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.
N1 < N2 is defined as N1 ∈ N2.
N + 1 is defined as N ∪ { N }.
N + 0 is defined as N.
N1 + (N2 + 1) is defined as (N1 + N2) + 1.
See also 53 in Linked Open Numbers