Compactness 2003-08-06 09:30:00 2003-08-06 09:31:14 Definition compactness % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A logic is said to be $(\kappa,\lambda)$-compact, if the following holds \begin{quote} 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. \end{quote} 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.