Nonreflexive, well founded sets and natural numbers

This paper presents a definition of a natural number and the corresponding development of arithmetic which does not use the axiom of foundation and does not use the axiom of infinity, or in any other way use an infinite (reflexive) class, to an audience with a modest background in set theory. Both the cardinal and ordinal properties of a natural number are apparent in this definition and the ensuing development indicates clearly how these two properties plus the fact that the sets used in constructing a natural number are well founded combine to give the natural numbers their properties. The importance of using well founded sets at the most elementary level of the foundations of mathematics is thereby made clear and the groundwork laid for the introduction of the axioms of foundation and infinity.