The Natural Numbers, = {1, 2, 3, …}, can be built from five basic axioms, as follows:

1. There is a natural number 1; that is, 1 ∈ .

2. If n ∈ , then there is another number called the successor of n and written as S(n) that is also a natural number; that is: n ∈ S(n) ∈ .

3. Two different numbers have different successors; that is: m n S(m) S(n).

4. Every natural number but 1 is the successor of some (other) number; that is:

n ∈ , n 1 m ∈ S(m) = n.

5. The Principle of Mathematical Induction: All natural numbers are formed in this way—either 1, or a successor. More formally,

If there is a subset S of natural numbers such that 1 ∈ S and n ∈ S S(n) ∈ S, then S is the set of Natural Numbers, that is S = .

Define addition: n + 1 = S(n).

Can prove addition follows commutative and associative laws.

Define subtraction to be S(n) – 1 = n.