Ordinal arithmetic is a branch of mathematical logic that deals with the addition, multiplication, and exponentiation of ordinals. Ordinals are a generalization of natural numbers that extend the concept of "size" or "position" beyond finite sets to infinite sets. They are used to describe the order type of well-ordered sets, which are sets in which every non-empty subset has a least element. ### Basic Concepts 1.
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Ordinal arithmetic defines addition, multiplication and exponentiation of ordinals through order types or transfinite recursion. The usual operations need not be commutative. Natural sum and natural product instead combine Cantor normal forms and are commutative; the Hessenberg natural sum is the natural addition operation.