STRUCTURE THEORY OF<i>L</i>(ℝ,<i>μ</i>) AND ITS APPLICATIONS

Trang, Nam

doi:10.1017/jsl.2014.65

Cited by 7 publications

(5 citation statements)

References 12 publications

(55 reference statements)

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“…In fact, a similar result holds for ω α Woodin cardinals and determinacy of analytic games of length ω 1+α for all 1 < α < ω and all limit ordinals ω ≤ α < ω 1 . For details see Theorems 2.2.1 and 2.3.11 in [Tr13], as well as [Tr14] and [Tr15].…”

Section: Introductionmentioning

confidence: 99%

The Consistency Strength of Long Projective Determinacy

Aguilera¹,

Müller²

2019

J. symb. log.

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We determine the consistency strength of determinacy for projective games of length ω 2 . Our main theorem is that Π 1 n+1 -determinacy for games of length ω 2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn(A), the canonical inner model for n Woodin cardinals constructed over A, satisfies A = R and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this.We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω 2 with payoff in R Π 1 1 or with σ-projective payoff. The pointclass of all σ-projective sets is the smallest pointclass closed under complements, countable unions, and projections, where countable unions refer to sets which are subsets of the same product space. Moreover, we as usual identify R and ω ω.2 In fact, an argument as in [AMS, Proposition 2.7] with a more careful analysis of the complexity of the payoff sets (using projective determinacy) shows that these two determinacy hypotheses are equivalent. 3 We would like to thank the referee for asking whether (4) and (5) could be added to Corollary 1.2; see also [AgMu].

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Section: Introductionmentioning

confidence: 99%

The Consistency Strength of Long Projective Determinacy

Aguilera¹,

Müller²

2019

J. symb. log.

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“…It comes down to defining T M as in Definition 2.2. Working in the 20 , where f is a surjection from R onto M, we need to see that the genericity iteration that defines T M terminates in less than Θ many steps. Suppose not, letting T ∈ N be the corresponding tree of length Θ + 1.…”

Section: The Maximal Model Of "Admentioning

confidence: 99%

“…, where G ⊆ Col(ω, R) 19 By results of [10], M Ω,♯ 1 generically interprets Ω for (P, Σ) being a G-Ω-suitable pair or a hod pair with Σ having branch condensation and is Ω-fullness preserving. 20 By "Σ", we mean the set {(T , β) :…”

Section: The Maximal Model Of "Admentioning

confidence: 99%

“…The first and most important example is X = R. The theory ZFC + "there is a measurable cardinal" is equiconsistent with the theory ZF + DC + "ω 1 is R-strongly compact." (For a proof of the forward direction, see Trang [20]. The reverse direction is proved by noting that ω 1 is ω 1 -strongly compact, hence measurable, and considering an inner model L(µ) where µ is a measure on ω 1 .…”

Section: Introductionmentioning

confidence: 99%

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Determinacy from strong compactness of $ω_1$

Trang¹,

Wilson²

2016

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In the absence of the Axiom of Choice, the "small" cardinal ω 1 can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say that ω 1 is Xstrongly compact (where X is any set) if there is a fine, countably complete measure on ℘ ω 1 (X). Working in ZF + DC, we prove that the ℘(ω 1 )-strong compactness and ℘(R)-strong compactness of ω 1 are equiconsistent with AD and AD R + DC respectively, where AD denotes the Axiom of Determinacy and AD R denotes the Axiom of Real Determinacy. The ℘(R)-supercompactness of ω 1 is shown to be slightly stronger than AD R + DC, but its consistency strength is not computed precisely. An equiconsistency result at the level of AD R without DC is also obtained.

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“…[17], [15], [16] made some progress in resolving the conjecture by exploring consistency strength and structural consequences of various fragments of supercompactness of ω 1 .…”

Section: Introductionmentioning

confidence: 99%

On supercompactness of $ω_1$

Ikegami,

Trang

2019

Preprint

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This paper studies structural consequences of supercompactness of ω1 under ZF. We show that the Axiom of Dependent Choice (DC) follows from "ω1 is supercompact". "ω1 is supercompact" also implies that AD + , a strengthening of the Axiom of Determinacy (AD), is equivalent to AD R . It is shown that "ω1 is supercompact" does not imply AD. The most one can hope for is Suslin co-Suslin determinacy. We show that this follows from "ω1 is supercompact" and Hod Pair Capturing (HPC), an inner-model theoretic hypothesis that imposes certain smallness conditions on the universe of sets. "ω1 is supercompact" on its own implies that every Suslin co-Suslin set is the projection of a determined (in fact, homogenously Suslin) set. "ω1 is supercompact" also implies all sets in the Chang model have all the usual regularity properties, like Lebesgue measurability and the Baire property.

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STRUCTURE THEORY OFL(ℝ,μ) AND ITS APPLICATIONS

Cited by 7 publications

References 12 publications

The Consistency Strength of Long Projective Determinacy

The Consistency Strength of Long Projective Determinacy

Determinacy from strong compactness of $ω_1$

On supercompactness of $ω_1$

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