March 15
Chris Lambie-Hanson,
Czech Academy of Sciences
Squares, ultrafilters and forcing axioms
A uniform ultrafilter $U$ over a cardinal $\kappa > \omega_1$ is called indecomposable if, whenever $\lambda \lt\kappa$ and $f:\kappa \rightarrow \lambda$, there is a set $X \in U$ such that $f[X]$ is countable. Indecomposability is a natural weakening of $\kappa$-completeness and has a number of implications for, e.g., the structure of ultraproducts. In the 1980s, Sheard answered a question of Silver by proving the consistency of the existence of an inaccessible but not weakly compact cardinal carrying an indecomposable ultrafilter. Recently, however, Goldberg proved that this situation cannot hold above a strongly compact cardinal: If $\lambda$ is strongly compact and $\kappa \geq \lambda$ carries an indecomposable ultrafilter, then $\kappa$ is either measurable or a singular limit of countably many measurable cardinals. We prove that the same conclusion follows from the Proper Forcing Axiom, thus adding to the long list of statements first shown to hold above a strongly compact or supercompact cardinal and later shown also to follow from PFA. Time permitting, we will employ certain indexed square principles to prove that our results are sharp. This is joint work with Assaf Rinot and Jing Zhang.