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.)
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.
is weakly compact, then there are chains of well-founded elementary end-extensions of
[2] Assuming V = L, a cardinal is weakly compact iff it is 2-stationary.