Martin's maximum

In set theory, a branch of mathematical logic, Martin's maximum, introduced by Foreman, Magidor & Shelah (1988), 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) 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. It is a well known fact that forcing with a ccc notion of forcing preserves stationary subsets of ω₁, thus MM extends MA(${\displaystyle \aleph _{1}}$). 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 MM is called the maximal extension of Martin's axiom.

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