UniversalFormula 2002-08-23 21:00:08 2002-08-24 13:40:42 Definition universal function % 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 %\PMlinkescapeword{theory} If $\Phi$ is a class of $n$-ary relations with $\vec{x}$ as the only free variables, an $n+1$-ary formula $\psi$ is \emph{universal} for $\Phi$ if for any $\phi\in\Phi$ there is some $e$ such that $\psi(e,\vec{x})\leftrightarrow\phi(\vec{x})$. In other words, $\psi$ can simulate any element of $\Phi$. Similarly, if $\Phi$ is a class of function of $\vec{x}$, a formula $\psi$ is universal for $\Phi$ if for any $\phi\in\Phi$ there is some $e$ such that $\psi(e,\vec{x})=\phi(\vec{x})$.