“Martin's maximum”的意思、由来-开放百科全书

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Martin's maximum

释义

References
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In set theory, a branch of mathematical logic, Martin's maximum, introduced by {{harvtxt|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.

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 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.

References

1. ^Jech (2003) p.684
2. ^Jech (2003) p.685
3. ^Jech (2003) p.687

{{citation | last=Foreman | first= M. | author1-link=Matthew Foreman | last2=Magidor | first2= M. | author2-link=Menachem Magidor| last3= Shelah | first3= Saharon | author3-link=Saharon Shelah | title=Martin's maximum, saturated ideals, and nonregular ultrafilters. I. |journal=Ann. of Math. |volume= 127 |year=1988|issue= 1|pages= 1–47 |doi=10.2307/1971415|publisher=The Annals of Mathematics, Vol. 127, No. 1|jstor=1971415 |mr=0924672 | zbl=0645.03028}} [https://www.jstor.org/stable/1971520 correction]
{{Citation | last1=Jech | first1=Thomas | author1-link=Thomas Jech | title=Set Theory | edition=Third millennium| publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-44085-7 | year=2003 | zbl=1007.03002}}
{{citation | last=Moore | first=Justin Tatch | chapter=Logic and foundations: the proper forcing axiom | zbl=1258.03075 | editor1-last=Bhatia | editor1-first=Rajendra | title=Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. II: Invited lectures | location=Hackensack, NJ | publisher=World Scientific | isbn=978-981-4324-30-4| pages=3–29 | year=2011 | url=http://www.math.cornell.edu/~justin/Ftp/ICM.pdf}}

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