Suppose we have the Peano Axioms or can prove them using set theory. Suppose we have not yet developed beyond the axioms.
Successor is the function that gives us the next natural number. It is specified by the axioms. So 1=0′ = S(0) is an example.
We define a tail set as a set closed under successor.
We can define a head set as the complement of a tail set.
We can then talk about sets contained in a head set as being finite.
If we feel we have defined 1 to 1 correspondences, called bijections, then we can define a finite set as one that has a bijection to a subset of a head set.