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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(ω_1) which is Δ_1 definable with parameter a subset of ω_1. Our proof shows that if BPFA holds then any inner model of the universe of sets that correctly computes N_2 and also satisfies BPFA must contain all subsets of ω_1. 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

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