A family of covering properties

Abstract

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 1 .