The stationary set splitting game^{∗}

Paul B. Larson  Saharon Shelah

August 17, 2007

Abstract

The stationary set splitting game is a game of perfect information of length ω_{1} between two players, unsplit and split, in which unsplit chooses stationarily many countable ordinals and split tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to Σ_{2}^{2} maximality with a predicate for the nonstationary ideal on ω_{1}, and an example of a consistently undetermined game of length ω_{1} with payoff definable in the second-order monadic logic of order. We also show that the determinacy of the game is consistent with Martin’s Axiom but not Martin’s Maximum.

MSC2000: 03E35; 03E60

The stationary set splitting game (SG) is a game of perfect information of length ω_{1} between two players, unsplit and split. In each round α, unsplit either accepts 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_{1} many rounds have been played, split wins if unsplit has not accepted stationarily often, or if both of A and B are stationary.

In this note we prove that it is possible to force a winning strategy for either player in SG, or for neither, and we also show that the determinacy of SG is consistent with Martin’s Axiom but not Martin’s Maximum [4]. We also present two guessing principles, C_{s} (club for split) and D_{u} (diamond for unsplit), which imply the existence of winning strategies for split and unsplit, respectively (and are therefore incompatible; see Theorems 1.5 and 1.8). These principles may be of independent interest.

∗The work in this paper began during the Set Theory and Analysis program at the Fields Institute in the Fall of 2002. The first author is supported in part by NSF grant DMS-0401603, and thanks Juris Steprāns, Paul Szeptycki and Tetsuya Ishiu for helpful conversations on this topic. The research of the second author is supported by the United States-Israel Binational Science Foundation. This is the second author’s publication 902. Some of the research in this paper was conducted during a visit by the first author to Rutgers University, supported by NSF grant DMS-0600940.

1 Winning strategies

1.1 Strategies for split

A collection X of countable sets is stationary if for every function F: [⋃ X]^{<ω} → ⋃ 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_{1} the set of X ∈ X with X ∩ ω_{1} ∈ S is stationary. We note that a partial order P is said to be proper if forcing with P preserves the stationarity (in the sense above) of stationary sets from the ground model (see [11]).

The following statement holds in fine structural models such as L. It is a strengthening of the principle (+) used in [8]. Justin Moore has pointed out to us that his Mapping Reflection Principle [9] implies the failure of (+). We note also that in the statement of (+), “projective stationary” can be replaced with “club” without strengthening the statement. We do not know if that is the case for C+.

1.1 Definition. Let C+ be the statement that there exists a projective stationary set X consisting of countable elementary substructures of H(ℵ_{2}) such that for all X, Y in X with X ∩ ω_{1} = Y ∩ ω_{1}, either every for every club C ⊂ ω_{1} in X there is a club D ⊂ ω_{1} in Y with D ∩ X ⊂ C ∩ X, or for every for every club D ⊂ ω_{1} in Y there is a club C ⊂ ω_{1} 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 accepted by 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_{2}) witnessing C+, and for each α < ω_{1} let X_{α} be the set of X ∈ X with X ∩ ω_{1} = α. Let Z be the set of α < ω_{1} 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 the set of all X ∈ X_{α} such that X contains a stationary subset of ω_{1}, E_{X}, such that E_{X} ∩ α = E_{α}. If Y_{α} = ∅, put α ∈ A_{α+1}. Otherwise, since every club subset of ω_{1} in every member of Y_{α} intersects E_{α}, there cannot be two club subsets of ω_{1} in ⋃ Y_{α} , one disjoint from A_{α} and one disjoint from B_{α}, since some club subset of ω_{1} in ⋃ Y_{α} would be contained in both of these clubs. If any member of Y_{α} contains a club subset of ω_{1} disjoint from A_{α}, then put α in A_{α+1}, and if any member of Y_{α} contains a club subset of ω_{1} 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 this strategy, and let A and B be the corresponding play by split. Let C be a club subset of ω_{1} and supposing that E is stationary, fix X ∈ X containing E, A, B and C with X ∩ ω_{1} ∈ E ∩ C. Then if A ∩ C ∩ X ∩ ω_{1} = ∅, then X ∩ ω_{1} ∈ A ∩ C, and if B ∩ C ∩ X ∩ ω_{1} = ∅, then X ∩ ω_{1} ∈ B ∩ C, which shows that C does not witness that unsplit won this run of the game.

The following fact, in conjunction with Theorem 1.2, shows that Martin’s Axiom 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. partial orders.

Proof. Let P be a c.c.c. forcing and let X witness C+. Let γ be a regular cardinal greater than ℵ_{2} and 2^{|P |}. Let G ⊂ P be a V -generic filter, and let

X [G] = {X[G] ∩ H(ℵ_{2})^{V [G]} : X ≺ H(γ)^{V} , X ∩ H(ℵ_{2})^{V} ∈ X }.

Since every club subset of ω_{1} 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 stationary there. Fix a P -name ρ for a function from [H(ℵ_{2})^{V [G]}]^{<ω} to H(ℵ_{2})^{V [G]}. For any countable X ≺ H(γ) with X ∩ H(ℵ_{2}) ∈ X and ρ ∈ X, X[G] ∩ H(ℵ_{2})^{V [G]} is in X [G] and closed under the realization of ρ. Fix a P -name τ for a stationary subset of ω_{1} and a condition p ∈ P. Let S be the set of countable ordinals forced to be in τ by some condition below p. Then exist a countable X ≺ H(γ) with X ∩ H(ℵ_{2}) ∈ X, X ∩ ω_{1} ∈ S and ρ ∈ X and a condition q below p forcing that X[Ġ] ∩ ω_{1} (where Ġ is the name for the generic filter) is in the realization of τ. By genericity, then, X [G] is projective stationary.

We do not know how to force C+, however, and use a different principle to force the existence of a winning strategy for split.

1.4 Definition. Let C_{s} be the statement that there exist c_{α} (α < ω_{1} limit) such that each c_{α} is a sequence ⟨a_{β}^{α} : β < γ_{α}⟩ (for some countable γ_{α}) of cofinal subsets of α of orderype ω and

• for all limit α < ω_{1} and all β < β′ < γ_{α}, a_{β′}^{α} \ a_{β}^{α} is finite;

• for every club C ⊂ ω_{1} and every stationary E ⊂ ω_{1} there exists an a_{β}^{α} with α ∈ E such that a_{β}^{α} \ C is finite and a_{β}^{α} ∩ E is infinite.

The principle C_{s} also holds in fine structural models such as L. The winning strategy 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_{β}^{α} (α < ω_{1} 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_{β}^{α} intersects A_{α} infinitely and B_{α} finitely, then put α in B_{α+1}. If some a_{β}^{α} intersects B_{α} infinitely and A_{α} finitely, then put α in A_{α+1}. Since the a_{β}^{α}’s (β < γ_{α}) are ⊂-decreasing mod 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 this strategy, and let A and B be the corresponding play by split. Let C be a club subset of ω_{1} and supposing that E is stationary, fix a_{β}^{α} with α ∈ E such that a_{β}^{α} \ C is finite and a_{β}^{α} ∩ E is infinite. Then if A ∩ a_{β}^{α} is finite, then α ∈ A ∩ C, and if B ∩ a_{β}^{α} is finite, then α ∈ B ∩ C, which shows that C does not witness that unsplit won this run of the game.

A partial order P is said to be strategically ω-closed if there exists a function f: P^{<ω} → P(P) such that whenever ⟨p_{i} : i ≤ n⟩ is a finite descending sequence in P, f(⟨p_{i} : i ≤ n⟩) is a dense subset below p_{n} and, whenever ⟨p_{i} : i < ω⟩ is a descending sequence in P such that for each n there exists a j with

p_{j} ∈ f(⟨p_{i} : i ≤ n⟩),

the sequence has a lower bound in P. It is easy to see that strategic ω-closure is 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 is infinite. Given A ⊂ ω_{1}, let ℂ(A) be the partial order which adds a club subset of A by initial segments. We force C_{s} by first adding a potential C_{s}-sequence by initial segments, and then iterating to kill off every counterexample.

We refer the reader to [11] for background on countable support iterations of proper forcing.

Theorem 1.6. Suppose that CH and 2^{ℵ_{1}} = ℵ_{2} hold. Let P̄ = ⟨P_{η}, Q_{∼}_{η} : η < ω_{2}⟩ be a countable support iteration such that P_{0} is the partial order consisting of sequences ⟨c_{α} : α < δ limit⟩, for some countable ordinal δ, such that each c_{α} is a sequence ⟨a_{β}^{α} : β < γ_{α}⟩ (for some countable ordinal γ_{α}) of cofinal subsets of α of ordertype ω, deceasing by mod-finite inclusion (and P_{0} is ordered by extension). Suppose that the remainder of P̄ satisfies the following conditions.

• For each nonzero η < ω_{2} there is a P_{η}-name τ_{η} for a subset of ω_{1} such that if (τ_{η})_{G_{η}} (where G_{η} is the restriction of the generic filter to P_{η}) is stationary in the P_{η} extension and there exists a club C ⊂ ω_{1} in this extension such that no a_{β}^{α} with α ∈ τ_{G_{η}} captures the pair τ_{G_{η}}, C, then Q_{∼}_{η} is ℂ(ω_{1} \ (τ_{η})_{G_{η}}) (and otherwise, Q_{∼}_{η} is ℂ(ω_{1})).

• For every pair E, C of subsets of ω_{1} in any P_{η}-extension (η < ω_{2}), if E is stationary in this extension and C is club and no a_{β}^{α} with α ∈ E captures E, C, then there is a ρ ∈ [η, ω_{2}) such that if E is stationary in the P_{ρ} extension, then Q_{∼}_{ρ} is ℂ(ω_{1} \ E).

Then P̄ is strategically ω-closed, and C_{s} holds in the P̄-extension. Furthermore, in the P̄ extension, ◇(S) holds for every stationary S ⊂ ω_{1}.

Proof. Let X be a countable elementary submodel of H((2^{|P̄|)^{+}}) with P̄ ∈ X, let g be an X-generic filter contained in P̄ ∩ X. Let γ_{X∩ω_{1}} be the ordertype of X ∩ ω_{2}, and for each β < γ_{X∩ω_{1}}, let η_{β} be the βth member of X ∩ ω_{2}. For each β < γ_{X∩ω_{1}}, let a_{β}^{X∩ω_{1}} be a cofinal subset of X ∩ ω_{1} of ordertype ω such that, letting g_{η} denote the restriction of g to P_{η},

• for all β′ < β < γ_{X∩ω_{1}}, a_{β}^{X∩ω_{1}} \ a_{β′}^{X∩ω_{1}} is finite;

• a_{β}^{α} is eventually contained in every club subset of ω_{1} in X[g_{η_{β}}] and intersects infinitely every stationary subset of ω_{1} in every X[g_{η_{β′}}], β′ ∈ [β, γ_{X∩ω_{1}}).

It remains to see that we can extend g to a condition whose first coordinate is given by adding c_{X∩ω_{1}} = ⟨a_{β}^{α} : β < γ_{X∩ω_{1}}⟩ to the union of the first coordinates of the elements of g, and whose ηth coordinate, for each nonzero η ∈ X ∩ ω_{2} is the condition given by the union of {X ∩ ω_{1}} and the set of realizations of the ηth 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_{2} ∩ X, there is a P_{η}-name σ ∈ X for a club subset of ω_{1} 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_{β}^{α} with α ∈ (τ_{η})_{G_{η}}, then σ_{G_{η}} is such a C. However, the realizations of τ_{η} and σ by g are captured by a_{o.t.(η∩ω_{2})}^{X∩ω_{1}}, so g′_{η} forces that τ_{G_{η}}, σ_{G_{η}} is captured by a_{o.t.(η∩ω_{2})}^{X∩ω_{1}}. It follows that g′_{η} forces that either Q_{∼}_{η} is ℂ(ω_{1}), or X ∩ ω_{1} is not in τ_{G_{η}}. In either case, the union of the members of g ∩ Q_{∼}_{η} be can extended to a condition in Q_{∼}_{η} by adding {X ∩ ω_{1}}.

To see that ◇(S) holds for every stationary S ⊂ ω_{1} in the P̄ extension, fix such an S in the P_{α} extension for some α < ω_{2}. Since P̄ is (ω, ∞) distributive, there exists in this extension a set ⟨e_{β}^{δ} : δ, β < ω_{1}⟩ such that for every δ < ω_{1} and every x ⊂ δ there are uncountably many β such that e_{β}^{δ} = x. Then, letting T ∈ P(ω_{1})^{V [G_{α}]} be the set such that the realization of Q_{∼}_{α} is ℂ(T), Q_{∼}_{α} adds a ◇ sequence ⟨b_{δ} : δ ∈ S⟩ defined by letting b_{δ} be e_{β}^{δ}, where the βth element of T above β is the first element of the generic club for Q_{∼}_{α} above δ. To see that this is a ◇ sequence, note that since S is stationary in the P̄ extension, there are stationarily many elementary submodels X of any sufficiently large H(θ)^{V [G]} in this extension with X ∩ ω_{1} ∈ S. Then X ∩ (G/G_{α}) is a (X ∩ V [G_{α}], P̄/P_{α})- generic filter which can be extended to a condition in P̄/P_{α} by adding X ∩ω_{1} to each coordinate, and extended again to make any element of T \ ((X ∩ ω_{1}) + 1) the least element of the generic club for Q_{∼}_{α} above X ∩ ω_{1}. That ⟨b_{β} : β ∈ S⟩ is a ◇ sequence then follows by genericity.

Section 2 shows that proper forcing does not always preserve the existence of a winning strategy for split.

1.2 A strategy for unsplit

In this section we show that it is consistent for unsplit to have a winning strategy in SG. We do this via the following guessing principle.

1.7 Definition. Let D_{u} be the statement that there exists a diamond sequence ⟨σ_{α} : α < ω_{1}⟩ such that for every E ⊂ ω_{1} there is a club C ⊂ ω_{1} such that either

∀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.

Proof. Let ⟨σ_{α} : α < ω_{1}⟩ witness D_{u}. Play for unsplit by accepting α if and only if σ_{α} = A_{α}. At the end of the game, the set of α such that σ_{α} = A_{α} is stationary, 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 ⟨σ_{α} : α < ω_{1}⟩ by initial segments, and we then iterate to make this sequence witness D_{u}, iteratively forcing a club through the set of α < ω_{1} such that σ_{α} ≠ E ∩ α or α ∈ E for each E ⊂ ω_{1} such that the sets {α ∈ E | σ_{α} = E ∩ α} and {α ∈ ω_{1} \ E | σ_{α} = E ∩ α} are both stationary.

More specifically, we have the following. Given a sequence Σ = ⟨σ_{α} : α < ω_{1}⟩ such that each σ_{α} is a subset of α, and given E ⊂ ω_{1}, 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_{1} \ E such that σ_{α} = E ∩ α.

Theorem 1.9. Suppose that CH + 2^{ℵ_{1}} = ℵ_{2} holds, and let P̄ be a countable support iteration ⟨P_{α}, Q_{∼}_{α} : α < ω_{2}⟩ such that P_{0} is the partial order consisting of sequences ⟨σ_{β} : β < γ⟩, for some countable ordinal γ, such that each σ_{β} is a subset of β, ordered by extension. Let Σ be the sequence added by P_{0} and suppose that the remainder of P̄ satisfies the following conditions.

• Each Q_{∼}_{α} is either ℂ(ω_{1}) or ℂ(ω_{1} \ B(Σ, E)) for some E ⊂ ω_{1} such that A(Σ, E) and B(Σ, E) are both stationary.

• For every E ⊂ ω_{1} in any P_{α}-extension (α < ω_{2}) there is a γ ∈ [α, ω_{2}) such that if A(Σ, E) and B(Σ, E) are both stationary in the P_{γ} extension,

then Q_{∼}_{γ} is ℂ(ω_{1} \ B(Σ, E)).

Then P̄ is strategically ω-closed, and in the P̄-extension, D_{u} holds. Furthermore, in the P̄ extension, ◇(S) holds for every stationary S ⊂ ω_{1}.

Proof. The iteration P̄ is clearly strategically ω-closed, since for any countable X ≺ H((2^{|P̄|})^{+}) and any (X, P̄)-generic filter g contained in X, one can extend g to a condition by making σ_{X∩ω_{1}} unequal to the realization by g of any name in X for a subset of ω_{1}, and adding X ∩ ω_{1} to all the clubs being added by the Q_{∼}_{α}’s, α ∈ X ∩ ω_{2}. It is clear also that in the P̄-extension there is no E ⊂ ω_{1} 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_{1}, we first note the following.

Claim 1. Suppose that E ⊂ ω_{1} is a member of the P_{α} extension, for some α < ω_{2}, and A(Σ, E) is stationary in this extension. Then A(Σ, E) remains stationary in the P̄ extension.

Note that A(Σ, E) has countable intersection with B(Σ, F), for every F ⊂ ω_{1}. Fix X ≺ H(((2^{|P̄|})^{+})^{V})^{V [G_{α}]} (where G_{α} is the restriction of the generic filter G to P_{α}) with X ∩ ω_{1} ∈ A(Σ, E) and A(Σ, E) ∈ X. Then any (X, P̄/P_{α})- generic filter contained in X can be extended to a condition by adding X ∩ ω_{1} to the clubs being added at every stage of P̄ after the first.

Similar reasoning shows the following two facts, which complete the proof that Σ witnesses D_{u} in the P̄ extension.

Claim 2. Suppose that E ⊂ ω_{1} is a member of the P_{α} extension, for some α < ω_{2}, 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_{1} which is forced to be unequal to any such subset in any P_{γ} extension, for any γ < α. Fix X ≺ H(((2^{|P̄|})^{+}))^{V} with τ ∈ X. Let g be an (X, P_{α})-generic filter, and note that the realization of τ↾(X ∩ ω_{1}) by g is different from the realizations of ρ↾(X ∩ ω_{1}) by g for any P_{γ}-name ρ ∈ X for a subset of ω_{1}, for any γ ∈ X ∩ α. It follows that adding the realization of τ↾(X ∩ ω_{1}) by g to the union of the first coordinate projection of g gives a condition in P_{0} forcing that X ∩ ω_{1} is not in any Σ(B, ρ_{G_{γ}}), for any for any P_{γ}-name ρ ∈ X for a subset of ω_{1}, for any γ ∈ X ∩ α. Therefore, we can add X ∩ ω_{1} to the clubs being added in every other stage of P̄ in X ∩ α, and get a condition extending every condition in g.

Claim 3. Suppose that E ⊂ ω_{1} is a member of the P_{α} extension, for some α < ω_{2}, and A(Σ, E) is nonstationary in this extension. Then B(Σ, E) remains stationary in the P̄ extension.

This is similar to the previous claims, noting that every subsequent stage of P̄ forces a club though the complement of a set with countable intersection with B(Σ, E).

The proof that ◇(S) holds for every stationary S ⊂ ω_{1} in the P̄ extension is (literally) the same as in the proof of Theorem 1.6.

Note that that the iterations P̄ in Theorems 1.6 and 1.9 are strategically ω-closed.

1.3 Σ_{2}^{2} maximality

The statements that split and unsplit have winning strategies in SG are each Σ_{2}^{2} in a predicate for NS_{ω_{1}}, and they are obviously not consistent with each other. Woodin (see [6]) has shown that if there is a proper class of measurable Woodin cardinals, then there exists in a forcing extension a transitive class model of ZFC satisfying all Σ_{2}^{2} sentences ϕ such that ϕ + CH can be forced over the ground model. The results here show that this result cannot be extended to include a predicate for NS_{ω_{1}}. This was known already, in that ◇^{∗} (in the sense of [7]) and “the restriction of NS_{ω_{1}} to some stationary set is ℵ_{1} dense” were both known to be consistent with ◇ (the second of these is due to Woodin, uses large cardinals and 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 consistent with “◇(S) holds for every stationary set S ⊂ ω_{1}.”

1.4 A determined variation

There are many natural variations of SG. We show that one such variation is determined.

Theorem 1.10. Let G be the following game of length ω_{1}. In round α, player I puts α into one of two sets E_{0} and E_{1}, and player II puts α into one of two sets A_{0} and A_{1}. After all ω_{1} rounds have been played, II wins if one of the following pairs of set are both stationary.

• E_{0} ∩ A_{0} and E_{0} ∩ A_{1}

• E_{1} ∩ A_{0} and E_{1} ∩ A_{1}

Then II has a winning strategy in G.

Proof. Let B_{00}, B_{01}, B_{10} and B_{11} be pairwise disjoint stationary subsets of ω_{1}. In round α, if α is in B_{ij}, let II put α in A_{i} if I put α in E_{0} and in A_{j} otherwise. Then after all ω_{1} many rounds have been played, suppose that A_{i} ∩ E_{0} is nonstationary. Then B_{i0} and B_{i1} are both contained in E_{1} modulo NS_{ω_{1}}, which means that E_{1} ∩ A_{0} and E_{1} ∩ A_{1} are both stationary. Similarly, if A_{i} ∩ E_{1} is nonstationary then B_{0i} and B_{1i} are both contained in E_{0} modulo NS_{ω_{1}}, which means that E_{0} ∩ A_{0} and E_{0} ∩ A_{1} are both stationary.

2 Indeterminacy from forcing axioms

The axiom PFA^{+2} says that whenever P is a proper partial order, D_{α} (α < ω_{1}) are dense subsets of P and σ_{1}, σ_{2} are P -names for stationary subsets of ω_{1}, there is a filter G ⊂ P such that G ∩ D_{α} ≠ ∅ for each α < ω_{1}, and such that {α < ω_{1} | ∃p ∈ G p⊩α̌ ∈ σ_{i}} is stationary for each i ∈ {1, 2}. Theorems 1.6 and 1.9 together show that PFA^{+2} implies the indeterminacy of SG. Furthermore, a straightforward argument shows that the following statement implies the nonexistence of a winning strategy for unsplit in SG, where Add(1, ω_{1}) is the partial order that adds a subset of ω_{1} by initial segments : for any pair σ_{1},σ_{2} of Add(1, ω_{1})-names for stationary subsets of ω_{1}, there is a filter G ⊂ Add(1, ω_{1}) realizing both σ_{1} and σ_{2} as stationary sets. This statement is trivially subsumed by PFA^{+2}, but also holds in the collapse of a sufficiently large cardinal to be ω_{2}, and thus is consistent with CH.

The axiom Martin’s Maximum [4] says that whenever P is a partial order such that forcing with P preserves stationary subsets of ω_{1} and D_{α} (α < ω_{1}) are dense subsets of P, there is a filter G ⊂ P such that G ∩ D_{α} ≠ ∅ for each α < ω_{1}.

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 result of a generic run of SG where unsplit plays by Σ (the partial order consists of countable partial plays where unsplit plays by Σ, ordered by extension). If the complement of E has stationary intersection with every stationary subset of ω_{1} in the ground model, one can force to kill the stationarity of E in such a way that the induced two step forcing preserves stationary subsets of ω_{1} and produces a run of SG where unsplit plays by Σ and loses. If the complement of E does not have stationary intersection with some stationary F ⊂ ω_{1} in the ground model, then there is a partial run of the game p and a name τ for a club such that p forces that E will contain F ∩ τ_{G}. Then there exists in the ground model a run of SG extending p in which unsplit plays by Σ and loses: split picks a pair of disjoint stationary subsets F_{0}, F_{1} of F, and plays so that

• for every α < ω_{1}, some initial segment of the play forces some ordinal greater than α to be in τ,

• whenever unsplit accepts α ∈ F, split puts α in A if α ∈ F_{0} and puts α ∈ B if α ∈ F_{1}.

Now fix a strategy Σ for split in SG, and generically add a regressive function f on ω_{1} by initial segments. Let E^{α} = f^{−1}(α) and let A^{α}, B^{α} be the responses given 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_{1} and stationary sets S, T in the ground model such that (S ∩ E^{α}) \ A^{α} and (T ∩ E^{α}) \ B^{α} are both nonstationary. Then there is a condition p in our forcing (i.e., a regressive function on some countable ordinal) such that p forces that (S ∩ E^{α}) ⊂ A^{α} and (T ∩ E^{α}) ⊂ B^{α}, modulo nonstationarity (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_{1}, at successor stages extending to add a new element to the realization of τ, and at limit stages (when for some β < ω, f↾β has been decided and f(β) has not, and β is forced by f↾β to be a limit member of the realization of τ) extending so that f(β) = α if and only if β ∈ S. Then the run of SG corresponding to f^{−1}(α) is winning for unsplit, since the corresponding set B^{α} is nonstationary.

If there exist no such α, S, T, there is a function h on ω_{1} 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. Then Martin’s Maximum applied to the corresponding two step forcing produces a run of SG (the run for any f^{−1}(α) which is stationary) where split plays by Σ and loses.

Theorem 2.1 leads to the following question.

2.2 Question. Does the Proper Forcing Axiom imply that SG is not determined?

The following question is also interesting. The consistency of the ℵ_{1}-density of NS_{ω_{1}} (relative to the consistency of AD^{L(ℝ)}) is shown in [13].

2.3 Question. Does the ℵ_{1}-density of NS_{ω_{1}} decide the determinacy of SG?

3 MLO games

The second-order Monadic Logic of Order (MLO) is an extension of first-order logic with logical constants =, ∈ and ⊂ and a binary symbol < as the only non-logical constant, allowing quantification over subsets of the domain. Every ordinal is a model for MLO, interpreting < as ∈.

Given an ordinal α, an MLO game of length α is determined by an MLO formula 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 whether these two sets satisfy the formula in α.

Büchi 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_{1} whose determinacy is independent of ZFC.

We thank Assaf Rinot for pointing out to us the connection between SG and MLO games.

References

[1] J.R. Büchi, L.H. Landweber, Solving sequential conditions by finite-state strategies, Trans. Am. Math. Soc. 138 (Apr. 1969), 295-311

[2] Q. Feng, T. Jech, Projective stationary sets and a strong reflection principle, J. London Math. Soc. 58 (1998), 271-283

[3] M. Foreman, Ideals and generic elementary embeddings, Handbook of Set Theory, Foreman, Kanamori eds., to appear

[4] M. Foreman, M. Magidor, S. Shelah, Martin’s Maximum. saturated ideals, and non-regular ultrafilters. Part I, Annals of Mathematics, 127 (1988), 1-47

[5] T. Jech, 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_{2}^{2}-maximality, homogeneity and the stationary tower, preprint

[7] K. Kunen, Set Theory, North-Holland, 1980

[8] P. Larson, The canonical function game, Archive Math. Logic 44 (2005), # 7, 817-827

[9] J. Moore, Set Mapping Reflection, J. Math. Logic 5 (2005), # 1, 87-98

[10] A. Rabinovich, The Church problem for countable ordinals, preprint, 2007 [11] S. Shelah, Proper and improper forcing, Perspectives in Mathematical

Logic, Springer-Verlag, Berlin, 1998

[12] A. Shomrat, Uniformization problems in the monadic theory of countable ordinals, manuscript, 2006

[13] W.H. Woodin, The axiom of determinacy, forcing axioms and the nonstationary ideal, de Gruyter Series in Logic and its Applications 1, Walter de Gruyter & Co., Berlin, 1999

Department of Mathematics and Statistics, Miami University, Oxford, Ohio

45056, USA; Email: larsonpb@muohio.edu

The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, Jersalem 91904, Israel

Department of Mathematics, Hill Center - Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019,

USA; Email: shelah@math.huji.ac.il