BPFA and projective well-orderings of the reals

Abstract

If the bounded proper forcing axiom BPFA holds and ω₁=ω₁^L, then there is a lightface Σ¹₃ well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of “David's trick.” We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface Σ¹₄, for many “consistently locally certified” relations R on ℝ. This is accomplished through a use of David's trick and a coding through the Σ₂ stable ordinals of L.