Chris Lambie-Hanson: A forcing axiom deciding the generalized Souslin Hypothesis

Mathematical logic seminar – Oct 3 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Chris Lambie-Hanson
Department of Mathematics
Bar-Ilan University

Title:     A forcing axiom deciding the generalized Souslin Hypothesis

Abstract:

Given a regular, uncountable cardinal $\kappa$, it is often desirable to be able to construct objects of size $\kappa^+$ using approximations of size less than $\kappa$. Historically, such constructions have often been carried out with the help of a $(\kappa,1)$-morass and/or a $\diamondsuit(\kappa)$-sequence.
We present a framework for carrying out such constructions using $\diamondsuit(\kappa)$ and a weakening of Jensen’s $\square_\kappa$. Our framework takes the form of a forcing axiom, $\textrm{SDFA}(\mathcal P_\kappa)$. We show that $\textrm{SDFA}(\mathcal P_κ)$ follows from the conjunction of $\diamondsuit(\kappa)$ and our weakening of $\square_\kappa$ and, if $\kappa$ is the successor of an uncountable cardinal, that $\textrm{SDFA}(\mathcal P_\kappa)$ is in fact equivalent to this conjunction. We also show that, for an infinite cardinal $\lambda$, $\textrm{SDFA}(\mathcal P_{\lambda^+})$ implies the existence of a $\lambda^+$-complete $\lambda^{++}$-Souslin tree. This implies that, if $\lambda$ is an uncountable cardinal, $2^\lambda =\lambda^+$, and Souslin’s Hypothesis holds at $\lambda^{++}$, then $\lambda^{++}$ is a Mahlo cardinal in $L$, improving upon an old result of Shelah and Stanley. This is joint work with Assaf Rinot.

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