Semi-proper forcing, remarkable cardinals, and Bounded Martin's Maximum
✍ Scribed by Ralf Schindler
- Publisher
- John Wiley and Sons
- Year
- 2004
- Tongue
- English
- Weight
- 150 KB
- Volume
- 50
- Category
- Article
- ISSN
- 0044-3050
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✦ Synopsis
Abstract
We show that L(ℝ) absoluteness for semi‐proper forcings is equiconsistent with the existence of a remarkable cardinal, and hence by [6] with L(ℝ) absoluteness for proper forcings. By [7], L(ℝ) absoluteness for stationary set preserving forcings gives an inner model with a strong cardinal. By [3], the Bounded Semi‐Proper Forcing Axiom (BSPFA) is equiconsistent with the Bounded Proper Forcing Axiom (BPFA), which in turn is equiconsistent with a reflecting cardinal. We show that Bounded Martin's Maximum (BMM) is much stronger than BSPFA in that if BMM holds, then for every X ∈ V , X^#^ exists. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)