Incompatible bounded category forcing axioms

Abstract

We introduce bounded category forcing axioms for well-behaved classes $\Gamma$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{{\lambda }_{\Gamma }^{+}}$ modulo forcing in $\Gamma$, for some cardinal ${\lambda }_{\Gamma }$ naturally associated to $\Gamma$. These axioms naturally extend projective absoluteness for arbitrary set-forcing--in this situation ${\lambda }_{\Gamma }=\omega$--to classes $\Gamma$ with ${\lambda }_{\Gamma }>\omega$. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms, but can be forced under mild large cardinal assumptions on $V$. We also show the existence of many classes $\Gamma$ with ${\lambda }_{\Gamma }={\omega }_1$, and giving rise to pairwise incompatible theories for $H_{{\omega }_2}$.

arXiv admin note: substantial text overlap with arXiv:1805.08732