Forcing Axioms, Approachability, and Stationary Set Reflection

Cox, Sean

doi:10.1017/jsl.2020.4

Cited by 9 publications

(28 citation statements)

References 31 publications

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“…The purpose of the present note is to prove the following variant of Theorem 1.1 involving Π Cox [2] proved that DRP IC is strictly stronger than DRP internal . This was obtained by forcing over a model of a strong forcing axiom in a way that preserved DRP internal while killing DRP IC (in fact killing RP IC ; the argument owed much to Krueger [10]).…”

Section: Introductionmentioning

confidence: 93%

Section: Preliminariesmentioning

confidence: 99%

“…We consider variants of the Diagonal Reflection Principle introduced in Cox [4] and [2]. We use the following definition, which by Cox-Fuchs [5] is equivalent to the definitions from [4] and [2]: Definition 2.4. DRP internal asserts that for every sufficiently large regular θ, there are stationarily many W ∈ ℘ ω 2 (H θ ) such that:…”

Section: Preliminariesmentioning

confidence: 99%

“…4 is simply the diagonal version of an internal variant of WRP introduced in Fuchino-Usuba [8] (see Cox [2] for a discussion).…”

Section: Preliminariesmentioning

confidence: 99%

“…Stationary Logic) substructure of size ≤ ω 1 " (LST) carries large cardinal consistency strength. 2 The quoted statement above is now typically called the Löwenheim-Skolem-Tarski (LST) property of Stationary Logic. 3 Fuchino et al recently proved that LST is equivalent to a version of the Diagonal Reflection Principle introduced in Cox [4]: Theorem 1.1 (Fuchino-Maschio-Sakai [7]).…”

Section: Introductionmentioning

confidence: 99%

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The $Π^1_1 \! \! \downarrow$ Löwenheim-Skolem-Tarski property of Stationary Logic

Cox

2020

Preprint

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proved that the Löwenheim-Skolem-Tarski (LST) property of Stationary Logic is equivalent to the Diagonal Reflection Principle on internally club sets (DRPIC) introduced in [4]. We prove that the restriction of the LST property to (downward) reflection of Π 1 1 formulas, which we call the Π 1 1 ↓-LST property, is equivalent to the internal version of DRP from [2]. Combined with results from [2], this shows that the Π 1 1 ↓-LST Property for Stationary Logic is strictly weaker than the full LST Property for Stationary Logic, though if CH holds they are equivalent.

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Section: Introductionmentioning

confidence: 93%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…4 is simply the diagonal version of an internal variant of WRP introduced in Fuchino-Usuba [8] (see Cox [2] for a discussion).…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The $Π^1_1 \! \! \downarrow$ Löwenheim-Skolem-Tarski property of Stationary Logic

Cox

2020

Preprint

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Forcing axioms and the complexity of non-stationary ideals

Cox

Lücke

2022

Monatsh Math

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We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $$\omega _2$$ ω 2 and its restrictions to certain cofinalities. Our main result shows that the strengthening $$\mathrm{{MM}}^{++}$$ MM + + of Martin’s Maximum does not decide whether the restriction of the non-stationary ideal on $$\omega _2$$ ω 2 to sets of ordinals of countable cofinality is $$\Delta _1$$ Δ 1 -definable by formulas with parameters in $$\mathrm{{H}}(\omega _3)$$ H ( ω 3 ) . The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on $$\omega _2$$ ω 2 and strong forcing axioms that are compatible with $$\mathrm{{CH}}$$ CH . Finally, we answer a question of S. Friedman, Wu and Zdomskyy by showing that the $$\Delta _1$$ Δ 1 -definability of the non-stationary ideal on $$\omega _2$$ ω 2 is compatible with arbitrary large values of the continuum function at $$\omega _2$$ ω 2 .

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The Diagonal Strong Reflection Principle and Its Fragments

Cox¹,

Fuchs

2023

J. symb. log.

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A diagonal version of the strong reflection principle is introduced, along with fragments of this principle associated with arbitrary forcing classes. The relationships between the resulting principles and related principles, such as the corresponding forcing axioms and the corresponding fragments of the strong reflection principle, are analyzed, and consequences are presented. Some of these consequences are “exact” versions of diagonal stationary reflection principles of sets of ordinals. We also separate some of these diagonal strong reflection principles from related axioms.

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Forcing Axioms, Approachability, and Stationary Set Reflection

Cited by 9 publications

References 31 publications

The $Π^1_1 \! \! \downarrow$ Löwenheim-Skolem-Tarski property of Stationary Logic

The $Π^1_1 \! \! \downarrow$ Löwenheim-Skolem-Tarski property of Stationary Logic

Forcing axioms and the complexity of non-stationary ideals

The Diagonal Strong Reflection Principle and Its Fragments

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