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 \(\kappa\) is Subcompact then \(\square_\kappa\) fails. We extend Jensen's result to: If \(\kappa\) is \(\alpha_+\)-Subcompact (and \(\kappa\) is at most \(\alpha\)) then \(\square_\alpha\) (indeed \(\square_{\alpha<\kappa}\) fails. Moreover, forcing shows that this is best possible: One can preserve all instances of \(\alpha\)-Subcompactness as well as very LARGE cardinals (like \(\omega\)-Superstrongs) and force \(\square_\alpha\) to hold everywhere not ruled out by the previous result. We found similar results with \(\square_\alpha\) replaced by Stationary Reflection at \(\alpha^+\) (on small cofinalities) and with \(\alpha^+\) Subcompactness replaced by \(\alpha^{++}\) Subcompactness. The proof of this latter result, unlike the proofs of the earlier ones, caused us some worries.
What Andrew and I talked about last week
27.05.2010 15:00 - 16:30
Organiser:
KGRC
Location:
SR 101, 2. St., Währinger Str. 25