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 is equiconsistent with the Bounded Proper Forcing Axiom , which in turn is equiconsistent with a reflecting cardinal. We show that Bounded Martin's Maximum is much stronger than BSPFA in (...) that if BMM holds, then for every X ∈ V , X# exists. (shrink)