In mathematics, specifically in set theory, 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 κ {\displaystyle \kappa } , or more generally on any set. For a cardinal κ {\displaystyle \kappa } , it can be described as a subdivision of all of its subsets into large and small sets such that κ {\displaystyle \kappa } itself is large, the empty set and all singletons { α } {\displaystyle \{\alpha \}} with α ∈ κ {\displaystyle \alpha \in \kappa } are small, complements of small sets are large and vice versa. The intersection of fewer than κ {\displaystyle \kappa } 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 Stanisław Ulam in 1930.
Definition Formally, a measurable cardinal is an uncountable cardinal number κ {\displaystyle \kappa } such that there exists a κ {\displaystyle \kappa } -additive, non-trivial, 0-1-valued measure μ {\displaystyle \mu } on the power set of κ {\displaystyle \kappa } . Here, κ {\displaystyle \kappa } -additive means that for every λ < κ {\displaystyle \lambda <\kappa } and every λ {\displaystyle \lambda } -sized collection { A β } β < λ {\displaystyle \{A_{\beta }\}_{\beta <\lambda }} of pairwise disjoint subsets A β ⊆ κ {\displaystyle A_{\beta }\subseteq \kappa } , we have
μ ( ⋃ β < λ A β ) = ∑ β < λ μ ( A β ) {\displaystyle \mu {\Big (}\bigcup _{\beta <\lambda }A_{\beta }{\Big )}=\sum _{\beta <\lambda }\mu (A_{\beta })} . Equivalently, κ {\displaystyle \kappa } is a measurable cardinal if and only if it is an uncountable cardinal with a κ {\displaystyle \kappa } -complete, non-principal ultrafilter. This means that the intersection of any strictly less than κ {\displaystyle \kappa } -many sets in the ultrafilter is also in the ultrafilter. Equivalently, κ {\displaystyle \kappa } is measurable if it is the critical point of a non-trivial elementary embedding of the universe V {\displaystyle V} into a transitive class M {\displaystyle M} and κ M ⊆ M {\displaystyle ^{\kappa }M\subseteq M} . This equivalence is due to Jerome Keisler and Dana Scott, and uses the ultrapower construction from model theory. Since V {\displaystyle 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.
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