Semi‐proper forcing,remarkable cardinals,and Bounded Martin's Maximum

We show that L(?) absoluteness for semi‐proper forcings is equiconsistent with the existence of a remarkable cardinal, and hence by [6] with L(?) absoluteness for proper forcings. By [7], L(?) 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)