Displaying similar documents to “Some Remarks on Tall Cardinals and Failures of GCH”

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.

Indestructible Strong Compactness and Level by Level Equivalence with No Large Cardinal Restrictions

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.

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...

Indestructibility, strongness, and level by level equivalence

Arthur W. Apter (2003)

Fundamenta Mathematicae

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We construct a model in which there is a strong cardinal κ whose strongness is indestructible under κ-strategically closed forcing and in which level by level equivalence between strong compactness and supercompactness holds non-trivially.

The Wholeness Axioms and the Class of Supercompact Cardinals

Arthur W. Apter (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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We show that certain relatively consistent structural properties of the class of supercompact cardinals are also relatively consistent with the Wholeness Axioms.

Level by Level Inequivalence, Strong Compactness, and GCH

Arthur W. Apter (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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We construct three models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In the first two models, below the supercompact cardinal κ, there is a non-supercompact strongly compact cardinal. In the last model, any suitably defined ground model Easton function is realized.

More Easton theorems for level by level equivalence

Arthur W. Apter (2012)

Colloquium Mathematicae

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We establish two new Easton theorems for the least supercompact cardinal that are consistent with the level by level equivalence between strong compactness and supercompactness. These theorems generalize Theorem 1 in our earlier paper [Math. Logic Quart. 51 (2005)]. In both our ground model and the model witnessing the conclusions of our present theorems, there are no restrictions on the structure of the class of supercompact cardinals.

Mixed Levels of Indestructibility

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...

A generic theorem in cardinal function inequalities

Alejandro Ramírez-Páramo (2008)

Colloquium Mathematicae

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We establish a general technical result, which provides an algorithm to prove cardinal inequalities and relative versions of cardinal inequalities.

Universal Indestructibility is Consistent with Two Strongly Compact Cardinals

Arthur W. Apter (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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We show that universal indestructibility for both strong compactness and supercompactness is consistent with the existence of two strongly compact cardinals. This is in contrast to the fact that if κ is supercompact and universal indestructibility for either strong compactness or supercompactness holds, then no cardinal λ > κ is measurable.

Singular Failures of GCH and Level by Level Equivalence

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.