well ordered set WellOrderedSet 2001-10-17 00:54:56 2005-07-26 22:38:49 Definition well-ordering \usepackage{graphicx} %\usepackage{xypic} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand{\mathbb}[1]{\mathbbmss{#1}} \newcommand{\figura}[1]{\begin{center}\includegraphics{#1}\end{center}} \newcommand{\figuraex}[2]{\begin{center}\includegraphics[#2]{#1}\end{center}} A \emph{well-ordered} set is a totally ordered set in which every nonempty subset has a least member. An example of well-ordered set is the set of positive integers with the standard order relation $(\mathbbmss{Z}^+,<)$, because any nonempty subset of it has least member. However, $\mathbbmss{R}^+$ (the positive reals) is not a well-ordered set with the usual order, because $(0,1)=\{x:0<x<1\}$ is a nonempty subset but it doesn't contain a least number. A \textbf{well-ordering} of a set $X$ is the result of defining a binary relation $\leq$ on $X$ to itself in such a way that $X$ becomes well-ordered with respect to $\leq$.