The Tree Property at $\omega _2$ and Bounded Forcing Axioms
Volume 63 / 2015
Bulletin Polish Acad. Sci. Math. 63 (2015), 207-216
MSC: Primary 03E35; Secondary 03E65.
DOI: 10.4064/ba8038-1-2016
Published online: 19 January 2016
Abstract
We prove that the Tree Property at $\omega _2$ together with $\mathrm {BPFA}$ is equiconsistent with the existence of a weakly compact reflecting cardinal, and if $\mathrm {BPFA}$ is replaced by $\mathrm {BPFA}(\omega _1)$ then it is equiconsistent with the existence of just a weakly compact cardinal. Similarly, we show that the Special Tree Property for $\omega _2$ together with $\mathrm {BPFA}$ is equiconsistent with the existence of a reflecting Mahlo cardinal, and if $\mathrm {BPFA}$ is replaced by $\mathrm {BPFA}(\omega _1)$ then it is equiconsistent with the existence of just a Mahlo cardinal.
