# induction axiom

An induction axiom^{} specifies that a theory includes induction^{}, possibly restricted to specific formulas^{}. IND is the general axiom of induction:

$$\varphi (0)\wedge \forall x(\varphi (x)\to \varphi (x+1))\to \forall x\varphi (x)\text{for any formula}\varphi $$ |

If $\varphi $ is restricted to some family of formulas $F$ then the axiom is called F-IND, or F induction. For example the axiom ${\mathrm{\Sigma}}_{1}^{0}$-IND is:

$$\varphi (0)\wedge \forall x(\varphi (x)\to \varphi (x+1))\to \forall x\varphi (x)\text{where}\varphi \text{is}{\mathrm{\Sigma}}_{1}^{0}$$ |

Title | induction axiom |
---|---|

Canonical name | InductionAxiom |

Date of creation | 2013-03-22 12:56:51 |

Last modified on | 2013-03-22 12:56:51 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 7 |

Author | Henry (455) |

Entry type | Definition |

Classification | msc 03F35 |

Synonym | IND |

Synonym | -IND |

Synonym | axiom of induction |