# 2014 seminar talk: The Tree Property for $\omega_2$ and Bounded Forcing Axioms

Talk held by Victor Torres Perez (TU Wien) at the KGRC seminar on 2014-06-26.

### Abstract

We prove that the Tree Property for $\omega_2$ together with $\mathrm{BPFA}(\omega_1)$ is equiconsistent with the existence of a weakly compact cardinal. Also, we prove that the tree property for $\omega_2$ together with $\mathrm{BPFA}$ is equiconsistent with the existence of a weakly compact cardinal which is also reflecting. Similarly, we show that the Special Tree Property for $\omega_2$ together with $\mathrm{BPFA}(\omega_1)$ is equiconsistent with the existence of a Mahlo cardinal and the special tree property for $\omega_2$ together with $\mathrm{BPFA}$ is equiconsistent with the existence of a Mahlo cardinal which is also reflecting.

This is a joint work with Sy Friedman.