Abstract:

A long-standing open question is whether Baumgartner's result Con(supercompact) implies Con(PFA) can be reversed. The traditional approach to this problem is core model theory, which at present cannot yield consistency strengths past Woodin cardinals. In joint work with Peter Holy, based on some earlier work of Neeman, I'll show that it is consistent that there is a proper class of subcompact cardinals with PFA failing in all proper forcing extensions of the universe. This "quasi lower bound" provides some evidence that the consistency strength of PFA is beyond subcompactness. I'll also mention analogous work of Viale-Weig, which is based on entirely different methods.

Generalised Amoeba forcing and measurability

Amoeba for Silver which does not add Cohen reals, and we discuss some applications and open questions about Σ12-measurability. 2

Internality in model theory and category theory

Abstract:

I will explain the model theoretic notion of internality, which provides conditions for the automorphism group of a structure over a reduct to be definable (in the original structure). I will then explain how the assumptions can be stated in a general categorical framework, and state a weak version of the main theorem, that holds in the categorical setting.