Displaying similar documents to “The consistency strength of the tree property at the double successor of a measurable cardina”

Strong compactness, measurability, and the class of supercompact cardinals

Arthur W. Apter (2001)

Fundamenta Mathematicae

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We prove two theorems concerning strong compactness, measurability, and the class of supercompact cardinals. We begin by showing, relative to the appropriate hypotheses, that it is consistent non-trivially for every supercompact cardinal to be the limit of (non-supercompact) strongly compact cardinals. We then show, relative to the existence of a non-trivial (proper or improper) class of supercompact cardinals, that it is possible to have a model with the same class of supercompact cardinals...

Measurable cardinals and the cofinality of the symmetric group

Sy-David Friedman, Lyubomyr Zdomskyy (2010)

Fundamenta Mathematicae

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Assuming the existence of a P₂κ-hypermeasurable cardinal, we construct a model of Set Theory with a measurable cardinal κ such that 2 κ = κ ⁺ ⁺ and the group Sym(κ) of all permutations of κ cannot be written as the union of a chain of proper subgroups of length < κ⁺⁺. The proof involves iteration of a suitably defined uncountable version of the Miller forcing poset as well as the “tuning fork” argument introduced by the first author and K. Thompson [J. Symbolic Logic 73 (2008)].

The Tree Property at ω₂ and Bounded Forcing Axioms

Sy-David Friedman, Víctor Torres-Pérez (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove that the Tree Property at ω₂ together with BPFA is equiconsistent with the existence of a weakly compact reflecting cardinal, and if BPFA is replaced by BPFA(ω₁) then it is equiconsistent with the existence of just a weakly compact cardinal. Similarly, we show that the Special Tree Property for ω₂ together with BPFA is equiconsistent with the existence of a reflecting Mahlo cardinal, and if BPFA is replaced by BPFA(ω₁) then it is equiconsistent with the existence of just a Mahlo...

The relative consistency of some consequences of the existence of measurable cardinal numbers

K. A. Bowen

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CONTENTSIntroduction..................................................................................................................................................5I. Topological forcing..................................................................................................................................81.1. Preliminaries.......................................................................................................................................81.2. Languages,...

HOD-supercompactness, Indestructibility, and Level by Level Equivalence

Arthur W. Apter, Shoshana Friedman (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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In an attempt to extend the property of being supercompact but not HOD-supercompact to a proper class of indestructibly supercompact cardinals, a theorem is discovered about a proper class of indestructibly supercompact cardinals which reveals a surprising incompatibility. However, it is still possible to force to get a model in which the property of being supercompact but not HOD-supercompact holds for the least supercompact cardinal κ₀, κ₀ is indestructibly supercompact, the strongly...

Some applications of Sargsyan's equiconsistency method

Arthur W. Apter (2012)

Fundamenta Mathematicae

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We apply techniques due to Sargsyan to reduce the consistency strength of the assumptions used to establish an indestructibility theorem for supercompactness. We then show how these and additional techniques due to Sargsyan may be employed to establish an equiconsistency for a related indestructibility theorem for strongness.

Stationary reflection and level by level equivalence

Arthur W. Apter (2009)

Colloquium Mathematicae

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We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional “inner model like” properties. In particular, in this model, the class of Mahlo cardinals reflecting stationary sets is the same as the class of weakly compact cardinals, and every regular Jónsson cardinal is weakly compact. On the other hand, we force and construct a model for the level by level equivalence between strong compactness...

Continuous tree-like scales

James Cummings (2010)

Open Mathematics

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Answering a question raised by Luis Pereira, we show that a continuous tree-like scale can exist above a supercompact cardinal. We also show that the existence of a continuous tree-like scale at ℵω is consistent with Martin’s Maximum.

A note on strong compactness and resurrectibility

Arthur Apter (2000)

Fundamenta Mathematicae

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We construct a model containing a proper class of strongly compact cardinals in which no strongly compact cardinal ĸ is ĸ + supercompact and in which every strongly compact cardinal has its strong compactness resurrectible.