Generic Absoluteness Revisited

TL;DR

The paper extends the landscape of generic absoluteness by linking Recurrence Axioms with Laver-generic large cardinals and the Ground Axiom. It generalizes Viale’s Σ_2-absoluteness from MM^{++} with Woodin cardinals to broader forcing classes via (P, H(κ))_{Σ_2} RcA^+ combined with tightly P-Laver-generically huge κ, while showing this framework often contradicts GA. It also develops a rich hierarchy of restricted Recurrence Axioms and Maximality Principles, clarifying when RcA^+ aligns with MP and when higher levels split, and establishes both Σ_2 and related absoluteness results under Laver-genericity, sometimes with weaker consistency strength than full Laver-hugeness. The work highlights delicate interactions between forcing axioms, grounds, and large cardinals, and poses open questions about the precise boundaries and potential equivalences between Laver-generated frameworks and standard forcing axioms.

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 $\mathcal{H}(\aleph_2)^{\mathsf{V}}\prec_{Σ_2}\mathcal{H}(\aleph_2)^{\mathsf{V}[\mathbb{G}]}$ 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 $(\mathcal{P},\mathcal{H}(κ))_{Σ_2}$-${\sf RcA}^+$ (which is a fragment of a reformulation of the Maximality Principle for $\mathcal{P}$ and $\mathcal{H}(κ)$), 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 $(\mathcal{P},\mathcal{H}(κ))_Γ$-${\sf RcA}^+$ can be different from the corresponding fragments of Maximality Principle ${\sf MP}(\mathcal{P},\mathcal{H}(κ))_Γ$ for $Γ=Π_2$.