Surreal number

Description

The surreal numbers are built in stages, called days. A surreal number is written $x = \{ L \mid R \}$, where $L$ and $R$ are sets of surreal numbers built on earlier days. The condition is that no element of $L$ is greater than or equal to any element of $R$. The order is defined at the same time. The rule is: $x \le y$ holds when no left option of $x$ is $\ge y$ and no right option of $y$ is $\le x$. Two forms $x$ and $y$ name the same number when $x \le y$ and $y \le x$.

On day $0$, only $0 = \{ \mid \}$ exists, with both sets empty. On day $1$, one gets $1 = \{ 0 \mid \}$ and $-1 = \{ \mid 0 \}$. On day $2$, one gets $2 = \{ 1 \mid \}$, $\tfrac12 = \{ 0 \mid 1 \}$, and their negatives. After infinitely many days one gets $\omega = \{ 0, 1, 2, \dots \mid \}$, which is larger than every natural number. One also gets $\varepsilon = \{ 0 \mid 1, \tfrac12, \tfrac14, \dots \}$, which is positive but smaller than every positive real number.

Addition and multiplication are defined by recursion on the options. With them the surreal numbers satisfy all the axioms of an ordered field. The one caveat is size: they form a proper class, not a set, so strictly they are an ordered field only in this class sense. This field contains a copy of the real numbers and a copy of every ordinal number. It is not Archimedean, since $\omega$ exceeds every natural number, and $\omega \cdot \varepsilon = 1$. It is a real closed field.