Surreal number

The surreal numbers form an ordered field built recursively from cuts $\{L\mid R\}$, where every already constructed left option is smaller than every right option. The earliest examples are $0=\{\mid \}$, $1=\{0\mid \}$, $-1=\{\mid 0\}$ and $1/2=\{0\mid 1\}$. A cut names the simplest number between its options; different cuts can name the same number.

Allowing construction through arbitrary ordinals introduces both infinite numbers, such as $\omega =\{0,1,2,\dots \mid \}$, and positive infinitesimals, such as $1/\omega$. All real numbers embed in the resulting ordered field, but its entire collection is a proper class. Unlike ordinal arithmetic, surreal multiplication and addition obey the usual commutative field laws.