aa r X i v : . [ m a t h . L O ] S e p GAMES WITH FILTERS
MATTHEW FOREMAN, MENACHEM MAGIDOR, AND MARTIN ZEMAN
Abstract.
This paper has two parts. The first is concerned with a variant ofa family of games introduced by Holy and Schlicht, that we call
Welch games .Player II having a winning strategy in the Welch game of length ω on κ isequivalent to weak compactness. Winning the game of length 2 κ is equivalentto κ being measurable. We show that for games of intermediate length γ , IIwinning implies the existence of precipitous ideals with γ -closed, γ -dense trees.The second part shows the first is not vacuous. For each γ between ω and κ + , it gives a model where II wins the games of length γ , but not γ + . Thetechnique also gives models where for all ω < γ ≤ κ there are κ -complete,normal, κ + -distributive ideals having dense sets that are γ -closed, but not γ + -closed. September 10, 20201.
Introduction
Motivated by ideas of generalizing properties of the first inaccessible cardinal ω ,Tarski [16] came up with the idea of looking at uncountable cardinals κ such that L κκ -compactness holds for languages of size κ . This became the definition of a weakly compact cardinal. Hanf [7], showed that weakly compact cardinals areMahlo. Work of Keisler [11] and Keisler and Tarski [12] showed: Theorem.
Let κ be an uncountable inaccessible cardinal. Then the following areequivalent to weak compactness: (1) Whenever R ⊆ V κ there is a transitive set X and S ⊆ X such that h V κ , ∈ , R i ≺ h X, ∈ , S i . (2) If B ⊆ P ( κ ) is a κ -complete Boolean subalgebra with |B| = κ and F is a κ -complete filter on B , then F can be extended to a κ -complete ultrafilteron B . Items 1 and 2 are clearly implied by their analogues for measurable cardinals:(1 ′ ) There is an elementary embedding of V into a transitive class M .(2 ′ ) There is a non-atomic, κ -complete ultrafilter on P ( κ ). Holy-Schlicht Games.
This paper concerns several of a genre of games originat-ing in the paper [8] of Holy and Schlicht and explored by Nielsen and Welch [13].The following small variant of the Holy-Schlicht-Nielsen game was suggested to usby Welch.Players I and II alternate moves:
Matthew Foreman was supported in part by NSF grant DMS-1700143.Menachem Magidor was supported by ISF grant 684-17. I A A . . . A α A α +1 . . .II U U . . . U α U α +1 . . .where hA δ : δ < ℓ ≤ γ i is an increasing sequence of κ -complete subalgebras of P ( κ )of cardinality κ and h U δ : δ < ℓ i is sequence of uniform κ -complete filters, each U α is a uniform ultrafilter on A α and α < α ′ imples that U α ⊆ U α ′ . We assumewithout loss of generality that A contains all singletons. Player I goes first atlimit stages. The game continues until either player II can’t play or the play haslength γ . We denote this game by G Wγ . The winning condition.
Player II wins if the game continues through all stagesbelow γ .There are two extreme cases: γ ≤ ω and γ = 2 κ . Using item (2) of the charac-terization of weakly compact cardinals, one sees easily that if κ is weakly compactthen II wins the game of length ω . At the other end if κ is measurable one canfix in advance a κ -complete uniform ultrafilter U on P ( κ ) and at stage α play U α = U ∩ A α .It is remarked in Nielsen-Welch [13] that player II having a winning strategy inthe game of length ω + 1, then there is an inner model with a measurable cardinal.This motivated the following: Welch’s Question.
Welch asked whether Player II having a winning strategy inthe game of length ω implies the existence of a non-principal precipitous ideal.The main result of this paper is that if 2 κ = κ + and Player II can win the gameof length ω + 1 then there is a uniform normal precipitous ideal on κ .We note here that for γ a limit, there is an intermediate property between“Player II wins the game of length γ ” and “Player II wins the game of length γ + 1.”. It is the game G ∗ γ of length γ that is played the same way the originalWelch game G Wγ , but with a different winning condition: For Player II to win,there must be an extension of S α<γ U α to a uniform κ -complete ultrafilter on the κ -complete subalgebra of P ( κ ) generated by S α<γ A α . Precipitous ideals.
We are fortunate Welch’s question leads to a number of morerefined results, namely whether the game implies the existence of precipitous idealsand of what kind. We begin discussing a strong hypothesis:A κ -complete, uniform ideal I on κ such that the Boolean algebra P ( κ ) / I has the κ + -chain condition is called a saturated ideal .It follows from results of Solovay in [14] that if I is a saturated ideal on κ then I is precipitous. Thus to show that there is a non-principal precipitous ideal on κ it suffices to consider the case where κ does not carry a saturated ideal.The most direct answer to Welch’s question is given by the following theorem: Theorem 1.1.
Assume that κ is inaccessible, κ = κ + and that κ does not carrya saturated ideal. If Player II has a winning strategy in the game G ∗ ω , then there isa uniform normal precipitous ideal on P ( κ ) . We could omit “uniform” and simply require A to include the co- <κ subsets of κ and U toextend the co- <κ -filter. Recall that an ideal I on a set X is precipitous if for all generic G ⊆ P ( X ) / I the genericultrapower V X /G is well-founded. See [9] or [3]. AMES WITH FILTERS 3
We recall that a normal uniform ideal on κ is κ -complete. As a corollary weobtain: Corollary.
Suppose that Player II wins either G ∗ ω or G Wγ for any γ ≥ ω + 1 , thenthere is a uniform normal precipitous ideal on κ . While this is the simplest statement, the proof gives a lot more information.
Theorem 1.2.
Assume that κ is inaccessible and κ = κ + and that κ does notcarry a saturated ideal. Let γ > ω be a regular cardinal less than κ + . If Player IIhas a winning strategy in the Welch game of length γ , there is a uniform normalideal I on κ and a set D ⊆ I + such that: (1) ( D, ⊆ I ) is a downward growing tree of height γ , (2) D is closed under ⊆ I -decreasing sequences of length less than γ (3) D is dense in I + .In fact, it is possible to construct such a set D where (1) and (2) above hold withthe almost containment ⊆ ∗ in place of ⊆ I . How does precipitousness arise?
In [6], Galvin, Jech and Magidor introducedthe following game of length ω . Fix an ideal I . Players I and II alternate playingI A A . . . A n A n +1 . . .II B B . . . B n B n +1 . . .With A n ⊇ B n ⊇ A n +1 and each A n , B n ∈ I + . Player II wins the game if T n B n = ∅ . We will call this game the Ideal Game . They proved the followingtheorem.
Theorem. [6]
Let I be a countably complete ideal on a set X . Then I is precipitousif and only if Player I does not have a winning strategy. In the proof of Theorem 1.1, we construct an ideal I and show that Player IIhas a winning strategy in the ideal game. In Theorem 1.2, the existence of a denseset D closed under descending ω -sequences immediately gives that Player II hasa winning strategy in the ideal game. (See [2] for some information about therelationship between games and dense closed subsets of Boolean Algebras.) Theproofs of both Theorem 1.1 and Theorem 1.2 are in Section 4. Is this vacuous?
So far we haven’t addressed the question of the existence ofstrategies in the Welch games if κ is not measurable. We answer this with thefollowing theorem. We use the terminology regarding closure and distributivityproperties of forcing posets from [1]. Theorem 1.3.
Assume κ is measurable and V = L [ E ] is a fine structural extendermodel. Then there is a generic extension in which κ is inaccessible, carries nosaturated ideals (in particular, κ is non-measurable) and for all regular κ ≥ γ > ω : (a) There is a uniform normal ideal J γ over κ which is ( κ + , ∞ ) -distributiveand such that P ( κ ) / J γ has a dense γ -closed subset but not a γ + -closedsubset. (b) Player II has a winning strategy S γ in G Wγ such that S γ is not included inthe winning strategy for Player II in G Wγ + defined the same way. (c) There is an ideal I γ as in Theorem 1.2. MATTHEW FOREMAN, MENACHEM MAGIDOR, AND MARTIN ZEMAN
We give a proof of (a) in the above theorem in Section 5. The existence of awinning strategy for Player II in G Wγ follows from Theorem 1.4. Clause (b) showsthat if the ideal J γ is chosen carefully then one can determine the exact length ofthe game in which Player II has a winning strategy. The proof of clause (b) is atthe end of Section 5. Clause (c) is a direct consequence of Theorem 1.2. Theorem 1.4.
Assume
GCH . Let γ ≤ κ be uncountable regular cardinals and J γ be a uniform κ -complete ideal over κ which is ( κ + , ∞ ) -distributive and such that P ( κ ) / J γ has a dense γ -closed subset. Then Player II has a winning strategy ingame G Wγ . A proof of Theorem 1.4 is at the end of Section 4. A slight modification of theproof of Theorem 1.3 gives us the following variant of the theorem.
Theorem 1.5.
Assume κ is measurable, γ ≤ κ is regular uncountable and V = L [ E ] is a fine structural extender model. Then there is a generic extension in which κ is inaccessible, carries no saturated ideals (in particular, κ is non-measurable) and(a) and (c) of Theorem 1.3 hold. Additionally, the generic extension satisfies thefollowing strengthening of clause (b) in Theorem 1.3. (b) Player II has a winning strategy S γ in G Wγ but does not have any winningstrategy in G Wγ + . A proof of Theorem 1.5 is at the end of Section 52.
Hopeless Ideals
In this section we review the notion of a hopeless ideal in certain general context,and toward the end of the section we will narrow our focus to the context of games.Fix an inaccessible cardinal κ . Assume F is a function with domain say R suchthat for every r ∈ R the value F ( r ) is a sequence of length ξ r of the form(1) F ( r ) = hA ri , U ri | i < ξ r i where(i) A ri ⊆ P ( κ ) and(ii) U ri is a κ -complete ultrafilter on the κ -algebra of subsets of κ generated by S j ≤ i A rj .(iii) The sequence h U ri | i < ξ r i is monotonic with respect to the inclusionwhenever r ∈ R .(iv) (Density) For every r ∈ R , j < ξ r and B ∈ [ P ( κ )] ≤ κ there is s ∈ R suchthat F ( r ) ↾ j = F ( s ) ↾ j and B ⊆ A sj .One can also formulate a variant with normal ultrafilters U ri . Given a κ -algebra A of subsets of κ , when we write h A i | i < κ i ∈ A we mean that A ∈ A where A is the image of the set {h i, η i ∈ κ × κ | η ∈ A i } under the natural isomorphism h : ( κ × κ, < mlex ) → ( κ, ∈ ) where < mlex is the maximo-lexicographical ordering of κ × κ . We then say that the κ -algebra A is normal iff for every A ∈ A all sections( h − [ A ]) i where i < κ as well as the diagonal intersection ∆ i<κ A i are elements of A . Finally we say that an ultrafilter U on a normal κ -algebra A is normal iff forevery sequence h A i | i < κ i ∈ A ,(2) ( ∀ i < κ )( A i ∈ U ) = ⇒ ∆ i<κ A i ∈ U For the the variant with normal ultrafilters we then require, instead of (ii) above,that
AMES WITH FILTERS 5 (ii) ′ U ri is a κ -complete normal ultrafilter on the normal κ -algebra of subsets of κ generated by S j ≤ i A rj .If (ii) ′ is satisfied we say that F is normal. Notice that there is no need to modifyclause (iv), as normal κ -algebras are able to decode κ -sequences of subsets of κ which are coded by subsets of κ via G¨odel’s pairing function. Thus, in (iv), insteadfamilies B ∈ [ P ( κ )] ≤ κ we could consider just sets B ∈ P ( κ ). Definition 2.1.
Given a function F as above, we define the ideal I ( F ) as follows. (3) I ( F ) = the set of all A ⊆ κ such that A / ∈ U ri for any i < ξ r and any r ∈ R .The ideal I ( F ) is called the hopeless ideal on P ( κ ) induced by F .It follows immediately that ∅ ∈ I ( F ) and I ( F ) is downward closed under inclu-sion. That I ( F ) is a κ -complete ideal follows from the density condition imposedon F in (1); even a proof that I ( F ) is an ideal, that is, I ( F ) is closed under finiteunions, seems to need a (weak form of) density requirement. Proposition 2.2.
Given a function F as in (1), the ideal I ( F ) is κ -complete. Ifall ultrafilters U ri are uniform then I ( F ) is uniform. If additionally F is normalthen I ( F ) is normal.Proof. We first verify κ -completeness of I ( F ). By the discussion above, it sufficesto check that I ( F ) is closed under unions of cardinality < κ . If h A η | η < ξ i issuch that ξ < κ and A = S η<ξ A j / ∈ I ( F ), then there is some r ∈ R and some i < ξ r such that A ∈ U ri . By the density condition, there is some s ∈ R such that A sj = A rj and U sj = U rj for all j ≤ i , and { A j | j < ξ } ⊆ A si +1 . In particular, A ∈ U si ⊆ U si +1 and all sets A η , η ≤ ξ are in the κ -algebra generated by S j ≤ i +1 A j .By κ -completeness of U si +1 then A η ∈ U si +1 for some η < ξ , hence A η / ∈ I ( F ).The proof of normality of I ( F ) for normal F is the same, with ∇ η<κ A η in place of S η<ξ A η . The conclusion on uniformity of I ( F ) follows by an easy straigthforwardargument from the definition of I ( F ). ⊣ Now assume G is a game of perfect information two players play, and S is astrategy for Player II in G . Denote the set of all runs (under a run we mean acomplete play) of G according to S by R S and assume every r ∈ R S is associatedwith a sequence of fragments A ri ⊆ P ( κ ) and ultrafilters U ri , so that we have afunction F = F S with domain R S as in (1) satisfying (i) – (iv) or the normalvariant thereof. In all games we will consider in this paper we may view function F S , under slight abuse of notation, as the identity function(4) F S ( r ) = r, where the members of sequence r correspond to rounds in G . Consistently with (1),these rounds are represented as ordered pairs where left components are families A ri ,possibly under some simple coding, played by Player I, and the right componentesare ultrafilters U ri played by strategy S . As the strategy S makes it clear whichgame is played, we suppress writing G explicitly in our notation. If P is a positionin G played according to S we let(5) R S ,P = the set of all r ∈ R S extending P and(6) F S ,P = F S ↾ R S ,P MATTHEW FOREMAN, MENACHEM MAGIDOR, AND MARTIN ZEMAN
We are now ready to define the central object of our interest.
Definition 2.3.
Assume G is a game of perfect information played by two players, S is a strategy for player II in G , and there is an assignment F S with domain R S .Consider a position P in G according to S . We define I ( S , P ) = I ( F S ,P ) to be the hopeless ideal with respect to S conditioned on P . The ideal I ( S , ∅ ) iscalled the unconditional hopeless ideal with respect to S . We will briefly write I ( S )for I ( S , ∅ ).When the strategy S is clear from the context we suppress referring to it, andwill talk briefly about the “hopeless ideal conditioned on P ” and the “unconditionalhopeless ideal”. By Proposition 2.2 we have the following. Proposition 2.4.
Given a game G of limit length, a strategy S for Player IIin G and a position P as in Definition 2.3, the ideal I ( S , P ) is κ -complete. Ifall ultrafilters U ri associated with F S ,P are uniform then I ( S , P ) is uniform. Ifmoreover F S ,P is normal, then I ( S , P ) is normal as well.Proof. This boils down to verifying clauses (i) – (iv) for F S ,P . The verificationis straightforward, but we would like to emphasize the important point that thedensity requirement (iv) holds in the case of games of limit lengths. ⊣ Games we Play
In this section we introduce a sequence of games G k closely related to Welch’sgame G Wγ . The last one will be G , and we will be able to show that if S is awinning strategy for II in this game of sufficient length then we can construct awinning strategy S ∗ for Player II in G such that I ( S ∗ ) is precipitous and more,depending on the length of the game and the payoff set.To unify the notation, we let G of length γ be the Welch’s game G Wγ . Thus, arun of the game continues until either Player II cannot play or else until γ roundsare played. The set of all runs of G of length γ is denoted by R γ . As usual withthese kinds of games, a set B ⊆ R γ is called a payoff set. We say that Player IIwins the run of the game G of length γ with payoff set B , briefly G ( B ), if she isable to play γ rounds and the resulting run is an element of B . If B = R γ we drop B in G ( B ) and write briefly G . With this notation, the game G ∗ γ is just the game G ( Q γ ) of length γ where(7) Q γ = The set of all runs h α i , U i | i < γ i ∈ R γ such that there isa κ -complete ultrafilter on the κ -algebra generated by S i<γ A i extending all U i , i < γ .As already discussed in the introduction, existence of a winning strategy for Player IIin game G ( Q γ ) of length γ is a strengthening of the requirement that Player II hasa winning strategy in G of length γ , and this strengthening seems to be minimalamong those which increase the consistency strength. See (c) below for more tech-nical formulation. As we will see, the consistency strength increases significantly inthe case γ = ω , which is of primary interest. Here are some trivial observations.(TO1) G ( Q γ ) is the same game as G whenever γ is a successor ordinal, so awinning strategy for Player II in G ( Q γ ) gives us something new only when γ is a limit. AMES WITH FILTERS 7 (TO2) A winning strategy for Player II in G ( Q γ ) is a winning strategy for Player IIin G , but the converse may not be true in general.(TO3) If S is a winning strategy for Player II in G of length > γ then the restric-tion of S to positions of length < γ is a winning strategy for Player II in G ( Q γ ) of length γ .(TO4) Given ξ < κ and sequences hA i | i < ξ i and h U i | i < ξ i where A i ⊆ P ( κ )and U i is a κ -complete (and normal – see below for the definition) ultrafilteron the (normal – see below for the definition) κ -algebra of subsets of κ generated by S j ≤ i A j such that U i ⊆ U j whenever i ≤ j , there is at mostone κ -complete (and normal) ultrafilter U on the (normal) κ -algebra B of subsets of κ generated by S i<ξ A i which extends all U i . Thus, if wechanged the rules of G to require that Player II goes first at limit stagesthen Player II has a winning strategy in this modified G if and only ifPlayer II has a winning strategy in the original game G .In what follows we will consider θ a regular cardinal much larger than κ , andfix a well-ordering of H θ which we denote by < θ . We augment our language of settheory by a binary relation symbol denoting this well-ordering, and work in thislanguage when taking elementary hulls of H θ . We will thus work with the structure( H θ , ∈ , < θ ), but will suppress the symbols denoting ∈ and < θ in our notation.The common background setting for the games we are going to describe is acontinuous sequence h N α | α < κ + i of elementary substructures of H θ such thatfor all α < κ + the following hold.(a) κ + 1 ⊆ N α and card ( N α ) = κ ,(b) <κ N α +1 ⊆ N α +1 ,(c) h N ξ | ξ < α ′ i ∈ N α whenever α ′ < α , and(d) P ( κ ) ⊆ S α<κ + N α Clearly (d) implies that 2 κ = κ + , which we stated as an assumption in Theorems 1.1and 1.2. On the other hand, if 2 κ = κ + then it is easy to construct a sequence h N α | α < κ + i as above. Definition 3.1 (The Game G − ) . The rules of the game G − are as follows. • Player I plays an increasing sequence of ordinals α i < κ + . • Player II plays an increasing sequence of uniform κ -complete ultrafilters U i on P ( κ ) M where M = N α i +1 . • Player I plays first at limit stages.A run of G − of length γ ≤ κ + continues until Player II cannot play or else until itreaches length γ . Player II wins the run in G − of length γ iff the length of the runis γ .Payoff sets R γ and Q γ for G − are defined analogously as for G . So R γ consistsof all runs of G − of length γ , and Q γ consists of all runs h α i , U i | i < γ i ∈ R γ suchthat there is a κ -complete ultrafilter on the κ -algebra generated by P ( κ ) ∩ N α , where α = sup i<γ α i , extending all U i , i < γ . If γ is a limit ordinal then P ( κ ) ∩ N α = S i<γ ( P ( κ ) ∩ N α i +1 ) where α = sup i<γ α i .Thus, the definition of R γ and Q γ for G − is actually an instance of that for G .Notice also that P ( κ ) ∩ N α +1 is a κ -algebra of sets for any α < κ + , and so is N α whenever α is of cofinality κ . Hence if γ is a successor ordinal or cf ( γ ) = κ thenthe κ -algebra generated by P ( κ ) ∩ N α in the above definition is equal to P ( κ ) ∩ N α MATTHEW FOREMAN, MENACHEM MAGIDOR, AND MARTIN ZEMAN where α = sup i<γ α i . If γ is a limit ordinal of cofinality < κ then this κ -algebraof sets is strictly larger in the sense of inclusion. Finally let us stress that remarksanalogous to remarks (TO1) – (TO4) for games G and G ( Q γ ) stated below thedisplayed formula (7) also hold for G − and G − ( Q γ ). Proposition 3.2.
Assuming κ = κ + and γ ≤ κ + is an infinite regular cardinal,the following hold. (a) Player II has a winning strategy in G − of length γ iff Player II has a winningstrategy in G of length γ . (b) Player II has a winning strategy in G − ( Q γ ) of length γ iff Player II has awinning strategy in G ( Q γ ) of length γ .Proof. This is an easy application of auxiliary games. Regarding (a), if S is awinning strategy for Player II in G then S induces a winning strategy for Player IIin G − the output of which at step i is the same as the output of S at step i in theauxiliary game G where Player I plays P ( κ ) ∩ N α i +1 at step i ; here α i is the moveof Player I in G − at step i . For the converse we proceed similarly; this time if S − is a winning strategy for Player II in G − and Player I plays A i at step i in G thenPlayer I plays α i = the least α > α i ′ for all i ′ < i such that A i ⊆ N α +1 in the auxiliary game G − . It is straightforward to verify that this choice of strategiesalso works in the case of games with payoff sets Q γ in (b). ⊣ Definition 3.3 (The Game G ) . The rules of G are exactly the same as those of G − with the only difference that the ultrafilters U i played by Player II are additionallyrequired to be normal with respect to N α i +1 .As before, the payoff set R γ is defined for G the same way as for G and G − ,that is, R γ consists of all runs of G of length γ . For G we define a payoff set W γ as follows. W γ = the set of all h α i , U i | i < γ i ∈ R γ such that if h X i | i < γ i is asequence satisfying X i ∈ U i for all i < γ then T i<γ X i = ∅ . Notice that W γ = ∅ whenever γ ≥ κ , so the game G ( W γ ) is of interest onlyfor γ < κ . The existence of a winning strategy for Player II in G ( W ω ) of length ω seems to be just what is needed to run the proof of precipitousness of hopeless ideal I ( S ∗ ) in Section 4; see Proposition 4.5. As we will see shortly, the existence of sucha winning strategy follows from the existence of a winning strategy for Player II in G − ( Q ω ) of length ω . In the case of G we will not make use of a payoff set for G that would be an analogue of what was Q γ for G and G − , so we will not introduceit formally.Let us also note that the somewhat abstract notion of normality of an ultrafilteron A i = P ( κ ) ∩ N α i +1 introduced in Section 2 is identical with the usual notion ofnormality with respect to the model N α i +1 where it is required that U i is closedunder diagonal intersections of sequences h A ξ | ξ < κ i ∈ N α i +1 such that A ξ ∈ U i for all ξ < κ . Proposition 3.4. (Passing to normal measures.) The following correspondencesbetween the existence of winning strategies for G − and G hold. AMES WITH FILTERS 9 (a)
Let γ ≤ κ + be an infinite regular cardinal. If Player II has a winningstrategy in G − of length γ then Player II has a winning strategy in G oflength γ . (So in fact we have “iff” here, as the converse holds trivially.) (b) If Player II has a winning strategy in G − ( Q ω ) of length ω then Player IIhas a winning strategy in G ( W ω ) of length ω . We could formulate and prove (b) in this lemma for general γ in place of ω (astraightforward generalization of the proof below works), but we will not make anyuse of such a generalization, and moreover, the case γ = ω seems to be of primaryinterest. Proof.
We begin with some conventions and settings. Let M α be the transitivecollapse of N α . We will work with models M α in place of N α . This is of coursenot necessary, but it is intuitively easier to work with transitive models rather thanintransitive ones. And, since κ + 1 ⊆ N α , we have P ( κ ) N α = P ( κ ) ∩ N α = P ( κ ) ∩ M α = P ( κ ) M α , so switching to M α will cause no harm.If U is an M -ultrafilter over κ we denote the internal ultrapower of M by U by Ult ( M, U ), and if U is clear from the context then simply just M ∗ . Recall that Ult ( M, U ) is formed using all functions f : κ → M which are elements of M . If U is κ -complete then Ult ( M, U ) is well-founded, and we will always consider ittransitive; moreover the critical point of the ultrapower map π U : M → Ult ( M, U )is precisely κ . Recall also that U is normal if and only if κ = [ id ] U , that is, κ isrepresented in the ultrapower by the identity map. As M | = ZFC − (under ZFC − wemean ZFC without the power set axiom), the Lo´s Theorem holds for all formulae,hence the ultrapower embedding π U is fully elementary. Finally recall that the M -ultrafilter derived from π U , which we denote by U ∗ , is defined by(8) X ∈ U ∗ ⇐⇒ κ ∈ π U ( X )and U ∗ is normal with respect to M .Assume α < α ′ and U , resp. U ′ is a κ -complete M α -ultrafilter, resp. M α ′ -ultrafilter over κ such that U = U ′ ∩ M α . We have the following diagram:(9) M α ′ π U ′ / / Ult ( M α ′ , U ′ ) M α π U / / σ O O Ult ( M α , U ) σ ′ O O Here σ : M α → M α ′ is the natural map arising from collapsing of N α and N α ′ , and σ ′ is the natural embedding of the ultrapowers defined by(10) [ f ] U [ σ ( f )] U ′ ;notice that cr ( σ ) = κ + M . Using the Lo´s theorem, it is easy to check that thediagram is commutative, σ ′ is fully elementary, and σ ′ ↾ κ = id ↾ κ . It follows that(11) σ ′ ( κ ) ≥ κ. Given a set X ∈ P ( κ ) ∩ M α ,(12) X ∈ U ∗ ⇐⇒ κ ∈ π U ( X ) ⇐⇒ σ ′ ( κ ) ∈ σ ′ ( π U ( X )) = π U ′ ( σ ( X )) = π U ′ ( X ) . Thus, using (11) combined with (12),(13) U ∗ ( U ′ ) ∗ = ⇒ σ ′ ( κ ) > κ Notice that since U ∗ , ( U ′ ) ∗ are ultrafilters on the respective models, the statement U ∗ ⊆ ( U ′ ) ∗ can be equivalently expressed as U ∗ = ( U ′ ) ∗ ∩ M α .A position P of the game G − has the form(14) P = h α Pi , U Pi | i < β P i where α Pi are moves of player I and U Pi are moves of player II, and we will drop thesuperscripts P if there is no danger of confusion. Given an infinite regular cardinal γ and a strategy S for player II in the game G − of length ≥ γ , let Z γ be the set ofall positions in G − of length < γ according to S . On Z γ we define a binary relation R γ as follows. Given two positions P, Q ∈ Z γ , we let(15) P R γ Q if and only if the following conditions are met:(a) β Q < β P and Q = P ↾ β Q ,(b) Letting P = h α i , U i | i < β P i , there is an i ∗ such that β Q ≤ i ∗ < β P and U ∗ h = U ∗ i ∗ ∩ M α h +1 for unboundedly many h < i ∗ .If i is a successor ordinal then (b) simply means that U ∗ i − = U ∗ i ∩ M α i − +1 , butlet us note that one also needs to consider the case where i is a limit ordinal. Claim 3.5.
Assume one of the following holds (a) γ > ω and S is a winning strategy for Player II in G − of length γ . (b) γ = ω and S is a winning strategy for Player II in G − ( Q γ ) of length γ .Then R γ is well-founded.Proof. Assume for a contradiction h P n | n ∈ ω i is an infinite descending chain in R γ , and let β n = β P n . Let P be the concatenation of all P n ; then P ∈ Z γ if γ > ω and P is a run of G − according to S if γ = ω ; the latter follows from the fact thatall P n are finite in this case. Say P is of the form P = h α i , U i | i < β i . Let α = sup i<β α i . Since S is a winning strategy for Player II, there is a κ -completeultrafilter U on a κ -algebra of subsets of κ which contains P ( κ ) ∩ N α and extendsall U i , i < β . The ultrafilter U is the response of S at the next step in the gameafter P if γ > ω , and its existence is guaranteed by the requirements on the payoffset if γ = ω . The important point for us here is that the ultrapower Ult ( M α , U )exists and is well-founded. AMES WITH FILTERS 11
Similarly as in (9), we obtain a diagram(16) M α π U / / Ult ( M α , U )... O O ... O O M α j +1 π Uj / / O O Ult ( M α j +1 , U j ) O O M α i +1 π Ui / / σ i,j O O Ult ( M α i +1 , U i ) σ ′ i,j O O whenever i < j < β . It is a routine, using the definition of the maps σ ′ i,j , to checkthat all these diagrams commute. Since the chain h P n | n ∈ ω i is descending in R γ ,for each n ∈ ω r { } we can pick indices i n , ℓ n such that the following hold for allsuch n .(a) β n − ≤ i n < β n (b) ℓ n < i n ≤ ℓ n +1 (c) U ∗ ℓ n = U ∗ i n ∩ M α ℓn +1 .Here recall that U ∗ ℓ n is the normal M α ℓn +1 -measure associated with U ℓ n similarlyas in (8), and accordingly for U ∗ i n . Now if n < n ′ then, with (13) in mind, we obtain σ ′ ℓ n ,ℓ n ′ ( κ ) = σ ′ i n ,ℓ n ′ ( σ ′ ℓ n ,i n ( κ )) ≥ σ ′ ℓ n ,i n ( κ ) > κ Keeping just the leftmost and rightmost term in this formula and applying σ ′ ℓ n ′ ,β gives σ ′ ℓ n ,β ( κ ) > σ ′ ℓ n ′ ,β ( κ ), so we obtain a descending sequence of ordinals σ ′ ℓ ,β ( κ ) > σ ′ ℓ ,β ( κ ) > σ ′ ℓ ,β ( κ ) > · · · > σ ′ ℓ n ,β ( κ ) > · · · , a contradiction. ⊣ Call an ordinal i ∗ as in (b) in the definition (15) of R γ a drop . The well-foundedness of R γ guarantees that every position P ∈ Z γ has only finitely manydrops, as if h i n | n ∈ ω i is an increasing sequence of drops in P then h P ↾ i n | n ∈ ω i constitutes an infinite decreasing sequence in R γ .Now assume S is as in (a) or (b) in Claim 3.5, and fix an R γ -minimal P ∈ Z γ .Let i P be the largest drop in P if P does have a drop, and i P = 0 otherwise. Bythe minimality of P , if P ′ ∈ Z γ extends P then i P ′ = i P ; in other words, P ′ hasno drops above i P , hence ( U P ′ i ) ∗ ⊆ ( U P ′ i ′ ) ∗ whenever i P ≤ i < i ′ . This allows usto define a winning strategy S P for player II in G of length γ . For α < β P denoteby i ( α ) the least i ≥ i P such that α ≤ α Pi . Given a run Q of G of length < γ according to S P and a legal move α of player I, the strategy S P responds as follows:(17) S P ( Q a h α i ) = ( ( U Qi ( α ) ) ∗ ∩ N α +1 if α < β P ( S ( Q a h α i )) ∗ otherwise Since i P is the last drop of Q a h α, S ( Q a h α i ) i , we see that ( U Qi ) ∗ ⊆ ( S ( Q a h α i )) ∗ for all i < β Q . This verifies that S P is a strategy for player II in G . That S P iswinning follows immediately from the fact that S is winning.It remains to complete the proof of (b) of Proposition 3.4. So assume γ = ω , S is a winning strategy for player II on G − ( Q γ ) of length ω and P is as above.Consider a run h α i , U ∗ i | i < ω i according to S P and a sequence h X i | i < ω i such that X i ∈ U ∗ i for all i < ω . Thisrun is associated with a run Q = h α i , U i | i < ω i of G − according to S . If α i ≤ α Pβ P − for all i ∈ ω the conclusion follows easily andwe leave this case to the reader.From now on assume α i > α Pβ P − for some i < ω . Let α = sup i ∈ ω α i and U bea κ -complete ultrafilter on the κ -algebra generated by P ( κ ) ∩ N α extending all U i , i ∈ ω . The existence of such an ultrafilter is guaranteed by the fact that S is awinning strategy for Player II in G − ( Q γ ). Now consider the system of commutingdiagrams as in (16) associated with Q and U ; so i, j ≤ ω here. Notice first thatthere are only finitely many i < ω such that σ ′ i,i +1 ( κ ) > κ , as if h i n | n ∈ ω i werean increasing sequence of such i then σ ′ i ,ω ( κ ) > σ ′ i ,ω ( κ ) > σ ′ i ,ω ( κ ) > · · · > σ ′ i n ,ω ( κ ) > · · · which would contradict the well-foundedness of Ult ( M α , U ).We can thus fix an i ∗ < ω such that the following three requirements are met:(a) α i ∗ > α Pβ P − .(b) σ i,i +1 ( κ ) = κ for all i ≥ i ∗ , hence σ ′ i,ω ( κ ) = σ ′ i ′ ,ω ( κ ) whenever i ∗ ≤ i < i ′ .(c) i ∗ ≥ i P .It is not hard to see that σ i,ω ( κ ) = κ for all i ≥ i ∗ , but we will not need this fact.For i < ω let i ′ = i ∗ if i < i ∗ and i ′ = i otherwise. The definition of S P guaranteesthat κ ∈ π U i ′ ( X i ) as X i ∈ U ∗ i ′ . Applying σ ′ i ′ ,ω to both sides of this statement, weobtain σ ′ i ′ ,ω ( κ ) ∈ σ ′ i ′ ,ω ( π U i ′ ( X i )) = π U ( σ i ′ ,ω ( X i )) = π U ( X i )for all i < ω . Let f ∈ M α be a function representing σ i ′ ,ω in Ult ( M α , U ) where i ≥ i ∗ ; recall that σ i ′ ,ω does not depend on i ≥ i ∗ by (b) above. Lo´s Theoremapplied to this situation then yields A i = { ξ < κ | f ( ξ ) ∈ X i } ∈ U. The intersection of all sets A i is nonempty by κ -completeness of U , so we can picksome ξ ∗ ∈ T i ∈ ω A i . Then f ( ξ ∗ ) ∈ X i for all i ∈ ω , witnessing that T i ∈ ω X i = ∅ . ⊣ Definition 3.6 (The Game G ) . The rules of the game G are as follows. • Player I plays an increasing sequence of ordinals α i < κ + as before. • Player II plays sets Y i ⊆ κ such thatthe following are satisfied. (i) Y j ⊆ ∗ Y i whenever i < j , and (ii) Letting U i = { X ∈ P ( κ ) ∩ N α i +1 | Y i ⊆ ∗ X } , the family U i is a uniformnormal ultrafilter on P ( κ ) ∩ N α i +1 . • Player I goes first at limit stages.
AMES WITH FILTERS 13
A run of G of length γ ≤ κ + continues until Player II cannot play or else until itreaches length γ .Payoff sets R γ and W γ for the game G are defined analogously to those for G . So R γ consists of all runs in G of length γ and W γ consists of all those runs h α i , Y i | i < γ i ∈ R γ such that if h X i | i < γ i is a sequence satisfying X i ∈ N α i +1 and Y i ⊆ ∗ X i for all i < γ then T i<γ X i = ∅ . Note that Y i / ∈ N α i +1 , so Y i = Y j whenever i < j in (i) and Y i = X whenever X ∈ N α i +1 in (ii). Note also that since the ultrafilters U i are required to be uniform,the sets Y i are unbounded in κ . As with G , we will not make any use of what wouldbe an analogue of payoff set Q γ . Proposition 3.7.
Assume γ ≤ κ + is an infinite regular cardinal. (a) Player II has a winning strategy in G of length γ iff Player II has a winningstrategy in G of length γ . (b) Player II a winning strategy in G ( W γ ) of length γ iff Player II has awinning strategy in G ( W γ ) of length γ .Proof. Regarding the non-trivial direction in (a), assume Player II has a winningstrategy S in G of length γ . We describe a winning strategy S ′ for Player II in G of length γ . Given a position P = h α i , Y i | i < j i of length j < γ played according to the strategy S ′ , and a legal move α j by Player I,the strategy S ′ responds as follows. Player II first plays auxiliary steps • α ′ j = the least ordinal α ′ such that α ′ ≥ α j , α ′ > α ′ i for all i < h , and Y i ∈ N α ′ +1 for all i < j , and • U j = S ( ¯ P a h α ′ j i ) where ¯ P = h α ′ i , U i | i < j i and U i is as in clause (b) inDefinition 3.6.Then, letting h X ξ | ξ < κ i be the < θ -least (recall that < θ is the well-ordering of H θ fixed at the beginning of this section; see the paragraphs immediately aboveDefinition 3.1) enumeration of U j of length κ , S ′ ( P a h α j i ) = ∆ ξ<κ X ξ . That the definition of U j makes sense follows from the fact that ¯ P a h α ′ j i is a positionin G according to S , which in turn follows by our choice of α ′ j to be larger thanall α ′ i , i < j , and the fact that ¯ P is a position in G according to S , using astraightforward induction. That S ′ ( P a h α j i ) ⊆ ∗ Y i for all i < j follows from theabove requirement on α ′ j that Y i ∈ N α ′ j +1 for all i < j . Finally, that U i ⊆ U j forall i < j follows from the choice of α ′ j to be at least α j .Regarding (b), it follows immediately that the strategy S ′ is a winning strategyfor Player II in G ( W γ ) of length γ , granting that S is a winning strategy forPlayer II in G ( W γ ) of length γ . The converse is, similarly as in (a), trivial. ⊣ Strategy S ∗ Consider a winning strategy S for Player II in G of length γ and a position P in G according to S . Given a set X ∈ I ( S , P ) + , there may exist different runs of G extending P which witness that X is I ( S , P )-positive. This causes difficultiesin proving that I ( S , P ) has strong properties like precipitousness or existence of a dense subset with high degree of closure. To address this issue, we construct awinning strategy S ∗ for Player II in G of length γ such that for each position Q in G according to S ∗ and each X ∈ I ( S ∗ , Q ) there is a unique run witnessing that X is I ( S ∗ , Q )-positive, and show that using S ∗ one can run natural arguments thatprove precipitousness of I ( S ∗ , Q ) and existence of a dense subset with high degreeof closure, thus proving Theorems 1.1 and 1.2. The point of introducing G is toobtain a formulation of the task which is suitable for construction of strategy S ∗ .Recall from the introduction that when we talk about saturated ideals over κ ,we always mean uniform κ -complete and κ + -saturated ideals over κ . The resultsin this section are formulated under the assumption of non-existence of a normalsaturated ideal over κ , as this allows to fit the results together smoothly. That theresults actually constitute a proof of Theorem 1.2, which is stated under a seeminglystronger requirement on non-existence of a saturated ideal over κ , is a consequenceof the following proposition. Proposition 4.1.
Given a regular cardinal κ > ω , the following are equivalent. (a) κ carries a saturated ideal. (b) κ carries a normal saturated ideal.Proof. Clause (a) follows from (b); saturation does not play any role here. A stan-dard elementary argument shows that any uniform normal ideal over κ is κ -complete.To see that (b) follows from (a), assume I is a saturated ideal over κ . Let P bethe poset ( I + , ⊆ ) and ˙ U be a P -term for the normal V -ultrafitler over κ derivedfrom the generic embedding j G : V → M G associated with Ult ( V , G ) where G is( P , V )-generic. Let I ∗ ∈ V be the ideal over κ defined by a ∈ I ∗ ⇐⇒ (cid:13) V P ˇ a / ∈ ˙ U .
A standard argument shows that I ∗ is a uniform normal ideal over κ . To see that I ∗ is saturated, we construct an incompatibility-preserving map e : ( I ∗ ) + → I + . Let f : κ → κ be a function in V which represents κ in Ult ( V , G ) whenever G is ( P , V )-generic. As I is saturated, such a function can be constructed by standard means.(However, this is not an essential use of saturation of I . Instead we could workbelow a condition x ∈ P which forces a given f ∈ V to represent κ in Ult ( V , G ) andembed the corresponding ( I ∗ x ) + into I + below x .) Notice that for every a ∈ P ( κ ) V and every ( P , V )-generic G , a ∈ ˙ U G ⇐⇒ κ ∈ j G ( a ) ⇐⇒ [ f ] G ∈ [ c a ] G ⇐⇒ e ( a ) def = { ξ < κ | f ( ξ ) ∈ a } ∈ G It follows from these equivalences that indeed e ( a ) ∈ I + whenever a ∈ ( I ∗ ) + .To see that e is incompatibility preserving, we prove the contraposition. Assume e ( a ) , e ( b ) are compatible, so e ( a ) ∩ e ( b ) ∈ I + . Let G be ( P , V )-generic such that a ( a ) ∩ e ( b ) ∈ G . Then e ( a ) , e ( b ) ∈ G , so a, b ∈ ˙ U G by the above equivalences. Butthen a ∩ b ∈ ˙ U G , which tells us that a ∩ b ∈ ( I ∗ ) + . ⊣ We are now ready to formulate the main techincal result of this section.
Proposition 4.2.
Assume there is no normal saturated ideal over κ . Let γ ≤ κ + be an infinite regular cardinal and S be a winning strategy for Player II in G oflength γ . Then there is a tree T ( S ) which is a subtree of the poset ( P ( κ ) , ⊇ ∗ ) suchthat the following hold. AMES WITH FILTERS 15 (a)
The height of T ( S ) is γ and T ( S ) is < γ -closed. Here “ < γ -closed” meansthat if b is a branch through T ( S ) of length < γ then there is a node in T ( S ) above b . (b) If Y, Y ′ ∈ T ( S ) are ⊆ ∗ -incomparable then Y, Y ′ are almost disjoint. (c) There is an assignment Y P Y assigning to each Y ∈ T ( S ) a position P Y in G of successor length according to S in which the last move by Player IIis Y ; we denote the last move of Player I in P Y by α ( Y ) . The assignment Y P Y has the following property: Y ′ ⊂ ∗ Y = ⇒ P Y ′ is an extension of P Y and α ( Y ) < α ( Y ′ ) .Here Y ′ ⊂ ∗ Y abbreviates the conjunction Y ′ ⊆ ∗ Y ∧ Y ′ = Y . (d) If b is a branch of T ( S ) of length < γ , let P b = S Y ∈ b P Y ; clearly P b is aposition in G according to S . Then the set of all immediate successors of b in T ( S ) is of cardinality κ + , and the assignment Y α ( Y ) is injectiveon this set. Recall the notion of “almost disjoint” in clause (b): Sets
Y, Y ′ ∈ P ( κ ) are almostdisjoint iff Y ∩ Y ′ is bounded in κ . Clause (d) in the above definition treats bothsuccessor and limit cases for ¯ γ . The successor case in (d) simply says that if Y ∈ T ( S ) then the conclusions in (d) apply to the set of all immediate successorsof Y in T ( S ). Proof.
The tree T ( S ) is constructed by induction on levels. Limit stages of thisconstruction are trivial: If ¯ γ < γ is a limit and we have already constructed initialsegments T γ ∗ of T ( S ) of height γ ∗ for all γ ∗ < ¯ γ so that the trees T γ ∗ satisfy(b) – (d) and T γ ′ end-extends T γ ∗ whenever γ ∗ < γ ′ then it is easy to see that T ¯ γ = S γ ∗ < ¯ γ T γ ∗ is a tree with tree ordering ⊇ ∗ end-extending all T γ ∗ , γ ∗ < ¯ γ whichsatisfies (b) – (d). We will thus focus on the successor stages of the construction.Assume ¯ γ < γ and T ( S ) is constructed all the way below level ¯ γ ; our task nowis to construct the ¯ γ -th level of T ( S ). Let b be a cofinal branch through this initialsegment of T ( S ), so b is of length ¯ γ . We construct the set of immediate successorsof b in T ( S ), along with the assignment Y P Y on this set, as follows. As weare assuming there is no normal saturated ideal over κ , we can pick an antichain A in I ( S , P b ) + of cardinality κ + . For each X ∈ A there is a position Q X in G of successor length < γ according to S extending P b such that the last move byPlayer II in Q X is almost contained in X . For the sake of definability we can letthis position to be < θ -least, where recall that < θ is the well-ordering of H θ fixedat the beginning of this section.Now construct the set h Y ξ | ξ < κ + i of all immediate successors of b in T ( S )recursively as follows. Assume ξ < κ + and we have already constructed the set h Y ¯ ξ | ¯ ξ < ξ i along with the assignment Y ¯ ξ P Y ¯ ξ with the desired properties. Sinceeach model N β is of cardinality κ , we can pick a set X ∈ A which is not an elementof any N α ( Y ¯ ξ )+1 where ¯ ξ < ξ . Now let Player I extend Q X by playing an ordinal α such that(18) { X } ∪ { Y ¯ ξ | ¯ ξ < ξ } ⊆ N α +1 , which is a legal move in G following Q X , and let Y be the response of strategy S to Q X a h α i . To make the things definable, let us require Player I to play minimalsuch α . We let Y ξ be this Y and P Y = Q X a h α, Y i . Notice that Y ξ ⊆ ∗ X , as Y ξ , being played according to S , is almost contained in the last move by Player II in Q X . We show:(19) Any two sets Y = Y ′ on the ¯ γ -th level are almost disjoint.If Y, Y ′ are above two distinct cofinal branches, say b = b ′ , Y ∈ b and Y ′ ∈ b ′ , thenthis follows immediately from the induction hypothesis: Letting Z , resp. Z ′ be theimmediate successor of b ∩ b ′ in b , resp. b ′ , we have Y ⊆ ∗ Z and Y ′ ⊆ ∗ Z ′ , andthe induction hypothesis tells us that Z, Z ′ are almost disjoint. Now assume Y, Y ′ are above the same branch b ; without loss of generality we may assume Y = Y ξ and Y ′ = Y ξ ′ in the above enumeration and ξ ′ < ξ . By the above construction,we have a set X ′ ∈ A such that X ′ = X and P Y ′ witnesses X ′ ∈ I ( S , P b ) + , thatis, X ′ ∈ N α ( Y ′ )+1 and Y ′ ⊆ ∗ X ′ . If Y ⊆ ∗ Y ′ then Y ⊆ ∗ X ∩ X ′ , thus witnessing X ∩ X ′ ∈ I ( S , P b ) + . This contradicts the fact that A is an antichain in I ( S , P b ) + .It follows that Y ∗ Y ′ . Now for every Z ∈ N α ( Y )+1 the set Y is either almostcontained in or else almost disjoint from Z . As Y ′ ∈ N α ( Y )+1 by our choice of α ( Y )in (18), necessarily Y is almost disjoint from Y ′ . This proves (19).To verify that (b) – (d) hold after adding the immediate successors of a singlebranch b as described in the previous paragraph, notice that (c) and (d) immediatelyfollow from the construction just described, so all we need to check is clause (b)and the fact that ⊇ ∗ is still a tree ordering after adding the entire ¯ γ -th level. Butclause (b) follows from the combination of (19) with the induction hypothesis andthe fact that every set on the ¯ γ -th level is almost contained in some set on anearlier level. Finally, that adding the ¯ γ -th level keeps ⊇ ∗ a tree ordering followsfrom clause (b). More generally, any collection X ⊆ P ( κ ) which satisfies (b) with X in place of T ( S ) has the property that the set of all Y ′ ∈ X which are ⊇ ∗ -predecessors of a set Y ∈ X is linearly ordered under ⊇ ∗ . What now remains is tosee that clause (a) holds, but this is immediate once we have completed all γ stepsof the construction. ⊣ The new strategy S ∗ for Player II in G is now obtained by, roughly speaking,playing down the tree T ( S ). More precisely: Definition 4.3.
Assume γ ≤ κ + is an infinite regular cardinal, S is a winningstrategy for Player II in G of length γ , and T ( S ) is a subtree of the poset ( P ( κ ) , ⊇ ∗ ) satisfying (a) – (d) in Proposition 4.2. We define a strategy S ∗ for Player II in G of length γ associated with T ( S ) recursively as follows.Assume P = { ( α i , Y i ) | i < j } is a position in G of length j < γ according to S ∗ . Denote the corresponding branchin T ( S ) by b P , that is, b P = { Y i | i < j } . If α j is a legal move of Player I in G at position P then S ∗ ( P a h α j i ) = the unique immediate successor Y of b P in T ( S ) with minimal possible α ( Y ) ≥ α j .Here recall that α ( Y ) is the last move of Player I in P Y . As an immediate consequence of the properties of T ( S ) we obtain: AMES WITH FILTERS 17
Proposition 4.4.
Let γ ≤ κ + be an infinite regular cardinal and assume T ( S ) isas in Proposition 4.2. Then S ∗ is a winning strategy for Player II in G of length γ .Moreover, if r ∗ = h α i , Y i | i < γ i is a run of G of length γ according to S ∗ then r = [ i<γ P Y i is a run of G of length γ according to S . We are now ready to give a proof of Theorem 1.1. If there is a normal saturatedideal over κ then there is nothing to prove. Otherwise Player II has a winning strat-egy in G ( W ω ) of length ω , as follows from Propositions 3.2(b), 3.4(b) and 3.7(b).The conclusion in Theorem 1.1 then follows from a more specific fact we prove,namely from Proposition 4.5 below. In the proof of this proposition we will makeuse of the criterion on precipitousness in terms of the ideal game, see Section 1. Proposition 4.5.
Assume there is no normal saturated ideal over κ . Let • S be a winning strategy for Player II in G ( W ω ) of length ω . • Let S ∗ be the winning strategy constructed from S as in Definition 4.3.Then Player I does not have a winning strategy in the ideal game G ( I ( S ∗ )) . Con-sequently, the ideal I ( S ∗ ) is precipitous.Proof. Assume S I is a strategy for Player I in the ideal game G ( I ( S ∗ ))). Weconstruct a run in G ( I ( S ∗ )) according to S I which is winning for Player II. Oddstages in this run will come from positions in G played according to S ∗ ; moreprecisely, they will be tail-ends of sets on those positions. So suppose Q = h X , X , X , X . . . , X n − i is the finite run of G ( I ( S ∗ )) constructed so far, and β , Z , β , Z , · · · β n − , Z n − is the associated auxiliary run of G according to S ∗ such that Z i ⊆ ∗ X i and X i +1 = the longest tail-end of Z i that is contained in X i for all i < n . Let X n be the response of S I to Q in G ( I ( S ∗ )). As X n ∈ I ( S ∗ ) + ,there is a finite position in G according to S ∗ where the last move of Player II is a setalmost contained in X n and, letting Z n be this set, we also have X n ∈ N α ( Z n )+1 .As the sets Z n constitute an ⊆ ∗ -decreasing chain of nodes in T ( S ), positions P Z n extend P Z m whenever m < n , so r = [ n ∈ ω P Z n . is a run in G of length ω according to S , by Proposition 4.4. Say r = h α i , Y i | i ∈ ω i . For each i ∈ ω let X ′ i = X n where n is such that lh ( P Z n ) ≤ i < lh ( P Z n +1 ).Then \ n ∈ ω X n = \ n ∈ ω X n = \ i ∈ ω X ′ i = ∅ . Here the equality on the left comes from the fact that the sets X n , n ∈ ω constitutean ⊆ -descending chain, and the inequality on the right follows from the fact that X ′ i ∈ N α i +1 and Y i ⊆ ∗ X ′ i for all i ∈ ω , and that S is a winning strategy forPlayer II in G ( W ω ) of length ω ; see the last paragraph in Definition 3.6. ⊣ The following proposition gives a proof of Theorem 1.2. Recall that all back-ground we have developed so far was under the assumption that κ is inaccessibleand 2 κ = κ + . Also recall that by trivial observation (TO3) at the beginning ofSection 3 and results in Section 3, if Player II has a winning strategy in G oflength γ > ω then Player II has a winning strategy in G ( Q ω ) of length ω and in G ( W ω ) of length ω , as well as in G of length γ whenever γ is regular. By a similarargument, if Player II has a winning strategy in G of length γ > ω then Player IIhas a winning strategy in G ( W ω ) of length ω . Thus, under the assumptions ofTheorem 1.2, the assumptions of Proposition 4.6 below are not vacuous. Proposition 4.6.
Assume there is no normal saturated ideal over κ . Let γ ≤ κ + be an uncountable regular cardinal. Assume further that S and S ∗ are strategies asin Proposition 4.5, with γ in place of ω .Then Player I does not have a winning strategy in the ideal game G ( I ( S ∗ )) .Consequently, the ideal I ( S ∗ ) is precipitous. Additionally, T ( S ) is a γ -closed densesubset of I ( S ∗ ) + .Proof. The proof that Player I does not have a winning strategy in G ( I ( S ∗ )) followsfrom Proposition 4.5 and from the fact that a winning strategy for Player II in G of length γ > ω guarantees the existence of a winning strategy for Player II in G ( W ω ).That T ( S ) is a γ -closed dense subset of I ( S ∗ ) + follows immediately from theproperties of T ( S ) and from the definitions of S ∗ and I ( S ∗ ). ⊣ Proof of Theorem 1.4.
Consider a ( κ + , ∞ )-distributive ideal J γ over κ such thatthere is a dense γ -closed set D ⊆ ( I + , ⊆ ). We work inside H θ for a sufficiently large θ and will use the well-ordering < θ introduced at the beginning of Section 3 to definea winning strategy ¯ S γ for Player II in G Wγ . As usual, ¯ S γ is defined inductively onthe length of runs.So assume A , U , A , U , . . . , A j , U j , . . . be a run of G Wγ according to ¯ S γ for j < i . Along the way, we define auxiliary moves X j played by Player II; these moves are elements of D , constitute a descendingchain in the ordering by ⊆ , and for each j < i ,(20) X j (cid:13) P J γ ˙ G ∩ ˇ A j = ˇ U j . At step i < γ
Player I plays a κ -algebra A i on κ of cardinality κ extending all A j , j < i . As D is γ -closed and i < γ , there is an element X ∈ D below all X j , j < i . If G is a ( P J γ , V )-generic filter such that X ∈ G then by (20), U j ⊆ G whenever j < i . Since P J γ is ( κ + , ∞ )-distributive and A i ∈ V is of cardinality κ ,the intersection G ∩ A i is an element of V , and is a uniform κ -complete ultrafilteron A i extending all U j where j < i . This is then forced by some condition X ′ ∈ G AMES WITH FILTERS 19 such that X ′ ⊆ X j for all j < i . As D is dense in P J γ , X ′ can be chosen to be anelement of D . The following is thus not vacuous. We define X i = the < θ -least element Y of D such that Y ⊆ X j for all j < i and there is a U ∈ V satisfying Y (cid:13) P Jγ ˙ G ∩ ˇ A i = ˇ U and U i = the unique U ∈ V such that X i (cid:13) P J γ ˙ G ∩ ˇ A i = ˇ U .Letting ¯ S γ ( hA j , U j | j < i i a hA i i ) = U i , it is straightforward to verify that ¯ S γ is a winning strategy for Player II in G Wγ . ⊣ The Model
In this section we give a construction of a model where the following holds.(21) κ is inaccessible and carries no saturated idealsand(22) For every regular uncountable γ ≤ κ there is anideal J γ on P ( κ ) as in clause (a) in Theorem 1.3.The model is a forcing extension of a universe V in which the following is satisfied.(A) GCH .(B) U is a normal measure on κ .(C) h T α,ξ | ξ < α + i is a disjoint sequence of stationary subsets of α + ∩ cof ( α )whenever α ≤ κ is inaccessible.(D) Assume V [ K ] is a generic extension via a set-size forcing which preserves κ + , and – j ′ : V → M ′ is a class elementary embedding where M ′ is transitive,and – hh T ′ α,ξ | ξ < α + i | α ≤ j ( κ ) is inaccessible in M ii == j ′ ( hh T α,ξ | ξ < α + i | α ≤ κ is inaccessible i )Then V , M ′ agree on what H κ + is and T ′ κ,ξ = T κ,ξ whenever ξ < κ + .We will informally explain the purpose of the sets T α,ξ before we begin with theconstruction of the model. These sets are not needed for the construction of ideals J γ in Theorem1.3, but only for the proof that κ does not carry a saturated ideal inour model. To understand this proof, it suffices to accept (D) as a black box, thatis, it is not necessary to understand how the system of sets T α,ξ is constructed.Proper class models satisfying (A) – (D) are known to exist. If there is a properclass inner model with a measurable cardinal then any K c -construction (see forinstance [15] for K c -constructions of models with Mitchell-Steel indexing of exten-ders, and [19] for K c -constructions with Jensen’s λ -indexing) performed inside sucha model gives rise to a fine structural proper class model satisfying (A) – (D). Wewill sketch a proof of this fact below in Proposition 5.1.There some similarity in the argument in Proposition 5.1 of the existence of asequence of mutually disjoint stationary subsets T κ,ξ of κ + which behave nicelywith respect to the ultrapower by a normal ultrafilter on κ to a similar claim [5] inwhere it is proved that one can have such sequence of stationary sets in L [ U ].A K c -construction as above gives rise to a model K c of the form L [ E ] where E = h E α | α ∈ On i is such that each E α either codes an extender in a way made precise, or E α = ∅ . Additionally, a model of this kind admits a detailed finestructure theory. There is an entire family of such models, so called fine structuralmodels; the internal first order theory of these models is essentially the same, up tothe large cardinal axioms. We now list some notation, terminology and general factswhich will be used for the proof of (C) and (D). Clauses (A) and (B) follow fromthe construction of the model, and their proofs can be found in [15] or [19]. In fact,each proper initial segment of the model is acceptable in the sense of fine structuretheory. We omit the technical definition here and merely say that acceptability isa local form of GCH , and is proved as the model is constructed.From now on Assume W = L [ E ] is a fine structural model. We often write E W in place of E to emphasize that E is the extender sequence of W .FS1 W || α is the initial segment of W of height ωα with the top predicate, thatis, W || α = ( J Eα , E ωα ).FS2 If α is a cardinal of W then E α = ∅ . Thus, in this case W || α = ( J Eα , ∅ )and we identify this structure with J Eα .FS3 If µ is a cardinal of W then the structure W || µ calculates all cardinals andcofinalities ≤ µ the same way as W . This is a consequence of acceptability.FS4 β ( τ ) is the unique β such that τ is a cardinal in W || β but not in W || ( β +1).FS5 ̺ stands for the first projectum; that ̺ ( W || β ) ≤ α is equivalent to sayingthat there is a surjective partial map f : α → J Eβ which is Σ -definable over W || β with parameters.FS6 (Coherence.) If i : W → W ′ is a Σ -preserving map in possibly some outeruniverse of W such that κ is the critical point of i and τ = κ + W then E W ′ ↾ τ = E W ↾ τ .FS7 (Cores.) Assume α is a cardinal in W and N is a structure such that ̺ ( N ) = α and there is a Σ -preserving map π of N into a level of W suchthat π ↾ α = id . Let p N be the < ∗ -least finite set of ordinals p such thatthere is a set a ⊆ α which is Σ ( N )-definable in the parameter p satisfying a ∩ α / ∈ N . Here < ∗ is the usual well-ordering of finite sets of ordinals,that is, finite sets of ordinals are viewed as descending sequences and < ∗ is the lexicographical ordering of these sequences. Let X be the Σ -hull of α ∪{ p N } and σ : ¯ N → N be the inverse of the collapsing isomorphism. Then ρ ( ¯ N ) = α , models ¯ N, N agree on what P ( α ) is, and π is Σ -preservingand maps ¯ N cofinally into N . In this situation, ¯ N is called the core of N and σ is called the core map.FS8 (Condensation lemma.) Assume α is a cardinal in W and N, ¯ N , π and σ are as in FS7. Then ¯ N is a level of W , that is, ¯ N = W || β for some β . Proposition 5.1.
There is a formula ϕ ( u, v, w ) in the language of extender modelssuch that the following holds. If W = L [ E ] is a fine structural extender model, α is an inaccessible cardinal of W and ξ < α + , letting T α,ξ = { τ ∈ α + ∩ cof ( α ) | W || α + W | = ϕ ( τ, α, ξ ) } , each T α,ξ is a stationary subset of α + ∩ cof ( α ) in W , and T α,ξ ∩ T α,ξ ′ = ∅ whenever ξ = ξ ′ . Moreover, the sequence (( T α,ξ | ξ < α + ) | α ≤ κ is inaccessible in W ) satisfies clause (D) above with W in place of V .Proof. Since the definition of ( T α,ξ | ξ < α + ) takes place inside W || α + W , anytwo extender models W, W ′ such that α + W = α + W ′ and E W ↾ α + = E W ′ ↾ α + calculate this sequence the same way (here α + stands for the common value of the AMES WITH FILTERS 21 cardinal successor of α in both models). Now if V = W and j, is as in (D) abovethen T ′ α,ξ = { τ ∈ α + M ′ ∩ cof ( α ) | M ′ || α + M ′ | = ϕ ( τ, α, ξ ) } , whenever α ≤ j ′ ( κ ) is inaccessible in M ′ , so to see that T ′ κ,ξ = T κ,ξ for all ξ < κ + it suffices to prove that κ + M ′ = κ + V and E V ↾ κ + = E M ′ ↾ κ + (where again κ + stands for the common value of the cardinal successor of κ in V and M ′ ). Regardingthe former, the inequality κ + V ≤ κ + M ′ is entirely general and follows from the factthat P ( κ V ) ⊆ P ( κ ) M ′ . The reverse inequality follows from the assumption thatthe generic extension preserves κ + , so κ + V remains a cardinal in M ′ . The latter isthen a consequence of he coherence property FS6.It remains to come up with a formula ϕ such that the sets T α,ξ are stationary in W for all α, ξ of interest, and pairwise disjoint. Here we make a more substantialuse of the fine structure theory of W . Given an inaccessible α and a ξ < α + , letting(23) T α,ξ def = the set of all τ ∈ α + ∩ cof ( α ) such that ̺ ( W || β ( τ )) = α and W || β ( τ ) has ξ + 1 cardinals above α ,it is clear that T α,ξ ∩ T α,ξ ′ whenever ξ = ξ ′ . Then it suffices to we show that(24) T α,ξ is stationary in W ,as we can then take ϕ be the defining formula for the system ( T α,ξ ) α,ξ .The first step toward the proof of (24) is the following observation.(25) Assume ν > α is regular in W , p ∈ W || ν and X is the Σ -hull of α ∪ { p } in W || ν . Let ν X = sup( X ∩ ν ). Then cf W ( ν X ) = α . Proof.
Obviously, γ = cf W ( ν X ) ≤ α . Assume for a contradiction that γ < α .Let h ν i | i < γ i be an increasing sequence converging to ν X such that ν i ∈ X for every i < γ . For each such i pick a j i ∈ ω and an ordinal η i < α such that ν i = h W || ν ( j i , h η i , p i ) where h W || ν is the standard Σ -Skolem function for W || ν .Here W || ν is of the form h J Eν , ∅ i (see FS2), and we identify it with the structure J Eν . The Skolem function h W || ν has a Σ -definition of the form ( ∃ w ) ψ ( w, u , u , v )where ψ is a ∆ -formula in the language of extender models. (The standard Σ -Skolem function has a uniform Σ -definition, which means that it has s definingΣ -formula which defines a Σ -Skolem function h N over every acceptable structure N . However, the argument below does not make use of uniformity of the definition.)Since ν > α is regular,(26) ( ∃ ¯ ν ) (cid:16) J E ¯ ν | = ( ∀ i < γ )( ∃ w )( ∃ v ) ψ ( w, j i , h η i , p i , v ) (cid:17) Since the statement in (26) is Σ , there is some such ¯ ν with J E ¯ ν ∈ X . To justifythis note that the sequences h η i | i < γ i and h j i | j < γ i are elements of X as J Eα ⊆ X , and we can view these sequences as parameters in the formula in (26).Fix such an ordinal ¯ ν . Now consider i < γ such that ν i > ω ¯ ν . Using (26) pick z and ν ∗ in J E ¯ ν such that J E ¯ ν | = ψ ( z, j i , h η i , p i , ν ∗ ). Since ψ is ∆ , we actuallyhave J Eν | = ψ ( z, j i , h η i , p i , ν ∗ ), which tells us that ν ∗ = h W || ν ( j i , h η i , p i ) = ν i . As ν i > ω ¯ ν , this is a contradiction. This completes the proof of (25). ⊣ Now let C be a club subset of α + , X be the Σ -hull of α ∪ { C, ξ, α + ξ +1 } in W || α ξ +2 , N be the transitive collapse of X , and π : N → W || α ξ +2 be the inverseof the collapsing isomorphism. Let further τ = X ∩ α + = cr ( π ). Then τ > ξ as α ∪ { ξ } ⊆ X . It is a standard fact that cf W ( τ ) = cf W (sup( X ∩ On )) (and canbe proved similarly as (25) above). Now cf W (sup( X ∩ On )) = α by (26), hence cf W ( τ ) = α . Moreover τ ∈ C as C is closed and τ is a limit point of C . Thus, theproof of (23) will be complete once we show that ̺ ( W || β ( τ )) = α and W || β ( τ ))has ξ + 1 cardinals above κ . We first look at the set of cardinals in N .By acceptability, the structures W || α + ξ +1 and W || α ξ +2 agree on what is acardinal below α ξ +1 . It follows that in W || α ξ +2 , the statement“The order type of the set of cardinals in the interval ( α, α ξ +1 ) is ξ ”can be expressed in a Σ -way as(27)“The order type of the set of cardinals above α in the structure W || α ξ +1 is ξ .Since π is Σ -preserving and cr ( π ) = τ , this Σ -statement can be pulled back to N via π . Also by the Σ -elementarity of π we have π − ( α + ξ +1 ) is the largest cardinalin N . Then, using acceptability in N , we conclude:(28) The order type of the set of cardinals above κ in N is ξ + 1.By construction, the Σ -Skolem function of N induces a partial surjection of α onto N . Then ̺ ( N ) ≤ α by FS5. Since α is a cardinal in W , we conclude ̺ ( N ) = α .Let ¯ N be the core of N and σ : ¯ N → N be the core map. By FS7, ̺ ( ¯ N ) = α and P ( α ) ¯ N = P ( α ) N , so in particular τ = α + N = α + ¯ N . By FS8, ¯ N = W || β forsome β . Since ̺ ( ¯ N ) = α , FS5 implies β = β ( τ ). To see that ¯ N = W || β ( τ ) has ξ + 1 cardinals above α , first notice that, since by FS7 the map σ is cofinal, thelargest cardinal in N must be in the range of σ . This along with (28) providesa Σ -definition of ξ in N from parameters in rng( σ ). The point here is that wecan reformulate the notion of cardinal in N below α + ξ +1 as the cardinal in thesense of the structure N || α + ξ +1 , similarly as in (27). It follows that ξ ∈ rng( σ ),and since ξ < α + N we have ξ < cr ( σ ). Then, using the Σ -reformulation of (28)one more time, we conclude that α + η ∈ rng( σ ) for every η ≤ ξ , which means that W || β ( τ ) = ¯ N has ξ + 1 cardinals above α . This completes the proof of (23) andthereby the proof of Proposition 5.1. ⊣ Two main tools we will use to construct the forcing to build our model are clubshooting with initial segments, and adding non-reflecting stationary sets with initialsegments. We then use variations of standard techniques for building ideals usingelementary embeddings. The background information on the first two can be foundin [1] and on ideal constructions in [4], but we review the relevant facts for thereader’s convenience.Recall that if S ⊆ λ + is a stationary set (here we assume λ is a cardinal) thenthe club shooting poset CS ( S ) consists of closed bounded subsets of λ + which arecontained in S , and is ordered by end-extension. In general, this poset may nothave good preservation properties, but if T is sufficiently large then it is known tobe highly distributive. Proposition 5.2 (See [1]) . Assume λ is inaccessible and T is a subset of λ + suchthat T ∩ α is non-stationary in α whenever α < λ + . Then the following hold. (a) CS ( λ + r T ) is ( λ + , ∞ ) -distributive, that is, it does not add any new function f : λ → V . In particular, generic extensions of V via CS ( λ + r T ) agreewith V on all cardinals and cofinalities ≤ λ + , and on what H λ + is. AMES WITH FILTERS 23 (b) If γ ≤ λ is regular and T ⊆ λ + ∩ cof ( γ ) then CS ( λ + r T ) has a dense setwhich is γ -closed but it does not have a dense set which is γ + -closed. (c) If G is ( CS ( λ + r T ) , V ) -generic then C G = S G is a closed unboundedsubset of λ + such that C G ⊆ λ + r T . (d) If G is as in (c) and S ⊆ λ + r T is stationary in V then S remainsstationary in V [ G ] . Given a regular cardinal µ , the poset NR ( µ ) for adding a non-reflecting stationarysubset of µ consists of functions p : α → { , } such that α < µ and(a) letting S p = { ξ < α | p ( ξ ) = 1 } , for every ¯ α ≤ α there is a closed unbounded set c in ¯ α such that S p ∩ c = ∅ .The variant NR ( µ, γ ) of the poset NR ( µ ) for adding a non-reflecting stationarysubset of µ ∩ cof ( γ ), where γ < µ is regular, consists of conditions p ∈ NR ( µ ) whichconcentrate on µ ∩ cof ( γ ), that is, they additionally satisfy the requirement(b) p ( ξ ) = 0 whenever cf ( ξ ) = γ .We will make use of these posets in the special case where µ is of the form λ + .For this reason we formulate the next proposition for cardinals of the form λ + ,although it is true for any regular µ > ω . Proposition 5.3 (See [1]) . Assume γ ≤ λ where γ is regular and λ is a cardinal.Then the following hold. (a) Both NR ( λ + ) and NR ( λ + , γ ) are strategically λ + -closed. In particular, both NR ( λ + ) and NR ( λ + , γ ) preserve stationarity of stationary subsets of λ + ,are ( λ + , ∞ ) -distributive, so they do not add any new function f : λ → V ,and generic extensions of V via these posets agree with V on all cardinalsand cofinalities ≤ λ + and on what H λ + is. (b) NR ( λ + , γ ) is γ -closed but not γ + -closed. (c) If G is ( NR ( λ + , γ ) , V ) -generic then S G = S { S p | p ∈ G } is a non-reflectingstationary subset of λ + ∩ cof ( γ ) . (d) If G is ( NR ( λ + ) , V ) -generic then S G = S { S p | p ∈ G } is a non-reflectingstationary subset of λ + such that S G ∩ λ + ∩ cof ( γ ) is stationary for allregular γ < λ + . Although both posets NR ( λ + , γ ) and CS ( λ + r T ) have low degree of closurein general, the composition of NR ( λ + , γ ) followed by adding a closed unboundedsubset of the complement of the generically added non-reflecting stationary set hasa high degree of closure. This is an important feature which will be used in theconstruction of our model. Proposition 5.4.
Assume λ is a cardinal, γ ≤ λ is regular, and ˙ S is the canonical NR ( λ + , γ ) -term for the generic non-reflecting stationary subset of λ + ∩ cof ( γ ) . Thenthe composition NR ( λ + , γ ) ∗ CS ( λ + r ˙ S ) has a dense λ + -closed subset. Namely, D = the set of all ( p, ˙ c ) ∈ NR ( λ + , γ ) ∗ CS ( λ + r ˙ S ) | p (cid:13) ˙ c = ˇ c for some closed unbounded set c ⊆ dom( p ) is such a dense set.In particular, this composition preserves stationarity of stationary subsets of λ + . Let(29) j : V → M be the ultrapower embedding by U where M is transitive; here U is as in (B) above.Then κ is the critical point of j . We now describe in an informal way the key stepsof the forcing construction.We first describe an initial attempt at our model where the ideals J γ as inTheorem 1.3 exist and then explain why a modification is needed. We begin withiterative adding a non-reflecting stationary subset S α of α + ∩ cof ( γ ) followed byshooting a closed unbounded set through its complement at each inaccessible α < κ .At each such step α the ordinal γ ≤ α is chosen generically in the lottery style.The iteration uses Easton supports. At step κ we add generically a non-reflectingstationary subset S κ,γ of κ + ∩ cof ( γ ) for all γ ≤ κ . For the moment, let P ∗ bethis poset. Given a ( P ∗ , V )-generic filter G ∗ , one can extend the map j : V → M to j H : V [ G ∗ ] → M [ H ∗ ]. This allows to define the ideals J γ in V [ G ∗ ] in thestandard way to consist of sets that never become elements of the normal measuresover V [ G ∗ ] derived from j H , for all possible H . Using the closure properties ofthe tail-end of j ( P ∗ ) above κ and the fact that V [ G ∗ ] contains the first part of thecomposition NR ( κ + , γ ) ∗ CS ( κ + r ˙ S κ,γ ), one can construct the generic H ∗ inside ageneric extension of V [ G ∗ ] via CS ( κ + r S κ,γ ), which allows to show that P ( κ ) / J γ has a dense subset isomorphic to CS ( κ + r S κ,γ ).What remains to show is that κ is not measurable in V [ G ∗ ] or, more generally,that in V [ G ∗ ] the cardinal κ does not carry a κ + -saturated ideal. It is not clearthat the construction above will achieve this, even if U is a measure of Mitchellorder zero. This is the place where forcing over a fine structural model is relevant(obviously, the construction described in the previous paragraph can be carried outover any V satisfying GCH which carries a normal measure over κ ). One option howto proceed would be to pick non-reflecting stationary sets S α in some canonical wayinstead of adding them generically, but it is not clear to us how to do this. Anotherapproach is to code the sets S α using some canonically defined stationary sets thestationarity of which is sufficiently absolute. We will follow this approach, and thisis the place where the sets T α,ξ from (C) and (D) at the beginning of this sectionare used. We modify poset P ∗ in that at each step α < κ we code the genericallyadded set S α by forcing with CS ( α + r (cid:16)S ξ ∈ α + r S α T α, ξ +1 ∪ S ξ ∈ S α T α, ξ (cid:17) ). Thiswill code the set S α as follows: ξ ∈ S α ⇐⇒ T α, ξ +1 is stationary and T α, ξ is non-stationary.Then, given a map j ′ : V [ G ] → M [ H ], this will allow us to decode the set S κ,γ bylooking at which sets T κ,ξ are stationary. The absoluteness of stationarity of T κ,ξ between V , M and various generic extensions will play a critical role here. We nowmake the things technically precise.Given a cardinal α and a set X ⊆ α + , let(30) T α ( X ) = [ ξ ∈ X T α, ξ ∪ [ ξ / ∈ X T α, ξ +1 We let ( P α | α ≤ κ + 1)be the forcing iteration satisfying the following. AMES WITH FILTERS 25
FI-1 Conditions in each P α are partial functions p with dom( p ) contained ininaccessibles below α such that dom( p ) ∩ β is bounded in β whenever β ≤ α is inaccessible.FI-2 If p ∈ P α and ¯ α ∈ dom( p ) then p (¯ α ) = ( γ p (¯ α ) , w p (¯ α )) ∈ H ¯ α + (see the remarks below the definition of the iteration for explanations) isan ordered pair such that γ p (¯ α ) ∈ R ¯ α = { γ ≤ ¯ α | γ is regular uncountable } , and w p (¯ α ) is a P ¯ α -term for a condition in Q ¯ α def = NR (¯ α + , γ p (¯ α )) ∗ CS (¯ α + r ˙ T ¯ α ) ∗ CS (¯ α + r ˙ S ¯ α )where ˙ S ¯ α is the canonical NR (¯ α + , γ p (¯ α ))-term for the generically addednon-reflecting stationary set S ¯ α and ˙ T ¯ α is the canonical NR (¯ α + , γ p (¯ α ))-term for T ¯ α ( S ¯ α ).The ordering on P α is defined in the standard way, that is,FI-3 p ≤ q iff the following hold:(i) dom( p ) ⊇ dom( q ) and(ii) for every ¯ α ∈ dom( q ):(a) γ p (¯ α ) = γ q (¯ α ) and(b) p ↾ ¯ α (cid:13) P ¯ α “ w p (¯ α ) extends w q (¯ α ) in ˙ Q ¯ α ”where under p ↾ ¯ α we really mean p ↾ (dom( p ) ∩ ¯ α ).Strictly speaking, in order that P α is an iteration according to the official defini-tion we would need to consider γ p (¯ α ) a P α -term for an ordinal from R ¯ α . However,using conditions as in FI-2 simplifies the matters, and these conditions can beproved to constitute a dense subset of the official iteration that uses P α -terms forordinals from R ¯ α .Consider an inaccessible α < κ . Since the poset NR ( α + , γ p ( α )) is α + -distributive,all conditions in posets CS ( α + r T α ) and CS ( α + r S α ) built in the generic extensionvia NR ( α + , γ p ( α )) are in the ground model. (Recall S α is the generic non-reflectingstationary set added by NR ( α + , γ p ( α )).) Since CS ( α + r T α ) is α + -distributive inany such generic extension, CS ( α + r S α ) is the same poset in both the generic ex-tension via NR ( α + , γ p ( α )) and via NR ( α + , γ p ( α )) ∗ CS ( α + r ˙ T α ), and its conditionsare in the ground model. From this we conclude that(31) Forcing with Q α = NR ( α + , γ p ( α )) ∗ CS ( α + r ˙ T α ) ∗ CS ( α + r ˙ S α )over the ground model is equivalent to forcing with CS ( α + r T α ) × CS ( α + r S α ) ≃ CS ( α + r S α ) × CS ( α + r T α )over the generic extension via NR ( α + , γ p ( α )).Then, with a little bit of work, we can view w p ( α ) as a triple(32) w p ( α ) = ( s p ( α ) , a p ( α ) , c p ( α ))where s p ( α ) is a P α -term for a condition in NR (¯ α + , γ p (¯ α )), a p ( α ) is a P α -term fora condition in CS ( α + r T α ), and c p ( α ) is a P α -term for a condition in CS (¯ α + r S α ).Of course, a p ( α ) and c p ( α ) are connected with s p ( α ) in the natural way; we leavethe details here to the reader. And, to be precise, it is a dense set of conditions in NR ( α + , γ p ( α )) ∗ CS ( α + r ˙ T α ) ∗ CS ( α + r ˙ S α ) that can be viewed this way. The fact that w p ( α ) can be represented as in (32) justifies that the requirement in FI-2 that p ( α ) ∈ H α + is legitimate.The verification that w p ( α ) can be represented as in (32) uses, additionally tothe above remarks, also the important fact that for any P α -term ˙ x such that (cid:13) P α “ ˙ x is a bounded subset of α + ” one can construct a canonical P α -term ˙ x ′ ∈ H α + such that (cid:13) P α ˙ x = ˙ x ′ . This fact follows from(33) If α ≤ κ is inaccessible then P α ⊆ V α , so in particular card ( P α ) = α which, in turn, is a consequence of the requirement in FI-2 that p ( α ) ∈ H α + . Allof this, along with the property(34) If α ≤ κ is regular then P α is α -c.c.,which also relies on the requirement in that p ( α ) ∈ H α + , is proved by standardinduction on α . From (34) we conclude:(35) If α ≤ κ is regular then P α preserves regularity of both α and α + .By (31), if α is inaccessible then the α -th step Q α of the iteration can be viewedas the composition NR ( α + , γ p ( α )) ∗ CS ( α + r ˙ S α ) ∗ CS ( α + r ˙ T α )By Proposition 5.4, the composition NR ( α + , γ p ( α )) ∗ CS ( α + r ˙ S α ) has an α + -closed dense subset. Since the set T α concentrates on points of cofinality α , byProposition 5.2(b), the poset CS ( α + r T α ) is α -closed in any generic extension via NR ( α + , γ p ( α )) ∗ CS ( α + r ˙ S α ). It follows that(36) (cid:13) P α Q α has an α -closed dense subset.Now given an ordinal δ < κ , the poset P δκ is the top part of the iteration P κ above δ , that is, we only apply Q α at inaccessibles ≥ δ . More technically, the poset P δκ isdefined using FI-1 – FI-3 above with the only amendment that in FI-1 we requiredom( p ) ∩ δ = ∅ . Clearly all of the above preservation properties can be establishedfor P δκ in place of P κ , which together with (36) allows to conclude:(37) P δκ has an α -closed dense subset where α is the least inaccessible ≥ δ .The above closure property will be important in establishing properties of the ideal J γ among other things; for the moment we just record a preservation property of P κ . For regular δ , using the Factor Lemma we have P κ ≃ P δ ∗ ˙ P δκ . This along with(37) and (35) allows us to conclude:(38) P κ preserves all cardinals and cofinalities ≤ κ ,hence all cardinals and cofinalities.Now define a poset P ∗ as follows(39) P ∗ = P κ ∗ NR ( κ + ) ∗ CS ( κ + r ˙ T )where T = T κ ( S ) and S is the generic non-reflecting stationary added via NR ( κ + ).We claim that any generic extension via P ∗ produces a model as in Theorem 1.3.We will first focus on the proof of the following proposition. Proposition 5.5.
In any generic extension via P ∗ all cardinals and cofinalities arepreserved, κ remains inaccessible, and for each regular γ ≤ κ there is a uniformnormal ( κ + , ∞ ) -distributive ideal J γ such that P ( κ ) / J γ has a dense γ -closed set,but no dense γ + -closed set. AMES WITH FILTERS 27 By GCH in V , any generic extension via P κ satisfies 2 κ = κ + , so in any suchgeneric extension the poset NR ( κ + ) has cardinality κ + . Using the strategic closureof NR ( κ + ) we conclude that 2 κ = κ + in the generic extension via P κ ∗ NR ( κ + ), so CS ( κ + r T ) has cardinality κ + in any such generic extension. All of this combinedwith the distributivity properties of NR ( κ + ) and CS ( κ + r T ), yields(40) P ∗ preserves all cardinals and cofinalities and also GCH .Here the conclusion on
GCH below κ follows by a reasoning similar to that for theconclusion 2 κ = κ + above. Preservation of GCH above κ is close to trivial.Now return to the map j from (29). Let G be ( P κ , V )-generic and G ′ = G ′ ∗ G ′ be( NR ( κ + ) ∗ CS ( κ + r T, V [ G ])-generic. (Here T, S are as in the previous paragraph,so in particular T = T κ ( S ).) Fix a γ ∈ R κ (see FI-2). Notice that the map π : NR ( κ + ) → NR ( κ + , γ ) defined by dom( π ( s )) = dom( s ) and π ( s )( ξ ) = (cid:26) s ( ξ ) if cf ( ξ ) = γ cf ( ξ ) = γ is a surjective projection, so G κ, = π [ G ′ ] is a ( NR ( κ + , γ ) , V [ G ])-generic filter and S κ def = the subset of κ + ∩ cof ( γ ) with characteristic function S G κ, is the generic non-reflecting stationary set added by forcing with NR ( κ + , γ ). Clearly S κ = S ∩ cof ( γ ). By (33) and (34), M [ G ] is closed under κ -sequences in V [ G ] andmodels M [ G ] , V [ G ] agree on what H κ + is. It follows that models M [ G ] , V [ G ] agreeon what NR ( κ + , γ ) is. Since NR ( κ + ) is ( κ + , ∞ )-distributive,(41) M [ G, G κ, ] is closed in V [ G, G ′ ] under κ -sequences.To see this note that any f : κ → On which is in V [ G, G ′ ] must be in V [ G ],and therefore by M [ G ] by the κ -closure property of M [ G ] discussed above. Inparticular, M [ G, G κ, ] and V [ G, G ′ ] agree on what H κ + and CS ( κ + r T κ ( S κ )) are.Let C ∈ V [ G, G ′ ] be a closed unbounded subset of κ + r T added by CS ( κ + r T )over V [ G ]. Notice that T κ ( S κ ) ∈ M [ G, G κ, ] and C ∩ T κ ( S κ ) = ∅ . From the pointof view of V [ G, G ′ ] there are only κ + many dense subsets of CS ( κ + r T κ ) whichare in M [ G, G κ, ] so we can construct a ( CS ( κ + , T κ ( S κ )) , M [ G, G κ, ])-generic filter G κ, ∈ V [ G, G ′ ] as follows. In V [ G, G ′ ] fix an enumeration h D β | β < κ + i of densesubsets of CS ( κ + r T κ ( S κ )) which are in M [ G, G κ, ]. Now recursively on β < κ + construct a descending chain h c β , c ′ β | β < κ + i in CS ( κ + r T κ ( S κ )) as follows. • Let c ′ = ∅ . • Given c ′ β , pick c β ∈ D β such that c β ≤ c ′ β in CS ( κ + r T κ ( S κ )). • Given c β , let c ′ β +1 = c β ∪{ γ β +1 } where γ β +1 is the least element of C largerthan max( c β ). • If β is a limit let c ′ β = (cid:16)S ¯ β<β c ′ ¯ β (cid:17) ∪{ γ β } where γ β = sup { max( c ′ ¯ β ) | ¯ β < β } .To see that this works, notice that for every β < κ + both c β and c ′ β are elementsof M [ G, G κ, ], which is verified inductively on β . The only non-trivial step in theinduction is to see that c ′ β ∈ M [ G, G κ, ] for β limit, but this follows from the κ -closure property of M [ G, G κ, ] verified above. That each c β is in CS ( κ + r T κ ( S κ )) istrivial, and that that each c ′ β is in CS ( κ + r T κ ( S κ )) follows from the fact that γ β ∈ C for all β < κ + . This last fact is trivial for successor β , relying on the unboundednessof C in κ + . That γ β ∈ C for limit β follows from the fact that C is closed. Now let G κ, be the filter on CS ( κ + r T κ ( S κ )) generated by the sequence h c β | β < κ + i ;it is clear that G κ, ∈ V [ G, G ′ ] and is ( CS ( κ + r T κ ( S κ )) , M [ G, G κ, ])-generic.Consider a ( CS ( κ + r S κ ) , V [ G, G ′ ])-generic filter H . Letting G κ = G κ, ∗ G κ, ,the filter G κ ∗ H is ( NR ( κ + , γ ) ∗ CS ( κ + r ˙ T κ ( S κ )) ∗ CS ( κ + r ˙ S κ ) , M [ G ])-generic. Itfollows that G ∗ G κ ∗ H is ( j ( P κ ) ↾ ( κ +1) , M )-generic. By the Factor Lemma appliedinside M [ G, G κ , H ], the quotient j ( P κ ) /G ∗ G κ ∗ H is isomorphic to the iteration P κ +1 j ( κ ) as calculated in M [ G, G κ , H ]; recall the definition of P δκ in the paragraphimmediately above (37). Let µ be the least inaccessible of M above κ . Using (37),we conclude that M [ G, G κ , H ] satisfies the following:(42) j ( P κ ) /G ∗ G κ ∗ H has a dense µ -closed subset.Since NR ( κ + ) ∗ CS ( κ + r ˙ T ) ∗ CS ( κ + r ˙ S κ ) is κ + -distributive in V [ G ],(43) M [ G, G κ , H ] is closed under κ -sequences in V [ G, G ′ , H ].This can be proved the same way we proved (41). Now working in V [ G, G ′ , H ],since the cardinality of H M [ G,G κ ,H ] j ( κ + ) is κ + , we have an enumeration h D β | β < κ + i of all dense subsets of j ( P κ ) /G ∗ G κ ∗ H which are in M [ G, G κ , H ]. Then, using(37), (42) and the fact that µ > κ + , we can construct a descending sequence h p β | β < κ + i each proper initial segment of which is an element of M [ G, G κ , H ]and such that p β ∈ D β for all β < κ + . Letting K be the filter on j ( P κ ) /G ∗ G κ ∗ H generated by this sequence, K is ( j ( P κ ) /G ∗ G κ ∗ H, M [ G, G κ , H ])-generic and K ∈ V [ G, G ′ , H ]. Since K can be viewed as a ( j ( P κ ) , M )-generic filter, we canextend j to an elementary map j H,K : V [ G ] → M [ G, G κ , H, K ] defined in the usualway by j H,K ( ˙ x G ) = j ( ˙ x ) K whenever ˙ x ∈ V is a P κ -term; recall that j H,K ( G ) = K .Since K can always be constructed inside V [ G, G ′ , H ], there is a CS ( κ + r S )-term˙ K ∈ V [ G, G ′ ] such that ˙ K H is ( j ( P κ ) /G ∗ G κ ∗ H, M [ G, G κ , H ])-generic whenever H is ( CS ( κ + r S ) , V [ G, G ′ ])-generic. We then let j H be as follows.(44) j H = j H, ˙ K H : V [ G ] → M [ ˙ K H ] . Additionally, we have a CS ( κ + r S )-term ˙ U ∈ V [ G, G ′ ] such that ˙ U H is the normal V [ G ]-measure over κ derived from j H , that is,(45) ˙ U H = { x ∈ P ( κ ) V [ G ] | x ∈ j H ( κ ) } whenever H is a ( CS ( κ + r S ) , V [ G, G ′ ])-generic filter. It is a standard fact that(46) M [ ˙ K H ] = Ult ( V [ G ] , ˙ U H ) and j H : V [ G ] → M [ ˙ K H ]is the associated ultrapower map.Since the composition NR ( κ + ) ∗ CS ( κ + r ˙ T ) ∗ CS ( κ + r ˙ S κ ) is κ + -distributive in V [ G ], the models V [ G ] and V [ G, G ′ ] agree on what P ( κ ) is, so ˙ U H is also a normal V [ G, G ′ , H ]-measure over κ . Since ˙ U H ∈ V [ G, G ′ , H ] we record that(47) κ is measurable in V [ G, G ′ , H ].Although we will not need this conclusion in the proofs of our main theorems, itmay be of interest to see that measurability can be introduced by adding a closedunbounded set through the complement of a non-reflecting stationary set. This isthe case, as we will prove below that κ is not measurable in V [ G, G ′ ]. AMES WITH FILTERS 29
We now define the ideal J γ on P ( κ ) in V [ G, G ′ ] in the standard way to consist ofall sets x ∈ P ( κ ) which will never make it into ˙ U H for any ( CS ( κ + r S κ ) , V [ G, G ])-generic filter H . In other words, for every x ∈ P ( κ ) V [ G,G ′ ] ,(48) x ∈ J γ ⇐⇒ (cid:13) V [ G,G ′ ] CS ( κ + r S κ ) ˇ x / ∈ ˙ U , which also shows that J γ ∈ V [ G, G ′ ]. Standard arguments show that J γ is auniform normal ideal on P ( κ ) in V [ G, G ′ ]. Here recall that S κ ⊆ κ + ∩ cof ( γ ) where γ was fixed at the beginning; this will be crucial in determining closure properties of P ( κ ) / J γ . The main tool for analyzing properties of J γ is the following proposition. Proposition 5.6. In V [ G, G ′ ] there is a dense embedding e : CS ( κ + r S κ ) → P ( κ ) / J γ . Proof. In V , fix an assignment x f x where x ∈ M and f x : κ → V is such that(49) x = [ f x ] U = j ( f x )( κ ) . The poset CS ( κ + r S κ ) in the generic extension M [ G, G κ ] can be viewed as thequotient ( j ( P κ ) ↾ ( κ + 1)) /G ∗ G κ , so we can consider conditions CS ( κ + r S κ ) aselements of M ordered the same way as conditions in j ( P κ ). Hence each suchcondition p is represented in the ultrapower by U by the function f p .Next, recall that at each inaccessible α < κ , the α -th step Q α of the iteration P κ is a composition of three posets where the last one is CS ( α + r S α ); see FI-2.The α -th component of the generic filter G is then of the form G α, ∗ G α, ∗ h ( α )where h ( α ) is ( CS ( α + r S α ) , V [ G ↾ α ∗ G α, ∗ G α, ])-generic. The function h is thusan element of V [ G ] and represents the filter H in the ultrapower by ˙ U H , that is, H = j H ( h )( κ ); see (46).Then for any p ∈ CS ( κ + r S κ ) we have the following:(50) p ∈ H ⇐⇒ j H ( f p )( κ ) ∈ j H ( h )( κ ) ⇐⇒ a p def = { α < κ | f p ( α ) ∈ h ( α ) } ∈ ˙ U H . We show that in V [ G, G ′ ], the map e : CS ( κ + r S κ ) → P ( κ ) / J γ defined by(51) e ( p ) = [ a p ] J γ is a dense embedding. The proof is a standard variant of the duality argument,which we include for the reader’s convenience. We write briefly [ a ] for [ a ] J γ .To see that e is order-preserving, consider p ≤ q in CS ( κ + r S κ ). By theabove remarks on the ordering of the quotient, we have p ≤ q in j ( P κ ), hence j ( f p )( κ ) ≤ j ( f q )( κ ) in j ( P κ ). It follows that b p,q def = { ξ < κ | f p ( ξ ) ≤ f q ( ξ ) } ∈ U, and so b p,q ∈ ˙ U H whenever H is a ( CS ( κ + r S κ ) , V [ G, G ′ ])-generic filter. It followsthat κ r b p,q ∈ J γ . Since a p r a q ⊆ κ r b p,q , we have [ a p ] ≤ J γ [ a q ].To see that the map e is incompatibility preserving, we prove the contraposition.Assume p, q ∈ CS ( κ + r S κ ) are such that a p ∩ a q ∈ J + γ . It follows that there issome ( CS ( κ + r S κ ) , V [ G, G ′ ])-generic H such that a p ∩ a q ∈ ˙ U H . Then a p ∈ ˙ U H and a q ∈ ˙ U H . Using (50) we conclude that p, q ∈ H . Hence p, q are compatible.To see that e is dense, assume that a ∈ J + γ . It follows that there is some( CS ( κ + r S κ ) , V [ G, G ′ ])-generic filter H such that a ∈ ˙ U H . So there is some p ∈ H such that(52) p (cid:13) V [ G,G ′ ] CS ( κ + r S κ ) ˇ a ∈ ˙ U .
Now for every ( CS ( κ + r S κ ) , V [ G, G ′ ])-generic filter H we have a p ∈ ˙ U H = ⇒ p ∈ H = ⇒ a ∈ ˙ U H . Here the first implication follows from (50) and the second implication from (52).We thus conclude that a p r a / ∈ ˙ U H whenever H is a ( CS ( κ + r S κ ) , V [ G, G ′ ])-genericfilter, which means that a p r a ∈ J γ , or equivalently, [ a p ] ≤ J γ [ a ]. ⊣ We can now complete the proof of Proposition 5.5. By Proposition 5.6, this boilsdown to look at properties of the poset CS ( κ + r S κ ) in V [ G, G ′ ]. The ( κ + , ∞ )-distributivity follows from Proposition 5.2(a). The existence of a dense γ -closed setas well as the non-existence of a dense γ + -closed set follows from Proposition 5.2(b)and the fact that S κ ⊆ κ + ∩ cof ( γ ). The latter additionally relies on the fact that S κ is stationary in V [ G, G ′ ], which follows from the discussion in the paragraph im-mediately above (41) and Proposition 5.2(d). The point here is that a poset with adense γ + -closed set cannot destroy the stationarity of a stationary set concentratingon ordinals of cofinality γ .The last major step toward the proof of Theorem 1.3 is the following proposition. Proposition 5.7. κ does not carry a saturated ideal in a generic extension via P ∗ .Proof. Assume for a contradiction that κ does carry a saturated ideal in V [ G, G ′ ]where G, G ′ are as above. Denote this ideal by I , and let L be a ( P I , V [ G, G ′ ])-generic filter where P I is the poset ( I + , ⊆ ). Let further j ′ : V [ G, G ′ ] → N bethe generic embedding associated with the ultrapower Ult ( V [ G, G ′ ] , L ). Letting M ′ = j ′ ( M ) and ( K, K ′ ) = j ′ ( G, G ′ ), we have N = M ′ [ K, K ′ ].Now look at the κ -th step of the iteration j ′ ( P κ ). Obviously j ′ ( P κ ) ↾ κ = P κ and K ∩ P κ = G . Let γ ∈ R M ′ κ = R κ (See FI-2) be chosen by the filter K . The κ -thstep is thus forcing with NR ( κ + , γ ) ∗ CS ( κ + r ˙ T ( S κ )) ∗ CS ( κ + r ˙ S κ )over M [ G ]. The κ -th component K κ of K has the form K κ, ∗ K κ, ∗ K κ, . Theset S κ is the generic non-reflecting stationary subset of κ + ∩ cof ( γ ) added by K κ, over M [ G ], that is, the characteristic function of S κ is the union S K κ, . Since S K κ, ∈ M [ K ] ⊆ V [ G, G ′ , L ] is a closed unbounded subset of κ + disjoint from S κ ,the set S κ is non-stationary in V [ G, G ′ , L ].By (D) at the beginning of this section, inside M ′ [ K ], the generic filter K κ, codes the set S κ as follows. Given an ordinal ξ ∈ κ + ∩ cof ( γ ), ξ ∈ S κ ⇐⇒ T κ, ξ +1 is stationary and T κ, ξ is non-stationary.Recall that the filter G ′ κ, codes the set S in V [ G, G ′ ] the same way, that is, ξ ∈ S ⇐⇒ T κ, ξ +1 is stationary and T κ, ξ is non-stationary.It follows that for every ξ ∈ κ + ∩ cof ( γ ), ξ ∈ S κ = ⇒ T κ, ξ is non-stationary in M ′ [ K ]= ⇒ T κ, ξ is non-stationary in V [ G, G ′ , L ]= ⇒ T κ, ξ is non-stationary and T κ, ξ +1 is stationary in V [ G, G ′ , L ]= ⇒ ξ ∈ S Here the third implication follows from the fact that in V [ G, G ′ ], exactly one of T κ, ξ , T κ, ξ +1 is stationary whenever ξ < κ + . If T κ, ξ were stationary in V [ G, G ′ ] AMES WITH FILTERS 31 then it would remain stationary in V [ G, G ′ , L ] as P I is κ + -c.c. Hence T κ, ξ +1 isstationary in V [ G, G ′ ], and, again by the κ + -c.c. of P I , it remains stationary in V [ G, G ′ , L ]. Similarly we verify the implication ξ / ∈ S κ = ⇒ ξ / ∈ S . Alltogetherwe then conclude that S κ = S ∩ cof ( γ ). But then, by Proposition 5.3(d), S κ isstationary in V [ G, G ′ ]. Then, again by the κ + -c.c. of P I , S κ remains stationary in V [ G, G ′ , L ], a contradiction. ⊣ At this point we give a proof of Theorem 1.3(b) in that we construct the strategy S γ in the generic extension V [ G, G ′ ] via P ∗ using ideal J γ defined in (48). Thisstrategy will be a slight variation on strategy ¯ S γ constructed in the proof of Theo-rem 1.4 at the end of Section 4. Essentially the same proof will then show that S γ isa winning strategy in G Wγ . We show that S γ is not included in any winning strategy S ′ for Player II in the game G Wγ + by constructing a run of length γ according to S γ to which no strategy S ′ ⊇ S γ for Player II in G γ + is able to respond. The mainpoint here is that there is a dense embedding e : CS ( κ + r S κ ) → P ( κ ) / J γ where S κ is the non-reflecting stationary subset of κ + ∩ cof ( γ ) in (48). We define S γ in H θ using < θ , similarly as in the case of ¯ S γ . The use of conditions X ∈ P in theproof of Theorem 1.4 will be replaced with the use of conditions of the form e ( c ) for c ∈ CS ( κ + r S κ ). The strategy S γ will make a reference to models N α introducedat the beginning of Section 3. Among other things, this will give Player I somecontrol of ordinals of the form max( c ) where c ∈ CS ( κ + r S κ ). The strategy is thendefined inductively as follows. Given a run¯ r = ( A i , U i | i < ξ )according to S γ , a descending sequence h c i | i < ξ i of conditions in CS ( κ + r S κ )which constitute auxiliary moves by Player II and a legal move A ξ of Player Ifollowing ¯ r , let α ( ξ ) = the least α < κ + such that A ξ ⊆ N α and c ξ = the < θ -least condition c ∈ CS ( κ + r S κ ) such that c ≤ c i for all i < ξ , max( c ) ≥ α ( ξ ) and e ( c ) decides the value ˙ G ∩ A ξ .We then set S γ (¯ r a hA ξ i ) = the unique U such that e ( c ξ ) (cid:13) P J γ ˙ G ∩ ˇ A ξ = ˇ U . Proposition 5.8.
Let J γ be the ideal in V [ G, G ′ ] constructed in (48) and S γ bethe winning strategy for Player II in G Wγ defined above. Then S γ is not included inthe winning strategy S ′ for Player II in the game G Wγ + by the same definition. Infact, there is a run r = ( A i , U i | i < γ ) of G Wγ of length γ according to S γ such that S ′ for Player II in G γ + is not able torespond to r a hA γ i whenever A γ is a legal move of Player I in G Wγ + following r .Proof. To construct r , we will work in H θ , similarly as in the proof of Theorem 1.4.Let Z be a an elementary substructure of H θ of cardinality κ such that <γ Z ⊆ Z , δ def = sup( Z ∩ κ + ) ∈ S κ , and all objects of interest are elements of Z . In particular, κ, S κ , J γ , e , the sequence( N α | α < κ + ) and S γ are elements of Z . The existence of such a Z is guaranteedby GCH . Finally pick a normal sequence h δ i | i < γ i cofinal in δ contained in Z .We will make use of the following fact.In Z we now build the run r inductively as follows. Assume¯ r = ( A i , U i | i < ξ )is the run constructed so far where ξ < γ . This run is accompanied by a descendingsequence of auxiliary moves ( c i | i < ξ ) in the poset CS ( κ + r S κ ) such thatmax( c i ) ≥ δ i and e ( c i ) (cid:13) P J γ ˙ G ∩ ˇ A i = ˇ U i whenever i < ξ .At round ξ , let Player I plays A ξ ∈ Z such that A ξ N δ ξ . Since Z is closedunder < γ -sequences and all A i , U i and c i for i < ξ are elements of Z , so is thesequence ( A i , U i , c i | i < ξ ) a hA ξ i . And, since S γ ∈ Z , the response U ξ alongwith the auxiliary move c ξ of S γ to this sequence are also in Z . In particular, δ ≥ max( c ξ ) ≥ α ( ξ ) ≥ δ ξ . This completes the construction of r .Now let c = S i<γ c i . Then sup( c ) = δ ∈ S κ , so the descending chain ( c i | i < γ )does not have a lower bound in CS ( κ + r S κ ). Assume S ′ is as in the statementof the proposition. If S ′ ( r a hAi ) were defined for some legal move A of Player Ifollowing r , then it would provide a lower bound on the sequence ( c i | i < γ ) in CS ( κ + r S κ ). This is a contradiction. ⊣ Finally we give a proof of Theorem 1.5.
Proof of Theorem 1.5.
The forcing poset P κ is in this case defined similarly as inthe proof of Theorem 1.3(a), with the only difference that the ordinals γ p are notchosen generically, but are equal to γ . Thus, if p ∈ P κ then for each inaccessible α ∈ dom( p ), the component p ( α ) is a P α -term for a condition in Q α = NR ( α + , γ ) ∗ CS ( α + r ˙ T α ) ∗ CS ( ˙ S α )where where ˙ S α is the canonical NR ( α + , γ )-term for the generically added non-reflecting stationary set S α and ˙ T α is the canonical NR (¯ α + , γ )-term for T α ( S α ); see(30). The poset P ∗ is then defined by P ∗ = P κ ∗ NR ( κ + , γ ) ∗ CS ( κ + r ˙ T κ )where S κ , ˙ S κ , T κ , ˙ T κ are defined analogously as S α , ˙ S α , T α , ˙ T α above. Letting G ∗ G ′ be an ( P ∗ , V )-generic for P ∗ , this time we can take G κ = G ′ as the κ -th step inthe iteration consists of P κ -terms for Q α . Then j ( P κ ) ↾ ( κ + 1) /G ∗ G ′ is forcingequivalent to CS ( κ + r S κ ), so in V [ G, G ′ ] we can define the ideal J γ as in the proofof Theorem 1.3 and construct a dense embedding e : CS ( κ + r S κ ) → P ( κ ) / J γ asin that proof. Using the ideal J γ we then define the strategy S γ for Player II in G Wγ as in the proof of Theorem 1.4 and using the same argument prove that thatit is a winning strategy. The same proof as before also yields that in V [ G, G ′ ], κ does not carry a saturated ideal. The point here is that if j ′ : V [ G, G ] → N is ageneric embedding coming from a precipitous ideal on κ then N = M ′ [ K, K ′ ] forsome generic filters K, K ′ where M ′ = j ′ ( V ) and K is ( j ′ ( P κ ) , M ′ )-generic. The κ -th step in the iteration j ′ ( P κ ) is again Q α , as the models V and M ′ agree onwhat H κ + is. As before, using the coding system ( T κ,ξ | ξ < κ + ) we show thatthe non-reflecting stationary set S κ added by ( NR ( κ + , γ ) , V [ G ])-generic filter G ′ is the same as as the non-reflecting set added by ( NR ( κ, γ ) , M ′ [ G ])-generic K κ, .As before we then get a contradiction by arguing that S κ must be nonstationary in V [ G, G ′ ], as the κ -th component of K adds a club subset throught the complementof S κ on the one side, and stationary in V [ G, G ′ ], as κ -c.c. posets preserve thestationarity of stationary subsets of κ + on the other side.We now show that in the model V [ G, G ′ ] there is no winning strategy for Player IIin G Wγ + . Assume for a contradiction there is a winning strategy S for Player II in G ∗ γ + . By Theorem 1.2, there is a precipitous ideal I over κ such that the poset P I = ( I + , ⊆ ) contains a dense γ + -closed subset D such that ⊆ I restricted to D is a tree ordering. Actually, we will not need the last part on tree ordering.Let j ∗ : V [ G, G ′ ] → N = M ∗ [ K, K ′ ] be a generic embedding coming from some( P I , V [ G, G ′ ])-generic G ∗ . Similarly as above, M ∗ = j ∗ ( V ) an K is ( j ∗ ( P κ ) , M [ K ])-generic. Now we proceed exactly as above with j ′ and show that the generic non-reflecting stationary set S κ added by G ′ is the very set generically added at the κ -thstep of j ∗ ( P κ ) by K κ . This step did not make use of saturation of the ideal in theabove argument. Also, as before we argue that the third component of K κ makes S κ non-stationary, so S κ must be non-stationary also in V [ G, G ′ , G ∗ ]. Now the factthat P I has a γ + -closed dense subset implies that P I preserves the stationarity ofstationary sets concentrating on ordinals of cofinality γ (even though κ + may becollapsed in V [ G, G ′ , G ∗ ]). This yields the desired contradiction. ⊣ An Open Problem
In this section we raise a question we don’t know the answer to.
An Ulam Game
Consider the following variant of the cut-and-choose game oflength ω derived from games introduced by Ulam in [17] (see [10]). I ( A , A ) ( A , A ) . . . ( A n , A n ) ( A n +10 , A n +11 ) . . .II B B . . . B n B n +1 . . .At stage 0 Player I plays a partition ( A , A ) of κ . At stage n ≥ B n be either A n or A n , and plays B n . At stage n ≥ A n +10 , A n +11 ) of B n . The winning condition for Player II is that | T n ∈ ω B n | ≥ Observation:
If Player II has a winning strategy in the game G ∗ ω , thenPlayer II has a winning strategy in the Ulam game.This is immediate: Player II follows her strategy in the Auxiliary game against theBoolean Algebras A n generated by { A i , A i : i ≤ n } . In the game G ∗ ω she thenplays as B n whichever of A n or A n belongs to µ n . By the winning condition on G ∗ ω , T n ∈ ω B n belongs to a κ -complete, uniform filter. Hence | T n B n | = κ > G ∗ ω ” is that of a measurable cardinal.What is unclear is the relationship between the Ulam Game and the WelchGame. For example, the following is open: Velickovic [18] calls these Mycielski games
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Department of Mathematics, University of California at Irvine, Irvine, CA 92697-3875, USA
E-mail address : [email protected] Institute of Mathematics, Hebrew university of Jerusalem, Jerusalem 91904, Israel
E-mail address : [email protected] Department of Mathematics, University of California at Irvine, Irvine, CA 92697-3875, USA
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