The bounded proper forcing axiom and well orderings of the reals

We show that the bounded proper forcing axiom $\BPFA$ implies that there is a well-ordering of ${\mathcal P}(\w_1)$ which is $\Delta_1$ definable with parameter a subset of $\omega_1$. Our proof shows that if $\BPFA$ holds then any inner model of the universe of sets that correctly computes $\al2$ and also satisfies $\BPFA$ must contain all subsets of $\w_1$. 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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Published 1 January 2006