CircularReasoning 2006-07-25 16:28:42 2007-05-30 03:10:49 Definition % 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 {\sl Circular reasoning\/} is an attempted proof of a statement that uses at least one of the following two things: \begin{itemize} \item the statement that is to be proven \item a fact that relies on the statement that is to be proven \end{itemize} Such proofs are not valid. As an example, below is a faulty proof that the \PMlinkname{well-ordering principle implies the axiom of choice}{WellOrderingPrincipleImpliesAxiomOfChoice}. The step where circular reasoning is used is surrounded by brackets [ ]. Let $C$ be a collection of nonempty sets. By the well-ordering principle, each $S \in C$ is well-ordered. [For each $S \in C$, let $<_S$ denote the well-ordering of $S$.] Let $m_S$ denote the least member of each $S \in C$ with respect to $<_S$. Then a choice function $\displaystyle f \colon C \to \bigcup_{S \in C} S$ can be defined by $f(S)=m_S$. The step surrounded by brackets is faulty because it actually uses the axiom of choice, which is what is to be proven. In the step, for each $S \in C$, an ordering is chosen. This cannot be done in general without appealing to the axiom of choice.