One way of doing that, 'prior' to Peano's axioms is this: define "0" to be the empty set. (Are you going to argue about whether that can exist?) Define "1" to be the set whose only member is the empty set- that is the set whose only member is "0". Define "2" to be the set whose only members are "0" and "1". In general, given any "number" n, which is now a set of objects, define its successor to be the set whose members are n

**and** all members of n:

.

That will be our set N and the "successor function" in Peano's axioms.

(This includes "0" in N as Peano initially did. Of course, we would get exactly the same result defining "1" to be the empty set and continuing from there.)