reserved to this filter, while we shall denote by F1 the filter on K that we intend to construct in

v CG, HiI.

6.4. Idea of the proof. We now fix a cardinal i > K+ and, letting Gn = G *HA be P * Hi-generic over V, shall construct in I/ [ G, H,] a premeasure F1 over K. We first define, in V, the following set of conditions. R will be the set of all bijections g : X -+ A, where X c I., A c z, and 1x1 < K (hence A E .I), ordered by inverse inclusion. Clearly, R is < K-directed closed with greatest lower bounds in V, and a R-generic set K yields

canonically a bijection m : A -+ z defined by m = U K, such that, whenever X E V is such that X c A and 1x1 d K, then g = mix E K. We shall be interested in the set of conditions (C - (0)) x R, shortly denoted by C x R, and the strategy of the proof will be as follows. A C x R-generic set G2 x K over V will yield a P * Qz-generic set G1 over I/ and a V [ GJ-normal ultrafilter D over P( rc) n V [GA]. F, will be the intersection of all these ultrafilters D, G2 x K running through all the C x R-generic sets over V which yield the same fixed GA. This will, of course, require that B, can be seen as a complete subalgebra of the boolean completion of C x R. The ultrafilter D itself will be the B(F) generic set over V[G, H’] associated with the C-generic set G2 over V, where we set G * H’ = .f - ’ (G,),f: B, + C being as in 6.3, and this will be possible because we will show that Pi V[G, H’] = P(lc)n V[G, H,].

6.5. We now proceed to the necessary technical lemmas. We let G2 x K be C x R-

generic over I/ and set G * H’ =f - ’ (G2), which is P * Q’-generic over I/.

Lemma 1. V[G2, K] is a generic extension of V[G, H’] by a set ofconditions which is

strongly K+-distributive in V[G, H’].

Proof. Let F E V[G, H’] be the filter on K constructed in 6.3. We know that, in V[G, H’], B(F) admits a dense, K +-closed subset, say E. Moreover, if D is any E-generic set over V[G, H’], then V[G, H’][D] = V[G,]. On the other hand, R is K+-closed in V; hence, by Lemma 5 of Section 5, strongly K+-distributive in I/ [ G, H’],

since P * Q’ has the K+-antichain condition in I/. Hence, using Lemma 8 of Section 5, we see that E x R is strongly K ‘-distributive in V[ G, H’]. But V[G2, K] = V[ G, H’] [D x K] is a generic extension of I/ [G, H’] by the set of conditions ExR. 0

Lemma 2. Assume that G2 x K is C x R-generic over V, and set G * H’ = f -‘(G,).

Then there is, in V[G,,K], a (QI)“[‘l -generic set H, over V [G], such that On”nV[G,H’] =On”nV[G,H,].

198 J.-P. Levinskil Annals of Pure and Applied Logic 72 (1995) 177-212

Proof. Set m = lJ K, which is a bijection from A onto r. The map #, defined on ( QI)YIGl

by the formula 4(q) = q * m is, inside V[ GZ, K J, an isomorphism from this set onto

(QJvfG’, since, by the K+-antichain condition of P in V, every Y E V[G] such that Y c On and 1 Y 1 < K is included in some X E I/ such that 1 X 1 < IC. Hence, we set HA = gh"(H') and show that HA is (QA)“tGl -generic over V[G]. Let A E V[G] be a maximal antichain in (Q) ‘[‘I Since IAl < FC, we can find some set X c A, such that .

XE I/, /Xl d K and A c(Q~)“[‘]. Setting A = m”(X) we see that, since mix E V c V[G], the restriction of $ to (QA)i’tG1 yields an isomorphism from this set onto ( Qx)“tG1, which itself belongs to V[G], hence, in V, an isomorphism $ from P * QA onto P * Qx, such that $“(G * (H’n QA)) = G * (HA n Qx). Hence, HA n Qx is Qx-generic over V[G], and A n HA # 0. H;, is then defined, and proven to be (QA)“tG1 -generic over V[G]. We still have to

prove that V[G, H'] and I/ [ G, HA] have the same K-sequences of ordinals. By Lemma 1, this amounts to showing that On”nV[G,H’] c V[G,HJ. Hence, let a E On” n I’[ G, H']. By the K+-antichain condition of P * Q’ in I/, there exists some set A E I/ such that A c z, [Al d K and a E V[G, H'nQA]. With the notations and by the arguments of the proof of the existence of HA, we see that, setting X = m-‘(A), UE V[G,H'nQJ= V[G,H;,nQ,]c V[G,HJ •l

Remark. Hence, by Lemmas 1 and 2, V [G, H,] is < K-closed in I/ [ Gz, K], which means that every K-sequence of elements of T/CC, H,] which lies in V[ Gz, K], already liesin V[G,H2].

6.6. Now, we proceed in a standard algebraic way. We let the same expression G * HA be a term of IJ”~ s denoting the P * Q,-generic set over I/ that we just defined from the C x R-generic set and, letting as usual B(P) denote the boolean completion of any set P, define a map e: BA -+ B(C x R) setting e(r) = /I r E G *HA IIBccxR). It is a standard fact that e is a complete boolean morphism, and we now show that e is one-one. Lettingr=(p,g)EP*Q1,Xci:besuchthat/XI~rcandrEP*Qx,andg:X~A be any bijection from X onto some subset A of z, set T’ = (p, q 0 g - ‘) E P * QA c P * Q' andc=i(r)~b,=(p,q~g-‘,l,l)r\bAECA. By Section 4, c # 0, hence we can let G2 x K be some C x R-generic set over V such that (kA(c),g) E G2 x K. Keeping the notations of the proof of Lemma 2, we see that (p, q 0 g- ’ ) E G * H', that g c m = lJ K, and that hence r E G * HA. Hence, certainly, // I E G * HA /I BfCx R, # 0, which means that e(r) # 0. We have hence represented BA as a complete subalgebra of S(C x R), and we know from 6.5 that the quotient B(C x R)/e”(Bn) is < x-distributive in VBA.

6.7. Construction of Ft. Let G * HA be a fixed P* HA-generic V. We are ready to define a filter FE over K in V[G, HA]. Let Gz x K be any C x R-generic set over V such that e”(G * H,) c G2 x K, set G * H' =fS1(GZ x K) and let D be the B(F)-generic set over V [ G, H'] corresponding to Gz, as in 6.3. Since P(K)nV[G,H'] = P(K)n V[G, H,], D is a V [G, HA]-normal, V [G, H,J-ultrafilter over K. Let still the letter D be a term of VR which denotes this ultrafilter, where we set

J.-P. Levinski/Annais of Pure and Applied Logic 72 (1995) 177-212 199

R = ~(CxR)/B~ = ~(~xR)/e”(G*~~),andset,in V[G,IlJ,Fi = fE c ~jll-&&13f. Fi is obviously, in V[G, ZZ,], a normal ultrafilter over K (the proof of this fact goes as in 5.2), and we are left to show that B(F,) is < ~-distributive there. It is clearly, due to

Lemma 2, enough to show that the B(F,)-generic sets over V [G, H,] are precisely the ultrafilters Z? that we just constructed. To this end, we define, as usual, a map 5, : B(FI ) -+ R by +([E&,) = // E E I> IIR, which is obviously a one-one boolean morphism, and it is enough to show that 4 is complete. It will not be harder to show that

(fi is dense, which characterizes B(F,).

Claim 1. Im(4) is dense in R.

Proof of Claim 1. Let s E Z?. We seek some E c IC, E E V [ G, HA], such that E E (F, )’ and j/E E D IjR < s. In order to achieve this, let us first find some (c,g) E C x R such that the following conditions hold: (1) in R, 0 -=z (c,g)/G* HA d s, (2) if we set A = Im(g), then there exists a (necessarily unique) c’ E CA such that

This can be done, because the set of (c, g)/G * Hi. satisfying {2) is dense in R. Letting now X = Dam(g), we define as usual an isomorphism Z,: BA -+ Rx by the formula Z9( p, q) = (p, q 0 g), for (p, q) E P * QA. Clearly, Zg E V. Now, c’ E CA c j(B,), and we can hence find some function h E (BA)‘n Y such that c’ = [hlIi (where U is the ultrafiiter on K in V we started from). We finally set E = (~~-‘(~g)-‘(G* HA) =

(h)-‘(Zg)-t{G*H,).

Subclaim 1. E E (F,)+.

Proof of Subclaim 1. Keeping our notations, we have to find a C x R-generic set G2 x K over V, such that e”(G * H,) c Gz x K and E E D. To do this, we let Gt x K be any C x R-generic set over V which extends G * HA and contains (c, g). Since c E GZr (~)~~‘(G~(H’nQ~))~D. Since gEK, G*(H’nQ~)=(Zg)-‘(G*(~~nQ~)), and hence E E D. 0

Observe, although we do not need this fact, that we have proven that j}E E D llR >, (c,g)/G * HA. We are now going to prove the converse, i.e. that /I E E D jiR Q (c,g)/G * HA d s. Hence, let Gz x K be any C x R-generic set over V ex-

tending G * HA and containing a representative of 11 E E D II? We have got to show that

(c>g) E Gz x R.

Subclaim 2. Assume that g f K. Then c E Gz.

Proof of Subclaim 2. This goes as the proof of Subclaim I, the notations of which we keep. Since g E K, G * (ti’nQA) = (Zg)- ‘(G * (H, nQx)). Since E E D, it follows that (~‘)-‘(G*(~‘nQ~))~~, which exactly means that CEG~. 0

200 J.-P. LeuinskilAnnals of Pure and Applied Logic 72 (1995) 177-212

Hence, in order to finish the proof of Claim 1 and of Theorem I, we are left to prove the following.

Subclaim 3. g E K.

Proof of Subclaim 3. Assume this is not the case, and set m = u K, n = mix (where X still denotes the domain of g), S = Im(n) and t = bin-‘. Hence, SeJ, t is a one-one map from S onto A c r, and, since g$K, we see that t # IdIs. We keep the notations of Section 4, where we defined, associated with the two maps r and Id/s, two

elements b, and bs ini( from which we showed in Lemma 5 of Section 4 that they are incompatible, as soon as t # IdIs. It is hence enough to use our other assumptions in order to prove that b, A bs > 0.

Since RED, we get that E = ~~~‘(Zg)-‘(G*~~) = h-‘~Z~)-‘(G*~‘) =(Z,oh)-’ (G * H’) E D. This means that k,( [It 0 h]n) E GZ, which implies that [Z, 0 Zr$, A bs > 0 in j(S,). It is then enough to prove that [I,0 h]” 6 b,. To do this, we look at the isomorphism j(Z,):j(Rjr,) -+ j(Bs) and observe that, by Section 4, Lemma 3(l), b, = j(Z,)(b,). On the other hand, since c’ E CA, we have c1 < bA, which implies that [I1 u^ h JU = j(Z,)(c’) < j(Z~)(b~) = b,, as was to be shown.

This completes the proof of Claim 1 and hence of Theorem 1. [7

Remark. We could replace the function 0, = (p,)’ by any “Easton-like” function, as in Section 4, and still get the result.

6.8. More closure properties.

Definition 1. A strong prevneasure on K is a normal filter Fi on K such that B(F,) is strongly x+-distributive. K is strongly premesurable iff there exists over K a strong premeasure.

We shall improve Theorem 1 by showing the following.

Theorem 2. TheJilter F, constructed in the proof of Theorem 1 is a strong premeasure on K.

Proof. In order to proceed, we give an equivalent construction of Fi, based on the fact that C x R = (P * Q’ * B(F)) x R (due to Section 3, Lemma 4) = P * [R x Q’] * B(F) (with the obvious identification of terms).

6.9. Hence, keeping the notations of this paragraph, let G be P-generic over I’, K be R-generic over i/ [G J, and HA be {Q~)“[‘]-generic over P’[G, K]. Set m = U K. m is a bijection from d onto r which yields, inside V[G,K], an isomorphism

I(I : (QAVrG1 -+ (QA)“‘~“, defined by 11/(q) = q 0 m, actually already because

(Qr) “[‘I = (Qr)“tG*K1 and (QI)“tG1 = (Q1)YtG*K1, since, by Easton’s lemma, R is K'- distributive in I’[ G].


This means that we can define in V an isomorphism 4 1 :

P * [R x QT] -+ P * [R x QJ, given on a dense set by 4, (p, g, q) = (p, g, q 0 g). More- over, the term B(F) E Vp*Q’ can be seen as a term of VP*[RxQA1, which allows us to extend & to an isomorphism & : P * [Qr x R] * B(F) -+ P * [Qn x R] * B(F) or, with the allowable identifications, to an isomorphism 4 : [P * Q’ * B(F)] x R +

[P * QA] * [R* B(F)], where we decide to use B(F) as a term of V [G, HAIR as well. Let us now denote by t the canonical complete inclusion from P* QA into

CP * Qnl * CR * W)l, g iven by t(p, q) = (p, q, 1,l). Since HA is constructed from H’ as

in Lemma 2, and since P * Q’ * B (F) is essentially C, we see that t 0 4 = e, where e is as defined in 6.6. This implies that B(F,) can, in V[G, H,], be densely embedded in R * B(F), and we are left to show that this set is strongly rC+-distributive in V[G, HA].

However,

(1) By Lemma 5 of Section 5 R is strongly rc+-distributive in V[G, H,].

(2) If K is R-generic over V[G,HJ, then V[G,H,,K] = V[G,H’,K].

(3) B(F) admits, in V [G, H’], a dense set E, which is < K-closed. (4) Since R is K+-distributive in V[G, H’], E remains d K-closed, hence strongly

rc+-distributive, in V[G, H’, K] = V[G, HA, K].

(5) By Lemma 7 of Section 5, R * B(F) is then strongly Kf-distributive in V[G, H,], as was to be shown. 0

Remark. Why, would the reader ask, not just define F, by the previous construction? The answer is that the density lemma (Claim 1 of 6.7) is then more unpleasant to prove.

6.10. The reader might be interested in how to define directly a weakly regular Kunen-Paris family (B,, IZ, C) yielding the filter Fi in VBA. We shall shortly indicate the construction.

S~~S={X~AI(XI<K} and,forXES,setFx={g:X-+rlgisone-one}.For X ES and g E Fx, let b, ~j(Bx) be defined as in Section 4. For X ES, let Ox be the (complete) ideal of j(Bx) defined by Ox = {q ~j(&)) for all g E Fx, b, A q = 0}, set Cx = j(&)/Ox and let Zi’x:j(Bx) -+ Cx be the (complete) quotient map.

Let X, Y ES be such that X c Y. One proves, as in Lemma 4 of Section 4, that Ox = 0 y n j( B,), and can hence define a complete monomorphism kxr : C, -+ C, by the formula kxr(IZx(q)) = Ilu(q). Finally, we let C be the direct limit of the family ((Cx)xEs, (kxr)x c y) and ZZ:j(B,) -+ C be the direct limit of the family (Ux)x.s. This works.

7. Restoring the GCH

7.1. We have seen, assuming the consistency of the theory “ZFCM”, how to obtain a model of “ZFC + the critical cardinal is strongly premeasurable”. The question

202 J.-P. LeuinskilAnnals qf Pure and Applied Logic 72 (1995) 177-2/2

remains open whether we can find a model of ZFC where the critical cardinal K bears a quasi-measure, as defined in Section 2, i.e. a normal filter F such that B(F) admits a dense, K+-closed subset. We shall answer this question positively, assuming however the consistency of a theory strictly stronger than “ZFCM” itself. In order to do this, we shall start with a model of “ZFC + K is measurable + 2” > K+“, restore the GCH beneath K, and check that a quasi-measure over K exists in the generic extension.

Theorem 1. Assume that V + “ZFC”. Then, there exists an iteration (Pa),< On such that

(1) whenever K is strongly Mahlo, P, satisfies the K-antichain condition, is of cardinality K and kp, “(va <x)(2” = a+ )“,

(2) Itp, GCH, where P, is the direct limit of(P,),,,,, (3) if V + “tz is ~easurable~‘, then VP” + % is quasi-measurable”, (4) if V+ “K is measurable”, then VP* /== “lc is measurable”.

7.2. Remarks. (1) The definition of a class P, restoring GCH and preserving measurability has been done long ago in [7]. Hence, we include point (4) of Theorem 1

only because the P, of [7] does not satisfy the K-antichain condition. (2) One can show that forcing with P, and P, also preserves the following

properties of K:

(a) K is n l,,-indescribable (for any fixed n), (b) K is weakly ineffable, (c) K iS ineffable, (d) K is completely ineffable. We shall omit the proofs.

7.3. Definition of the iteration. The iteration is classical, and due to Silver. We shall hence just sketch the construction and quote its main properties.

Let f6&0~ denote the strictly increasing enumeration of all regular cardinals. For two infinite cardinals A,@, let L(h,p) = (p : /: + p / IpI < d) denote the L&y collapse. We define, by induction on c1 E On, two sequences (Pa)a<on and (a,),,onr the first one being the iteration itself and the second one a sequence of terms such that, for all @EOn,a,EVP=and VpX. “a8 is an infinite cardinal”, as follows. (1) PO = {S} and a, = o,

(2) Pa+ 1 = P, * L( (aa)+, 2”=)), where the second term is computed in VP%,

(3) a,, 1 is the term of V ‘* which is forced over P,, 1 to be the successor of a,, (4) if lim(a), a, is forced over P, to be the supremum of all the a<, for < < ct, (5) if c1 is a limit ordinal and a < pn, P, is the inverse limit of the family (P,), <II while, if

CC = pII, then P, is the direct limit of this family.

7.4. Properties of the iteration. (1) If K is strongly inaccessible, then, for c( < K, 1 P,j -C K, and hence 1 P,l = K.

(2) For all a,a, is forced over P, to be larger than or equal to K,; hence, for all a,/I with c( K 8, P,/P, is at least < &-closed in VP,.

J.-P. Leainski/Annals of Pure and Applied Logic 72 (1995) 177-212 203

(3) If K is strongly Mahlo, then (a) P, has the tc-antichain condition, (b) VP,+ “(‘dcr <K)(2’ = x’)“.

(4) VP, I_ GCH. The verifications are left to the reader.

7.5. Proof of ~~eo~e~ 1, point (3). Let, in V, U be a normal measure on K and define, as usual,j: V -+ M = V’/U. Set P = P,, let B denote the boolean completion of P, let G be P-generic over I/ and, in I/CC], let F be the filter over K generated by U. By 3.2, B(F) is isomorphic to j(B)/B =j(B)/j”G. Butj(P) = P* PrK.j(,)[ and jlp = IdIp, since

P c V,, hence, P’ = [PtK,j(x)11 M is densely embedded in B(F). But P’ is d K-closed in

M[G], hence in V[G], since M[G] is < K-closed in V[G]. •l

7.6. Proof qf Theorem 1, point (4). Since forcing with P,,, ocf over VP*+’ adds no subset of K+ and VP’ I= GCH, we have got to prove that K is measurable in VP,*‘. Let, as before, j: V --+ M = V “JU and P = P,. Let G be P-generic over V and F denote the filter generated over K by U in V[G]. As we have just seen, B(F) admits, in V[G],

a dense, < K-closed subset, say E. Now, let H be L(K ‘, 2”)-generic over V [ G]. Since V[G] is 6 ic-closed in V[G, H], V[G,H]+ “F is a normal filter on K and E is 6 K-closed in B(F)“. Hence, in V [G, H], 2” = K + and K is quasi-measurable, which

implies that K is measurable there. 0

7.7. As consequence, we naturally get the following.

Theorem 2. Assume that the theory “ZFC + there exists a cardinal K of Mitchell order

K ++” is consistent. Then, so is the theory “ZFC + the critical cardinal is quasi- measurable”.

For, by Silver-Woodin-Gitik, from the assumptions, one can get a model of “ZFC + there exists a measurable cardinal K such that 2” > K+“.

It remains hence unsettled whether the critical cardinal being quasi-measurable is a strictly stronger theory than “ZFCM”, or not.

8. .Non-measurability, premeasurability and GCH

8.1. Assume that GCH holds. Then, strong premeasurability implies measurability. We now show how to obtain a model of ZFC + GCH, where there is some premeasurable cardinal K which is not measurable. If B is a boolean algebra and i is a cardinal, “B is A-saturated” will mean the same as “B satisfies the A-antichain condition”.


Theorem 1. Assume that Vk “ZFC + GCH + K is measurable”. Then, there exists

a generic extension of V satisfying “ZFC + GCH + K is premeasurable, but non- measurable”. Moreover, in this extension, we can Jind on K a normalfilter F1 such that

B(F,) is K+-distributive and K+-saturated.

8.2. Remarks. (1) In this model, there will be no rc-complete filter F on K such that B(F) is K-saturated, for, if such a filter existed, then, either K would be measurable, or there would exist a K-Suslin tree, which is impossible, since K is Ramsey; hence weakly compact.

(2) Kunen and Paris constructed in [S, Theorem 4.4) a model in which there exists some normal, K+-saturated filter on K, and K is weakly compact, non-measurable. However, their model can contain no premeasure on K.

8.3. Discussion of the model and idea of the proof: Let us start with I/ + “ZFC + GCH”. Let U be a normal measure on K and j: V -+ M = V’/V. We shall try to find our premeasure F on K in a weakly regular extension of V, the simplest to try being a K-CC extension of V. Hence, assume that P has the K-CC, is defined as the direct limit of an iteration (Pa),<,, where, for a < K, P,, 1 = P, * QII, for some term

Qa E VP,, and (P,l < K, and that PM = P”. Hence, jlp: P + j(P) is complete and, if we denote by F the filter on K generated by U in VP, then B(F) admits P* = [QK * PIK,iCK)I]M as dense subset. P* has to be K+-distributive in VP, but not strongly, for K would then be measurable. This implies that direct Easton forcing, adding one subset of each a+, for a < K, would not work. For then, we see that P* is !c+-closed in V; hence strongly ic+-distributive in VP, by Lemma 5 of Section 5.

Hence, assume that G is P-generic over V, and let us look at the usual method for proving that K is not measurable in V [Cl. We assume it is, let D be a normal measure on K in V[G], and build i: V[G] -(V[G]” n V[G])/D = N. We say “i(G) is i(P)- generic over i(V), but i(P) = [P* QK * P,K,i(T)~li’“‘) and, since Pi(“) = P”, i(G) yields

a [ Q,]““‘-generic set H over i( V) [ G]; but then, H is [ QK] ‘(“)-generic over V [ G], and H E V [G], a contradiction”. In order to make this argument work, we need some conditions to be fulfilled by the terms Qa. (1) First we need that P ‘(“) = P” 3 which will follow from the obvious fact that

(i(V), = V,. (2) Then we need H to be [Q,]““‘-generic over V[G], which will be achieved if

[Q ]iu’) belongs to V[G], satisfies there the K+-antichain condition and if ([QK]i@‘)tG1)rn V[G] c i(V)[G].

In order to achieve (2), we first need some conditions on V ensuring that i(V) is a < rc-closed class in V, which we shall obtain by assuming that V = L[ U] (for, in this case, i(V) will be an iterate of V, and indeed a finite iterate, for if not, Solovay’s “0 dagger” would exist, as recalled in [S, Lemma 3.21). Then we need that [Q,]io’)rG1 has the K+-antichain condition in V[G]. and not only in i( V)[G], which implies that we need an absolute enough definition of the terms Qol.

J.-P. LeainskilAnnals of Pure and Applied Logic 72 (1995) 177-212 205

As much for the non-measurability of K in the generic extension, and we address now the question of B(F) being K+-distributive in V[G]. This implies first that

CQJ “@I has to be K+-distributive in V[G], which, together with being K+-saturated there, leaves few choice besides a K’ -Suslin tree. On the other hand, if H is [QK]“tG1-

generic over V[G], then [PIK,JcK,r] MtG.H1 has to be K’-distributive in V[G, H], which

we can scarcely ensure but by demanding it to be 6 rc-Closed in M[G, H], hence in I/ [G, H], since M[G, H] itself will be < ic-closed in V [G, H]. This means that QZ should be forced over P, to be an < a-closed, cc+-Suslin tree.

8.4. Diamond sequences. Let t be a regular cardinal. We have got to review shortly how to construct a z-closed, z ‘-Suslin tree from some O,--type sequence, in order to adapt it to the construction of enough absolute Suslin trees.

Let hence i, be a regular cardinal, such that i 2 w, , and let us recall the following principle. O*(1) o there exists a sequence (Sa)a<l such that (1) for all c( < 2, IS,1 d c( + 0, (2) for every X c 1, there exists some set C club in 1 such that, for all c( E C, X n c1 E S,.

Since we have got to construct Suslin trees, not merely in V, but in generic extensions, we shall need, when 1 is a successor cardinal, a slightly reinforced form of O*(i,). Hence, assume that i = tt and set H, = {al ITC(a)] < 5}, where TC(a) denotes the transitive closure of a. We set O**(r+) * there exists a sequence (S,),,,< such that (1) for all c1 < T’, (S, I SG z, S, is transitive, S, satisfies ZFC and H, E S,, (2) for all X c H, x 7+, there exists some club set C in T+, such that, for all CI E C,

Xn(H,xa)ES,.

8.5. S&in trees. Now, let T be a regular cardinal, such that z cc = 7 (let us remark that it is esily seen that, in this case, the principles O**(z ’ ) and O*(T + ) are equivalent), and let (S,),,,+ be a fixed O*(z + )-sequence. We recall how to use it in order to construct a normal, z-closed, T+-Suslin tree, say T, with underlying set 7 ‘. We will define T by induction on CI < t ‘, defining at stage tl the set T Ia = (x E T 1 the order of x in T is < a}, using for this purpose a fixed well-order on P(t ’ ). Hence, assume that, for all 5 < a, TI, has been defined. (1) We set TJ, = 0.

(2) If lim(a), we set T loL = U { T le I( < cc},

(3) Assume that ;x = 5 + 1, and let us define T6 = {x E T 1 the order of x in T is r}.

Case 1: 5 = q + 1. We just appoint to each x E T,, two successors x0, x1 in T,, taking, say, the first two elements of zf not already used.

Case 2: 5 is a limit ordinal and cof(5) < 7. Since t<’ = 5, TI, admits at most 5 cofinai branches, and we appoint to each such branch b a unique successor xb in T,.

Case 3: 5 is a limit ordinal and cof(5) = t. We observe that, since TI, is r-closed and IS,] < z, we can, for each x E T Ii, find a cofinal branch b, through T I<, such that x E b, and, whenever A E S, is a maximal antichain in T I<, then b,n A # 8. Setting

206 J.-P. Levinski JAnnals of Pure and Applied Logic 7.2 (1995) 177-212

B = {b, 1 x E TIC}, we construct Tc by appointing to each b E B a unique successor xb at level 5.

Thefactthat T=u(TJ,la<z’}isaz-closed,t’- Suslin tree is then standard to verify. Observe also that, in order to perform this construction, we only need the principle 0( { a < z+ ) cof(a) = r}) to hold. However, it would be more work to extract such a sequence from the 0*-one than to use the latter directly. Let us also make the following observation.

Remark 1. Assume that W is an inner model, i.e. W is transitive, contains the ordinals and satisfies ZFC. Assume that, in V, r is a cardinal such that z<’ = z and that (7+)” = (z+)~. Let (S,),,,+ E W be a O*(r*)-sequence both in V and in W, and assume that a well-order ofP(z+)n W is fixed in W. Then, it is obvious that the Suslin tree obtained using this O*(r+)-sequence and the given well-order is the same, whether constructed in V or in W.

8.6. Back to the construction. Still assume that V = L[ U], where U is a normal measure on K. We have got to define an iteration ((P,, QblOlcK) where, for almost all c1 modulo U, Qa is, in VP*, an a-closed, a+-Suslin tree. Moreover, QK should be a K-closed, IC+-Suslin tree both in VP and NP, where P = P, and N is any finite iterate of V. To be sure that the O**(K+)-sequence we use to construct QK lies in every such N, we shall take it in the intersection of all iterates of V, namely the core model K.

Hence, we must, as first step, recall how to construct O**(a+)-sequences in K. They will do for each a < IC, since (HK)K = V,. However, for K i a, then might fail to be what they ought to. We will, hence, have to enlarge them, in order to include a O**(a+)-sequence in L[P~], whenever pa happens to be a normal measure on a in L[p,]. This, however, has to be done inside K, which does not know about any possible larger model where a could be measurable. So, we first recall how to get O**(r + + )-sequence in K, then how to get 0**(7 + +)-sequences in L[p] when p is a normal measure on z in L[p], and finally we put the two of them together, using to do this a lemma of [4].

8.7. Condensation in K. We recall some facts about condensation in K, needed to construct the sequences, which we prove for lack of a reference. We assume hence some familiarity with the core model, and refer the reader to [3] if necessary. As in [3, Chapter 143, we let D c On x On be the canonical predicate coding all o-mice, i.e. D = {(y, y’) I for some w-mouse M at y, y’ E C,} (recall that a mouse M is at y if y is the measurable cardinal in the sense of M). The formula “D(x,y)” is absolute between transitive models of ZFC- containing wl, and K = L [D]. For a E On, we set K, = L,[D]. We recall that, whenever ,I is an infinite cardinal in K, K1 = (I!!,#. This, in itself, implies that K k GCH.

Lemma 1. Assume that 1 is an inaccessible cardinal. Then, there exists a set El, closed and cojinal in A, such that, for all a E EL, there is no w-mouse at a.

J.-P. Levinski/Annnls of Pure and Applied Logic 72 (1995) 177-212 20-l

Proof. Set Se = (a < i, / there exists an w-mouse at ~3, and assume that So is stationary in 1. For c1 E So, let M, be an o-mouse at 01 and the N, be the core mouse such that M, = (N& (the wth iterate of N,). Say, N, is a mouse at some y(a) < ~1. By Fodor, there exists a stationary set S1 c So and an ordinal y such that, for all c( E Si, y(a) = y.

For c1 E Sr, let a(a) c y be the standard code of N,. Since 2y < A, there exists some stationary set St c S1 and some a c On such that, for all CI E Sz, U(U) = a. Hence, for all cc,fi E Sz, y(g) = y(/?), a(a) = a(P); hence N, = N,, and a = /I, a contradiction. 0

Remark. Obviously, the club set Ei of Lemma 1 can be taken in K, since the whole argument can be carried out there.

Lemma 2. Assume that A is an uncountable regular cardinal. Let El E K be us in Lemma 1 if2 is inaccessible and set El = i, otherwise. Let ;C+ be the successor of I

in K and assume that XI(KI+, E,EJ is such that IX/ < 1 and Xn,I is transitive.

Let 71: J? --P X be the transitive collapse of X, and set 5 = _+4f n On. Then, &? = K,.

Proof. Let D* be defined in .A by the formula defining D n(2’ x A+) in Kh+. We have got to show that D* = Dn(r x <). The direction D* c Dn(5 x <) is clear, because every mouse in the sense of 4 is a real mouse, and iteration is an absolute process. In order to prove the converse inclusion, let us first set c1 = X n A. cz is then the critical point of n, and n(a) = A. We shall distinguish three cases.

(1) D n(cc x 5) c D*. This follows immediately from c1 being the critical point of rr. (2) Dn(]a, <[I x 5) c D*. Assume that N is an o-mouse at some y, with a < y < t.

We have got to show that N E J?. Let f be a bijection from a onto y which belongs to A, and let ME & be a mouse at some 6 > y such that f~ M. Let 6’ be a regular cardinal such that 0 > M and MO, NB be the iterates of M, N at 0. If Me c NO, then f E Nor so f E N, hence N l= “y is not a cardinal”, a contradiction. Hence, NB E: MO, so CN,, being definable in Ne belongs to M,. Hence, CN E MB, which implies that CN E M c &, which in turn implies that N E JY, as was to be shown.

(3) D n( {a) x 5) c LP. This is clear if I is inaccessible, since, in this case, CI E E2 * Dn( {a) x 5) = 8. Assume hence that ,4 is a successor, say J. = t+, and let N be an w-mouse at a, from which we shall show that it belongs to A’. Let a E _&‘, a c t x 01, code a family (fV)r<g<ar where each f, is a bijection from z onto q and let M be a mouse in 4 at some y 2 51 such that a E M. As in point (2), let 8 > E be a regular cardinal, and Me, Ne be the iterates of M, N at 0. If M, c Ne, then a E N, implying that c1 is the successor of r in N. But then o! is not a cardinal in the first iterate of N, hence not a cardinal in N itself, a contradiction. From which it follows that Ne E MB, and we conclude that N E Jz’ in the same manner as in point (2). q

Remark. Although we gave the full lemma, we shall oniy use it in the case where i is a successor cardinal.

8.8. O**(z+)-sequences in K. In this section we work in the theory ZFC + V = K. We let z be an infinite cardinal and will define the canonical O**(r +)-sequence.


Assume that 7<u <7+. Let 6(c() be the least ordinal 6 such that CI < 6 and Kd /= “ZFC- + V = K + CI is not a cardinal”, and set T, = Kdfaj. Obviously 1 T,I d 7, and we shall show that (T,),,,+ is a 0**(7+ )-sequence. Let X c H, x 7+ and, for t<Cl<t+ ,setA,=Hull({X}ua,K,+~).

Set C = {a < z’jA,n~~ = cc}, which is club in 7+. For tl~ C, let 7~,:&~--,,4, be the transitive collapse of& and set <(cr) = OnnAa. We know that .Y& c Ktc,,. On the other hand, for a E C, Xn(H, x a) = (x,)-l(X) E ~2’~. It is then enough to prove that, for CI E C, ((cr) d 6(u). Assume that 6(a) 6 ((a). Then, by Lemma 2, KdCaj c ,A;l,, hence ,le,+ “a is not a cardinal”, a contradiction.

8.9. 0**(7+)-sequences in L[p]. We now assume that V = L[U], where U is a normal measure on 7, and shall define a canonical O**(r +)-sequence in L[ U]. We recall that, by Silver, U c L,+ [U]. Now, for CI such that 7 < CI < 7+, set U, = U n L,[ U],

let cr(~1) be the least ordinal Q such that U, is not an ultrafilter on P(t)nL,[ U,],

L,[ U,] k ZFC- and H, E L,[ U,], and set R, = L,(,)[U,]. We claim that (R,),,,t is a O**(t+)-sequence in L[U]. To see this, let X c H, x 7+ and, for 7 < a < t+, set A, = Hull({X}ucr,L,++ [VI). Set C = (a < 7+ IA,n7+ = a}, which is club in 7’. For a E C, let 71,: ,le, + A, be the transitive collapse of A,, and set ~(a) = Onn&‘I,. Hence, for some pa, J& = L+)[pJ and, since rc,l, = Idl, and n,(a) = t+, we see that p. = U,, and hence J& = &(,)[ U,]. From this it follows that it is enough to prove, for o! E C, that a(a) < a(a), since Xn(H, x CC) = (zJl(X) E Me. But, if we assumed that a(a) d I, then U, would not be an ultrafilter in ME, a contradiction.

8.10. “Absolute” O**(z+)-sequences. Now we put 8.8 and 8.9 together in order to obtain a 0 ** (7 + )-sequence in K which is also such a sequence in L [p], whenever p is a normal measure on 7 in L[p]. Hence, for this section, we do not assume anything about V, except ZFC. In order to proceed, we need the following.

Definition 1. A structure A = (A, W, y) is called strongly iterable if it satisfies the following: (1) A is transitive and rudimentarily closed, (2) forallaEA, WnaEA, (3) y is an ordinal, y E A, and W is an A-normal, A-ultrafilter over y, (4) if{Xili<o}~Vand{Xi~i<w)cW,then~(Xili<w}#~.

Now we have the following lemma.

Lemma 3. Assume that V = K and that u < 7 + is such that K, < K,+. Then, there

exists at most one W such that (Kd, W,7) is strongly iterable.

Proof. This is “Lemma 2.7” of [4, p. 743. The hypothesis cof(u) > o of [4] is not needed here, because of condition (4) in Definition 1 above. lJ

J.-P. Levinski J Annals of Pure and Applied Logic 72 (I 995) 177-212 209

Hence, let us stay inside K and define, for 7 < c( < 7+, an ordinal 6(c() as follows. 6(a) is the least 6 satisfying the following two conditions:

(1) c1 < 6 and Kd+ “ZFC- + V = K + CI is not a cardinal”. (2) Assume that K,< K,+ and that W is such that (K,, W,t) is strongly iterable;

then, letting 0 be the least ordinal such that W is not an ultrafilter on P(7)n L,[ W] and L,[ W] k ZFC-, we have L,[ W] c Ks.

By Lemma 3, 6(a) < 7+ for all CI < 7’ and hence, setting S, = Kdtaj, we see that

IS,1 6 7. By 8.8, (S,),<,+ is a 0**(7+)-sequence in K, since T, c S,. On the other hand, assume that U is a normal measure on 7 in L [U], and let us show that (S,),, rt isaO**(r+)-sequenceinL[U].SinceL,+[U] =K,+thereisinL[U]aclubsetCin 7+ such that, for all c( E C, K, = L,[U] and K,<K,+. But then, for c( E C, R, c S,, and we are done.

We will call this sequence the canonical O**(z+)-sequence.

8.11. Construction ofthe set of conditions. We are now ready to construct the iteration

(P&on. It will be constructed entirely inside K, and nothing is assumed about I/ at that point. To simplify the notation, we shall for the moment omit superscripts, 7+ meaning ( 7+)K, etc. Hence, for every infinite cardinal 7, let (S,,,),,,+ denote the canonical O**(z+)-sequence and let (pJacon denote the strictly increasing enumer-

ation of all inaccessible cardinal. We define an iteration (PalaEon as follows:

(1) P, = (0). (2) For c( = pa, P, is the direct limit of (P,), <a and its inverse limit if c( is a limit ordinal

but CI < pa. (3) We now assume that P, is defined and intend to define P,, 1. We first observe that,

either 1 P,I < pa, or c( = pd, in which case 1 PaI = pa. In both cases, setting pa = 7, we see that P, c K, (with an appropriate choice of supports). Hence, let G be Pa-generic over I/. For each l < 7+, P, E S,.,, G is Pa-generic over S,, <, and we can form the ZFC--model S,,,[G].

Claim 1. (1) (S,,,[Gl)6<,+ is a O*(z+)-sequence in K[G].

(2) Zf U is a normal measure on 7 in N = L[U], then (S,,,[G])r<r+ is a O*(z+)- sequence in N [ G].

Proof. We prove (1). The proof of (2) is identical. Let X E K [ G] be such that X c 7 +, let X E KPx be a term such that X=XG, set Z={(p,<)~P,xz+l p Ik “r E %“} c K, x 7+ and let C be a club set in 7+ such that, for all 4 E C, Z, = Zn(K, x <) E S,,,. It is clear that, for all 5 E C, Xnt = (Z,), E S,,,[G]. 0

We can now finish the definition of the iteration, setting P,, I = P, * Q=, where Q. is the r-closed, t+-Suslin tree constructed in KP*, using the well-order which is defined from G and the canonical well-order of K.

210 J.-P. Levinski/Annals of Pure and Applied Logic 72 (1995) 177-212

8.12. Properties of the iteration. Assume for a moment that I/ = K. Then, the iteration has the following usual properties. For CI < /I, P,JP, is < p,-closed in VP,. For

M = pa, P, c K,. For CI Mahlo, P, has the U-CC. Forcing with P, preserves cofinalities and the GCH.

8.13. Construction of the model. We are now ready to construct the model, as sketched in 8.3. So we assume that V = L [ U J, where U is a normal measure on K, and construct j: I/ -+ M = V”/U as usual. We shall force over V with the set of conditions P = P,. Let us observe that V, = K, and that hence the properties 8.12 of P are true in V as well as in K. Let now G be P-generic over V and, in V[G], let F={XCK1%EU,S c X} be the filter generated over K by U. We intend to show that F is a premeasure on K, and that K is not measurable in V [G]. Set Q = (QK)M and P’ = PjK,j(K)cs We know that B(F) admits Q * P’ as dense subset.

Claim 2. Qc is K ‘-distributive in V [ G].

Proof. Since KM = K, we see that Q = (Q.)“, From which we get Claim 2, using Claim l(2). 0

Claim 3. Let H be Q-generic over V[G]. P’ is then < K-closed in V [G, H].

Proof. P’ is k-closed in K [ G, H], hence in M [G, H], since ( Vj/j(K))M = ( Vj(K,))K, hence in V [G, H], since M [G, H] is, because of the K + -antichain condition of P, + 1 in V. I-J

From these two claims, it follows that B(F) is ic+-distributive in V[G] which proves first part of Theorem 1. Observe that they also imply that B(F) admits, in V[G], a dense, < K-closed subset. In order to prove that K is not measurable in V[G], we proceed as sketched in 8.3. Assume it is, let D be a normal ultrafilter on K,

and form i: V[G] -+ (V[G]“n V[G])/D = N[i(G)], where N = i(V). Since KN = K,

we have (P,+ I)N = P,+ 1, and hence i(G) yields a Q-generic set over N [G], say H. On the other hand, N must be an iterate of V, which implies that it has the same subsets of K, hence the same bounded subsets of K + (as we quoted earlier, N must, in this case, be a finite iterate of V, hence is 6 K-closed in V, but we do not need this full fact here). Hence, V [ G] has the same bounded subsets of K + as N [ G]. In particular, since Q has the ic+-antichain condition in V [G], H is Q-generic over V [ G], a contradiction.

We now have to find in V[G] a normal filter Pi on K such that B(Fi) is K+-distributive and K+-saturated. Let H * G’ be Q * PI-generic over V[G]. Since the latter set is dense in B(F) and rc+-distributive in V[G], K is measurable in V [G, H, G’]. But 2“ = K + holds in V [G, H], which entails, thanks to Claim 3, that K is measurable in V [G, H]. Let hence B(Q) be the boolean completion of Q in V [ G], let D E (V [G])Q be a term which is forced over Q to be a normal ultrafilter on rc, define


4:P(rc)nV[G]+B(Q) by 4(X)= (IXEDII~(~), and set Pi ={XEP(K)~V[G]I kB(Q) “X E D”}. F1 is obviously a normal filter on K, and 4 yields a one-one boolean morphism II/: B(FI) + B(Q). Since B(Q) is K+-saturated, so is B(F,). On the other hand, since PI is normal, $ is < k--complete, hence complete, since B(F,) is K+-

saturated. This implies that B(FI) is ti+-distributive, as was to be shown. Observe also that B(F,) is complete, as boolean algebra.

Remark. F, is of course different from F, since B(F) is equivalent to Q * P’, itself, because of Claim 3 and the fact thatj(ic) < (2”)’ = K+ +, being, in V[G], equivalent to Q x C, where C is the Cohen set for adding a subset of K+.

8.14. Another possible proof. We also could have taken, instead of reverse, direct Eastern forcing. This means that we proceed as follows. For every inaccessible cardinal r in K, we let Qr be the r-closed, z+-Suslin tree constructed in K from the canonical O*(r+)-sequence, and let P be the product, with Easton supports, of the family (P,),,,. Let us check shortly that the proof goes through in this case, keeping our notations. (1) Since IPI = K and Q has the K+-cc, P x Q has the JC+-cc, which implies that, for any

P-generic set G over V, Q has the K+-CC in V[G], and hence is there a K+-Suslin tree, and in particular is K+-distributive.

(2) P’ is f K-closed in K, hence in M, hence in I/. (3) By points (l), (2) and Lemma 5 of Section 5, P’ is strongly rc+-distributive in

V [ G, H], whenever H is Q-generic over V [ G]. (4) By points (1) and (3) B(F) is K+-distributive in V[G]. (5) By points (4) and (3) K is measurable in V[G, H].

(6) By points (5) and (I), B( F1 ) is K +-distributive and Ic + -saturated in V [ G]. (7) The proof of the non-measurability of K in V[G] goes through, due to point (1).

This proof is shorter only in that it does not use Claim 1. On the other hand, it does not yield a dense < K-closed subset of B(F).

References

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filters and large cardinals

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