Dependent Choice, Properness, and Generic Absoluteness

TL;DR

This paper demonstrates that Dependent Choice suffices for proper forcing theory and generic absoluteness in the presence of large cardinals, establishing ZF+DC as a foundational framework for classical mathematics and set theory.

Contribution

It shows that Dependent Choice is enough for proper forcing and generic absoluteness, extending the foundational role of ZF+DC in mathematics and set theory.

Findings

Dependent Choice suffices for proper forcing theory.

Dependent Choice enables generic absoluteness with large cardinals.

ZF+DC provides a robust foundation for classical analysis and set theory.

Abstract

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 DC-preserving symmetric submodels of forcing extensions. Hence, ZF+DC 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 ZF, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in ZF+DC and ZFC. Our results confirm ZF+DC 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.