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

Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps [Yesterday][prev-2] 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…