The road to epsilon-zero: Infinite Nim as a coin-moving game

Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps Nim always ends, even with infinite ordinals In the previous articles I talked about the game of Nim, a very simple game for two players: There are some piles of beans Players alternate turns A legal move is to take any number of beans from one pile Whoever takes the last bean wins I wrote about how Num could be extended to include certain types of “infinite” piles while still remaining a sensible game. This involved introducing green tokens that could be replaced with any number of beans, then square tokens that could be replaced with any number of green tokens and beans, and so on. Rather than think about an infinite family of different kinds of tokens, there's a simple way to make them all the same sort of thing. Imagine a game where the board is a track of squares, extending to the right (and to the right only) as far as needed. Let's number the squares: the leftmost one is , then and so on. On some of the…