compactness

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

If $\mathrm{\Phi }$ is a set of sentences of cardinality less than or equal to $\kappa$ and all subsets of $\mathrm{\Phi }$ of cardinality less than $\lambda$ are consistent, then $\mathrm{\Phi }$ 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
Canonical name	Compactness
Date of creation	2013-03-22 13:49:34
Last modified on	2013-03-22 13:49:34
Owner	Aatu (2569)
Last modified by	Aatu (2569)
Numerical id	5
Author	Aatu (2569)
Entry type	Definition
Classification	msc 03B99
Defines	compactness