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 2ω = ω2 and that [Formula: see text] satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that □ fails for all regular κ > ω1.

We show that Bounded Forcing Axioms (for instance, Martin's Axiom, the Bounded Proper Forcing Axiom, or the Bounded Martin's Maximum) are equivalent to principles of generic absoluteness, that is, they assert that if a $\Sigma_1$ sentence of the language of set theory with parameters of small transitive size is forceable, then it is true. We also show that Bounded Forcing Axioms imply a strong form of generic absoluteness for projective sentences, namely, if a $\Sigma^1_3$ sentence with parameters is forceable, then (...) it is true. Further, if for every real x, $x^{\sharp}$ exists, and the second uniform indiscernible is less than $\omega_2$ , then the same holds for $\Sigma^1_4$ sentences. (shrink)

We analyze the notion of guessing model, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinal axioms, ranging from supercompactness to rank-to-rank embeddings. The majority of these large cardinal properties can be defined in terms of suitable elementary embeddings j:Vγ→Vλ. One key observation is that such embeddings are uniquely determined by the image structures j[Vγ]≺Vλ. These structures will be the prototypes guessing models. We shall show, (...) using guessing models M, how to prove for the ordinal κM=jM) many of the combinatorial properties that we can prove for the cardinal j) using the structure j[Vγ]≺Vj. κM will always be a regular cardinal, but consistently can be a successor and guessing models M with κM=ℵ2 exist assuming the proper forcing axiom. By means of these models we shall introduce a new structural property of models of PFA: the existence of a “Laver function” f:ℵ2→Hℵ2 sharing the same features of the usual Laver functions f:κ→Hκ provided by a supercompact cardinal κ. Further applications of our analysis will be proofs of the singular cardinal hypothesis and of the failure of the square principle assuming the existence of guessing models. In particular the failure of square shows that the existence of guessing models is a very strong assumption in terms of large cardinal strength. (shrink)

We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of an uplifting cardinal.