Weakly compact cardinal

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 cannot be proven from the standard axioms of set theory. (Tarski originally called them 'not strongly incompact' cardinals.) 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 cannot be proven from the standard axioms of set theory. (Tarski originally called them 'not strongly incompact' cardinals.) 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. Every weakly compact cardinal is a reflecting cardinal, and is also a limit of reflecting cardinals. This means also that weakly compact cardinals are Mahlo cardinals, and the set of Mahlo cardinals less than a given weakly compact cardinal is stationary. The following are equivalent for any uncountable cardinal κ: A language Lκ,κ is said to satisfy the weak compactness theorem if whenever Σ is a set of sentences of cardinality at most κ and every subset with less than κ elements has a model, then Σ has a model. Strongly compact cardinals are defined in a similar way without the restriction on the cardinality of the set of sentences.