Terminal Absoluteness of Collapse Forcings

TL;DR

Under certain large cardinal assumptions, collapse forcings are sufficient to capture the full extent of generic absoluteness, making them terminal in the study of forcing invariance.

Contribution

The paper proves that collapse forcings are sufficient for projective generic absoluteness under large cardinal hypotheses, showing their terminal role in forcing invariance.

Findings

Collapse forcings can realize any forcing extension under certain conditions.

Under large cardinal hypotheses, projective generic absoluteness for collapse forcings equals that for all forcings.

At low projective levels, the result holds in ZFC without additional hypotheses.

Abstract

Generic absoluteness is the phenomenon that certain truths in the set-theoretic universe remain stable under forcing expansions. A classical result by Kripke asserts that every complete Boolean algebra completely embeds into a countably generated one, implying that any forcing extension can be realised inside one obtained via a collapse forcing. This observation raises a deeper question: are all forcing notions truly necessary when studying projective generic absoluteness, or does a particular class of forcing notions suffice to capture the same level of invariance? Here we show that, under suitable large cardinal hypotheses, projective generic absoluteness for collapse forcings is indeed equivalent to absoluteness for arbitrary forcings; and we discuss the necessity of these hypotheses, showing that at a low projective level the result holds in ZFC. Thus, we reveal the terminality of collapse forcings since they capture the full robustness of the universe under forcing extensions.