A maximal bounded forcing axiom

Abstract

AbstractAfter presenting a general setting in which to look at forcing axioms, we give a hierarchy of generalized bounded forcing axioms that correspond level by level, in consistency strength, with the members of a natural hierarchy of large cardinals below a Mahlo. We give a general construction of models of generalized bounded forcing axioms. Then we consider the bounded forcing axiom for a class of partially ordered sets Γ1 such that, letting Γ0 be the class of all stationary-set-preserving partially ordered sets, one can prove the following:(a) Γ0 ⊆ Γ1,(b) Γ0 = Γ1 if and only if NSω1 is ℵ1-dense.(c) If P ∉ Γ1, then BFA({P}) fails.We call the bounded forcing axiom for Γ1Maximal Bounded Forcing Axiom (MBFA). Finally we prove MBFA consistent relative to the consistency of an inaccessible Σ2-correct cardinal which is a limit of strongly compact cardinals.