Proper forcing and remarkable cardinals II

Abstract

Abstract The current paper proves the results announced in [5]. We isolate a new large cardinal concept, “remarkability.” Consistencywise, remarkable cardinals are between ineffable and ω -Erdös cardinals. They are characterized by the existence of “0 # -like” embeddings; however, they relativize down to L . It turns out that the existence of a remarkable cardinal is equiconsistent with L (ℝ) absoluteness for proper forcings. In particular, said absoluteness does not imply determinacy.