Semi-proper forcing, remarkable cardinals, and Bounded Martin's Maximum

Bounded Martin's Maximum, Weak Erd} Os Cardinals, and Ac

We prove that a form of the Erd} os property (consistent with V = LH! 2 ] and strictly weaker than the Weak Chang's Conjecture at !1), together with Bounded Martin's Maximum implies that Woodin's principle AC holds, and therefore 2 @ 0 = @2. We also prove that AC implies that every function f : !1 ! !1 is bounded by some canonical function on a club and use this to produce a model of the Bounde...

Bounded Martin's Maximum, Weak Erdös Cardinals and psi AC

Generic Vopěnka's Principle, remarkable cardinals, and the weak Proper Forcing Axiom

We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures C of the same type, there exist B 6= A in C such that B elementarily embeds into A in some set-forcing extension. We show that, for n ≥ 1, the Generic Vopěnka’s Principle fragment for Πn-definable classes is equiconsistent with a proper class of n-remarkable cardi...

Forcing axioms and projective sets of reals

This paper is an introduction to forcing axioms and large cardinals. Specifically, we shall discuss the large cardinal strength of forcing axioms in the presence of regularity properties for projective sets of reals. The new result shown in this paper says that ZFC + the bounded proper forcing axiom (BPFA) + “every projective set of reals is Lebesgue measurable” is equiconsistent with ZFC + “th...

Hierarchies of Forcing Axioms, the Continuum Hypothesis and Square Principles

I analyze the hierarchies of the bounded and the weak bounded forcing axioms, with a focus on their versions for the class of subcomplete forcings, in terms of implications and consistency strengths. For the weak hierarchy, I provide level-by-level equiconsistencies with an appropriate hierarchy of partially remarkable cardinals. I also show that the subcomplete forcing axiom implies Larson’s o...