Ordinal Arithmetic

Abstract

For natural numbers we used the recursion theorem to define the arithmetic operations, and, subsequently, we proved that those operations are related to the operations of set theory in various desirable ways. Thus, for instance, we know that the number of elements in the union of two disjoint finite sets E and F is equal to #(E) + #(F). We observe now that this fact could have been used to define addition. If m and n are natural numbers, we could have defined their sum by finding disjoint sets E and F, with #(E) = m and #(F) = n, and writing m + n = #(E ∪ F).