Determinacy from strong reflection

Steel, John R.; Zoble, Stuart

doi:10.1090/s0002-9947-2013-06058-8

Cited by 5 publications

(13 citation statements)

References 24 publications

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“…. For this it's enough to show µ ⊆ F. 10 We note that there is a canonical nameṄ for N and the proof above gives a condition of the form (0, U ) forcing thatṄ has ω 2 Woodin cardinals.…”

Section: The Equiconsistency and Structure Theory Of L(r µ)mentioning

confidence: 96%

“…Contradiction. 10 Letting λ = sup i,j δ i j and γ i = sup j<ω δ i j , by the construction of N , there is a H ⊆ Col(ω, < λ) generic over N such that R * H = R V . To see this, it suffices to see that every…”

Section: The Equiconsistency and Structure Theory Of L(r µ)mentioning

confidence: 99%

“…Finally, with regard to constructing L(R, µ) in a core model induction, the following is still open (cf. [10]), where NS is the nonstationary ideal on ω 1 .…”

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confidence: 99%

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STRUCTURE THEORY OFL(ℝ,μ) AND ITS APPLICATIONS

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2015

J. symb. log.

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In this paper, we explore the structure theory ofL(ℝ,μ) under the hypothesisL(ℝ,μ) ⊧ “AD +μis a normal fine measure on” and give some applications. First we show that “ ZFC + there existω2Woodin cardinals”1has the same consistency strength as “ AD +ω1is ℝ-supercompact”. During this process we show that ifL(ℝ,μ) ⊧ AD then in factL(ℝ,μ) ⊧ AD+. Next we prove important properties ofL(ℝ,μ) including Σ1-reflection and the uniqueness ofμinL(ℝ,μ). Then we give the computation of full HOD inL(ℝ,μ). Finally, we use Σ1-reflection and ℙmaxforcing to construct a certain ideal on(or equivalently onin this situation) that has the same consistency strength as “ZFC+ there existω2Woodin cardinals.”

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Section: The Equiconsistency and Structure Theory Of L(r µ)mentioning

confidence: 96%

Section: The Equiconsistency and Structure Theory Of L(r µ)mentioning

confidence: 99%

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STRUCTURE THEORY OFL(ℝ,μ) AND ITS APPLICATIONS

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2015

J. symb. log.

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“…N ); 22 We abuse notation here, and will continue to do so in the future. Technically, we should write F ↾HC.…”

Section: Brief Remarks On S-constructionsmentioning

confidence: 99%

PFA and guessing models

Trang

2016

Isr. J. Math.

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This paper explores the consistency strength of The Proper Forcing Axiom (PFA) and the theory (T) which involves a variation of the Viale-Weiß guessing hull principle. We show that (T) is consistent relative to a supercompact cardinal. The main result of the paper is Theorem 0.2, which implies that the theory "AD R + Θ is regular" is consistent relative to (T) and to PFA. This improves significantly the previous known best lower-bound for consistency strength for (T) and PFA, which is roughly "AD R + DC".satisfying the hypothesis of the previous sentence is called κ-approximated by X. So a hull X is κ-guessing if whenever a ∈ X and whenever b ⊆ a is κ-approximated by X, then b is X-guessed.In this paper, we study the strength of the following theories • The Proper Forcing Axiom (PFA);• (T): there is a cardinal λ ≥ 2 ℵ 2 such that the set {X ≺ (H λ ++ , ∈) | |X| = ℵ 2 , X ω ⊆ X, ω 2 ⊂ X, and X is ω 2 -guessing } is stationary.Guessing models in [26] are ω 1 -guessing in the above notations. It's not clear that the theory (T) is consistent with PFA (in contrast to Viale-Weiß principle ISP(ω 2 ), which asserts the existence of stationary many ω 1 -guessing models of size ℵ 1 of H λ for all sufficiently large λ). However, it's possible that (T) is a consequence of or at least consistent with a higher analog of PFA.The outline of the paper is as follows. In section 1, we review some AD + facts that we'll be using in this paper. In section 2, using a Mitchell-style forcing, we prove Theorem 0.1. Con(ZFC + there is a supercompact cardinal) ⇒ Con(T).Of course, it is well-known that PFA is consistent relative to the existence of a supercompact cardinal. Theorem 0.4 suggests that it's reasonable to expect PFA and (T) are equiconsistent.Recall, for an infinite cardinal λ, the principle λ asserts the existence of a sequence C α | α < λ + such that for each α < λ + ,

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“…In this language, [31, Theorems 9.78 and 9.84] study (H2, N3)-mouse reflection; cf. also [28]. We shall be concerned with (K2, oo)-mouse reflection in what follows.…”

Section: J(x)mentioning

confidence: 99%

Woodin's axiom (*), bounded forcing axioms, and precipitous ideals on ω₁

Claverie

Schindler

2012

J. symb. log.

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If the Bounded Proper Forcing Axiom BPFA holds, then Mouse Reflection holds at ℵ2 with respect to all mouse operators up to the level of Woodin cardinals in the next ZFC-model. This yields that if Woodin's ℙmax axiom (*) holds, then BPFA implies that V is closed under the “Woodin-in-the-next-ZFC-model” operator. We also discuss stronger Mouse Reflection principles which we show to follow from strengthenings of BPFA, and we discuss the theory BPFA plus “NSω1 is precipitous” and strengthenings thereof. Along the way, we answer a question of Baumgartner and Taylor, [2, Question 6.11].

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Determinacy from strong reflection

Abstract: Abstract. The Axiom of Determinacy holds in the inner model L(R) assuming Martin's Maximum for partial orderings of size c.

Cited by 5 publications

References 24 publications

STRUCTURE THEORY OFL(ℝ,μ) AND ITS APPLICATIONS

STRUCTURE THEORY OFL(ℝ,μ) AND ITS APPLICATIONS

PFA and guessing models

Woodin's axiom (*), bounded forcing axioms, and precipitous ideals on ω₁

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Determinacy from strong reflection

Abstract: Abstract. The Axiom of Determinacy holds in the inner model L(R) assuming Martin's Maximum for partial orderings of size c.

Cited by 5 publications

References 24 publications

STRUCTURE THEORY OFL(ℝ,μ) AND ITS APPLICATIONS

STRUCTURE THEORY OFL(ℝ,μ) AND ITS APPLICATIONS

PFA and guessing models

Woodin's axiom (*), bounded forcing axioms, and precipitous ideals on ω1

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Woodin's axiom (*), bounded forcing axioms, and precipitous ideals on ω₁