TITLE: Why does induction have to be an axiom?
QUESTION [29 upvotes]: I noticed that there is an axiom that says that if $S(n)\implies S(n+1)$, and $S(1)$ is true, then $\forall n \in \Bbb N, S(n).$
My question is why is this an axiom? why can't we derive this from the other axioms?

REPLY [18 votes]: The Peano Postulates describe what we want the natural numbers to look like. One thing we want is for the natural numbers to be one continuous stream
0, 1, 2, 3, ...

and not made up of several infinite sequences like
0, 1, 2, 3, ..., 0', 1', 2', ...

where every number has a successor and arithmetic works fine, but if you start counting somewhere in the first sequence you'll never arrive at any number in the second (primed) sequence.
The induction axiom forbids this situation by saying that for every number N, if you start counting from 0 you'll eventually reach N.