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6 Bulletin of the Polish Academy of Sciences. Mathematics

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Mixed Levels of Indestructibility

Apter A. W.

Bulletin of the Polish Academy of Sciences. Mathematics

2015

Vol. 63, no. 2

113--122

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 Cohen subset of κ.

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

Apter A. W.

Bulletin of the Polish Academy of Sciences. Mathematics

2015

Vol. 63, no. 3

185--194

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.

The Wholeness Axioms and the Class of Supercompact Cardinals

Apter A. W.

Bulletin of the Polish Academy of Sciences. Mathematics

2012

Vol. 60, no 2

101--111

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

A note on indestructibility and strong compactness

Apter A. W.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 3-4

191-197

If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ+, ∞)-distributive and λ is 2λ supercompact, then by a result of Apter and Hamkins [J. Symbolic Logic 67 (2002)], {δ < κ | δ is δ+ strongly compact yet δ is not δ+ supercompact} must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in which no cardinal δ > κ is 2δ = δ+ supercompact, κ's supercompactness is indestructible under κ-directed closed forcing which is also (κ+, ∞)-distributive, and for every measurable cardinal δ, δ is δ+ strongly compact if δ is δ+ supercompact.

A reduction in consistency strength for universal indestructibility

Apter A. W., Sargsyan G.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 1

1-6

We show how to reduce the assumptions in consistency strength used to prove several theorems on universal indestructibility.

Universal indestructibility is consistent with two strongly compact cardinals

Apter A. W.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 2

131-135

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.