Incompatible bounded category forcing axioms

Abstract

We introduce bounded category forcing axioms for well-behaved classes [Formula: see text]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [Formula: see text] modulo forcing in [Formula: see text], for some cardinal [Formula: see text] naturally associated to [Formula: see text]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [Formula: see text] — to classes [Formula: see text] with [Formula: see text]. 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 [Formula: see text]. We also show the existence of many classes [Formula: see text] with [Formula: see text] giving rise to pairwise incompatible theories for [Formula: see text].