The Natural Ring of Ordinals

The natural ring of ordinals is the discretely ordered ring generated by the ordinals in the natural arithmetic. The ring exhibits many attractive features, while also holding several surprises.

Let us consider the ordinals and their natural algebraic structure. I should like to introduce and explore what I call the natural ring of ordinals, the ring generated by the ordinals with the natural sum and product.

A ring is a certain kind of algebraic structure generalizing the familiar arithmetic of the integers ⟨ℤ,+,·⟩ to a more general or abstract realm of number objects, such as polynomials or matrices. In the natural ring of ordinals, for example, we shall find the ordinals as well as formal ordinal differences α ─ β. Ring theory is about leveraging our deep understanding of the integers to these more abstract objects and contexts, which can often retain many of the attractive features of integer arithmetic in a manner that fruitfully aids our thinking about them, often leading to insight. We can add and multiply polynomials, for example, and factor them; in certain contexts we can perform long division of polynomials with remainders in a manner that is exactly analogous to what we do in the integers with the Euclidean algorithm. In the natural ring of ordinals, we shall find a robust concept of prime number and indeed every number in this ring will factor uniquely as a product of primes—thus we shall discover an ordinal analogue of the fundamental theorem of arithmetic. This feature is completely lacking, in contrast, in the ring of omnific integers in the surreal numbers, which exhibits only weaker ring-theoretic features.

To be more specific about the definition, a ring consists of a realm of objects with accompanying concepts of addition and multiplication for them exhibiting certain regular features, namely, both addition and multiplication are associative; addition is commutative; there is an additive identity 0; every number has an additive inverse; and lastly, multiplication distributes over addition.

Examples of rings would include the ring of integers ⟨ℤ,+,·⟩, of course, with the usual arithmetic; but also modular arithmetic ℤ/nℤ with addition and multiplication modulo n; the ring ℤ[x] of polynomials in the indefinite variable x with integer coefficients; the polynomial rings ℤ[x, y, z,…] allowing additional variables; the various field structures, such as the rational field ℚ; the real field ℝ; the complex field ℂ; and the associated polynomial rings over these fields, such as ℚ[x, y, z,…]. These are all commutative rings, meaning that multiplication as well as addition is commutative, but there are also many noncommutative rings, such as the various matrix rings ℝ^2×2 or ℂ^n×n, and many others.

Meanwhile, the natural numbers ℕ with the usual addition and multiplication do not form a ring, since they lack additive inverses, but the natural numbers do form what is called a semiring, which drops that requirement. Similarly, the ordinals are not a ring, since they also lack additive inverses, although with the natural sum and product they do form a semiring. We shall ultimately expand the class of ordinals with ideal number objects that will represent the various possible ordinal differences α - β, thereby providing a presentation of the natural ring of ordinals, which has many attractive features.

I denote the natural ring of ordinals by ⟨Ord⟩ in order to suggest that it is generated by but not identical to the class of ordinals. The natural ring of ordinals, it turns out, is a discretely ordered commutative ring, just like the integers themselves, and it is moreover an integral domain, which means nonzero numbers never multiply to zero. Further, it will turn out that the natural ring of ordinals is a unique factorization domain, which means that we shall find in it a robust concept of prime number and every number will factor uniquely as a finite product of primes.

Perhaps the reader will be surprised to learn that the natural ring of ordinals can be realized isomorphically as a vast discretely ordered polynomial ring over the integers, arising in a presentation as a transfinite tower of rings extending endlessly upward with newly created indefinite variables x_α, one for each ordinal α, with each new generator x_α added on top, larger in the order than every polynomial using only earlier variables.

\(\newcommand\bminus{\mathbin{\textbf{-\!-}}}\newcommand\bplus{\mathbin{\textbf{+}}} ℤ\ \subseteq \ ℤ[x_0]\ \subseteq \ ℤ\bigl[x_0,x_1\bigr]\ \subseteq \ \cdots\ \subseteq \ ℤ\bigl[x_α\bigr]_{α < λ}\ \subseteq \ \cdots\ \subseteq \ ℤ\bigl[x_α\bigr]_{α\in\textup{Ord}}\ \cong\ ⟨\textup{Ord}⟩. \)

We shall also find a variety of useful normal forms for the elements of the natural ring of ordinals, such as the signed analogue of the Cantor normal form,

\( \newcommand\bminus{\mathbin{\textbf{-\!-}}}\newcommand\bplus{\mathbin{{⧾}}}ω^{α_n}\cdot k_n\bplus\cdots\bplusω^{α_0}\cdot k_0, \)

with the difference that we now allow the coefficients k_i to be arbitrary integers, including negative integers, in order to accommodate the ordinal differences. Numbers in the natural ring of ordinals can also be represented as a signed sum of distinct powers of 2:

\(\pm2^{α_n}\pm\cdots\pm2^{α_0}, \)

with α_n > ··· > α₀, although this representation is not unique, in light of such examples as 8 - 1 = 4 + 2 + 1.

The natural ring of ordinals, it turns out, is the same as the ring generated by the ordinal numbers inside the surreal number field—but it is strictly contained within the omnific integers.

To mention a few intriguing features, we shall eventually prove that √2 is irrational in the natural ring of ordinals, just as it is with the classical Pythagorean result in the integers. Of course, this would be the expected result, at least until one recalls that this wasn’t true in the omnific integers, since we saw earlier that √2 is rational with respect to the omnific integers and indeed every surreal number is rational with respect to the omnific integers. The number 2 is prime in the natural ring of ordinals and every number factors uniquely into primes, which might seem initially to give a solid basis for the concepts of even and odd in the natural ring of ordinals. However, these concepts are regrettably a little less successful there than in the integers, for matters go somewhat awry in regard to the order—we lose the uniformly regular even/odd pattern and indeed, there will be arbitrary long intervals in the natural ring of ordinals having no even numbers at all. This phenomenon can be seen as an instance of the failure of the Euclidean algorithm in the natural ring of ordinals—⟨Ord⟩ is not a Euclidean domain, nor even a principal ideal domain, nor is it Noetherian. Although we shall have a concept of greatest common divisor in the natural ring of ordinals, nevertheless the ordinal analogue of Bézout’s identity will fail—the greatest common divisor of two numbers will not always arise as a linear combination of them.

Eventually we shall also perform the quotient field construction and thereby construct the natural field of ordinals, as well as its real algebraic closure. Let us aim to explore all these ordinal number rings and fields together.

Let’s get into it!