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Proper forcing and remarkable cardinals. II. (English) Zbl 0985.03042
Summary: The current paper proves the results announced in Part I [Bull. Symb. Log. 6, No. 2, 176-184 (2000; Zbl 0960.03044)].
We isolate a new large cardinal concept “remarkability”. Consistencywise, remarkable cardinals are between ineffable and $$\omega$$-Erdős cardinals. They are characterized by the existence of “$$0^\sharp$$-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, the said absoluteness does not imply $$\Pi^1_1$$ determinacy.

##### MSC:
 3e+55 Large cardinals 3e+35 Consistency and independence results 3e+15 Descriptive set theory 3e+60 Determinacy principles
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##### References:
 [1] Set theory of the continuum pp 407– (1992) [2] Proper forcing (1982) · Zbl 0495.03035 [3] DOI: 10.1090/S0002-9947-00-02636-2 · Zbl 0960.03043 [4] Coding the universe (1982) · Zbl 0468.03031 [5] Commentationes Mathematical Universitatis Carolinae 39 pp 281– (1998) [6] Set theory (1978) [7] DOI: 10.2307/421205 · Zbl 0960.03044
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