Chang's model and covering properties

Sureson, Claude

doi:10.1016/0168-0072(89)90065-1

Cited by 3 publications

(1 citation statement)

References 9 publications

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“…Kunen [17] showed that C(L ω 1 ω 1 ) fails to satisfy the Axiom of Choice, if we assume the existence of uncountably many measurable cardinals (see Theorem 5.10 below). Sureson [38,39] Let α be an ordinal such that A ∈ L ′ α and ϕ(x, y) is absolute for L ′ α , C(L * ) (see e.g. [19,IV.7.5]).…”

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Inner Models from Extended Logics: Part 1

Kennedy¹,

Magidor

Väänánen³

2020

Preprint

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If we replace first order logic by second order logic in the original definition of Gödel's inner model L, we obtain HOD ([32]). In this paper we consider inner models that arise if we replace first order logic by a logic that has some, but not all, of the strength of second order logic. Typical examples are the extensions of first order logic by generalized quantifiers, such as the Magidor-Malitz quantifier ([23]), the cofinality quantifier ([34]), or stationary logic ([6]). Our first set of results show that both L and HOD manifest some amount of formalism freeness in the sense that they are not very sensitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between L and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals > ℵ 1 of V are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and independent of the cofinality in question. * The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme Mathematical, Foundational and Computational Aspects of the Higher Infinite supported by EPSRC Grant Number EP/K032208/1. The authors are grateful to John Steel, Philip Welch and Hugh Woodin for comments on the results presented here.

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confidence: 99%

Inner Models from Extended Logics: Part 1

Kennedy¹,

Magidor

Väänánen³

2020

Preprint

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Inner models from extended logics: Part 1

2020

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If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model [Formula: see text], we obtain the inner model of hereditarily ordinal definable (HOD) sets [33]. In this paper, we consider inner models that arise if we replace first-order logic by a logic that has some, but not all, of the strength of second-order logic. Typical examples are the extensions of first-order logic by generalized quantifiers, such as the Magidor–Malitz quantifier [24], the cofinality quantifier [35], or stationary logic [6]. Our first set of results show that both [Formula: see text] and HOD manifest some amount of formalism freeness in the sense that they are not very sensitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between [Formula: see text] and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals [Formula: see text] of [Formula: see text] are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and independent of the cofinality in question. We do not know whether this model satisfies the Continuum Hypothesis, assuming large cardinals, but we can show, assuming three Woodin cardinals and a measurable above them, that if the construction is relativized to a real, then on a cone of reals, the Continuum Hypothesis is true in the relativized model.

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Inner Models from Extended Logics: Part 2

Kennedy

Magidor

Väänánen

2020

Preprint

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We introduce a new inner model C(aa) arising from stationary logic. We show that assuming a proper class of Woodin cardinals, or alternatively MM ++ , the regular uncountable cardinals of V are measurable in the inner model C(aa), the theory of C(aa) is (set) forcing absolute, and C(aa) satisfies CH. We introduce an auxiliary concept that we call club determinacy, which simplifies the construction of C(aa) greatly but may have also independent interest. Based on club determinacy, we introduce the concept of aa-mouse which we use to prove CH and other properties of the inner model C(aa).

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Chang's model and covering properties

Cited by 3 publications

References 9 publications

Inner Models from Extended Logics: Part 1

Inner Models from Extended Logics: Part 1

Inner models from extended logics: Part 1

Inner Models from Extended Logics: Part 2

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