Natural Ordinal Addition

Five different self-standing accounts of natural addition in the ordinals, reflecting five different philosophical perspectives on how we should best undertake definitions with the ordinals.

Let us consider the ordinals under what is called the natural sum and the natural product, also known as the Hessenberg operations on ordinals. These operations exhibit many attractive algebraic properties, making them form the structure of a semiring—the natural semiring of ordinals, which I shall aim for us to explore.

Notably, the natural sum and product on ordinals are both commutative operations—unlike the standard ordinal arithmetic—and so the natural semiring of ordinals is a commutative semiring. In fact, the natural sum and product operations on ordinals are the same operations that the ordinals exhibit in the surreal numbers, which makes the natural semiring of ordinals a subsemiring of the surreal numbers.

I shall describe several independent and self-standing approaches to the natural sum and product—we shall ultimately have five separate accounts of each operation, which proceed from and express different philosophical perspectives on how we should best undertake mathematical definitions with the ordinals. One account of the natural sum, for example, offers a purely order-theoretic structuralist account, while another can be seen as motivated by essentially computational concerns—how to compute the sum and product values—and still another account adopts in effect a proof-theoretic perspective by presenting a formal transfinite recursion. Ultimately, of course, we shall prove that the various alternative accounts of the natural sum and product are equivalent—they all ultimately define the same ordinal operations of the natural sum and product.

This is a happy situation, therefore, since to have multiple independent accounts of the same underlying mathematical idea is often valuable for mathematical insight. The different but ultimately equivalent approaches to the topic enrich our mathematical understanding by stretching our knowledge in different but fruitful directions. Different perspectives suggest different avenues of generalization, and some perspectives can be more clarifying than others depending on the specific case.

Welcome to this series of essays on the ordinals and ordinal arithmetic—you can find them in the ordinal-arithmetic tag. In this essay, we introduce the natural ordinal arithmetic—the natural sum and the natural product. These are the same operations that the ordinals exhibit in the surreal-numbers. Please enjoy!

Let’s get into it.