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 $\omega$-Erdos cardinals. They are characterized by the existence of "O$^#$-like" embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L absoluteness for proper forcings. In particular, said absoluteness does not imply $\Pi^1_1$ determinacy.