# 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 $\Delta$ is a set of sentences of $L_{\kappa,\kappa}$ such that $|\Delta|=\kappa$ and any $\theta\subset\Delta$ with $|\theta|<\kappa$ is consistent then $\Delta$ is consistent.

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

Title weakly compact cardinal WeaklyCompactCardinal 2013-03-22 12:50:53 2013-03-22 12:50:53 Henry (455) Henry (455) 5 Henry (455) Definition msc 03E10 weakly compact CardinalNumber weakly compact cardinal weak compactness theorem