Martin's maximum 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. Forcing with a ccc notion of forcing preserves stationary subsets of ω1, thus extends . 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 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.
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.
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, ISBN978-981-4324-30-4, Zbl 1258.03075
martin, maximum, theory, branch, mathematical, logic, introduced, foreman, magidor, shelah, 1988, named, after, donald, martin, generalization, proper, forcing, axiom, itself, generalization, martin, axiom, represents, broadest, class, forcings, which, forcing. In set theory a branch of mathematical logic Martin s maximum introduced by Foreman Magidor amp 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 MM textstyle operatorname MM states that if D is a collection of ℵ 1 displaystyle aleph 1 dense subsets of a notion of forcing that preserves stationary subsets of w1 then there is a D generic filter Forcing with a ccc notion of forcing preserves stationary subsets of w1 thus MM textstyle operatorname MM extends MA ℵ 1 textstyle operatorname MA aleph 1 If P is not a stationary set preserving notion of forcing i e there is a stationary subset of w1 which becomes nonstationary when forcing with P then there is a collection D of ℵ 1 displaystyle aleph 1 dense subsets of P such that there is no D generic filter This is why MM 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 1 The proof uses Shelah s theories of semiproper forcing and iteration with revised countable supports MM textstyle operatorname MM implies that the value of the continuum is ℵ 2 displaystyle aleph 2 2 and that the ideal of nonstationary sets on w1 is ℵ 2 displaystyle aleph 2 saturated 3 It further implies stationary reflection i e if S is a stationary subset of some regular cardinal k w2 and every element of S has countable cofinality then there is an ordinal a lt k such that S a is stationary in a In fact S contains a closed subset of order type w1 Notes edit Jech 2003 p 684 Jech 2003 p 685 Jech 2003 p 687 References editForeman 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 03075See also editTransfinite number nbsp This set theory related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Martin 27s maximum amp oldid 1211748916, wikipedia, wiki, book, books, library,
