Beyond X \ Absoluteness

There have been many generalizations of Shoenfield's Theorem on the absoluteness of S sentences between uncountable transitive models of ZFC. One of the strongest versions currently known deals with Sf absoluteness conditioned on CH. For a variety of reasons, from the study of inner models and from simply combinatorial set theory, the question of whether conditional S2 absoluteness is possible at all, and if so, what large cardinal assumptions are involved and what sentence(s) might play the role of CH, are fundamental questions. This article investigates the possiblities for S2 absoluteness by extending the connections between determinacy hypotheses and absoluteness hypotheses.