Surreal numbers are a proper class worth of numbers defined by John Horton Conway, significant for the fact that they can be totally ordered (any two surreals can be compared) and contain many other significant number fields such as the real numbers, the ordinal numbers, and the hyperreal numbers.
Construction[]
Similar to the Von Neumann hierarchy, the hierarchy of surreals is constructed in stages, termed birthdays. On each successive birthday, or day, a particular surreal number is born. On Day 0 there is the simplest surreal, 0.
Every surreal number is denoted in the form of curly brackets holding two sets of sets divided by a pipeline or colon. There is the left set and the right set, denoted {L|R} or {L:R}. 0 is {{}:{}} but is often abbreviated as {:}.
[To be continued. . . . . . .]