BPFA and projective well-orderings of the reals

Journal of Symbolic Logic 76 (4):1126-1136 (2011)
If the bounded proper forcing axiom BPFA holds and ω 1 = ${\mathrm{\omega }}_{1}^{\mathrm{L}}$ , then there is a lightface ${\mathrm{\Sigma }}_{3}^{1}$ 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 ${\mathrm{\Sigma }}_{4}^{1}$ , for many "consistently locally certified" relations R on $\mathrm{\mathbb{R}}$ . This is accomplished through a use of David's trick and a coding through the ∑ 2 stable ordinals of L
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DOI 10.2178/jsl/1318338842
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D. A. Martin & R. M. Solovay (1970). Internal Cohen Extensions. Annals of Mathematical Logic 2 (2):143-178.
Justin Tatch Moore (2005). Set Mapping Reflection. Journal of Mathematical Logic 5 (01):87-97.

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