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.
Denote by [ X ] < ω 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 Φ : [ X ] n → [ X ] < ω , we say that a subset U of X is free (with respect to Φ ), if for any n -element subset V of U and any u ∈ U ∖ V , u ∉ Φ ( V ) ,. Kuratowski published in 1951 the following result, which characterizes the infinite cardinals of the form ℵ n .
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 ℵ n if and only if for every mapping Φ from [ X ] n to [ X ] < ω , there exists an ( n + 1 ) -element free subset of X with respect to Φ .
For n = 1 , Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.
