Andrés Eduardo CaicedoBoban Veli\v ckovi´c
Bounded Proper Forcing Axiom and Well Orderings of the Reals
Preprint series: ESI preprints

MSC:

03E65 Other hypotheses and axioms

Abstract: 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 H\"artig
quantifier is not lightface projective.
Keywords: proper forcing, well orderings, Hartig's quantifier