Martin's maximum

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. It represents the broadest class of forcings for which a forcing axiom is consistent.

Martin's maximum ${\textstyle (\operatorname {MM} )}$ states that if D is a collection of $\aleph _{1}$ 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, thus ${\textstyle \operatorname {MM} }$ extends ${\textstyle \operatorname {MA} (\aleph _{1})}$. 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 $\aleph _{1}$ dense subsets of (P,≤), such that there is no D-generic filter. This is why ${\textstyle \operatorname {MM} }$ 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.

${\textstyle \operatorname {MM} }$ 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 ≥ 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

• Foreman, M.; Magidor, M.; Shelah, Saharon (1988), "Martin's maximum, saturated ideals, and nonregular ultrafilters. I.", Annals of Mathematics, Second series, 127 (1): 1–47, doi:10.2307/1971415, JSTOR 1971415, MR 0924672, Zbl 0645.03028 correction
• Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics (Third millennium ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-44085-7, Zbl 1007.03002
• Moore, Justin Tatch (2011), "Logic and foundations: the proper forcing axiom", in Bhatia, Rajendra (ed.), Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. II: Invited lectures (PDF), Hackensack, NJ: World Scientific, pp. 3–29, ISBN 978-981-4324-30-4, Zbl 1258.03075

See also

• Transfinite number