You might consider the Peano Aximoms.
I'm typing the ones I know... I was wondering if there are any others... I do not know that many...
1. For any numbers m, n
m + n = n + m and mn = nm.
2. For any numbers m, n, k,
(m+n) + k = m + (n+k) and (mn)k = m(nk).
3. For any numbers m, n, k
m(n+k) = mn + mk.
4. There is a number 0 which has the property that, for any number n,
n + 0 = n.
5. There is a number 1 which has the property that, for any number n,
n x 1 = n.
6. For every number n, there is another number k such that
n + k = 0.
7. For any numbers m, n, k,
if k cannot equal 0 and kn = km, then m = n.
Any additions would be appreciated, I'm interested in reading as many as I can.
(They might look strange at first).