Mathematical logic seminar – Oct 30 2012

Time: 12:00 – 13:20

Room: Wean Hall 7201

Speaker: Chris Lambie-Hanson

Department of Mathematical Sciences

Carnegie Mellon University

Title: Covering Matrices and Squares, Part II

Abstract:

Covering matrices were introduced by Matteo Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. In particular, he showed that PFA implies that certain covering matrices exhibit strong covering and reflection properties. In this series of talks, I will construct counterexamples to these covering and reflection properties and investigate their relationships with square principles. This will lead to an examination of a variety of square principles intermediate between $\square_\kappa$ and $\square(\kappa^+)$. In the first lecture, I will introduce the notion of a covering matrix and present results about the existence of certain types of $\kappa$-covering matrices for $\kappa^+$.

In the first lecture, we showed that the existence of transitive, normal, uniform covering kappa-covering matrices for $\kappa^+$ follows from $\square_{\kappa, < \kappa}$ (but not from weak square).

In the second lecture, we will show that the converse fails by constructing a model in which there is a transitive, normal, uniform $\kappa$-covering matrix for $\kappa^+$ but in which $\square_{\kappa, <\kappa}$ fails. If time permits, we will begin a discussion of Todorcevic‘s rho-functions on square sequences and their use in constructing covering matrices.