The natural product of ordinals

Five self-standing equivalent accounts of the natural product of ordinals, reflecting five different philosophical perspectives on this beautiful, core feature of the ordinals.

Let us discover together the natural product of ordinals α ▪ β, also known as the Hessenberg product as well as the Hausdorff product and commonly also denoted by α ⊗ β or α ⊙ β, and indeed often enough denoted by simple juxtaposition αβ. Just as we did previously with the natural sum of ordinals, we shall have here several alternative but equivalent accounts of the natural product of ordinals—five independent accounts in all of the natural product. To my way of thinking, these different approaches to the concept proceed from and express various philosophical perspectives on how to interact with and understand the ordinals.

In particular, we shall have a purely order-theoretic account, the merge product α ▪ β, which I prefer to conceive as the principal semantic concept, although in mathematical practice this is less often given as the main definition; next a computational account I shall denote by α ⊗ β, based on the Cantor normal form, along with a closely related formal polynomial account α ⊛ β; after this, we shall have a definition of the natural product α ⊙ β by transfinite recursion; and finally, the multiplication of ordinals that arises in the surreal numbers α • β. Ultimately, we shall prove that all five notions are identical—they are different equivalent ways of looking at the same operation, the natural product, which in the end we shall often denote simply by αβ.

\(\newcommand\sqdot{\mathbin{\rule[0.5ex]{0.6ex}{0.6ex}}} \newcommand\dott{\mathbin{\scriptsize\bullet}} αβ = α \sqdot β = α ⊗ β = α ⊛ β = α ⊙ β = α \dott β. \)

The argument is subtle, certainly not routine, and so I shall be glad to give a slow, careful presentation here. I am especially glad to do so because to my way of thinking, this is a core result about the natural product, but unfortunately, the full result is not commonly available in one place—one finds it piecemeal, stated and proved only partially and indeed it is often stated without any proof.

So let’s get into the fine details of what I regard as a fundamental illuminating result on the nature of the ordinals, regarding one of the most beautiful and natural operations on the ordinals, the natural product.

This essay is part of a series of essays on the ordinals, to be found in the Ordinal Arithmetic tag. Some readers may find it helpful to review my previous essay on the operation of natural addition in the ordinals.