= 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 $0=\{\mid\}$, $1=\{0\mid\}$, $-1=\{\mid0\}$ and $1/2=\{0\mid1\}$. 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 $\omega=\{0,1,2,\ldots\mid\}$, 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.
Back to article page
