Terminal Absoluteness of Collapse Forcings

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.