# recursively axiomatizable theory

Let $T$ be a first order theory. A subset $\mathrm{\Delta}\subseteq T$ is a set of axioms for $T$ if and only if $T$ is the set of all consequences of the formulas^{} in $\mathrm{\Delta}$. In other words, $\phi \in T$ if and only if $\phi $ is provable using only assumptions^{} from $\mathrm{\Delta}$.

Definition. A theory $T$ is said to be finitely axiomatizable^{} if and only if there is a finite set^{} of axioms for $T$; it is said to be recursively axiomatizable if and only if it has a recursive set^{} of axioms.

For example, group theory is finitely axiomatizable (it has only three axioms), and Peano arithmetic^{} is recursivaly axiomatizable : there is clearly an algorithm that can decide if a formula of the language^{} of the natural numbers^{} is an axiom.

As an example of the use of this theorem, consider the theory of algebraically closed fields of characteristic^{} $p$ for any number $p$ prime or 0. It is complete^{}, and the set of axioms is obviously recursive, and so it is decidable.

Title | recursively axiomatizable theory |
---|---|

Canonical name | RecursivelyAxiomatizableTheory |

Date of creation | 2013-03-22 12:43:13 |

Last modified on | 2013-03-22 12:43:13 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 7 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 03C07 |

Defines | finitely axiomatizable |