Why isn't it necessary to postulate the existence of $1$?

Question

These are the Peano axioms, I'll focus on the second one now:

If $a$ is a number, the successor of $a$ is a number.

Basically, here is defined the successor function $S(n)=n+1$. My question is, how doesn't $S(n)$ depend on the existence of $1$? To get from $0$ to $1$ we need to have a sum, and a sum is between two or more numbers. So, since we are summing $0$ and $1$ to get $1$, aren't we already assuming that $1$ is a number? I've been thinking about it for some time, thanks in advance for clarification.

Answer 1

We don’t need a sum in order to postulate the existence of a successor function. In fact, we use the successor function to define addition: the successor function comes first.

You’re confusing the successor function as a primitive entity whose properties are determined by the axioms with its intended interpretation, which is indeed that $S(n)=n+1$. The intended interpretation is a large part of the reason for the choice of axioms defining the characteristics of the successor function, but the successor function itself is defined by those axioms, not by the interpretation that inspired them.

Answer 2

Numbers in this context would be taken to be whatever satisfies the axioms. So you have the first one, then its successor, then the successor of that, and so on. No operation of addition is defined in advance on this set. Rather, that comes later: one uses the sequence of numbers and the successor function to say what addition is. The successor function is not in this context defined as $x\mapsto x+1$. It is merely a function satisfying the axioms.