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This question already has an answer here:

The idea of mathematical induction makes perfect sense, because if a statement is true for n=1, and if the statement being true for an arbitrary natural number $m$ implies the statement is true for $m+1$, then we can say "OK, well 1 is true, so 2 is true, so 3 is true, etc".

Although it makes perfect sense, I was wondering if there exists a proof that mathematical induction is never wrong, that doesn't rely on mathematical induction.

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marked as duplicate by N. F. Taussig, saulspatz, Community May 5 at 10:52

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The following are equivalent: Mathematical induction, Strong Induction, and the Well ordering principle. (https://proofwiki.org/wiki/Equivalence_of_Well-Ordering_Principle_and_Induction)

The Well-ordering principle, in particular, is sometimes treated as an axiom. https://en.wikipedia.org/wiki/Well-ordering_principle

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