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.
MM implies that the value of the continuum is  and that the ideal of nonstationary sets on ω1 is -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.
- Jech (2003) p.684
- Jech (2003) p.685
- Jech (2003) p.687
- Foreman, M.; Magidor, M.; Shelah, Saharon (1988), "Martin's maximum, saturated ideals, and nonregular ultrafilters. I.", Ann. of Math., The Annals of Mathematics, Vol. 127, No. 1, 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, 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
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