Martin's maximum
# 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 (MM){\textstyle (\operatorname {MM} )} states that if D is a collection of $\aleph _{1}$ dense subsets of a notion of forcing that preserves stationary subsets of $ω_1,$ then there is a D-generic filter. Forcing with a ccc notion of forcing preserves stationary subsets of $ω_1,$ 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 $ω_1,$ 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 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 $\aleph _{2}$ and that the ideal of nonstationary sets on $ω_1$ is $\aleph _{2}$-saturated. It further implies stationary reflection, i.e., if S is a stationary subset of some regular cardinal κ ≥ $ω_2$ 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 $ω_1.$

