You might consider the Peano Aximoms.
RonL
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.
You might consider the Peano Aximoms.
RonL
In high school, the textbooks have an entire list of the axioms of arithmetic in the cover. But again! They are not axioms they are theorems. I am just not sure why books do that. Anyway, the standard axioms are 5, called Peano Axioms in the link CaptainBlank gave thee.
(They might look strange at first).