For natural numbers, the Von Neumann definition is usually used: A natural number N then is the set of all lower natural numbers, with 0 being the empty set.

N_{1} < N_{2} is defined as
N_{1} ∈ N_{2}.

N + 1 is defined as N ∪ { N }.

N + 0 is defined as N.

N_{1} + (N_{2} + 1) is defined as
(N_{1} + N_{2}) + 1.