There are some people claim that the induction principle is an axiom however there are a proof. Theorem: Let P a proposition in N.We suppose that exist a non negative integer n0 such that: H1: Pn0 is True H2:For every integer n greater or equal than n0 Pn==>Pn+1 Let S is the set of propositions for n that are false.We suppose S have at least two element an element then it have a least element k0>n0 and Pk0 is false and k0-1>n0 and Pk-1 is true but Pk0-1==>Pk0 so S is an empty set
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There are many possible starting points for the axiomatic development of mathematics.
In one popular starting point, namely Peano's axioms for the natural numbers, induction is an axiom. You can see this development in Tao's book "Analysis I".
When doing analysis, another popular starting point is the axioms for the real numbers, namely, the ordered field axioms plus the completeness axiom. In that case the natural numbers can be defined and induction can be proved as a theorem. You can see this development in the introduction to Apostol's "Mathematical Analysis".
Nowadays, it is quite popular to start instead from axioms for set theory, namely the ZF axioms or the ZFC axioms. In that case one can again define the natural numbers and induction can be proved as a theorem. You can see this development in set theory textbooks, which I'm less familiar with, but I do have a fondness for Paul Cohen's book "Set theory and the continuum hypothesis" in which you'll find this point of view.
I think that it is simply an axiom.