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 :)

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# Well-Ordering and induction

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