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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.

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.

Level by Level Inequivalence, Strong Compactness, and GCH

Apter A. W.

Bulletin of the Polish Academy of Sciences. Mathematics

2012

Vol. 60, no 3

201--209

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.

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.