Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties

Sharpe, I.; Welch, Philip D.

doi:10.1016/j.apal.2011.04.002

Cited by 31 publications

(50 citation statements)

References 16 publications

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“…The CC * from Doebler-Schindler [3] is yet another version which is much stronger and will not be considered here. 5 • "Strong Chang's Conjecture" from Woodin [24] A similar discrepancy appears in the use of the notation CC + in [24] and [15], though we will not deal with either of these versions. The "Strong Chang's Conjecture" of Foreman-Magidor-Shelah [6] is apparently weaker than the "Strong Chang's Conjecture" of Woodin [24].…”

Section: Preliminariesmentioning

confidence: 99%

“…Thus by Steel [17] and Jensen-Steel [11], the Doebler-Schindler version of CC * has consistency strength at least a Woodin cardinal; whereas all the versions of Chang's Conjecture considered in this paper can be forced from a measurable cardinal. [21] CC * Usuba [21] CC * * Torres Perez-Wu [20] CC * Doebler [2] CC * Shelah [16] version in XII Theorem 2.5 Sharpe-Welch [15] SCC Woodin [24] SCC (Def 9.101 part 2)…”

Section: Preliminariesmentioning

confidence: 99%

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Chang’s conjecture and semiproperness of nonreasonable posets

Cox

2018

Monatsh Math

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Let Q denote the poset which adds a Cohen real then shoots a club through the complement of [ω 2 ] ω V with countable conditions. We prove that the version of Strong Chang's Conjecture from [19] implies semiproperness of Q, and that semiproperness of Q-in fact semiproperness of any poset which is sufficiently nonreasonable in the sense of Foreman-Magidor [5]-implies the version of Strong Chang's Conjecture from [24] and [18]. In particular, semiproperness of Q has large cardinal strength, which answers a question of Friedman-Krueger [8]. One corollary of our work is that the version of Strong Chang's Conjecture from [19] does not imply the existence of a precipitous ideal on ω 1 .

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Chang’s conjecture and semiproperness of nonreasonable posets

Cox

2018

Monatsh Math

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“…Note that these cardinal need not be weakly compact and Woodin cardinals are stationary limits of ω 1 -iterable cardinals. Moreover, [29,Lemma 5.2] shows that ω 1 -Erdős cardinals are stationary limits of ω 1 -iterable cardinals. Theorem 1.9.…”

Section: Introductionmentioning

confidence: 99%

Σ₁(κ)-DEFINABLE SUBSETS OF H(κ⁺)

2017

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We study Σ 1 (ω 1 )-definable sets (i.e. sets that are equal to the collection of all sets satisfying a certain Σ 1 -formula with parameter ω 1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ 1 (ω 1 )-definable, the set of all stationary subsets of ω 1 is not Σ 1 (ω 1 )definable and the complement of every Σ 1 (ω 1 )-definable Bernstein subset of ω 1 ω 1 is not Σ 1 (ω 1 )-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ 1 (ω 1 )-definable wellordering of H(ω 2 ) and the existence of a ∆ 1 (ω 1 )-definable Bernstein subset of ω 1 ω 1 . We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ 1 (ω 1 )-definable uniformization of the club filter on ω 1 . Moreover, we prove a perfect set theorem for Σ 1 (ω 1 )-definable subsets of ω 1 ω 1 , assuming that there is a measurable cardinal and the non-stationary ideal on ω 1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin's Pmax-forcing. Finally, we also prove variants of some of these results for Σ 1 (κ)-definable subsets of κ κ, in the case where κ itself has certain large cardinal properties.2010 Mathematics Subject Classification. 03E45, 03E47, 03E55.

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“…By the choice of player I's strategy we get that p q 0 is both weakly amenable, and it's also countably complete by the rules of G κ` `1 pκq (it's even normal). Now Lemma 2.9 of Sharpe and Welch (2011) gives us a set of good indiscernibles I 0 P 0 -measure on κ. Let player II play I 0 in G I pκq.…”

Section: [Strategic]mentioning

confidence: 99%

“…The completely Ramsey cardinals are the cardinals topping the hierarchy defined in Feng (1990). A completely Ramsey cardinal implies the consistency of a Ramsey cardinal, see e.g., Theorem 3.51 in Sharpe and Welch (2011). We are going to use the following characterisation of the completely Ramsey cardinals, which is Lemma 3.49 in Sharpe and Welch (2011).…”

Section: [Strategic]mentioning

confidence: 99%

Games and Ramsey-Like Cardinals

Nielsen¹,

Welch²

2019

J. symb. log.

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We generalise the α-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals α to arbitrary ordinals α, and answer several questions posed in that paper. In particular, we show that α-Ramseys are downwards absolute to the core model K for all α of uncountable cofinality, that strategic ω-Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic α-Ramsey cardinals are equiconsistent with measurable cardinals for all α > ω. We also show that the n-Ramseys satisfy indescribability properties and use them to provide a game-theoretic characterisation of completely ineffable cardinals, as well as establishing further connections between the α-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990), and Sharpe and Welch (2011).

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Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties

Cited by 31 publications

References 16 publications

Chang’s conjecture and semiproperness of nonreasonable posets

Chang’s conjecture and semiproperness of nonreasonable posets

Σ₁(κ)-DEFINABLE SUBSETS OF H(κ⁺)

Games and Ramsey-Like Cardinals

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Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties

Cited by 31 publications

References 16 publications

Chang’s conjecture and semiproperness of nonreasonable posets

Chang’s conjecture and semiproperness of nonreasonable posets

Σ1(κ)-DEFINABLE SUBSETS OF H(κ+)

Games and Ramsey-Like Cardinals

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Σ₁(κ)-DEFINABLE SUBSETS OF H(κ⁺)