A family of covering properties

In the first part of this paper I present the main results of my Ph.D. thesis: several proofs of the singular cardinal hypothesis $\SCH$ are presented assuming either a strongly compact cardinal or the proper forcing axiom $\PFA$. To this aim I introduce a family of covering properties which imply both $\SCH$ and the failure of various forms of square. In the second part of the paper I apply these covering properties and other similar techniques to investigate models of strongly compact cardinals or of strong forcing axioms like $\MM$ or $\PFA$. In particular I show that if $\MM$ holds and all limit cardinals are strong limit, then any inner model $W$ with the same cardinals has the same ordinals of cofinality at most $\aleph_1$.

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Published 1 January 2008