Martin's maximum

In set theory, a branch of mathematical logic, Martin's maximum, introduced by 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 $(\operatorname)$ 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 $\operatorname$ extends $\operatorname(\aleph_1)$. 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 $\operatorname$ 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.

$\operatorname$ implies that the value of the continuum is $\aleph_2$ and that the ideal of nonstationary sets on ω₁ 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.

Notes

References

correction
*
*

See also

* Transfinite number

Forcing (mathematics)

{{settheory-stub