|Time:|| 3:30pm - 4:30 pm
Wean Hall 8220
Department of Mathematics
A forcing axiom deciding the generalized Souslin Hypothesis
Given a regular, uncountable cardinal κ, it is often desirable to be able to construct objects of size κ+ using approximations of size less than κ. Historically, such constructions have often been carried out with the help of a (κ, 1)-morass and/or a ◇(κ)-sequence. We present a framework for carrying out such constructions using ◇(κ) and a weakening of Jensen's principle ◻κ. Our framework takes the form of a forcing axiom, SDFA(Pκ). We show that SDFA(Pκ) follows from the conjunction of ◇(κ) and our weakening of ◻κ and, if κ is the successor of an uncountable cardinal, that SDFA(Pκ) is in fact equivalent to this conjunction. We also show that, for an infinite cardinal λ, SDFA(Pλ+) implies the existence of a λ+-complete λ++-Souslin tree. This implies that, if λ is an uncountable cardinal, 2λ = λ+, and Souslin's Hypothesis holds at λ++, then λ++ is a Mahlo cardinal in L, improving upon an old result of Shelah and Stanley. This is joint work with Assaf Rinot.