Forcing when there are large cardinals: An introduction
Sy-David Friedman (2009)
Acta Universitatis Carolinae. Mathematica et Physica
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Sy-David Friedman (2009)
Acta Universitatis Carolinae. Mathematica et Physica
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Sy-David Friedman (2010)
Acta Universitatis Carolinae. Mathematica et Physica
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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, 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.
Menachem Magidor (1978)
Fundamenta Mathematicae
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Arthur Apter (1984)
Fundamenta Mathematicae
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Arthur W. Apter (2015)
Bulletin of the Polish Academy of Sciences. Mathematics
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Starting from a supercompact cardinal κ, we force and construct a model in which κ is both the least strongly compact and least supercompact cardinal and κ exhibits mixed levels of indestructibility. Specifically, κ 's strong compactness, but not its supercompactness, is indestructible under any κ -directed closed forcing which also adds a Cohen subset of κ. On the other hand, in this model, κ 's supercompactness is indestructible under any κ -directed closed forcing which does not add...
Aleksandar Jovanović (1977)
Zbornik Radova
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Arthur Apter, James Henle (1991)
Fundamenta Mathematicae
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Todd Eisworth (2003)
Fundamenta Mathematicae
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We investigate the problem of when ≤λ-support iterations of < λ-complete notions of forcing preserve λ⁺. We isolate a property- properness over diamonds-that implies λ⁺ is preserved and show that this property is preserved by λ-support iterations. Our condition is a relative of that presented by Rosłanowski and Shelah in [2]; it is not clear if the two conditions are equivalent. We close with an application of our technology by presenting a consistency result on uniformizing colorings...
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.
F. Drake (1970)
Fundamenta Mathematicae
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