Generic Vopěnka’s Principle, remarkable cardinals, and the weak Proper Forcing Axiom

Bagaria, Joan; Gitman, Victoria; Schindler, Ralf

doi:10.1007/s00153-016-0511-x

Cited by 18 publications

(66 citation statements)

References 11 publications

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“…The equivalencies proved in theorem 9 can be seen as uniformizing some of the results of [BGS17]. To avoid the bifurcation there into the two cases, such as getting from gVP(Σ 2 ) either a proper class of remarkable cardinals or a proper class of virtual rank-into-rank cardinals, what was needed was to drop the requirement that λ < j(κ) for the embeddings, and then one gets a pure equivalence as above.…”

Section: Large Cardinal Characterizations Of the Generic Vopěnka Prinmentioning

confidence: 88%

A model of the generic Vopěnka principle in which the ordinals are not Mahlo

Gitman

Hamkins

2018

Arch. Math. Logic

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The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a ∆ 2definable class containing no regular cardinals. In such a model, there can be no Σ 2 -reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.The research of the second author has been supported by grant #69573-00 47 from the CUNY Research Foundation.Commentary concerning this paper can be made at http://jdh.hamkins.org/generic-vopenka-ord-not-mahlo.

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Section: Large Cardinal Characterizations Of the Generic Vopěnka Prinmentioning

confidence: 88%

A model of the generic Vopěnka principle in which the ordinals are not Mahlo

Gitman

Hamkins

2018

Arch. Math. Logic

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“…For any generic elementary embedding j as in the conclusion of Lemma 2.6, the restriction j ↾ V β is a generic elementary embedding from V β to V β , so its critical point is by definition a virtual rank-into-rank cardinal. The proof of Lemma 2.6 is similar to the proof of existence of virtual rank-into-rank cardinals from a related hypothesis by Bagaria, Gitman, and Schindler[2, Theorem 5.4].…”

mentioning

confidence: 82%

“…In Section 2 we will prove Theorems 1.3, 1.4, and 1.5. In Section 3 we will give an application involving the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler [2]. §2.…”

Section: Does the Existence Of A Non-σ 2 -Reflecting Weakly Remarkamentioning

confidence: 99%

“… Schindler [10] originally gave another definition that did not involve forcing but was otherwise more complicated. See Bagaria, Gitman, and Schindler[2, Proposition 2.4] for several equivalent forms of remarkability.…”

mentioning

confidence: 99%

“…If κ is a remarkable cardinal, then for every ordinal > κ and every set x ∈ V there is an ordinal¯ < κ and a generic elementary embedding j : V¯ → V such that j(crit(j)) = κ and having the additional property that x ∈ range(j): see Bagaria, Gitman, and Schindler [2,Propositions 2.3 and 3.2]. The same argument establishes the corresponding fact without the condition¯ < κ for weakly remarkable cardinals.…”

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confidence: 99%

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Weakly Remarkable Cardinals, Erdős Cardinals, and the Generic Vopěnka Principle

Wilson¹

2019

J. symb. log.

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We consider a weak version of Schindler's remarkable cardinals that may fail to be Σ 2 -reflecting. We show that the Σ 2 -reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and we show that the existence of a non-Σ 2 -reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an ω-Erdős cardinal. We give an application involving gVP, the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler. Namely, we show that gVP + "Ord is not ∆ 2 -Mahlo" and gVP(Π 1 ) + "there is no proper class of remarkable cardinals" are both equiconsistent with the existence of a proper class of ω-Erdős cardinals, extending results of Bagaria, Gitman, Hamkins, and Schindler. Remarkability and weak remarkabilityMany large cardinal properties can be defined in terms of elementary embeddings between set-sized structures. For example, extendibility is defined in terms of elementary embeddings between rank initial segments of V , and supercompactness admits a similar characterization by Magidor [9, Theorem 1]. Any large cardinal property defined in this way can be virtualized by weakening the existence of an elementary embedding to the existence of a generic elementary embedding, meaning an elementary embedding that exists in some generic extension of V (and whose domain and codomain are in V .) The large cardinal properties obtained in this way are known as virtual large cardinal properties (see Gitman and Schindler [6].) The first virtual large cardinals to be studied were the virtually supercompact cardinals, also known as the remarkable cardinals: Definition 1.1 (Schindler 1 ). A cardinal κ is remarkable if for every ordinal λ > κ there is an ordinalλ < κ and a generic elementary embedding j : Vλ → V λ such that j(crit(j)) = κ.We will consider a weak form of remarkability obtained by removing the conditionλ < κ, analogous to the weak form of virtual extendibility defined by Gitman and Hamkins [5, Definition 6]. We work in ZFC unless otherwise stated. Definition 1.2. A cardinal κ is weakly remarkable if for every ordinal λ > κ there is an ordinalλ and a generic elementary embedding j : Vλ → V λ such that j(crit(j)) = κ.

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Generic Vopěnka cardinals and models of ZF with few $$\aleph _1$$ ℵ 1 -Suslin sets

Wilson

2019

Arch. Math. Logic

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Generic Vopěnka’s Principle, remarkable cardinals, and the weak Proper Forcing Axiom

Cited by 18 publications

References 11 publications

A model of the generic Vopěnka principle in which the ordinals are not Mahlo

A model of the generic Vopěnka principle in which the ordinals are not Mahlo

Weakly Remarkable Cardinals, Erdős Cardinals, and the Generic Vopěnka Principle

Generic Vopěnka cardinals and models of ZF with few $$\aleph _1$$ ℵ 1 -Suslin sets

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