I was trying to define "natural numbers" consider my reasoning and maybe you can complete my construction.

Steps:

1)Let S be the set of all FINITE sets (note that S itself is infinite)

2)Consider the following Axiom (Axiom of Disjointness): For any finite set K there exists another set K' with the same cardinality as K having the property that the interestion of K and K' is {}. IMPORTANT- this axiom probably cannot be proven because we do not have a definition for a set.

3)Define a natural number as the cardinality of any element of set S.

4)Define addition of two natural numbers as the union of two disjoint sets.

5)Demonstrate that addition is well-defined by using the Axiom of Disjointness.

6)Define "zero" as the cardinality of the empty set.

This is were I will stop. Perhaps you can complete this construction. Maybe use Zorn's Lemma to show there is a minimal element, prove the Well-Ordering Theorem for this set and define partial ordering.