The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal

In The Axiom of Determinacy, Forcing Axioms and the Nonstationary Ideal [W1], Woodin constructs the partial order ℙmax, which in the presence of large cardinals yields a forcing extension of L(ℝ) where ZFC holds and the non-stationary ideal on ω1 (hereafter denoted NSω1) is ω2-saturated. The basic analysis of ℙmax forcing over L(ℝ) can be carried out assuming only the Axiom of Determinacy (AD). In the central result of this paper, we show that if one increases slightly the strength of the determinacy assumptions, then Chang's Conjecture—the assertion that every finitary algebra on ω2 has a subalgebra of order type ω1—holds in this extension as well. Specifically, we obtain:Corollary 4.6. Assume AD + V = L(ℝ, μ) + μ is a normal, fine measure on. Chang's Conjecture holds in any ℙmax-generic extension of L(ℝ).This technique for obtaining Chang's Conjecture is fairly general. We [Se] have adapted it to obtain Chang's Conjecture in the model presented by Steel and Van Wesep [SVW] and Woodin [Wl] has adapted it to his ℚmax forcing notion. In each of these models, as in the ℙmax extension, one forces over L(ℝ) assuming AD to obtain ZFC and NSω1 is ω2-saturated.By unpublished results of Woodin, the assumption for Corollary 4.6 is equiconsistent with the existence of ω2 many Woodin cardinals, and hence strictly stronger than ADL(ℝ). One would like to reduce this assumption to ADL(ℝ). Curiously, this reduction is not possible in the arguments for ℚmax or the Steel and Van Wesep model, and the following argument of Woodin suggests why it may not be possible for ℙmax either.

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THE AXIOM OF DETERMINACY, FORCING AXIOMS, AND THE NONSTATIONARY IDEAL (de Gruyter Series in Logic and its Applications 1) By W. HUGH WOODIN: 934 pp., DM298.00, ISBN 3-11-015708-X (Walter de Gruyter, Berlin, 1999).

Bulletin of the London Mathematical Society ◽

10.1017/s0024609301219286 ◽

2001 ◽

Vol 33 (4) ◽

pp. 492-512

Author(s):

P. D. Welch

Keyword(s):

Forcing Axioms ◽

Axiom Of Determinacy ◽

Nonstationary Ideal

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Generic Σ31 absoluteness

Journal of Symbolic Logic ◽

10.2178/jsl/1080938826 ◽

2004 ◽

Vol 69 (1) ◽

pp. 73-80 ◽

Cited By ~ 1

Author(s):

Sy D. Friedman

Keyword(s):

Regular Cardinal ◽

Forcing Axioms ◽

Proper Forcing ◽

Class Forcing ◽

The Universe ◽

Generic Extensions

In this article we study the strength of absoluteness (with real parameters) in various types of generic extensions, correcting and improving some results from [3]. (In particular, see Theorem 3 below.) We shall also make some comments relating this work to the bounded forcing axioms BMM, BPFA and BSPFA.The statement “ absoluteness holds for ccc forcing” means that if a formula with real parameters has a solution in a ccc set-forcing extension of the universe V, then it already has a solution in V. The analogous definition applies when ccc is replaced by other set-forcing notions, or by class-forcing.Theorem 1. [1] absoluteness for ccc has no strength; i.e., if ZFC is consistent then so is ZFC + absoluteness for ccc.The following results concerning (arbitrary) set-forcing and class-forcing can be found in [3].Theorem 2 (Feng-Magidor-Woodin). (a) absoluteness for arbitrary set-forcing is equiconsistent with the existence of a reflecting cardinal, i.e., a regular cardinal κ such that H(κ) is ∑2-elementary in V.(b) absoluteness for class-forcing is inconsistent.We consider next the following set-forcing notions, which lie strictly between ccc and arbitrary set-forcing: proper, semiproper, stationary-preserving and ω1-preserving. We refer the reader to [8] for the definitions of these forcing notions.Using a variant of an argument due to Goldstern-Shelah (see [6]), we show the following. This result corrects Theorem 2 of [3] (whose proof only shows that if absoluteness holds in a certain proper forcing extension, then in L either ω1 is Mahlo or ω2 is inaccessible).

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Increasing u2 by a stationary set preserving forcing

Journal of Symbolic Logic ◽

10.2178/jsl/1231082308 ◽

2009 ◽

Vol 74 (1) ◽

pp. 187-200

Author(s):

Benjamin Claverie ◽

Ralf Schindler

Keyword(s):

Regular Cardinal ◽

Nonstationary Ideal ◽

Stationary Set ◽

Precipitous Ideal ◽

Countable Structure

AbstractWe show that if I is a precipitous ideal on ω1 and if θ > ω1 is a regular cardinal, then there is a forcing ℙ = ℙ(I, θ) which preserves the stationarity of all I-positive sets such that in Vℙ, ⟨Hθ; ∈, I⟩ is a generic iterate of a countable structure ⟨M; ∈, Ī⟩. This shows that if the nonstationary ideal on ω1 is precipitous and exists, then there is a stationary set preserving forcing which increases . Moreover, if Bounded Martin's Maximum holds and the nonstationary ideal on ω1 is precipitous, then .

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