WRP^{∗}

Paul B. Larson

August 16, 2010

Given a nonempty set X, a set S ⊆ P(X) is said to be club if there exists a function F: X^{<ω} → X such that S = {Y ⊆ X | F [Y^{<ω}] ⊆ Y }, and stationary if it intersects every club subset of P(X). A set S is said to be ⊆-cofinal in a set T if for every X ∈ T there exists a Y ∈ S such that X ⊆ Y. Given a set X, an ideal I ⊆ P_{ℵ_{1}}(X) is fine if it contains every ⊆-noncofinal set, i.e., if for all x ∈ P_{ℵ_{1}}(X), {y ∈ P_{ℵ_{1}}(X) | x ⊈ y} ∈ I. The ideal I is normal if for all f: X → I, S_{f} = {x ∈ P_{ℵ_{1}}(X) | ∃a ∈ x s.t. x ∈ f(a)} ∈ I, i.e., if I is closed under diagonal unions. We say that a set S ⊆ P_{ℵ_{1}}(X) reflects to a set Y ⊆ X if S ∩ P_{ℵ_{1}}(Y) is stationary in P_{ℵ_{1}}(Y).

In his book on ℙ_{max} [2], Woodin defines the statements WRP_{n}^{∗}(ω_{2}). The natural generalization of these statement to an arbitrary regular cardinal κ ≥ ℵ_{2} is as follows.

0.1 Definition. Given a regular cardinal κ ≥ ℵ_{2} and an integer n ∈ ω, WRP_{n}^{∗}(ω_{2}) is the statement that there exists a normal, fine ideal I on P_{ℵ_{1}}(κ) such that

• for all stationary T ⊆ ω_{1}, {x ∈ P_{ℵ_{1}}(κ) | x ∩ ω_{1} ∈ T } ∉ I;

• for all S_{1}, …, S_{n} ⊆ P_{ℵ_{1}}(κ), if S ∉ I, then the set of X ⊆ κ such that

– ω_{1} ⊆ X;

– |X| = ℵ_{1};

– for all i ∈ {1, …, n}, S_{i} reflects to X

is stationary.

We write WRP^{∗}(κ) for WRP_{1}^{∗}(κ). Steel and Zoble [1] have shown that W RP_{2}^{∗}(ω_{2}) + “NS_{ω_{1}} is saturated” + 2^{ℵ_{0}} ≤ ℵ_{2} implies that the Axiom of Determinacy holds in L(ℝ).

Theorem 0.2. Let κ ≥ ℵ_{2} be a regular cardinal. Then WRP^{∗}(κ) holds.

Proof. Let I be the collection of S ⊆ P_{ℵ_{1}}(κ) for which the set of X ∈ [κ]^{ℵ_{1}} for which ω_{1} ⊆ X and S reflects to X is nonstationary. We claim that I satisfies the theorem. First of all, note that every ⊆-noncofinal subset of P_{ℵ_{1}}(κ) is nonstationary, and every nonstationary subset of P_{ℵ_{1}}(κ) is in I. Note also that I is closed under countable unions and subsets. Clearly, I satisfies the first conclusion of the definition of WRP^{∗}(κ). Let us check that I is closed under diagonal unions. Fix a function f: κ → I, and suppose towards a contradiction that S_{f} reflects to a stationary set of X ∈ [κ]^{ℵ_{1}} such that ω_{1} ⊆ X. Then we may fix such an X such that for all α ∈ X, f(α) does not reflect to X. Let ⟨x_{β} : β < ω_{1}⟩ be a continuous, ⊆-increasing sequence of countable sets with union X. By assumption, for a stationary set F of β ∈ ω_{1}, there exists an α ∈ x_{β} such that x_{β} ∈ f(α). Pressing down, we have a stationary set F′ for which a fixed α suffices, contradicting that assumption that f(α) does not reflect to γ.

References

[1] J. Steel, S. Zoble, Determinacy from strong reflection, preprint

[2] W.H. Woodin. The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. Logic and its Applications. de Gruyter, 1999.