Proper forcing and remarkable cardinals II

Journal of Symbolic Logic 66 (3):1481-1492 (2001)
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-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(R) absoluteness for proper forcings. In particular, said absoluteness does not imply Π 1 1 determinacy
Keywords Set Theory   Descriptive Set Theory   Proper Forcing   Large Cardinals
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