Martin's maximum

In set theory, a branch of mathematical logic, Martin's maximum, introduced by and named after Donald Martin, is a generalization of the proper forcing axiom, itself a generalization of Martin's axiom. It represents the broadest class of forcings for which a forcing axiom is consistent.

Martin's maximum $ (\operatorname{MM})$ states that if D is a collection of $$\aleph_1$$ dense subsets of a notion of forcing that preserves stationary subsets of ω1, then there is a D-generic filter. Forcing with a ccc notion of forcing preserves stationary subsets of ω1, thus $ \operatorname{MM}$ extends $ \operatorname{MA}(\aleph_1)$. If (P,≤) is not a stationary set preserving notion of forcing, i.e., there is a stationary subset of ω1, which becomes nonstationary when forcing with (P,≤), then there is a collection D of $$\aleph_1$$ dense subsets of (P,≤), such that there is no D-generic filter. This is why $ \operatorname{MM}$ is called the maximal extension of Martin's axiom.

The existence of a supercompact cardinal implies the consistency of Martin's maximum. The proof uses Shelah's theories of semiproper forcing and iteration with revised countable supports.

$ \operatorname{MM}$ implies that the value of the continuum is $$\aleph_2$$ and that the ideal of nonstationary sets on ω1 is $$\aleph_2$$-saturated. It further implies stationary reflection, i.e., if S is a stationary subset of some regular cardinal κ ≥ ω2 and every element of S has countable cofinality, then there is an ordinal α < κ such that S ∩ α is stationary in α. In fact, S contains a closed subset of order type ω1.