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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:
03E55 Large cardinals
03E35 Consistency and independence results
03E15 Descriptive set theory
03E60 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
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[7] DOI: 10.2307/421205 · Zbl 0960.03044
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