# theory

If $L$ is a logical language for some logic $\mathcal{L}$, a set $T$ of formulas^{} with no free variables^{} is called a *theory* (of $\mathcal{L}$). If $\mathcal{L}$ is a first-order logic, then $T$ is called a *first-order theory*.

We write $T\models \varphi $ for any formula $\varphi $ if every model $\mathcal{M}$ of $\mathcal{L}$ such that $M\models T$, $M\models \varphi $.

We write $T\u22a2\varphi $ is for there is a proof of $\varphi $ from $T$.

Remark. Let $S$ be an $L$-structure^{} for some signature^{} $L$. The *theory of $S$* is the set of formulas satisfied by $S$:

$$\{\phi \mid S\vDash \phi \},$$ |

and is denoted by $\mathrm{Th}(S)$.

Title | theory |
---|---|

Canonical name | Theory |

Date of creation | 2013-03-22 13:00:12 |

Last modified on | 2013-03-22 13:00:12 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 8 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 03B10 |

Classification | msc 03B05 |