Weakly Compact Cardinal
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High Quality Content by WIKIPEDIA articles! In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erd s & Tarski (1961); weakly compact cardinals are large cardinals, meaning that their existence can not be proven from the standard axioms of set theory. Formally, a cardinal is defined to be weakly compact if it is uncountable and for every function f: [ ] 2 {0, 1} there is a set of cardinality that is homogeneous for f. In this context, [ ] 2 means the set of 2-element subsets of , and a subset S of is homogeneous for f if and only if either all of [S]2 m...
High Quality Content by WIKIPEDIA articles! In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erd s & Tarski (1961); weakly compact cardinals are large cardinals, meaning that their existence can not be proven from the standard axioms of set theory. Formally, a cardinal is defined to be weakly compact if it is uncountable and for every function f: [ ] 2 {0, 1} there is a set of cardinality that is homogeneous for f. In this context, [ ] 2 means the set of 2-element subsets of , and a subset S of is homogeneous for f if and only if either all of [S]2 maps to 0 or all of it maps to 1. The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below. Weakly compact cardinals are Mahlo cardinals, and the set of Mahlo cardinals less than a given weakly compact cardinal is stationary.
