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In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivision of all of its subsets into large and small sets such that itself is large, and all singletons are small, complements of small sets are large and vice versa. The intersection of fewer than large sets is again large. It turns out that uncountable cardinals endowed with a two-valued measure are large cardinals whose existence cannot be proved from ZFC. The concept of a measurable cardinal was introduced by Stanislaw Ulam in 1930. == Definition == Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ. (Here the term ''κ-additive'' means that, for any sequence ''A''α, α<λ of cardinality λ<κ, ''A''α being pairwise disjoint sets of ordinals less than κ, the measure of the union of the ''A''α equals the sum of the measures of the individual ''A''α.) Equivalently, κ is measurable means that it is the critical point of a non-trivial elementary embedding of the universe ''V'' into a transitive class ''M''. This equivalence is due to Jerome Keisler and Dana Scott, and uses the ultrapower construction from model theory. Since ''V'' is a proper class, a technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called Scott's trick. Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principal ultrafilter. Again, this means that the intersection of any ''strictly less than'' κ-many sets in the ultrafilter, is also in the ultrafilter. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Measurable cardinal」の詳細全文を読む スポンサード リンク
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