Relative consistency results via strong compactness
Arthur Apter, James Henle (1991)
Fundamenta Mathematicae
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Arthur Apter, James Henle (1991)
Fundamenta Mathematicae
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F. Drake (1970)
Fundamenta Mathematicae
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Dima Sinapova (2015)
Fundamenta Mathematicae
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We present a new forcing notion combining diagonal supercompact Prikry forcing with interleaved extender based forcing. We start with a supercompact cardinal κ. In the final model the cofinality of κ is ω, the singular cardinal hypothesis fails at κ, and GCH holds below κ. Moreover we define a scale at κ which has a stationary set of bad points in the ground model.
Sy-David Friedman (2009)
Acta Universitatis Carolinae. Mathematica et Physica
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A. Wojciechowska (1972)
Fundamenta Mathematicae
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Arthur W. Apter (2014)
Bulletin of the Polish Academy of Sciences. Mathematics
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We construct a model for the level by level equivalence between strong compactness and supercompactness in which below the least supercompact cardinal κ, there is an unbounded set of singular cardinals which witness the only failures of GCH in the universe. In this model, the structure of the class of supercompact cardinals can be arbitrary.
Sy-David Friedman, Mohammad Golshani (2013)
Fundamenta Mathematicae
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Starting from large cardinals we construct a pair V₁⊆ V₂ of models of ZFC with the same cardinals and cofinalities such that GCH holds in V₁ and fails everywhere in V₂.
Sy-David Friedman (2010)
Acta Universitatis Carolinae. Mathematica et Physica
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Julius Barbanel (1985)
Fundamenta Mathematicae
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Menachem Magidor (1978)
Fundamenta Mathematicae
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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.
Arthur W. Apter (2013)
Bulletin of the Polish Academy of Sciences. Mathematics
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We investigate two global GCH patterns which are consistent with the existence of a tall cardinal, and also present some related open questions.
Arthur W. Apter (2015)
Bulletin of the Polish Academy of Sciences. Mathematics
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We construct a model for the level by level equivalence between strong compactness and supercompactness with an arbitrary large cardinal structure in which the least supercompact cardinal κ has its strong compactness indestructible under κ-directed closed forcing. This is in analogy to and generalizes the author's result in Arch. Math. Logic 46 (2007), but without the restriction that no cardinal is supercompact up to an inaccessible cardinal.