The well ordering theorem states that every non-empty set has a least element for some ordering (<). This means that if we take the set of natural numbers, by considering the Peano Axioms ONLY, we can find an order (<) (not necessarily the usual one) in which the set of natural numbers N has a least element (again, not necessarily 0).(adsbygoogle = window.adsbygoogle || []).push({});

Now, suppose we take a number system N', which satisfies the Peano Axioms except the Axiom of Induction. Due to the Well-Ordering Theorem, we can find an order "<" in which N' has a least element. Can we show that the Axiom of Induction does not hold under this relation or at least make any progress on that? Thanks :)

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Well-Ordering and induction

Loading...

Similar Threads - Ordering induction | Date |
---|---|

I Order of "Extracted Factors" in SPSS Factor Analysis | Jul 24, 2017 |

A Transcription from SQL to FOL (First Order Logic) | Jun 3, 2017 |

A First order logic : Predicates | Jun 1, 2017 |

B Subsets of Rational Numbers and Well-Ordered Sets | May 31, 2017 |

I Sets, Subsets, Possible Relations | Feb 23, 2017 |

**Physics Forums - The Fusion of Science and Community**