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,
PFA is equivalent to:
x A principle for rank 1 names (equivalently, nice names) for subsets
of !1.
x A principle for rank 2 names for sets of reals.
• lambda-bounded forcing axioms are equivalent to name principles. Bagaria’s
characterisation of 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.