In set theory, a branch of mathematical logic, Martin's maximum, introduced by 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.

states that if D 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. 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 dense subsets of (P,≤), such that there is no D 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.

implies that the value of the continuum and that the ideal of nonstationary sets on ω1-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 inα. In fact, S contains a closed subset of order typeω1.

Notes

References

• correction

See also

• Transfinite number