Ordinal arithmetic and natural arithmetic

Abstract

Starting with the definition of an ordinal number, we develop some of the elementary theory of ordinals, with a focus on ordinal arithmetic. We investigate some of the algebraic and order-theoretic properties of ordinal addition, multiplication, and exponentiation. We then make use of these properties to prove the basis representation theorem for ordinals. This in turn allows us to define first the Cantor normal form and then the operations of natural addition and natural multiplication for ordinals. We briefly compare the algebraic properties of conventional ordinal arithmetic to those of natural arithmetic. In the appendices we briefly discuss the construction of the least infinite ordinal as well as a proof of the transfinite recursion theorem for ordinals.