# compactness

A logic is said to be $(\kappa,\lambda)$-compact, if the following holds

If $\Phi$ is a set of sentences of cardinality less than or equal to $\kappa$ and all subsets of $\Phi$ of cardinality less than $\lambda$ are consistent, then $\Phi$ is consistent.

For example, first order logic is $(\omega,\omega)$-compact, for if all finite subsets of some class of sentences are consistent, so is the class itself.

Title compactness Compactness 2013-03-22 13:49:34 2013-03-22 13:49:34 Aatu (2569) Aatu (2569) 5 Aatu (2569) Definition msc 03B99 compactness