Forcing Axioms, the Uniformization and the Basis Property

Stefan Hoffelner

Published 2020-01-22Version 1

We show that there are models of MA where the boldface $\Sigma^1_3$-uniformization property holds. Further we show that BPFA and the assertion $\aleph_1$ is accessible to reals outright implies that the boldface $\Sigma^1_3$-uniformization property is true. Further we show that MA and the $\Pi^1_3$-basis property is consistent and that BPFA and $\aleph_1$ is accessible to reals outright implies that the $\Pi^1_3$-basis property is true.