In set theory, a branch of mathematical logic, Martin's maximum, introduced by Foreman, Magidor & Shelah (1988) and named after Donald Martin, is a stronger form of the proper forcing axiom, itself a stronger form of Martin's axiom. It represents the broadest class of forcings for which a forcing axiom is consistent. Martin's maximum ( MM ) {\textstyle (\operatorname {MM} )} states that if D is a collection of ℵ 1 {\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 MM {\textstyle \operatorname {MM} } extends MA ( ℵ 1 ) {\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 ℵ 1 {\displaystyle \aleph _{1}} dense subsets of (P,≤), such that there is no D-generic filter. This is why MM {\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. The proof uses Shelah's theories of semiproper forcing and iteration with revised countable supports.
MM {\textstyle \operatorname {MM} } implies that the value of the continuum is ℵ 2 {\displaystyle \aleph _{2}} and that the ideal of nonstationary sets on ω1 is ℵ 2 {\displaystyle \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.
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See also Transfinite number
