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| '''Kuratowski's free set theorem''', named after [[Kazimierz Kuratowski]], is a result of [[set theory]], an area of [[mathematics]]. It is a result which has been largely forgotten for almost 50 years, but has been applied recently in solving several [[lattice theory]] problems, such as the [[Congruence Lattice Problem]]. | | Alyson Meagher is the title her parents gave her but she doesn't like when individuals use her complete name. To perform lacross is one of the issues she loves most. Mississippi is where her home is but her husband wants them to transfer. He is an order clerk and it's some thing he truly enjoy.<br><br>Feel free to visit my page: real psychics ([http://black7.mireene.com/aqw/5741 Recommended Web site]) |
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| Denote by <math>[X]^{<\omega}</math> the [[Set (mathematics)|set]] of all [[Finite set|finite subsets]] of a set <math>X</math>. Likewise, for a [[positive integer]] <math>n</math>, denote by <math>[X]^n</math> the set of all <math>n</math>-elements subsets of <math>X</math>. For a [[Map (mathematics)|mapping]] <math>\Phi\colon[X]^n\to[X]^{<\omega}</math>, we say that a [[subset]] <math>U</math> of <math>X</math> is ''free'' (with respect to <math>\Phi</math>), if <math>u\notin\Phi(V)</math>, for any <math>n</math>-element subset <math>V</math> of <math>U</math> and any <math>u\in U\setminus V</math>. [[Kuratowski]] published in 1951 the following result, which characterizes the [[Infinity|infinite]] [[Cardinal number|cardinals]] of the form <math>\aleph_n</math>.
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| The theorem states the following. Let <math>n</math> be a positive integer and let <math>X</math> be a set. Then the [[cardinality]] of <math>X</math> is greater than or equal to <math>\aleph_n</math> if and only if for every mapping <math>\Phi</math> from <math>[X]^n</math> to <math>[X]^{<\omega}</math>,
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| there exists an <math>(n+1)</math>-element free subset of <math>X</math> with respect to <math>\Phi</math>.
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| For <math>n=1</math>, Kuratowski's free set theorem is superseded by [[Hajnal's set mapping theorem]].
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| == References ==
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| * [[Paul Erdős|P. Erdős]], [[András Hajnal|A. Hajnal]], A. Máté, [[Richard Rado|R. Rado]]: ''Combinatorial Set Theory: Partition Relations for Cardinals'', North-Holland, 1984, pp. 282-285.
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| * [[Kazimierz Kuratowski|C. Kuratowski]], ''Sur une caractérisation des alephs'', Fund. Math. '''38''' (1951), 14--17.
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| * John C. Simms: Sierpiński's theorem, ''Simon Stevin'', '''65''' (1991) 69--163.
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| {{settheory-stub}}
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| [[Category:Set theory]]
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