Set mapping reflection

In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2^omega = omega_2 and that L(P(omega_1)) satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that combinatorial principle Square(kappa) fails for all regular kappa > omega_1.Comment: 11 page

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 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

Definable MAD families and forcing axioms

We show that under the Bounded Proper Forcing Axiom and an anti-large cardinal assumption, there is a $\mathbf{\Pi}^1_2$ MAD family.Comment: 13 page

Strongly uplifting cardinals and the boldface resurrection axioms

We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.Comment: 24 pages. Commentary concerning this article can be made at http://jdh.hamkins.org/strongly-uplifting-cardinals-and-boldface-resurrectio