The road to epsilon-zero: Nim always ends, even with infinite ordinals

Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps Yesterday I talked about the game of Nim, which involves two players taking beans from several piles, and an extension that includes green tokens that behave a bit like infinite piles: When there's a pile with one or more green tokens, it's legal for a player to remove any or all of them, and then to add any number of beans to the pile. At first it might seem that Nim with -tokens could go on forever. Not so! If someone gives you a Nim position where all the piles contain beans, you can say ahead of time how long the game might last. A game starting with nim-heaps of size simply can't last more than 16 turns, because each turn removes at least one bean from a pile, and the game ends when someone takes the last bean. If the game starts with nim-heaps of size , you can't know how long it might last. If you guess it will be over in turns, the first player might prove you wrong by replacing the -token with a pile of…