1. 0 is a natural number.
2. For every natural number x, x = x. That is, equality is reflexive.
3. For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric.
4. For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive.
5. For all a and b, if b is a natural number and a = b, then a is also a natural number. That is, the natural numbers are closed under equality.
6. For every natural number n, S(n) is a natural number.
7. For all natural numbers m and n, m = n if and only if S(m) = S(n). That is, S is an injection.
8. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.
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