The bounded proper forcing Axiom and well orderings of the reals

Abstract

We show that the bounded proper forcing axiom BPFA implies that there is a well-ordering of P(ω1) which is Δ1 definable with parameter a subset of ω1. Our proof shows that if BPFA holds then any inner model of the universe of sets that correctly computes N2 and also satisfies BPFA must contain all subsets of ω1. We show as applications how to build minimal models of BPFA and that BPFA implies that the decision problem for the Härtig quantifier is not lightface projective. © International Press 2006.