Beyond $\underTildeΣ^2_1$ absoluteness

Abstract

There have been many generalizations of Shoenfield's Theorem on the absoluteness of $Σ^1_2$ sentences between uncountable transitive models of $\mathrm{ZFC}$. One of the strongest versions currently known deals with $Σ^2_1$ absoluteness conditioned on $\mathrm{CH}$. For a variety of reasons, from the study of inner models and from simply combinatorial set theory, the question of whether conditional $Σ^2_2$ absoluteness is possible at all, and if so, what large cardinal assumptions are involved and what sentence(s) might play the role of $\mathrm{CH}$, are fundamental questions. This article investigates the possiblities for $Σ^2_2$ absoluteness by extending the connections between determinacy hypotheses and absoluteness hypotheses.