The axiom 5 of Peano guarantees that 1 is not the successor of any natural number, however large this be?

Question

The fifth axiom of peano is the axiom of induction, I would like to know what happens in both cases, when zero is considered natural and when it is not. If this statement is true then why consider another axiom that says that 1 is not anyone's successor? I appreciate any help!

Answer 1

You want $0$ (if you include it) or $1$ (if you do not include $0$) not to be the successor of anything. Otherwise you might have all the integers in your model. That has to be an axiom of its own, not depend on induction. Where it is in the order of the axioms depends on the presentation.

Answer 2

Suppose that you just have the induction axiom and the three axioms that say that $0$ is a natural number, every natural number has a successor that is a natural number, and if two natural numbers have the same successor, then they are in fact the same natural number. (Replace $0$ by $1$ if you don’t include $0$ amongst the natural numbers.) Let $N=\{0\}$, and define the successor of $0$ to be $0$. This set $N$ with this successor function satisfies all of your axioms, but it obviously doesn’t match our intuitive notion of the natural numbers. The axiom that says that $0$ is not the successor of any natural number excludes this model.