Forcing consequences of PFA together with the continuum large

TL;DR

This paper introduces a novel forcing iteration technique using symmetric systems of structures, demonstrating that the forcing axiom for finitely proper posets can coexist with a large continuum.

Contribution

The authors develop a new method for forcing iterations with symmetric systems as side conditions, enabling compatibility of forcing axioms with a large continuum.

Findings

Forcing axiom for finitely proper posets is compatible with a large continuum.

New method for building forcing iterations with symmetric systems.

Proof of consistency results involving continuum size.

Abstract

We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using our method we prove that the forcing axiom for the class of all the small finitely proper posets is compatible with a large continuum.