(A substantial part of this post was rewritten since it was first posted. I managed to mangle things while editing, and the result was not particularly comprehensible: for example, in the original version of the post, I managed to delete the definition of “mex”, which continuing to use mex in several other definitions. I’ve tried to clear it up. Sorry for the confusion!)
This is actually a post in the surreal numbers series, even though it’s not going to look like one. It’s going to look like an introduction to another very strange system of numbers, called nimbers. But nimbers are a step on the path from
surreal numbers to games and game theory.
Nimbers come from a very old game called Nim. We’ll talk more about Nim later, but it’s one of the oldest strategy games known. The basic idea of it is that you have
a couple of piles of stones. Each turn, each player can take some stones from one of the piles. Whoever is left making the last move loses. It seems like a very trivial game. But it turns out that you can reduce pretty much every impartial game to some variation of Nim.
Analyzing Nim mathematically, you wind up finding that it re-creates the concept of ordinal numbers, which is what surreals are also based on. In fact, creating nimbers can end up re-creating the surreals. But that’s not what we’re going to do here: we’re going to create the nimbers and the basic nimber addition and multiplication
operations.