Forcing axioms via ground model interpretations

We study principles of the form: if a name σ is forced to have a certain property φ, then there is a ground model filter g such that ${\sigma }^g$ satisfies φ. 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 ${\omega }_1$.

  • A principle for rank 2 names for sets of reals.
• •λ-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 φ is a notion of largeness for subsets of ${\omega }_1$ (such as being unbounded, stationary or in the club filter) and corresponding forcing axioms.