Time:  3:30pm  4:30 pm 
Room: 
Wean Hall 8220

Speaker: 
Chris LambieHanson Department of Mathematics BarIlan University 
Title: 
A forcing axiom deciding the generalized Souslin Hypothesis

Abstract: 
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(P_{κ}). We show that SDFA(P_{κ}) follows from the conjunction of ◇(κ) and our weakening of ◻_{κ} and, if κ is the successor of an uncountable cardinal, that SDFA(P_{κ}) is in fact equivalent to this conjunction. We also show that, for an infinite cardinal λ, SDFA(P_{λ+}) implies the existence of a λ^{+}complete λ^{++}Souslin tree. This implies that, if λ is an uncountable cardinal, 2^{λ} = 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. 