LindstromsTheorem 2003-08-06 09:22:34 2005-04-14 19:27:00 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} % 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 One of the very first results of the study of model theoretic logics is a characterization theorem due to Per Lindstr\"om. He showed that the classical first order logic is the strongest logic having the following properties \begin{itemize} \item Being closed under contradictory negation \item Compactness \item L\"owenheim-Skolem theorem \end{itemize} also, he showed that first order logic can be characterised as the strongest logic for which the following hold \begin{itemize} \item Completeness (r.e. axiomatisability) \item L\"owenheim-Skolem theorem \end{itemize} The notion of ``strength'' used here is as follows. A logic $\mathbf{L}'$ is stronger than $\mathbf{L}$ or as strong if every class of structures definable in $\mathbf{L}$ is also definable in $\mathbf{L}'$.