On Mon, Oct 12, 2020, 9:50 AM Stephen J. Turnbull 
As far as what Steven discussed, the ordinal numbers have the same properties (except I've never heard of ω-1 in a discussion of ordinals, but it should work I think).  (Maybe the surreals are constructed from the ordinals as the reals are constructed from the cardinals?)

Not exactly. Cauchy sequences define Reals in terms of countably infinite sequences of Rational numbers. The Surreals are defined by binary trees of every transfinite length (not only countably infinite).

Basically, the right-most branch in the Surreal tree is simply the Cantor ordinals. But in the other paths were encounter things like infinitesimals and ω-1. Subtraction and division wind up defined over Surreals, unlike for regular transfinite ordinals.

"If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations."