The surreal numbers are a generalization of the reals. Each surreal number consists of two parts (called the left and right), each of which is a set of surreal numbers. For any surreal number $N$ , these parts can be called $N_L$ and $N_R$ . (This could be viewed as an ordered pair of sets, however the surreal numbers were intended to be a basis for mathematics, not something to be embedded in set theory.) A surreal number is written $N=\langle N_L\mid N_R\rangle$ .

Not every number of this form is a surreal number. The surreal numbers satisfy two additional properties. First, if $x\in N_R$ and $y\in N_L$ then $x\nleq y$ . Secondly, they must be well founded. These properties are both satisfied by the following construction of the surreal numbers and the $\leq$ relation by mutual induction:

$\langle\mid\rangle$ , which has both left and right parts empty, is $0$ .

Given two (possibly empty) sets of surreal numbers $R$ and $L$ such that for any $x\in R$ and $y\in L$ , $x\nleq y$ , $\langle L\mid R\rangle$ .

Define $N\leq M$ if there is no $x\in N_L$ such that $M\leq x$ and no $y\in M_R$ such that $y\leq N$ .

This process can be continued transfinitely, to define infinite and infinitesimal numbers. For instance if $\mathbb{Z}$ is the set of integers then $\omega=\langle \mathbb{Z}\mid \rangle$ . Note that this does not make equality the same as identity: $\langle 1\mid 1\rangle=\langle \mid\rangle$ , for instance.

It can be shown that $N$ is ``sandwiched'' between the elements of $N_L$ and $N_R$ : it is larger than any element of $N_L$ and smaller than any element of $N_R$ .

Addition of surreal numbers is defined by

$$N+M=\langle \{N+x\mid x\in M_L\}\cup\{M+x\mid y\in N_L\}\mid \{N+x\mid x\in M_R\}\cup\{M+x\mid y\in N_R\}\rangle$$

It follows that $-N=\langle -N_R\mid -N_L\rangle$ .

The definition of multiplication can be written more easily by defining $M\cdot N_L=\{M\cdot x\mid x\in N_L\}$ and similarly for $N_R$ .

Then

The integers of surreal mathematics are called the omnific integers. In general positive integers $n$ can always be written $\langle n-1\mid\rangle$ and so $-n=\langle \mid 1-n\rangle=\langle \mid (-n)+1\rangle$ . So for instance $1=\langle 0\mid\rangle$ .

In general, $\langle a\mid b\rangle$ is the simplest number between $a$ and $b$ . This can be easily used to define the dyadic fractions: for any integer $a$ , $a+\frac{1}{2}=\langle a\mid a+1\rangle$ . Then $\frac{1}{2}=\langle 0\mid 1\rangle$ , $\frac{1}{4}=\langle 0\mid \frac{1}{2}\rangle$ , and so on. This can then be used to locate non-dyadic fractions by pinning them between a left part which gets infinitely close from below and a right part which gets infinitely close from above.

Ordinal arithmetic can be defined starting with $\omega$ as defined above and adding numbers such as $\langle \omega\mid\rangle=\omega+1$ and so on. Similarly, a starting infinitesimal can be found as $\langle 0\mid 1,\frac{1}{2},\frac{1}{4}\ldots\rangle=\frac{1}{\omega}$ , and again more can be developed from there.