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| In the mathematical field of [[set theory]], '''Martin's axiom''', introduced by {{harvs|txt|authorlink=Donald A. Martin|first=Donald A.|last=Martin|last2=Solovay|first2=Robert M.|author2-link=Robert M. Solovay|year=1970}}, is a statement which is independent of the usual axioms of [[ZFC set theory]]. It is implied by the [[continuum hypothesis]], but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all cardinals less than the [[cardinality of the continuum]], '''c''', behave roughly like <math>\aleph_0</math>. The intuition behind this can be understood by studying the proof of the [[Rasiowa–Sikorski lemma]]. It is a principle that is used to control certain [[forcing (mathematics)|forcing]] arguments.
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| ==Statement of Martin's axiom==
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| For any cardinal '''k''', we define a statement, denoted by MA('''k'''):
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| <blockquote>For any [[partial order]] ''P'' satisfying the [[countable chain condition]] (hereafter ccc) and any family ''D'' of dense sets in ''P'' such that ''|D|'' ≤ '''k''', there is a [[filter (mathematics)|filter]] ''F'' on ''P'' such that ''F'' ∩ ''d'' is non-[[empty set|empty]] for every ''d'' in ''D''.</blockquote> | |
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| Since it is a theorem of ZFC that MA('''c''') fails, the Martin's axiom is stated as:
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| <blockquote>'''Martin's axiom (MA):''' For every '''k''' < '''c''', MA('''k''') holds.</blockquote>
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| In this case (for application of ccc), an antichain is a subset ''A'' of ''P'' such that any two distinct members of ''A'' are incompatible (two elements are said to be compatible if there exists a common element below both of them in the partial order). This differs from, for example, the notion of antichain in the context of [[tree (set theory)|trees]].
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| MA(<math>\aleph_0</math>) is simply true. This is known as the [[Rasiowa–Sikorski lemma]].
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| MA(<math>2^{\aleph_0}</math>) is false: [0, 1] is a [[compact space|compact]] [[Hausdorff space]], which is [[separable space|separable]] and so ccc. It has no [[isolated point]]s, so points in it are nowhere dense, but it is the union of <math>2^{\aleph_0}</math> many points.
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| ==Equivalent forms of MA(k)==
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| The following statements are equivalent to Martin's axiom:
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| * If ''X'' is a compact Hausdorff [[topological space]] which satisfies the ccc then ''X'' is not the union of '''k''' or fewer [[nowhere dense]] subsets.
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| * If ''P'' is a non-empty upwards ccc [[Partially ordered set|poset]] and ''Y'' is a family of cofinal subsets of ''P'' with ''|Y|'' ≤ '''k''' then there is an upwards directed set ''A'' such that ''A'' meets every element of ''Y''.
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| * Let ''A'' be a non-zero ccc [[Boolean algebra (structure)|Boolean algebra]] and ''F'' a family of subsets of ''A'' with ''|F|'' ≤ '''k'''. Then there is a boolean homomorphism φ: ''A'' → '''Z'''/2'''Z''' such that for every ''X'' in ''F'' either there is an ''a'' in ''X'' with φ(''a'') = 1 or there is an upper bound ''b'' for ''X'' with φ(''b'') = 0.
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| ==Consequences==
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| Martin's axiom has a number of other interesting [[combinatorial]], [[Mathematical analysis|analytic]] and [[topological]] consequences:
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| * The union of '''k''' or fewer [[null set]]s in an atomless σ-finite [[Borel measure]] on a [[Polish space]] is null. In particular, the union of '''k''' or fewer subsets of '''R''' of [[Lebesgue measure]] 0 also has Lebesgue measure 0.
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| * A compact Hausdorff space ''X'' with ''|X|'' < 2<sup>'''k'''</sup> is [[Compact space|sequentially compact]], i.e., every sequence has a convergent subsequence.
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| * No non-principal [[ultrafilter]] on '''N''' has a base of cardinality < '''k'''.
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| * Equivalently for any ''x'' in β'''N'''\'''N''' we have χ(''x'') ≥ '''k''', where χ is the [[character (topology)|character]] of ''x'', and so χ(β'''N''') ≥ '''k'''.
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| * MA(<math>\aleph_1</math>) implies that a product of ccc topological spaces is ccc (this in turn implies there are no [[Suslin line]]s).
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| * MA + ¬CH implies that there exists a [[Whitehead group]] that is not free; [[Saharon Shelah|Shelah]] used this to show that the [[Whitehead problem]] is independent of ZFC.
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| ==See also==
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| *Martin's axiom has generalizations called the [[proper forcing axiom]] and [[Martin's maximum]].
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| *Sheldon W.Davis has suggested that Martin's axiom is motivated by [[Baire category theorem]] in his book.<ref>Sheldon W. Davis, 2005, Topology, McGraw Hill, p.29, ISBN 0-07-291006-2.</ref>
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| == Notes ==
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| *{{cite book | last=Fremlin | first=David H. | title=Consequences of Martin's axiom | publisher=[[Cambridge University Press]]| location=Cambridge | year=1984 | isbn=0-521-25091-9 | others=Cambridge tracts in mathematics, no. 84}}
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| *[[Thomas Jech|Jech, Thomas]], 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. ISBN 3-540-44085-2.
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| *[[Kenneth Kunen|Kunen, Kenneth]], 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. ISBN 0-444-86839-9.
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| *{{citation|mr=0270904
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| |last=Martin|first= D. A.|last2=Solovay|first2= R. M.
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| |title=Internal Cohen extensions.
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| |journal=Ann. Math. Logic|volume= 2|year= 1970 |issue= 2|pages= 143–178|doi=10.1016/0003-4843(70)90009-4 }}
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| == References ==
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| <references/>
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| {{Set theory}}
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| [[Category:Axioms of set theory]]
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I'm a 43 years old and work at the college (Engineering).
In my spare time I teach myself Danish. I've been there and look forward to go there sometime near future. I love to read, preferably on my kindle. I like to watch Bones and American Dad as well as documentaries about nature. I love Auto racing.
Feel free to visit my blog post :: http://www.hostgator1centcoupon.info/