Kuratowski's free set theorem

Last updated June 13, 2019

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.

Kazimierz Kuratowski Polish mathematician

Kazimierz Kuratowski was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.

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Denote by $[X]^{<\omega }$ the set of all finite subsets of a set $X$ . Likewise, for a positive integer $n$ , denote by $[X]^{n}$ the set of all $n$ -elements subsets of $X$ . For a mapping $\Phi \colon [X]^{n}\to [X]^{<\omega }$ , we say that a subset $U$ of $X$ is free (with respect to $\Phi$ ), if for any $n$ -element subset $V$ of $U$ and any $u\in U\setminus V$ , $u\notin \Phi (V)$ . Kuratowski published in 1951 the following result, which characterizes the infinite cardinals of the form $\aleph _{n}$ .

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The theorem states the following. Let $n$ be a positive integer and let $X$ be a set. Then the cardinality of $X$ is greater than or equal to $\aleph _{n}$ if and only if for every mapping $\Phi$ from $[X]^{n}$ to $[X]^{<\omega }$ , there exists an $(n+1)$ -element free subset of $X$ with respect to $\Phi$ .

In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set $contains 3 elements, and therefore has a cardinality of 3. There are two approaches to cardinality - one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible.$

For $n=1$ , Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.

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References

P. Erdős, A. Hajnal, A. Máté, R. Rado: Combinatorial Set Theory: Partition Relations for Cardinals, North-Holland, 1984, pp. 282–285.
C. Kuratowski, Sur une caractérisation des alephs, Fund. Math. 38 (1951), 14–17.
John C. Simms (1991) "Sierpiński's theorem", Simon Stevin 65: 69–163.

Paul Erdős was a renowned Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. He was known both for his social practice of mathematics and for his eccentric lifestyle. He devoted his waking hours to mathematics, even into his later years—indeed, his death came only hours after he solved a geometry problem at a conference in Warsaw.

András Hajnal was a professor of mathematics at Rutgers University and a member of the Hungarian Academy of Sciences known for his work in set theory and combinatorics.

Richard Rado FRS was a German-born British mathematician whose research concerned combinatorics and graph theory. He was Jewish and left Germany to escape Nazi persecution. He earned two Ph.D.s: in 1933 from the University of Berlin, and in 1935 from the University of Cambridge. He was interviewed in Berlin by Lord Cherwell for a scholarship given by the chemist Sir Robert Mond which provided financial support to study at Cambridge. After he was awarded the scholarship, Rado and his wife left for the UK in 1933. He was appointed Professor of Mathematics at the University of Reading in 1954 and remained there until he retired in 1971.

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