strongly minimal StronglyMinimal 2003-02-11 08:29:48 2004-07-11 10:24:21 Definition strongly minimal minimal minimal % 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 Let $L$ be a first order language and let $M$ be an $L$-structure. Let $S$, a subset of the domain of $M$ be a definable infinite set. Then $S$ is {\em minimal} iff every definable $C \subseteq S$ we have either $C$ is finite or $S \setminus C$ is finite. We say that $M$ is {\em minimal} iff the domain of $M$ is a strongly minimal set. \medskip We say that $M$ is {\em strongly minimal} iff for every $N \equiv M$, we have that $N$ is minimal. Thus if $T$ is a complete $L$ theory then we say $T$ is {\em strongly minimal} if it has some model (equivalently all models) which is strongly minimal. \medskip Note that $M$ is strongly minimal iff every definable subset of $M$ is quantifier free definable in a language with just equality. Compare this to the notion of o-minimal structures.