PlanetMath: weakly compact cardinal

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weakly compact cardinal

(Definition)

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 $\vert\Delta\vert=\kappa$ and any $\theta\subset\Delta$ with $\vert\theta\vert<\kappa$ is consistent then $\Delta$ is consistent.

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

"weakly compact cardinal" is owned by Henry.

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