The Surreal Reals

I was reading Conway’s Book, book on the train this morning, and found something I’d heard people talk about, but that I’d never had time to read or consider in detail. You can use a constrained subset of the surreal numbers to *define* the real numbers. And the resulting formulation of the reals is arguably *superior* to the more traditional formulations of the reals via Dedekind cuts or Cauchy sequences.

First, let’s look at how we can create a set of just the real numbers using the

surreal construction. What we want to do is get a notion of *the simplest surreal number* that satisfies some condition. We need to toss out a few definitions first, to work our way towards it.

Suppose you have a surreal number x={X_{L}|X_{R}}. The members of X_{L} are called the *left options* of x, and the members of X_{R} are called the *right options* of x. The elements of the union of the left and right options are called the *options* of x.

So, take the surreal number x={X_{L}|X_{R}}. Suppose we have *another* number z which satisfies the following two conditions:

- ∀a∈X
_{L},z>a - >∀b∈X
_{R}z<b.

If none of the options of z also satisfy the two conditions above, then we say that

x=z, and z is the *simplest number* equal to x.

There’s a bit of a trick hidden in there. Why do those conditions mean that z is the simplest number? If no options of z can satisfy the rules, then that means that z is the number with the *earliest* birthday that fits between X_{L} and X_{R}. Think of an example: {1,2|4,5}. What number is that? It’s greater than 2, but less than 4. 7/2 obviously matches the two conditions: (7/2) is greater than 1 or two; and it’s less than four or five. But 7/2 is {3|4} is surreals; and 3 *also* satisfies the condition. But no options of 3 can possibly satisfy them. So that final restriction about the options of z guarantees that we get the number with the earliest possible birthday as the simplest number.

Next, we can define the integers in terms of the surreals. An integer is a surreal number whose simplest form has only integers as options, and for which at least one of its options sets is empty. So {1,2|} is an integer; {|-2,-3} is an integer, etc.

The set of real numbers then, consists of all numbers x={X_{L}|X_{R}} such that:

- There exists an integer z such that -z<x<z
- X
_{L}={a | a=x-(1/n)}_{n>0} - X
_{R}={a | a=x+(1/n)}_{n>0}

It should be pretty easy to see why that defines the reals. It’s bounded by the set of integers – so ω, which is beyond the range of integers is excluded. And it clearly includes the irrationals – the definition above is very similar to Dedekind cuts; you can clearly defined π in terms of a series of numbers getting ever closer on either side, so that only π is left in the gap.

So we’ve got a very simple definition of the real numbers in terms of the surreals. Why is this a good thing?

Conway makes a convincing argument, based on how you teach numbers. When you teach numbers in an abstract way, trying to build up to our common understanding of the reals, you end up working through a lot of proofs and a lot of arguments. The pain of many of those arguments is the amount of case-based reasoning you need to work with. For example, when you define the real numbers using Dedekind sections over the rationals, you need to consider four cases when you define multiplication (based on the signs of the numbers being multiplied). The more complex you get, the more cases: the associative law has 8 cases! Screwing up the cases, or missing a case, is the bane of advanced math students everywhere.

In the surreals, there’s *no real difference* between positive numbers and negative numbers. There’s no essential difference between integers and rationals. The distinctions become no less important, but less *consequential* in discussions of

the surreal reals: the case-based reasoning can go away, because in all cases, the definitions reduce to questions about set membership and empty sets.

I’m not saying that I’m convinced that the surreal numbers are the right way to teach reals. I’d want to actually try teaching all about abstract numbers from the perspective of surreals, and see how it goes. But I find the argument compelling enough that I’d be willing to try it, given the chance.