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.
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
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.
[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.
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