Surreal numbers #
The basic theory of surreal numbers, built on top of the theory of combinatorial (pre-)games.
A pregame is
Numeric if all the Left options are strictly smaller than all the Right options, and
all those options are themselves numeric. In terms of combinatorial games, the numeric games have
"frozen"; you can only make your position worse by playing, and Left is some definite "number" of
moves ahead (or behind) Right.
A surreal number is an equivalence class of numeric pregames.
In fact, the surreals form a complete ordered field, containing a copy of the reals (and much else besides!) but we do not yet have a complete development.
Order properties #
Algebraic operations #
We show that the surreals form a linear ordered commutative group.
One can also map all the ordinals into the surreals!
Multiplication of surreal numbers #
The proof that multiplication lifts to surreal numbers is surprisingly difficult and is currently missing in the library. A sample proof can be found in Theorem 3.8 in the second reference below. The difficulty lies in the length of the proof and the number of theorems that need to proven simultaneously. This will make for a fun and challenging project.
surreal_mul contains some progress on this proof.
- Define the field structure on the surreals.
- [Conway, On numbers and games][conway2001]
- [Schleicher, Stoll, An introduction to Conway's games and numbers][schleicher_stoll]
A pre-game is numeric if everything in the L set is less than everything in the R set, and all the elements of L and R are also numeric.
x ≤ y on numeric pre-games, in terms of
The definition of
x < y on numeric pre-games, in terms of
< two moves later.
Lift an equivalence-respecting function on pre-games to surreals.
Lift a binary equivalence-respecting function on pre-games to surreals.