aa r X i v : . [ m a t h . L O ] M a r The stationary set splitting game ∗ Paul B. Larson Saharon ShelahNovember 6, 2018
Abstract
The stationary set splitting game is a game of perfect information oflength ω between two players, unsplit and split , in which unsplit choosesstationarily many countable ordinals and split tries to continuously dividethem into two stationary pieces. We show that it is possible in ZFC toforce a winning strategy for either player, or for neither. This gives a newcounterexample to Σ maximality with a predicate for the nonstationaryideal on ω , and an example of a consistently undetermined game of length ω with payoff definable in the second-order monadic logic of order. Wealso show that the determinacy of the game is consistent with Martin’sAxiom but not Martin’s Maximum. MSC2000: 03E35; 03E60The stationary set splitting game ( SG ) is a game of perfect information oflength ω between two players, unsplit and split . In each round α , unsplit eitheraccepts or rejects α . If unsplit accepts α , then split puts α into one of two sets A and B . If unsplit rejects α then split does nothing. After all ω many roundshave been played, split wins if unsplit has not accepted stationarily often, or ifboth of A and B are stationary.In this note we prove that it is possible to force a winning strategy for eitherplayer in SG , or for neither, and we also show that the determinacy of SG isconsistent with Martin’s Axiom but not Martin’s Maximum [4]. We also presenttwo guessing principles, C s ( club for split ) and D u ( diamond for unsplit ), whichimply the existence of winning strategies for split and unsplit , respectively (andare therefore incompatible; see Theorems 1.5 and 1.8). These principles may beof independent interest. ∗ The work in this paper began during the Set Theory and Analysis program at the FieldsInstitute in the Fall of 2002. The first author is supported in part by NSF grant DMS-0401603,and thanks Juris Stepr¯ans, Paul Szeptycki and Tetsuya Ishiu for helpful conversations on thistopic. The research of the second author is supported by the United States-Israel BinationalScience Foundation. This is the second author’s publication 902. Some of the research in thispaper was conducted during a visit by the first author to Rutgers University, supported byNSF grant DMS-0600940. Winning strategies split
A collection X of countable sets is stationary if for every function F : [ S X ] <ω → S X there is an element of X closed under F . A set X of countable sets is projective stationary [2] if for every stationary S ⊂ ω the set of X ∈ X with X ∩ ω ∈ S is stationary. We note that a partial order P is said to be proper ifforcing with P preserves the stationarity (in the sense above) of stationary setsfrom the ground model (see [11]).The following statement holds in fine structural models such as L . It is astrengthening of the principle (+) used in [8]. Justin Moore has pointed out tous that his Mapping Reflection Principle [9] implies the failure of (+). We notealso that in the statement of (+), “projective stationary” can be replaced with“club” without strengthening the statement. We do not know if that is the casefor C +. Let C + be the statement that there exists a projective station-ary set X consisting of countable elementary substructures of H ( ℵ ) such thatfor all X , Y in X with X ∩ ω = Y ∩ ω , either every for every club C ⊂ ω in X there is a club D ⊂ ω in Y with D ∩ X ⊂ C ∩ X , or for every for every club D ⊂ ω in Y there is a club C ⊂ ω in X with C ∩ X ⊂ D ∩ X .Given a partial run of SG of length α , we let E α be the set of β < α acceptedby unsplit , and we let A α , B α be the partition of E α chosen by split . Theorem 1.2. If C + holds then split has a winning strategy in SG .Proof. Let X be a set of countable elementary submodels of H ( ℵ ) witnessing C +, and for each α < ω let X α be the set of X ∈ X with X ∩ ω = α . Let Z bethe set of α < ω such that X α is nonempty (since X is projective stationary,this set contains a club).Play for split as follows. In round α ∈ Z , if unsplit accepts α , let Y α be theset of all X ∈ X α such that X contains a stationary subset of ω , E X , such that E X ∩ α = E α . If Y α = ∅ , put α ∈ A α +1 . Otherwise, since every club subsetof ω in every member of Y α intersects E α , there cannot be two club subsets of ω in S Y α , one disjoint from A α and one disjoint from B α , since some clubsubset of ω in S Y α would be contained in both of these clubs. If any memberof Y α contains a club subset of ω disjoint from A α , then put α in A α +1 , and ifany member of Y α contains a club subset of ω disjoint from B α , then put α in B α +1 . If neither case holds, put α ∈ A α +1 .Let E be the play by unsplit in a run of SG where split has played by thisstrategy, and let A and B be the corresponding play by split . Let C be a clubsubset of ω and supposing that E is stationary, fix X ∈ X containing E , A , B and C with X ∩ ω ∈ E ∩ C . Then if A ∩ C ∩ X ∩ ω = ∅ , then X ∩ ω ∈ A ∩ C ,and if B ∩ C ∩ X ∩ ω = ∅ , then X ∩ ω ∈ B ∩ C , which shows that C does notwitness that unsplit won this run of the game.2he following fact, in conjunction with Theorem 1.2, shows that Martin’sAxiom is consistent with the existence of a winning strategy for split . Theorem 1.3.
The statement C + is preserved by forcing with c.c.c. partialorders.Proof. Let P be a c.c.c. forcing and let X witness C +. Let γ be a regularcardinal greater than ℵ and 2 | P | . Let G ⊂ P be a V -generic filter, and let X [ G ] = { X [ G ] ∩ H ( ℵ ) V [ G ] : X ≺ H ( γ ) V , X ∩ H ( ℵ ) V ∈ X } . Since every club subset of ω in V [ G ] contains one in V , in order to show that X [ G ] witnesses C + in V [ G ], it suffices to show that X [ G ] is projective stationarythere. Fix a P -name ρ for a function from [ H ( ℵ ) V [ G ] ] <ω to H ( ℵ ) V [ G ] . Forany countable X ≺ H ( γ ) with X ∩ H ( ℵ ) ∈ X and ρ ∈ X , X [ G ] ∩ H ( ℵ ) V [ G ] isin X [ G ] and closed under the realization of ρ . Fix a P -name τ for a stationarysubset of ω and a condition p ∈ P . Let S be the set of countable ordinalsforced to be in τ by some condition below p . Then exist a countable X ≺ H ( γ )with X ∩ H ( ℵ ) ∈ X , X ∩ ω ∈ S and ρ ∈ X and a condition q below p forcingthat X [ ˙ G ] ∩ ω (where ˙ G is the name for the generic filter) is in the realizationof τ . By genericity, then, X [ G ] is projective stationary.We do not know how to force C +, however, and use a different principle toforce the existence of a winning strategy for split . Let C s be the statement that there exist c α ( α < ω limit)such that each c α is a sequence h a αβ : β < γ α i (for some countable γ α ) of cofinalsubsets of α of orderype ω and • for all limit α < ω and all β < β ′ < γ α , a αβ ′ \ a αβ is finite; • for every club C ⊂ ω and every stationary E ⊂ ω there exists an a αβ with α ∈ E such that a αβ \ C is finite and a αβ ∩ E is infinite.The principle C s also holds in fine structural models such as L . The winningstrategy for split given by C s is very similar to the one given by C +. Theorem 1.5. If C s holds then split has a winning strategy in SG .Proof. Let a αβ ( α < ω limit, β < γ α ) witness C s . Play for split as follows.In round α , α a limit, if unsplit has accepted α and if some a αβ intersects A α infinitely and B α finitely, then put α in B α +1 . If some a αβ intersects B α infinitelyand A α finitely, then put α in A α +1 . Since the a αβ ’s ( β < γ α ) are ⊂ -decreasingmod finite, both cases cannot occur. If neither case occurs, put α in A α +1 .Let E be the play by unsplit in a run of SG where split has played by thisstrategy, and let A and B be the corresponding play by split . Let C be a clubsubset of ω and supposing that E is stationary, fix a αβ with α ∈ E such that a αβ \ C is finite and a αβ ∩ E is infinite. Then if A ∩ a αβ is finite, then α ∈ A ∩ C ,and if B ∩ a αβ is finite, then α ∈ B ∩ C , which shows that C does not witnessthat unsplit won this run of the game. 3 partial order P is said to be strategically ω -closed if there exists a function f : P <ω → P ( P ) such that whenever h p i : i ≤ n i is a finite descending sequencein P , f ( h p i : i ≤ n i ) is a dense subset below p n and, whenever h p i : i < ω i is adescending sequence in P such that for each n there exists a j with p j ∈ f ( h p i : i ≤ n i ) , the sequence has a lower bound in P . It is easy to see that strategic ω -closureis equal to the property that for every countable X ≺ H ((2 | P | ) + ) and every( X, P )-generic filter g contained in X there is a condition in P extending g .Let us say that a set a captures a pair E, C if a \ C is finite and a ∩ E isinfinite. Given A ⊂ ω , let C ( A ) be the partial order which adds a club subsetof A by initial segments. We force C s by first adding a potential C s -sequence byinitial segments, and then iterating to kill off every counterexample.We refer the reader to [11] for background on countable support iterationsof proper forcing. Theorem 1.6.
Suppose that CH and ℵ = ℵ hold. Let ¯ P = h P η , Q ∼ η : η < ω i be a countable support iteration such that P is the partial order consisting ofsequences h c α : α < δ limit i , for some countable ordinal δ , such that each c α is asequence h a αβ : β < γ α i (for some countable ordinal γ α ) of cofinal subsets of α ofordertype ω , deceasing by mod-finite inclusion (and P is ordered by extension).Suppose that the remainder of ¯ P satisfies the following conditions. • For each nonzero η < ω there is a P η -name τ η for a subset of ω suchthat if ( τ η ) G η (where G η is the restriction of the generic filter to P η ) isstationary in the P η extension and there exists a club C ⊂ ω in thisextension such that no a αβ with α ∈ τ G η captures the pair τ G η , C , then Q ∼ η is C ( ω \ ( τ η ) G η ) (and otherwise, Q ∼ η is C ( ω ) ). • For every pair
E, C of subsets of ω in any P η -extension ( η < ω ) , if E isstationary in this extension and C is club and no a αβ with α ∈ E captures E, C , then there is a ρ ∈ [ η, ω ) such that if E is stationary in the P ρ extension, then Q ∼ ρ is C ( ω \ E ) .Then ¯ P is strategically ω -closed, and C s holds in the ¯ P -extension. Furthermore,in the ¯ P extension, ✸ ( S ) holds for every stationary S ⊂ ω .Proof. Let X be a countable elementary submodel of H ((2 | ¯ P | ) + ) with ¯ P ∈ X ,let g be an X -generic filter contained in ¯ P ∩ X . Let γ X ∩ ω be the ordertype of X ∩ ω , and for each β < γ X ∩ ω , let η β be the β th member of X ∩ ω . For each β < γ X ∩ ω , let a X ∩ ω β be a cofinal subset of X ∩ ω of ordertype ω such that,letting g η denote the restriction of g to P η , • for all β ′ < β < γ X ∩ ω , a X ∩ ω β \ a X ∩ ω β ′ is finite; • a αβ is eventually contained in every club subset of ω in X [ g η β ] and in-tersects infinitely every stationary subset of ω in every X [ g η β ′ ], β ′ ∈ [ β, γ X ∩ ω ). 4t remains to see that we can extend g to a condition whose first coordinate isgiven by adding c X ∩ ω = h a αβ : β < γ X ∩ ω i to the union of the first coordinatesof the elements of g , and whose η th coordinate, for each nonzero η ∈ X ∩ ω isthe condition given by the union of { X ∩ ω } and the set of realizations of the η th coordinates of the members of g . We do this by induction on η , letting g ′ η be our extended condition in P η .For each η ∈ ω ∩ X , there is a P η -name σ ∈ X for a club subset of ω such that if, in the P η -extension ( τ η ) G η is stationary and there exists a club C such that τ G η , C is not captured by any a αβ with α ∈ ( τ η ) G η , then σ G η is sucha C . However, the realizations of τ η and σ by g are captured by a X ∩ ω o.t. ( η ∩ ω ) , so g ′ η forces that τ G η , σ G η is captured by a X ∩ ω o.t. ( η ∩ ω ) . It follows that g ′ η forces thateither Q ∼ η is C ( ω ), or X ∩ ω is not in τ G η . In either case, the union of themembers of g ∩ Q ∼ η be can extended to a condition in Q ∼ η by adding { X ∩ ω } .To see that ✸ ( S ) holds for every stationary S ⊂ ω in the ¯ P extension, fixsuch an S in the P α extension for some α < ω . Since ¯ P is ( ω, ∞ ) distributive,there exists in this extension a set h e δβ : δ, β < ω i such that for every δ < ω and every x ⊂ δ there are uncountably many β such that e δβ = x . Then, letting T ∈ P ( ω ) V [ G α ] be the set such that the realization of Q ∼ α is C ( T ), Q ∼ α adds a ✸ sequence h b δ : δ ∈ S i defined by letting b δ be e δβ , where the β th element of T above β is the first element of the generic club for Q ∼ α above δ . To see that thisis a ✸ sequence, note that since S is stationary in the ¯ P extension, there arestationarily many elementary submodels X of any sufficiently large H ( θ ) V [ G ] inthis extension with X ∩ ω ∈ S . Then X ∩ ( G/G α ) is a ( X ∩ V [ G α ] , ¯ P /P α )-generic filter which can be extended to a condition in ¯ P /P α by adding X ∩ ω toeach coordinate, and extended again to make any element of T \ (( X ∩ ω ) + 1)the least element of the generic club for Q ∼ α above X ∩ ω . That h b β : β ∈ S i isa ✸ sequence then follows by genericity.Section 2 shows that proper forcing does not always preserve the existenceof a winning strategy for split . unsplit In this section we show that it is consistent for unsplit to have a winning strategyin SG . We do this via the following guessing principle. Let D u be the statement that there exists a diamond sequence h σ α : α < ω i such that for every E ⊂ ω there is a club C ⊂ ω such thateither ∀ α ∈ C (( E ∩ α = σ α ) ⇒ α ∈ E )or ∀ α ∈ C (( E ∩ α = σ α ) ⇒ α E ) . Theorem 1.8. If D u holds then unsplit has a winning strategy in SG . roof. Let h σ α : α < ω i witness D u . Play for unsplit by accepting α if andonly if σ α = A α . At the end of the game, the set of α such that σ α = A α isstationary, and there is a club C such that either for all α in C , if σ α = A α ,then α is in A , or for all α in C , if σ α = A α , then α is in B . In either case, split has lost.Our iteration to force D u employs the same strategy as the iteration for C s .We first force to add a ✸ -sequence h σ α : α < ω i by initial segments, and wethen iterate to make this sequence witness D u , iteratively forcing a club throughthe set of α < ω such that σ α = E ∩ α or α ∈ E for each E ⊂ ω such that thesets { α ∈ E | σ α = E ∩ α } and { α ∈ ω \ E | σ α = E ∩ α } are both stationary.More specifically, we have the following. Given a sequence Σ = h σ α : α < ω i such that each σ α is a subset of α , and given E ⊂ ω , let A (Σ , E ) be the set of α ∈ E such that σ α = E ∩ α , and let B (Σ , E ) be the set of α ∈ ω \ E such that σ α = E ∩ α . Theorem 1.9.
Suppose that CH + ℵ = ℵ holds, and let ¯ P be a countablesupport iteration h P α , Q ∼ α : α < ω i such that P is the partial order consistingof sequences h σ β : β < γ i , for some countable ordinal γ , such that each σ β isa subset of β , ordered by extension. Let Σ be the sequence added by P andsuppose that the remainder of ¯ P satisfies the following conditions. • Each Q ∼ α is either C ( ω ) or C ( ω \ B (Σ , E )) for some E ⊂ ω such that A (Σ , E ) and B (Σ , E ) are both stationary. • For every E ⊂ ω in any P α -extension ( α < ω ) there is a γ ∈ [ α, ω ) such that if A (Σ , E ) and B (Σ , E ) are both stationary in the P γ extension,then Q ∼ γ is C ( ω \ B (Σ , E )) .Then ¯ P is strategically ω -closed, and in the ¯ P -extension, D u holds. Further-more, in the ¯ P extension, ✸ ( S ) holds for every stationary S ⊂ ω .Proof. The iteration ¯ P is clearly strategically ω -closed, since for any countable X ≺ H ((2 | ¯ P | ) + ) and any ( X, ¯ P )-generic filter g contained in X , one can extend g to a condition by making σ X ∩ ω unequal to the realization by g of any namein X for a subset of ω , and adding X ∩ ω to all the clubs being added by the Q ∼ α ’s, α ∈ X ∩ ω . It is clear also that in the ¯ P -extension there is no E ⊂ ω such that A (Σ , E ) and B (Σ , E ) are both stationary.To see that at least one of A (Σ , E ) and B (Σ , E ) is stationary for each E ⊂ ω ,we first note the following. Claim Suppose that E ⊂ ω is a member of the P α extension, for some α < ω , and A (Σ , E ) is stationary in this extension. Then A (Σ , E ) remainsstationary in the ¯ P extension. Note that A (Σ , E ) has countable intersection with B (Σ , F ), for every F ⊂ ω . Fix X ≺ H (((2 | ¯ P | ) + ) V ) V [ G α ] (where G α is the restriction of the generic6lter G to P α ) with X ∩ ω ∈ A (Σ , E ) and A (Σ , E ) ∈ X . Then any ( X, ¯ P /P α )-generic filter contained in X can be extended to a condition by adding X ∩ ω to the clubs being added at every stage of ¯ P after the first.Similar reasoning shows the following two facts, which complete the proofthat Σ witnesses D u in the ¯ P extension. Claim Suppose that E ⊂ ω is a member of the P α extension, for some α < ω , and not a member of the P γ extension, for any γ < α . Then A (Σ , E ) ∪ B (Σ , E ) is stationary in the P α extension. To see Claim 2, let τ be a P α -name for a subset of ω which is forcedto be unequal to any such subset in any P γ extension, for any γ < α . Fix X ≺ H (((2 | ¯ P | ) + )) V with τ ∈ X . Let g be an ( X, P α )-generic filter, and notethat the realization of τ ↾ ( X ∩ ω ) by g is different from the realizations of ρ ↾ ( X ∩ ω ) by g for any P γ -name ρ ∈ X for a subset of ω , for any γ ∈ X ∩ α .It follows that adding the realization of τ ↾ ( X ∩ ω ) by g to the union of thefirst coordinate projection of g gives a condition in P forcing that X ∩ ω isnot in any Σ( B, ρ G γ ), for any for any P γ -name ρ ∈ X for a subset of ω , forany γ ∈ X ∩ α . Therefore, we can add X ∩ ω to the clubs being added in everyother stage of ¯ P in X ∩ α , and get a condition extending every condition in g . Claim Suppose that E ⊂ ω is a member of the P α extension, for some α < ω , and A (Σ , E ) is nonstationary in this extension. Then B (Σ , E ) remainsstationary in the ¯ P extension. This is similar to the previous claims, noting that every subsequent stageof ¯ P forces a club though the complement of a set with countable intersectionwith B (Σ , E ).The proof that ✸ ( S ) holds for every stationary S ⊂ ω in the ¯ P extensionis (literally) the same as in the proof of Theorem 1.6.Note that that the iterations ¯ P in Theorems 1.6 and 1.9 are strategically ω -closed. Σ maximality The statements that split and unsplit have winning strategies in SG are each Σ in a predicate for N S ω , and they are obviously not consistent with each other.Woodin (see [6]) has shown that if there is a proper class of measurable Woodincardinals, then there exists in a forcing extension a transitive class model of ZFCsatisfying all Σ sentences φ such that φ + CH can be forced over the groundmodel. The results here show that this result cannot be extended to include apredicate for N S ω . This was known already, in that ✸ ∗ (in the sense of [7]) and“the restriction of N S ω to some stationary set is ℵ dense” were both known tobe consistent with ✸ (the second of these is due to Woodin, uses large cardinalsand is unpublished, though a related proof, also due to Woodin, appears in [3]).Our example is simpler and doesn’t use large cardinals; it also gives (we believe,for the first time) a counterexample consisting of two sentences each consistentwith “ ✸ ( S ) holds for every stationary set S ⊂ ω .”7 .4 A determined variation There are many natural variations of SG . We show that one such variation isdetermined. Theorem 1.10.
Let G be the following game of length ω . In round α , player I puts α into one of two sets E and E , and player II puts α into one of twosets A and A . After all ω rounds have been played, II wins if one of thefollowing pairs of set are both stationary. • E ∩ A and E ∩ A • E ∩ A and E ∩ A Then II has a winning strategy in G .Proof. Let B , B , B and B be pairwise disjoint stationary subsets of ω . In round α , if α is in B ij , let II put α in A i if I put α in E and in A j otherwise. Then after all ω many rounds have been played, suppose that A i ∩ E is nonstationary. Then B i and B i are both contained in E modulo N S ω , which means that E ∩ A and E ∩ A are both stationary. Similarly,if A i ∩ E is nonstationary then B i and B i are both contained in E modulo N S ω , which means that E ∩ A and E ∩ A are both stationary. The axiom PFA +2 says that whenever P is a proper partial order, D α ( α < ω )are dense subsets of P and σ , σ are P -names for stationary subsets of ω ,there is a filter G ⊂ P such that G ∩ D α = ∅ for each α < ω , and such that { α < ω | ∃ p ∈ G p (cid:13) ˇ α ∈ σ i } is stationary for each i ∈ { , } . Theorems 1.6and 1.9 together show that PFA +2 implies the indeterminacy of SG . Further-more, a straightforward argument shows that the following statement impliesthe nonexistence of a winning strategy for unsplit in SG , where Add (1 , ω ) is thepartial order that adds a subset of ω by initial segments : for any pair σ , σ of Add (1 , ω )-names for stationary subsets of ω , there is a filter G ⊂ Add (1 , ω )realizing both σ and σ as stationary sets. This statement is trivially subsumedby PFA +2 , but also holds in the collapse of a sufficiently large cardinal to be ω , and thus is consistent with CH.The axiom Martin’s Maximum [4] says that whenever P is a partial ordersuch that forcing with P preserves stationary subsets of ω and D α ( α < ω )are dense subsets of P , there is a filter G ⊂ P such that G ∩ D α = ∅ for each α < ω . Theorem 2.1.
Martin’s Maximum implies that SG is undetermined.Proof. Fix a strategy Σ for unsplit in SG , and let E , A , and B be the resultof a generic run of SG where unsplit plays by Σ (the partial order consists ofcountable partial plays where unsplit plays by Σ, ordered by extension). If the8omplement of E has stationary intersection with every stationary subset of ω in the ground model, one can force to kill the stationarity of E in such away that the induced two step forcing preserves stationary subsets of ω andproduces a run of SG where unsplit plays by Σ and loses. If the complementof E does not have stationary intersection with some stationary F ⊂ ω in theground model, then there is a partial run of the game p and a name τ for a clubsuch that p forces that E will contain F ∩ τ G . Then there exists in the groundmodel a run of SG extending p in which unsplit plays by Σ and loses: split picksa pair of disjoint stationary subsets F , F of F , and plays so that • for every α < ω , some initial segment of the play forces some ordinalgreater than α to be in τ , • whenever unsplit accepts α ∈ F , split puts α in A if α ∈ F and puts α ∈ B if α ∈ F .Now fix a strategy Σ for split in SG , and generically add a regressive function f on ω by initial segments. Let E α = f − ( α ) and let A α , B α be the responsesgiven by Σ to a play of E α by unsplit . Note that each E α will be stationary.Suppose that there exist an α < ω and stationary sets S , T in the groundmodel such that ( S ∩ E α ) \ A α and ( T ∩ E α ) \ B α are both nonstationary. Thenthere is a condition p in our forcing (i.e., a regressive function on some countableordinal) such that p forces that ( S ∩ E α ) ⊂ A α and ( T ∩ E α ) ⊂ B α , modulononstationarity (and so in particular S and T have nonstationary intersection).Let τ be a name for a club disjoint from ( S ∩ E α ) \ A α and ( T ∩ E α ) \ B α .Extend p to a filter f (identified with the corresponding function) realizing τ as a club subset of ω , at successor stages extending to add a new element tothe realization of τ , and at limit stages (when for some β < ω , f ↾ β has beendecided and f ( β ) has not, and β is forced by f ↾ β to be a limit member of therealization of τ ) extending so that f ( β ) = α if and only if β ∈ S . Then the runof SG corresponding to f − ( α ) is winning for unsplit , since the correspondingset B α is nonstationary.If there exist no such α , S , T , there is a function h on ω such that each h ( α ) ∈ { A α , B α } and the forcing to shoot a club through the set of β such that f ( β ) = α ⇒ β ∈ h ( α ) preserves stationary subsets of the ground model. ThenMartin’s Maximum applied to the corresponding two step forcing produces arun of SG (the run for any f − ( α ) which is stationary) where split plays by Σand loses.Theorem 2.1 leads to the following question. Does the Proper Forcing Axiom imply that SG is not deter-mined?The following question is also interesting. The consistency of the ℵ -densityof N S ω (relative to the consistency of AD L ( R ) ) is shown in [13]. Does the ℵ -density of N S ω decide the determinacy of SG ?9 MLO games
The second-order Monadic Logic of Order (MLO) is an extension of first-orderlogic with logical constants =, ∈ and ⊂ and a binary symbol < as the onlynon-logical constant, allowing quantification over subsets of the domain. Everyordinal is a model for MLO, interpreting < as ∈ .Given an ordinal α , an MLO game of length α is determined by an MLOformula φ with two free variables for subsets of the domain. In such a game,two players each build a subset of α , and the winner is determined by whetherthese two sets satisfy the formula in α .B¨uchi and Landweber [1] proved the determinacy of all MLO games of length ω . Recently, Shomrat [12] extended this result to games of length less than ω ω ,and Rabinovich [10] extended it further to all MLO games of countable length.The stationary set splitting game is an example of an MLO game of length ω whose determinacy is independent of ZFC.We thank Assaf Rinot for pointing out to us the connection between SG andMLO games. References [1] J.R. B¨uchi, L.H. Landweber,
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Set Theory , The third millennium edition, revised and expanded.Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003[6] R. Ketchersid, P. Larson, J, Zapletal, Σ -maximality, homogeneity and thestationary tower , preprint[7] K. Kunen, Set Theory , North-Holland, 1980[8] P. Larson,
The canonical function game , Archive Math. Logic 44 (2005),
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The Church problem for countable ordinals , preprint, 20071011] S. Shelah,
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Uniformization problems in the monadic theory of countableordinals , manuscript, 2006[13] W.H. Woodin,
The axiom of determinacy, forcing axioms and the nonsta-tionary ideal , de Gruyter Series in Logic and its Applications 1, Walter deGruyter & Co., Berlin, 1999Department of Mathematics and Statistics, Miami University, Oxford, Ohio45056, USA; Email: [email protected]
The Hebrew University of Jerusalem, Einstein Institute of Mathematics,Edmond J. Safra Campus, Givat Ram, Jersalem 91904, IsraelDepartment of Mathematics, Hill Center - Busch Campus, Rutgers, The StateUniversity of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019,USA; Email: [email protected]@math.huji.ac.il