weakly compact cardinal

Weakly compact cardinals are (large) infinite cardinals which have a property related to the syntactic compactness theorem for first order logic. Specifically, for any infinite cardinal $\kappa$, consider the language $L_{\kappa ,\kappa }$.

This language is identical to first logic except that:

• infinite conjunctions and disjunctions of fewer than $\kappa$ formulas are allowed

• infinite strings of fewer than $\kappa$ quantifiers are allowed

The weak compactness theorem for $L_{\kappa ,\kappa }$ states that if $\mathrm{\Delta }$ is a set of sentences of $L_{\kappa ,\kappa }$ such that $|\mathrm{\Delta }|=\kappa$ and any $\theta \subset \mathrm{\Delta }$ with is consistent then $\mathrm{\Delta }$ is consistent.

A cardinal is weakly compact if the weak compactness theorem holds for $L_{\kappa ,\kappa }$.