Notes on the Theory of Surreal Numbers

By David K. Zhang


Under Construction!

1. Introduction

The surreal numbers form a vast number system which is much larger than all familiar everyday number systems, such as the integers $\Z$, rationals $\Q$, and real numbers $\R$. The surreal number system not only includes all real numbers, but also infinite and infinitesimal numbers, such as $\aleph_0$ (the size of the set $\Z$) and $\mathfrak{c}$ (the size of the set $\R$).

In fact, the surreal number system includes all of the “sizes of infinity”, by containing all ordinal numbers (and hence all cardinal numbers). This means that there are so many surreal numbers, they cannot even be considered as a set; instead, they form a structure mathematicians call a proper class.

As one would expect from a number system, surreal numbers can be added, subtracted, multiplied, divided, and exponentiated. These operations enjoy many of the familiar properties they would in $\Q$ or $\R$; for example, addition and multiplication are associative, commutative, and distributive. Moreover, unlike the complex numbers $\C$, the surreal numbers are linearly ordered in a natural way. This gives the surreal numbers the rich algebraic structure of an exponential ordered field.

The study of surreal numbers was initiated by John Conway and Donald Knuth in the mid-1970's. Conway's primary interest was to use the surreal numbers as a tool to study two-player combinatorial games, such as chess and Go. The original definition of “surreal number” he formulated was suitable for this purpose, but highly abstract. (Conway thought of surreal numbers as proper-class-sized equivalence classes of certain recursive structures defined through transfinite induction.)

In these notes, we will study the surreal numbers through an alternative, more concrete viewpoint, developed by the late Harry Gonshor in his 1986 book, An Introduction to the Theory of Surreal Numbers. Gonshor's formulation (which is fully equivalent to Conway's) makes the surreal number system much easier to conceptualize and compute with, and better reveals their natural (tree-like) structure.

1.1. Prerequisites

The surreal numbers are so radically different from the other number systems of mathematics that virtually no knowledge of their algebra or analysis is required. The only necessary prerequite to studying the surreal numbers is a basic familiarity with the ordinal number system e.g., what an ordinal number intuitively “looks like,” how to tell a limit ordinal from a successor ordinal, and a working understanding of ordinal arithmetic.

2. Basic Definitions

Definition: A surreal number is a function from an initial segment of the ordinal numbers into a 2-element set, traditionally denoted by $\{+,-\}$. The proper class of all surreal numbers is denoted by $\No$.

Informally, we may think of a surreal number as a (possibly infinite) sequence of pluses and minuses, indexed by the ordinal numbers, which terminates at some ordinal. Note that the empty sequence is also included as a possibility.

Examples: The function $f: \{0, 1, 2, 3\} \to \{+,-\}$ defined by

\[\begin{aligned} f(0) &= - \\ f(1) &= + \\ f(2) &= + \\ f(3) &= - \end{aligned} \]

is a surreal number, corresponding to the sequence $({-}{+}{+}{-})$. Similarly, the sequence $g: \{1, \dots, \omega\} \to \{+,-\}$ of $\omega$ pluses followed a single minus is a surreal number. We will later see that $f$ corresponds to the rational number $-\frac{3}{8}$, while $g$ corresponds to the surreal number $\omega - 1$ (which is not equal to the surreal number $\omega$).

Definition: If $x$ is a surreal number, then the length of $x$, denoted by $\len(x)$, is the smallest ordinal number at which $x$ is undefined. We say that $x$ is simpler than $y$ if $\len(x) < \len(y)$. (Here, $<$ denotes the usual order relation on the ordinal numbers.)

In the preceding example, $f$ has a length of $4$, while $g$ has a length of $\omega+1$. The empty sequence, which will henceforth be denoted by $(\ )$, has a length of $0$.

Note: If we assume the standard (von Neumann) representation of an ordinal number as the set of all smaller ordinal numbers, we could equivalently consider a surreal number $x$ as a function from $\ell(x)$ into $\{+,-\}$. However, for stylistic reasons, I prefer to think of ordinal numbers as atomic entities, focusing only on their identity and ignoring their internal structure.

Notation: If a surreal number $x$ is undefined at $\alpha$, we will abuse notation and write $x(\alpha) = \square$. We will use $\square$ throughout as a placeholder representing an “empty value.”

Definition: If $x$ and $y$ are distinct surreal numbers, we write $x < y$ iff $x(\alpha) < y(\alpha)$, where $\alpha$ is the smallest ordinal number at which $x(\alpha)$ and $y(\alpha)$ differ. We compare $x(\alpha)$ and $y(\alpha)$ by defining $- < \square < +$.

It is easy to see that this defines a total order on $\No$, which is essentially the lexicographic (dictionary) ordering of strings from the alphabet $\{+,-\}$.

Example: The set of all surreal numbers of length $\le 2$ is ordered as follows:

\[(--) < (-) < (-+) < (\ ) < (+-) < (+) < (++). \]

Alternatively, we can visualize this ordering by arranging the surreal numbers into a tree:

TODO: Draw a picture of the surreal number tree.

Here, we see that $x < y$ iff $x$ is drawn to the left of $y$. This is such an important pictorial representation of $\No$ that it merits a discussion on its own right.

2.1. The Surreal Number Tree

Just as the real numbers naturally fall into a line, the surreal numbers naturally fall into a binary tree, in the following way:

TODO: Draw a bigger picture of the surreal number tree.

Here, we see that moving from a node to its left child corresponds to adding a minus to the end of the parent sequence, while moving to its right child corresponds to adding a plus. The lexicographic order described in the previous section is depicted horizontally, while the partial order of $\No$ by length is depicted vertically.

At this point, it will be instructive to see how the $\{+,-\}$ sequences depicted above correspond to “actual numbers” as we usually imagine them.

TODO: Draw a picture of the surreal number tree with actual numbers.

We see immediately the following general principles:

Thus, all dyadic rational numbers (i.e., those with denominator $2^n$) are represented by surreal numbers of finite length. At this point, it is still unclear how to interpret surreal numbers of infinite length, but we may observe that any real number which is not a dyadic rational can be written uniquely as the limit of a surreal sequence

\[a_0, a_1, a_2, a_3, \dots \]

where each $a_n \in \No$ is a surreal number of length $n$ obtained by appending a single sign to $a_{n-1}$. For example, we can write $\frac{1}{3}$ as the limit of the sequence

\[0, 1, \frac{1}{2}, \frac{1}{4}, \frac{3}{8}, \frac{5}{16}, \frac{11}{32}, \dots \]

allowing us to represent $\frac{1}{3}$ by the surreal number

\[({+}{-}{-}{+}{-}{+}\overline{{-}{+}}\cdots) \]

where the overline indicates the repeating part of the $\{+,-\}$-expansion. In this way, we can represent any real number by a surreal number of length at most $\omega$. One might imagine that the real line is obtained by drawing the surreal tree down to “level $\omega$,” and projecting the resulting points onto a horizontal axis, excluding the two “points at infinity” given by the sequences $({+}{+}{+}\cdots)$ and $({-}{-}{-}\cdots).$

Warning! The correspondence described in this section should, at present, be considered purely conjectural. It remains to be shown that this assignment of numbers to $\{+,-\}$ sequences has any algebraic validity for example, when addition of surreal numbers is defined in a later chapter, we will have to prove that the number here identified as $0$ is in fact the additive identity. Likewise, simple arithmetic facts, such as $1 + 1 = 2$ or $3 \div 6 = \frac{1}{2}$, should presently be considered unproven theorems. (They will, of course, turn out to be correct in due time.)

Nonetheless, it will be useful to keep this correspondence in mind, as it will provide the necessary intuititon and motivation for definitions and theorems to come.

2.2. Order Properties of $\No$

Before we can begin doing arithmetic in $\No$, it will first be necessary to establish some fundamental properties of the lexicographic order relation $<$. Our goal will be to establish the Canonical Representation Theorem, which allows us to recursively represent any given surreal number in terms of strictly simpler surreal numbers. This will allow us to construct inductively the operations of surreal arithmetic, in similar fashion to the usual definitions of natural- and ordinal-number arithmetic.

We begin with a lemma establishing the counterinuitive property that all subsets of $\No$ are bounded.

Lemma 1.
Let $F$ be any set of surreal numbers. Then $F$ has a unique simplest upper bound and a unique simplest lower bound.

Notation: We write $\sub(F)$ and $\slb(F)$ for the simplest upper bound and simplest lower bound of $F$, respectively.

Note that we must be careful to distinguish subsets of $\No$ from subclasses of $\No$. Clearly, $\No$ itself is unbounded, since there exist arbitrarily long ordinal sequences of the form $({+}{+}{+}\cdots)$. However, any set-sized subcollection of $\No$ is “too small” to contain all such sequences. This is essentially the intutition behind the following proof.

Proof. Let $F \subset \No$ be given. First, observe that if $\sub(F)$ exists, then it cannot contain any minuses. Indeed, if $\sub(F)$ were to contain a minus, then by deleting everything from that minus onward, we would obtain a strictly larger and strictly simpler surreal number, contradicting the definition of $\sub(F)$.

Hence, the problem of finding $\sub(F)$ reduces to choosing a sufficiently long sequence of pluses that upper-bounds $F$. By the well-ordering of the ordinals, if there exists one such sequence, then there exists such a sequence of minimal length. But clearly, there does exist one such sequence: simply take a sequence of pluses which is longer than all elements of $F$.