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 (MM) 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 MM 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 MM 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.

MM 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
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See also

* Transfinite number

Forcing (mathematics)

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