Kurt Gödel Research Center

2010 seminar talk: What Andrew and I talked about last week

Talk held by Sy-David Friedman (KGRC) at the KGRC seminar on 2010-05-27.

Abstract

Actually I will only tell you about the mathematics we discussed, saving our discussion of kangaroos for a later time. We looked at the interplay between Square/Stationary Reflection on the one hand, and Large Cardinal Axioms on the other. Solovay showed ages ago that supercompactness kills Square, improved recently by Jensen, who showed that if κ is Subcompact then Square_κ fails. We extend Jensen's result to: If κ is α⁺-Subcompact (and κ is at most α) then Square_α (indeed Square_α,<κ fails. Moreover, forcing shows that this is best possible: One can preserve all instances of α-Subcompactness as well as very LARGE cardinals (like ω-Superstrongs) and force Square_α to hold everywhere not ruled out by the previous result. We found similar results with Square_α replaced by Stationary Reflection at α⁺ (on small cofinalities) and with α⁺ Subcompactness replaced by α⁺⁺ Subcompactness. The proof of this latter result, unlike the proofs of the earlier ones, caused us some worries.