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($\mathbb{R}$) absoluteness for proper forcings. In particular, said absoluteness does not imply $\Pi^1_1$ determinacy.
Publié le : 2001-09-14
Classification:
Set Theory,
Descriptive Set Theory,
Proper Forcing,
Large Cardinals,
03E55,
03E15,
03E35,
03E60
@article{1183746573,
author = {Schindler, Ralf-Dieter},
title = {Proper Forcing and Remarkable Cardinals II},
journal = {J. Symbolic Logic},
volume = {66},
number = {1},
year = {2001},
pages = { 1481-1492},
language = {en},
url = {http://dml.mathdoc.fr/item/1183746573}
}
Schindler, Ralf-Dieter. Proper Forcing and Remarkable Cardinals II. J. Symbolic Logic, Tome 66 (2001) no. 1, pp. 1481-1492. http://gdmltest.u-ga.fr/item/1183746573/