# universal relation

If $\mathrm{\Phi}$ is a class of $n$-ary relations^{} with $\overrightarrow{x}$ as the only free variables^{}, an $n+1$-ary formula^{} $\psi $ is *universal ^{}* for $\mathrm{\Phi}$ if for any $\varphi \in \mathrm{\Phi}$ there is some $e$ such that $\psi (e,\overrightarrow{x})\leftrightarrow \varphi (\overrightarrow{x})$. In other words, $\psi $ can simulate any element of $\mathrm{\Phi}$.

Similarly, if $\mathrm{\Phi}$ is a class of function of $\overrightarrow{x}$, a formula $\psi $ is universal for $\mathrm{\Phi}$ if for any $\varphi \in \mathrm{\Phi}$ there is some $e$ such that $\psi (e,\overrightarrow{x})=\varphi (\overrightarrow{x})$.

Title | universal relation |
---|---|

Canonical name | UniversalRelation |

Date of creation | 2013-03-22 12:58:18 |

Last modified on | 2013-03-22 12:58:18 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 6 |

Author | Henry (455) |

Entry type | Definition |

Classification | msc 03B10 |

Synonym | universal |

Defines | universal function |