A maximal bounded forcing axiom

After 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 G such that, letting G be the class of all stationary-set-preserving partially ordered sets, one can prove the following: (a) G ? G, (b) G = G if and only if NS is N-dense. (c) If P ? G, then BFA({P}) fails. We call the bounded forcing axiom for G Maximal Bounded Forcing Axiom (MBFA). Finally we prove MBFA consistent relative to the consistency of an inaccessible S-correct cardinal which is a limit of strongly compact cardinals.