Making the supercompactness of κ indestructible under κ-directed closed forcing

Laver, Richard

doi:10.1007/bf02761175

Cited by 291 publications

(283 citation statements)

References 4 publications

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“…For this we fix our increasing sequence of supercompact cardinals κ n | n < ω . Let F n | n < ω be a sequence of Laver functions [7] for the κ n s. Definition 2.8. Let R ω be the inverse limit of the sequence R n | n < ω where R n = def Q 0 * Q 1 * · · · * Q n−1 .…”

Section: Preliminaries On the Iterationmentioning

confidence: 99%

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A model of Cummings and Foreman revisited

Unger¹

2014

Annals of Pure and Applied Logic

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Abstract. This paper concerns the model of Cummings and Foreman where from ω supercompact cardinals they obtain the tree property at each ℵn for 2 ≤ n < ω. We prove some structural facts about this model. We show that the combinatorics at ℵ ω+1 in this model depend strongly on the properties of ω 1 in the ground model. From different ground models for the CummingsForeman iteration we can obtain either ℵ ω+1 ∈ I[ℵ ω+1 ] and every stationary subset of ℵ ω+1 reflects or there are a bad scale at ℵω and a non-reflecting stationary subset of ℵ ω+1 ∩ cof(ω 1 ). We also prove that regardless of the ground model a strong generalization of the tree property holds at each ℵn for n ≥ 2.

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Section: Preliminaries On the Iterationmentioning

confidence: 99%

“…Then we let V be the extension of V 0 by the Laver preparation [7] for κ 0 . Note that each κ n is still supercompact in V .…”

Section: A Bad Scale and Non-reflecting Stationary Setmentioning

confidence: 99%

A model of Cummings and Foreman revisited

Unger¹

2014

Annals of Pure and Applied Logic

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“…In an early appeal to the full global reflection properties available at a supercompact cardinal Baumgartner iteratively took care of the emerging proper partial orders along a special diamond-like sequence that anticipates all possibilities. Laver [1978] had formulated such a sequence, the "Laver diamond", toward establishing what has become a useful result for forcing theory; in a forcing extension he made a supercompact cardinal "indestructible" by any further forcing from a substantial, useful class of forcings. PFA became a widely applied forcing axiom, showcasing Shelah's concept, but beyond that, it would itself become a pivotal hypothesis in the large cardinal context.…”

Section: Forcing Consistency Results Through the 1970smentioning

confidence: 99%

Large Cardinals with Forcing

Kanamori¹

2012

Handbook of the History of Logic

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“…We must draw on several facts from the literature. The first is widely known: The second fact is a result of Laver [4].…”

Section: Reflection In Extensions By the Cmb Posetmentioning

confidence: 98%

Stationary sets added when forcing squares

Levine

2018

Arch. Math. Logic

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Current research in set theory raises the possibility that κ,<λ can be made compatible with some stationary reflection, depending on the parameter λ. The purpose of this paper is to demonstrate the difficulty in such results. We prove that the poset S(κ, < λ), which adds a κ,<λ -sequence by initial segments, will also add non-reflecting stationary sets concentrating in any given cofinality below κ. We also investigate the CMB poset, which adds * κ in a slightly different way. We prove that the CMB poset also adds non-reflecting stationary sets, but not necessarily concentrating in any cofinality.

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Making the supercompactness of κ indestructible under κ-directed closed forcing

Cited by 291 publications

References 4 publications

A model of Cummings and Foreman revisited

A model of Cummings and Foreman revisited

Large Cardinals with Forcing

Stationary sets added when forcing squares

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