# universal relation

If $\Phi$ is a class of $n$-ary relations with $\vec{x}$ as the only free variables, an $n+1$-ary formula $\psi$ is 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 $\psi(e,\vec{x})=\phi(\vec{x})$.

Title universal relation UniversalRelation 2013-03-22 12:58:18 2013-03-22 12:58:18 Henry (455) Henry (455) 6 Henry (455) Definition msc 03B10 universal universal function