Regrettable Failures in the Natural Ring of Ordinals

The natural ring of ordinals has unique prime factorization, but other natural features go awry—there are consecutive odd numbers, entire intervals of primes, GCDs failing Bézout’s identity, and more.

In recent essays we have been investigating the natural ring of ordinals ⟨Ord⟩, the mathematical system of numbers generated by the ordinals with the operations of natural sum and natural product, enabling not only addition and multiplication of these numbers, but also subtraction. In this ring we may speak sensibly of the numbers:

\( \newcommand\bminus{\mathbin{\textbf{─}}}\newcommand\bplus{\mathbin{\textbf{+}}} \omega\bminus 7,\qquad ω^{\omega^2}\bplus\omega^{\omega+5}\bminus\omega^3,\qquad\text{ and }\qquad \omega^3\bminus\omega^{\omega^\omega}.\)

In the previous essay, The Natural Ring of Ordinals Has Prime Factorization, we proved that this ring is an integral domain and indeed it is a unique factorization domain—every number factors uniquely as a finite product of primes. Thus, the ordinal analogue of the fundamental theorem of arithmetic holds in the natural ring of ordinals. Find the whole essay series in the ordinals tag.

Today, I should like to discuss several features in this ring that regrettably do not work out as one might have hoped or expected.

• The concept of even goes awry. Not every odd number has the form 2a＋1.

• The alternating even/odd pattern fails in the natural ring of ordinals. There are consecutive odd numbers!

• Indeed, there are arbitrarily long chains of consecutive odd numbers.

• The Collatz conjecture fails badly in the natural ring of ordinals.

• There are consecutive infinite prime numbers.

• There are arbitrarily long intervals of consecutive prime numbers.

• In this sense, the prime pair conjecture holds in the natural ring of ordinals in a very strong formulation.

• Greatest common divisors exist, but they do not always fulfill Bézout’s identity, by which the GCD is represented as a linear combination:

  \(\newcommand\bminus{\mathbin{\textbf{─}}}\newcommand\bplus{\mathbin{\textbf{+}}} \text{gcd}(a,b)=ra\bplus sb.\)

• The natural ring of ordinals ⟨Ord⟩ is not a Euclidean domain—we cannot always divide with remainder.

• We therefore cannot reliably implement the Euclidean algorithm.

• Indeed, the natural ring of ordinals is not a principal ideal domain, nor is it a Noetherian ring (these concepts will be explained).

• The natural ring of ordinals is not an integer part of the surreal field. There are arbitrarily large intervals in the surreal field containing no member of the natural ring of ordinals.

Let’s get into it!