Separating club-guessing principles in the presence of fat forcing axioms

We separate various weak forms of Club Guessing at $\omega_1$ in the presence of $2^{\aleph_0}$ large, Martin's Axiom, and related forcing axioms. We also answer a question of Abraham and Cummings concerning the consistency of the failure of a certain polychromatic Ramsey statement together with the continuum large. All these models are generic extensions via finite support iterations with symmetric systems of structures as side conditions, possibly enhanced with $\omega$-sequences of predicates, and in which the iterands are taken from a relatively small class of forcing notions. We also prove that the natural forcing for adding a large symmetric system of structures (the first member in all our iterations) adds $\aleph_1$-many reals but preserves CH

Dependent choice, properness, and generic absoluteness

We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to -preserving symmetric submodels of forcing extensions. Hence, not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in and. Our results confirm as a natural foundation for a significant portion of classical mathematics and provide support to the idea of this theory being also a natural foundation for a large part of set theory