Incompatible bounded category forcing axioms

TL;DR

This paper introduces bounded category forcing axioms for well-behaved classes, extending projective absoluteness to higher levels and showing the existence of incompatible theories for certain initial segments of the universe.

Contribution

It defines new bounded category forcing axioms for classes Gamma, extending projective absoluteness beyond set-forcing, and demonstrates the existence of incompatible theories for initial segments of the universe.

Findings

Bounded category forcing axioms can decide the theory of initial segments of the universe.

These axioms extend projective absoluteness to classes with larger lambda_\u03b3.

Existence of many classes Gamma with incompatible theories for H_{omega_2}.

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 $H_{\omega_2}$.