Forcing axioms via ground model interpretations

Abstract

We study principles of the form: if a name $\sigma$ is forced to have a certain property $\phi$, then there is a ground model filter $g$ such that ${\sigma }^g$ satisfies $\phi$. 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 $\phi$ is a notion of largeness for subsets of ${\omega }_1$ (such as being unbounded, stationary or in the club filter) and corresponding forcing axioms.