THE EXTENDER ALGEBRA AND PRESERVING STATIONARY SETS

PAUL B. LARSON

Abstract. We give an instance when the extender algebra can preserve stationary subsets of ω_{1}. 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(ω_{2}) while preserving stationary subsets of ω_{1}. 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 ℙ_{max} 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^{ω} 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(ω_{1}) by subsets of ω: x ⊆ ω codes a ∈ H(ω_{1}) 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 Σ_{1}^{1}, 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(ω_{1}), 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.

The statement of our main theorem uses the notion of A-closure from Woodin’s Ω-logic. For our purposes, we need to know only that if A is a universally Baire function which is invariant in the codes and N is an A-closed model, then N is closed under the induced class function from the previous paragraph (see [8, 1]).

The proof of our main theorem uses another notion of iterability, that of producing wellfounded models under generic iterated embeddings using the stationary tower. The following fact is proved in [8].

Theorem 0.3. Suppose that Q is a transitive model containing ω_{1} in which countable ordinals δ < λ are a Woodin cardinal and a strongly inaccessible cardinal, respectively. Then Q and V_{λ}^{Q} are both iterable with respect to ℚ_{<δ}^{Q}.

Theorem 0.4. Suppose that • δ_{0} < δ_{1} are a Woodin cardinals below a measurable cardinal.

• A is a set of reals coding a function that takes each real x to a model M(x) of ZFC + T containing x and an iteration strategy for M(x), in a way that is invariant for some coding of elements of H(ω_{1}) by reals, such that A and ω^{ω} \ A are δ_{1}^{+}-weakly homogeneously Suslin;

• ϕ is a statement of the form “some rank initial segment of the universe satisfies ψ”, for some statement ψ;

• for every real r, ϕ holds in an A-closed model of ZFC containing r. Then for every set Z ∈ V_{δ_{1}} such that M(Z) is NS_{ω_{1}}-correct, ϕ can be forced over M(Z) by a forcing preserving stationary subsets of ω_{1}.

Proof. Let κ > δ_{1} be measurable, and let θ be a regular cardinal greater than 2^{κ}. Let λ be a strongly inaccessible cardinal between δ_{0} and δ_{1} with Z ∈ V_{λ}. Applying Lemma 0.2, let Y be an elementary submodel of H(θ) with κ, δ_{0}, δ_{1}, Z and A as members such that Y ∩ δ_{1} is uncountable, Y ∩ λ is countable, and such that there exist trees S and T on ω × γ (for some ordinal γ) in Y such that p[S ∩ Y] = A, p[T ∩ Y] = ω^{ω} \ A.

Let Q be the transitive collapse of Y and let S_{Q}, T_{Q}, κ_{Q}, δ_{0Q}, δ_{1Q}, λ_{Q} and Z_{Q} be the images of S, T, κ, δ_{0}, δ_{1}, λ and Z under this collapse. Let P denote V_{κ_{Q}}^{Q}. Since ω_{1} ⊂ Q, Q is iterable with respect to ℚ_{<δ_{0Q}}^{Q}, and therefore P is as well. Let N be a countable A-closed model of ZFC + ϕ with P as an element. Let γ be an ordinal such that V_{γ}^{N} ⊧ ψ. Let j: P → P′ be an iterated generic elementary embedding of length ω_{1}^{N} in N via ℚ_{<δ_{0Q}}^{P} such that P′ is correct about stationary subsets of ω_{1} in N ([8]). Thus induces an iteration of Q with the same generic filters, such that P′ is a rank initial segment of the corresponding final model Q′. We let j denote the entire embedding from Q to Q′.

Since Q′ is wellfounded and the projections of j(S_{Q}) and j(T_{Q}) are disjoint in Q′, they are disjoint in V as well. Since p[S] = p[S_{Q}] ⊂ p[j(S_{Q})] and p[T] = p[T_{Q}] ⊂ p[j(T_{Q})], it follows that p[j(S_{Q})] = p[S] and p[j(T_{Q})] = p[T]. Therefore, M(j(Z_{Q})) exists in Q′ and is coded by a real in the projection of p[jS_{Q}] in any Coll(ω, j(Z_{Q})) extension of Q′ (this is just to say that M(j(Z_{Q})) is definable in Q′ from j(S_{Q}) and j(Z_{Q}), which since M(Z) is definable in the same way from S and Z means that facts about M(j(Z_{Q})) true in Q′ will be true about M(Z) in H(θ)).

Since N is A-closed, M(j(Z_{Q})) is in N as well. Furthermore, M(j(Z_{Q})) is NS_{ω_{1}}- correct in P′ and thus in N. Let k: M(j(Z_{Q})) → M^{∗} be an elementary embedding in N with critical point greater than 2^{ω_{1}} such that V_{γ}^{N} is generic over M^{∗}. Then M^{∗} and M(j(Z_{P})) have the same P(ω_{1}), so M^{∗} is NS_{ω_{1}}-correct in N, so M^{∗} must be NS_{ω_{1}}-correct in M^{∗}[V_{γ}^{N}]. Furthermore,

V_{γ}^{M^{∗}[V_{γ}^{N}]} = V_{γ}^{N}.

Therefore, M^{∗}[V_{γ}^{N}] is an NS_{ω_{1}}-preserving forcing extension of M^{∗} satisfying ϕ, so M(j(Z_{Q})), M(Z_{Q}) and M(Z) all also have NS_{ω_{1}}-preserving forcing extensions satisfying ϕ. □

References

[1] J. Bagaria, N. Castells, and P.B. Larson. An Ω-logic primer. In Set Theory, CRM 2003-2004. to appear.

[2] I. Farah. The extender algebra. unpublished note, 2006.

[3] I. Farah and P.B. Larson. Absoluteness for universally Baire sets and the uncountable I. in preparation, 2005.

[4] G. Hjorth. Some applications of coarse inner model theory. Journal of Symbolic Logic, 62:337– 365, 1997.

[5] P. Larson and S. Shelah. Bounding by canonical functions, with CH. Journal of Mathematical Logic, 3:193–215, 2003.

[6] I. Neeman. The determinacy of long games, volume 7 of de Gruyter Series in Logic and its Applications. Walter de Gruyter GmbH & Co. KG, Berlin, 2004.

[7] J. Steel. An outline of inner model theory. In Handbook of Set Theory. to appear.

[8] W.H. Woodin. The Axiom of Determinacy, forcing axioms and the nonstationary ideal, volume 1 of de Gruyter Series in Logic and Its Applications. de Gruyter, 1999.

Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056, USA

   E-mail address: larsonpb@muohio.edu
   URL: http://www.users.muohio.edu/larsonpb/