Category forcings, $M{M}^{+++}$, and generic absoluteness for the theory of strong forcing axioms

Abstract

We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial segment of the universe of height a super compact which is a limit of super compact cardinals is a stationary set preserving partial order which forces $M{M}^{++}$ and collapses its size to become the second uncountable cardinal. Next we argue that any of the known methods to produce a model of $M{M}^{++}$ collapsing a superhuge cardinal to become the second uncountable cardinal produces a model in which the cutoff of the category of stationary set preserving forcings at any rank initial segment of the universe of large enough height is forcing equivalent to a presaturated tower of normal filters. We let $M{M}^{+++}$ denote this statement and we prove that the theory of $L(Or{d}^{{\omega }_1})$ with parameters in $P({\omega }_1)$ is generically invariant for stationary set preserving forcings that preserve $M{M}^{+++}$. Finally we argue that the work of Larson and Asperó shows that this is a next to optimal generalization to the Chang model $L(Or{d}^{{\omega }_1})$ of Woodin's generic absoluteness results for the Chang model $L(Or{d}^{\omega })$. It remains open whether $M{M}^{+++}$ and $M{M}^{++}$ are equivalent axioms modulo large cardinals and whether $M{M}^{++}$ suffices to prove the same generic absoluteness results for the Chang model $L(Or{d}^{{\omega }_1})$.

- to appear on the Journal of the American Mathemtical Society

References in corpus (1)

• Resurrection axioms and uplifting cardinals

Cited by in corpus (1)

• Corona Rigidity