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

Planet Haskell · Jul 20, 2026

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…