Reals constructible from many countable sets of ordinals

Paul B. Larson

March 20, 2005

Abstract We show that if there exist proper class many Woodin cardinals, then the set of reals x for which there is exists an ordinal α with {a ∈ P_{ω_{1}}(α) | x ∈ L[a]} stationary is countable. These results were announced in [2].

Given a real x and an ordinal α, we let C(x, α) denote the set

{a ∈ P_{ω_{1}}(α) | x ∈ L[a]}.

We let C denote the set of reals x for which there exists an ordinal α such that C(x, α) is club.

We assume some familiarity with the stationary tower ℚ_{<δ} (see [2]).

Theorem 0.1. Suppose that there exists a proper class of Woodin cardinals. Then for every real x and every ordinal α, C(x, α) is either club or nonstationary, and C is countable.

Proof. Fix a cardinal λ such that, for every real x,

• if there exists an ordinal α with C(x, α) stationary and costationary, then there is such an ordinal below λ,

• if there exists an ordinal α with C(x, α) club, then there is such an ordinal below λ.

Let G be V -generic for Coll(ω_{1}, λ), and note that the set of (x, α) such that C(x, α) is stationary (likewise, costationary, club) is the same in V and V [G]. Now let V [G][H] be a semi-proper forcing extension of V [G] in which u_{2} > λ.

If δ is a Woodin cardinal, then since u_{2} ≥ λ, j[λ] is the same whenever j is the elementary embedding induced by forcing with ℚ_{<δ}^{V [G][H]}. Fix a real x and an ordinal α < λ. Then x ∈ L[j[α]] if and only if C(x, α) is in the ℚ_{<δ}^{V [G][H]}-generic. Since for all generic filters x ∈ L[j[α]] is decided in the same way, C(x, α) must be either club or nonstationary.

Suppose that (x_{β}, α_{β}) (β < ω_{1}) are pairs such that each x_{β} is a distinct real, each α_{β} < λ and each C(x, α_{β}) is stationary. Since u_{2} > λ, there is a real z such that for each β < ω_{1} there is a bijection between ω_{1}^{V} and α_{β} in L[z]. Therefore, each x_{β} is in L[z], which is impossible.

By contrast, Gitik [1] has shown that for every club subset C of P_{ω_{1}}(ω_{2}) and every real z there exist a, b, c ∈ C such that z ∈ L[a, b, c].

The Axiom of Determinacy implies that the club filter is an ultrafilter on P_{ω_{1}}(ω_{2}) [4]. It follows that if AD holds in L(ℝ), then for each real x the set C(x, δ_{∼}_{2}^{1}) is either club or nonstationary. Using this fact in place of the stationary tower argument in the proof of Theorem 0.1, one can easily prove the following.

Theorem 0.2. If AD holds in L(ℝ) in every forcing extension by a forcing of the form Coll(ω_{1}, γ), then for every real x and every ordinal α, C(x, α) is either club or nonstationary, and C is countable.

The following then follows easily from the fact that the theory of L(ℝ) cannot be changed by set forcing in the presence of a proper class of Woodin cardinals.

Theorem 0.3. If there exists a proper class of Woodin cardinals, then every inner model which is correct about ω_{2} contains all the reals in C.

0.4 Question. What are the optimal hypothesis for the conclusions of Theorems 0.1 and 0.2?

0.5 Question. Does C have an independent characterization?

We imagine that the following is true for stronger mice, although we don’t know how much stronger.

Theorem 0.6. Suppose that there exists a Woodin cardinal, and u_{2} = ω_{2}. If x is a real and C(x, ω_{2}) contains a club, then C(x^{#}, ω_{2}) contains a club.

Proof. If δ is a Woodin cardinal and u_{2} = ω_{2}, then j[ω_{2}] is the same for all ℚ_{<δ}-generics, and so for each real x, C(x, ω_{2}) is either club or nonstationary. Fix a real x, and suppose that C(x, ω_{2}) is club. Then x ∈ L[j[ω_{2}]]. But then there is a nontrivial elementary embedding from L_{ω_{2}^{V}} [x] to L_{j(ω_{2}^{V} )}[x] in L[j[ω_{2}]], so x^{#} ∈ L[j[ω_{2}]] and thus C(x^{#}, ω_{2}) is club.

A simple analysis of Shelah’s forcing to make the nonstationary ideal on ω_{1} (NS_{ω_{1}}) saturated [3] shows that it cannot add any real from the ground model to C. Since the saturation of NS_{ω_{1}} plus the existence of a measurable cardinal implies that u_{2} = ω_{2} [5], we have the following corollary of Theorem 0.6.

Corollary 0.7. If there exist proper class many Woodin cardinals, then the set C is closed under sharps.

References

[1] M. Gitik, Nonsplitting subset of P_{κ}(κ^{+}), J. Symbolic Logic 50 (1985), no. 4, 881–894

[2] P. Larson, The stationary tower. Notes on a course by W. Hugh Woodin, American Mathematical Society University Lecture Series 32, 2004 [3] S. Shelah, Proper and improper forcing, second edition. Perspectives in

Mathematical Logic. Springer-Verlag, Berlin, 1998

[4] W.H. Woodin, AD and the uniqueness of the supercompact measures on P_{ω_{1}}(λ), Cabal seminar 79–81, 67–71, Lecture Notes in Math., 1019, Springer, Berlin, 1983.

[5] W.H. Woodin, The axiom of determinacy, forcing axioms, and the nonstationary ideal, DeGruyter Series in Logic and Its Applications, vol. 1, 1999