BPFA and Projective Well-Orderings of the Reals
If the bounded proper forcing axiom BPFA holds and ω1=ω1L, 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.
Caicedo, Andrés Eduardo and Friedman, Sy-David. (2011). "BPFA and Projective Well-Orderings of the Reals". Journal of Symbolic Logic, 76(4), 1126-1136. https://doi.org/10.2178/jsl/1318338842