UpwardLowenheimSkolemTheorem 2002-08-29 22:03:00 2004-02-09 17:07:02 Theorem % 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} \usepackage{amsthm} % 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 \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newcommand{\card}[1]{\lvert #1 \rvert} Let $L$ be a first-order language and let $\mathcal{A}$ be an infinite $L$-structure. Then if $\kappa$ is a cardinal with $\kappa\geq\operatorname{Max}(\card{\mathcal{A}}, \card{L})$ then there is an $L$-structure $\mathcal{B}$ such that $\card{\mathcal{B}} = \kappa$ and $\mathcal{A} \preccurlyeq \mathcal{B}$.