The omnific integers are strange and weird

We shall explore several surprising failures of the analogy between the omnific integers and the integers.

Join me in this next instance of my ongoing series of essays on the surreal numbers. Today, we shall look at the ways in which the omnific integers don’t seem so very much like the integers after all. Several core features of the integers simply do not hold in the omnific integers.

You can find all the other surreal-number essays in the tag surreal-numbers, and readers who wish to review basic notions should look at The Surreal Numbers, which offers a gentle introduction to this amazing and fantastical structure.

In the previous post, we had introduced the omnific integers as providing an analogue of the integers for the surreal number system—the motivating idea was to view the omnific integers as sitting inside the surreal numbers in just the same way that the integers sit inside the real numbers. We had explored various senses in which the omnific integers Oz indeed play that role. In particular, the omnific integers Oz form an integer part of the surreal field No, which means that they are a discretely ordered subring for which every unit interval [x, x + 1) in the surreal numbers contains exactly one omnific integer.

\(\text{Oz}\quad \subseteq \quad\text{No}\qquad\text{ as }\qquad\mathbb{Z}\quad \subseteq \quad\mathbb{R}\ ?\)

However, I should like now to investigate several senses in which, regrettably, this analogy breaks down. The fact of the matter is that in several important respects, the omnific integers simply lack the properties that one might otherwise want to regard as core features of the integer ring as it is situated in the real field.

To give a taste of what is coming, we will show that the theory of prime numbers in the omnific integers is much less satisfactory than in the integers—there is, for example, no omnific analogue of the fundamental theorem of arithmetic, according to which every positive integer has a unique prime factorization. The first-order induction scheme fails for the nonnegative omnific integers. The process of placing fractions into lowest terms does not work properly with omnific integers. Indeed, to press this point sharply, we will prove specifically that the number √2 is actually rational with respect to the omnific integers Oz. And the number π is rational as well, it turns out, and you will find that there are many additional surprising instances of this phenomenon. In addition, Fermat’s last theorem, famously proved by Andrew Wiles to hold in the integers, is not true in the omnific integers—there are nontrivial omnific integer solutions to x³ + y³ = z³. Ultimately, therefore, we will see that many natural arithmetic principles that hold in the integers nevertheless fail in the omnific integers.