Forcing axioms via ground model interpretations

TL;DR

This paper establishes a connection between forcing axioms and ground model interpretations through name principles, providing new characterizations and systematic studies of their implications in set theory.

Contribution

It introduces a general framework linking forcing axioms to name principles and characterizes various forcing axioms via these principles, including Bagaria's BFA.

Findings

Forcing axioms can be expressed as name principles.

PFA is equivalent to a name principle for rank 1 and 2 names.

Lambda-bounded forcing axioms correspond to specific name principles.

Abstract

We study principles of the form: if a name $\sigma$ is forced to have a certain property $\varphi$, then there is a ground model filter $g$ such that $\sigma^g$ satisfies $\varphi$. We prove a general correspondence connecting these name principles to forcing axioms. Special cases of the main theorem are: Any forcing axiom can be expressed as a name principle. For instance, $\mathsf{PFA}$ is equivalent to a principle for rank $1$ names (equivalently, nice names) for subsets of $\omega_1$, and a principle for rank $2$ names for sets of reals. Moreover, $\lambda$-bounded forcing axioms are equivalent to name principles. Bagaria's characterisation of $\mathsf{BFA}$ via generic absoluteness is a corollary. We further systematically study name principles where $\varphi$ is a notion of largeness for subsets of $\omega_1$ (such as being unbounded, stationary or in the club filter) and corresponding forcing axioms.