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 ${L}_{\kappa ,\kappa}$.
Title  weakly compact cardinal 

Canonical name  WeaklyCompactCardinal 
Date of creation  20130322 12:50:53 
Last modified on  20130322 12:50:53 
Owner  Henry (455) 
Last modified by  Henry (455) 
Numerical id  5 
Author  Henry (455) 
Entry type  Definition 
Classification  msc 03E10 
Synonym  weakly compact 
Related topic  CardinalNumber 
Defines  weakly compact cardinal 
Defines  weak compactness theorem 