Martin's maximum

In set theory, a branch of mathematical logic, Martin's maximum, introduced by Foreman, Magidor & Shelah (1988) 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 ${\textstyle (\operatorname {MM} )}$ states that if D is a collection of ${\displaystyle \aleph _{1}}$ 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 ${\textstyle \operatorname {MM} }$ extends ${\textstyle \operatorname {MA} (\aleph _{1})}$. 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 ${\displaystyle \aleph _{1}}$ dense subsets of (P,≤), such that there is no D-generic filter. This is why ${\textstyle \operatorname {MM} }$ is called the maximal extension of Martin's axiom.

The existence of a supercompact cardinal implies the consistency of Martin's maximum.[1] The proof uses Shelah's theories of semiproper forcing and iteration with revised countable supports.

${\textstyle \operatorname {MM} }$ implies that the value of the continuum is ${\displaystyle \aleph _{2}}$[2] and that the ideal of nonstationary sets on ω₁ is ${\displaystyle \aleph _{2}}$-saturated.[3] 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.