The Extender Algebra and Preserving Stationary Sets

We give an instance when the extender algebra can preserve stationary subsets of ω1. In particular, we show that for any model operator satisfying certain conditions (satisfied by the currently known minimal inner models for large cardinal statements), any Ω-consistent statement about a rank initial segment of the universe can be forced over canonical model containing H(ω2) while preserving stationary subsets of ω1. This is a variation of Theorem 10.13 of [8]. We will use the following phrasing of Woodin’s extender algebra theorem (see [7, 4, 6, 2]). The notion of iterability here and in the statement of our main theorem refers to the existence of iteration strategies for iteration trees of arbitrary length. Theorem 0.1. Let M be an iterable model, and let δ be a Woodin cardinal in M . Then for any set x and any λ < δ there is an elementary embedding j : M → M∗ with critical point greater than λ such that x is M∗-generic for a partial order in M∗ of cardinality j(δ). The following well-known fact is used to produce Pmax conditions from large cardinals, and will be used in our argument in almost the same way. Proofs appears in [5, 3]. Lemma 0.2. Let θ be a regular cardinal, suppose that T is a weakly homogeneous tree on ω×Z in H(θ), for some set Z. Let γ ≥ 2 be an ordinal such that there exists a countable collection Σ of γ-complete measures witnessing the weak homogeneity of T . Assume that there is a measurable cardinal in the interval (γ, θ). Then for every elementary submodel X of H(θ) of cardinality less than γ with T , Σ, γ ∈ X, there is an elementary submodel Y of H(θ) containing X such that Y ∩ θ is uncountable, Y ∩ γ = X ∩ γ and p[T ∩ Y ] = p[T ]. Fixing a recursive bijection π : ω×ω → ω, we use the following coding of elements of H(ω1) by subsets of ω: x ⊆ ω codes a ∈ H(ω1) if 〈ω, {(n,m) | π(n,m) ∈ x}〉 ∼= 〈{a} ∪ tc(a),∈〉, where tc(a) is the transitive closure of a. Under this coding, the relations “ ∈ ” and “=” are both Σ1, since permutations of ω can give rise to different codes for the same set. We say that a function f : P(ω) → P(ω) is invariant in the codes if whenever x and y code the some element of H(ω1), f(x) and f(y) do as well. Note that if a function f : P(ω) → P(ω) is universally Baire and invariant in the codes, it induces a class function from V to V : for any set Z in any H(κ), letting f∗ denote the extension of f in the Coll(ω, κ)-extension, the set coded by f∗(x) exists already in V , for x any subset of ω in this extension coding Z. Date: December 4, 2006.