The fine structure of the constructible hierarchy

Jensen, Ronald Björn

doi:10.1016/0003-4843(72)90001-0

Cited by 644 publications

(447 citation statements)

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“…Jensen [6] proved that for a cardinal ν, * ν holds if and only if there is a special ν + -Aronszajn tree. So in particular the failure of weak square can be seen as a weakening of the tree property at ν + .…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

A model of Cummings and Foreman revisited

Unger¹

2014

Annals of Pure and Applied Logic

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“…It is a direct consequence of a characterizing reflection property of weakly compact cardinals that if κ is weakly compact, then Refl(κ). Jensen [1972] raised interest in stationary set reflection by getting the converse in L as an initial dividend of his fine structure theory: If V = L, then κ is weakly compact iff Refl(κ). On the other hand, Kunen in 1972Kunen in (cf.…”

Section: Conjecture 15mentioning

confidence: 99%

Large Cardinals with Forcing

Kanamori¹

2012

Handbook of the History of Logic

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“…The κ principle, pronounced "square kappa," was introduced by Jensen in order to study the fine structure of Gödel's constructible universe L [3]. κ and similar principles imply the failure of many of the compactness properties entailed by large cardinals.…”

Section: Introductionmentioning

confidence: 99%

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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The fine structure of the constructible hierarchy

Cited by 644 publications

References 2 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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