I'm having a bit of trouble understanding why the principle of induction is included as one of Peano's axioms. It seems like it should not be independent of the others. Obviously it can be stated as:(adsbygoogle = window.adsbygoogle || []).push({});

- If a predicate P is true only of natural numbers, [itex]P(0)[/itex] is true, and also [itex]P(n)\rightarrow P(S(n))[/itex], then it is true exactly of the natural numbers.

Now, the first two of Peano's axioms define the set of natural numbers:

- [itex]0 \in \mathbb{N}[/itex]
- [itex]n\in \mathbb{N} \rightarrow S(n)\in\mathbb{N}[/itex]

The third axiom essentially states that no other objects are natural numbers.

Now presume T is the set of all objects for which the predicate P holds. Presume P is a predicate such that P(0), and that [itex]P(n)\rightarrow P(S(n))[/itex]. Then clearly:

- [itex]0 \in T[/itex]
- [itex]n\in T\rightarrow S(n)\in T[/itex]

This is identical to the definition of [itex]\mathbb{N}[/itex], so it follows that [itex]T\supset\mathbb{N}[/itex].

This appears to be a proof of the principle of mathematical inductions from the first three of Peano's axioms. Where did I go wrong?

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# Mathematical Induction

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