Displaying similar documents to “A note on strong compactness and resurrectibility”

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...

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.

Indestructibility, strong compactness, and level by level equivalence

Arthur W. Apter (2009)

Fundamenta Mathematicae

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We show the relative consistency of the existence of two strongly compact cardinals κ₁ and κ₂ which exhibit indestructibility properties for their strong compactness, together with level by level equivalence between strong compactness and supercompactness holding at all measurable cardinals except for κ₁. In the model constructed, κ₁'s strong compactness is indestructible under arbitrary κ₁-directed closed forcing, κ₁ is a limit of measurable cardinals, κ₂'s strong compactness is indestructible...

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.

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.

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.

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.

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.

Some Remarks on Tall Cardinals and Failures of GCH

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.

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...

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...