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 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(). 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 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.

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.

MM implies that the value of the continuum is [2] and that the ideal of nonstationary sets on ω1 is -saturated.[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.

References

  1. ↑ Jech (2003) p.684
  2. ↑ Jech (2003) p.685
  3. ↑ Jech (2003) p.687

See also

Transfinite number


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