Generic Absoluteness Revisited

TL;DR

This paper explores the relationship between recurrence axioms, large cardinal axioms, and generic absoluteness, showing that certain assumptions imply the negation of the Ground Axiom, contrasting with prior results.

Contribution

It generalizes Viale's theorem to broader classes of posets and demonstrates that these assumptions imply the negation of the Ground Axiom, revealing new interactions between large cardinals and generic absoluteness.

Findings

Certain assumptions imply the negation of the Ground Axiom.

Generalization of Viale's theorem to other classes of posets.

Fragments of Recurrence Axiom can differ from Maximality Principle.

Abstract

The present paper is concerned with the relation between recurrence axioms and Laver-generic large cardinal axioms in light of principles of generic absoluteness and the Ground Axiom. M. Viale proved that Martin's Maximum$^{++}$ together with the assumption that there are class many Woodin cardinals implies $\mathcal{H}(\aleph_2)^{\mathsf{V}}\prec_{\Sigma_2}\mathcal{H}(\aleph_2)^{\mathsf{V}[\mathbb{G}]}$ for a generic $\mathbb{G}$ on any stationary preserving $\mathbb{P}$ which also preserves Bounded Martin's Maximum. We show that a similar but more general conclusion follows from each of $(\mathcal{P},\mathcal{H}(\kappa))_{\Sigma_2}$-${\sf RcA}^+$ (which is a fragment of a reformulation of the Maximality Principle for $\mathcal{P}$ and $\mathcal{H}(\kappa)$), and the existence of the tightly $\mathcal{P}$-Laver-generically huge cardinal. While under "$\mathcal{P}=$ all stationary preserving posets", our results are not very much more than Viale's Theorem, for other classes of posets, "$\mathcal{P}=$ all proper posets" or "$\mathcal{P}=$ all ccc posets", for example, our theorems are not at all covered by his theorem. The assumptions (and hence also the conclusion) of Viale's Theorem are compatible with the Ground Axiom. In contrast, we show that the assumptions of our theorems (for most of the common settings of $\mathcal{P}$ and with a modification of the large cardinal property involved) imply the negation of the Ground Axiom. This fact is used to show that fragments of Recurrence Axiom $(\mathcal{P},\mathcal{H}(\kappa))_\Gamma$-${\sf RcA}^+$ can be different from the corresponding fragments of Maximality Principle ${\sf MP}(\mathcal{P},\mathcal{H}(\kappa))_\Gamma$ for $\Gamma=\Pi_2$.