O is a natural number

For every natural number n, its sucessor Sn is a natural number.

For every natural number n, Sn is not equal to 0;

For every natural number n, for every natural number m, if Sn = Sm, then n = m;

The principle of mathematical induction;

For every natural number n, for every natural number m, their sum n + m is a natural number;

For every natural number n, n + 0 = n;

For every natural number n, for every natural number m, n + (Sm) = S(n + m);

For every natural number n, for every natural number m, their product nm is a natural number;

For every natural number n, n0 = 0; and

For every natural number n, for every natural number m, n(Sm) = (nm) + n.