Having seen this before, if you were not given a set of axioms, then Peano's axioms may be helpful for this. They are:
1. Let S be a set such that for each element x of S there exists a
unique element x' of S.
2. There is an element in S, we shall call it 1, such that for every
element x of S, 1 is not equal to x'.
3. If x and y are elements of S such that x' = y', then x = y.
4. If M is any subset of S such that 1 is an element of M, and for
every element x of M, the element x' is also an element of M, then
M = S.
The way I have seen it done is thus:
As a matter of notation, we write 1' = 2, 2' = 3, etc. We define
addition in S as follows:
The element x + y is called the sum of x and y.
Now to prove that 1 + 1 = 2.
From 1: with x = 1, we see that 1 + 1 = 1' = 2.
Standard properties of addition - for example, x + y = y + x for all x
and y in S - can be proved by induction (which is based on Peano's
On the other hand, 1 + 1 = 3 for 1 sufficiently large