In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erdős & 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 κ is defined to be weakly compact if it is uncountable and for every function f: 2 → {0, 1} there is a set of cardinality κ that is homogeneous for f. In this context, 2 means the set of 2-element subsets of κ, and a subset S of κ is homogeneous for f if and only if either all of 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.
Some authors use a weaker definition of weakly compact cardinals, such as one of the conditions below with the condition of inaccessibility dropped.
Read more about Weakly Compact Cardinal: Equivalent Formulations
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