The road to epsilon-zero: ordinals as nim-heaps

Previously: Ordinal numbers and basic set theory We're going to get to in a long and roundabout way. First I want to talk about the game of Nim. Nim Nim is a very simple game for two players. There are some piles of beans, which are called nim-heaps. When it's your turn, you are allowed to remove as many beans as you like, as long as they are all in the same pile. Whoever takes the last bean wins. Nim with only one pile of beans is trivial, because whoever goes first can simply take all the beans from the one pile and win. And with two piles it's very simple. But with three or more piles it starts to be a little interesting. Consider the case where there are three nim-heaps, with 1, 2, and 3 beans respectively. The first player can't prevent the second player from taking the last bean. For a slightly less simple example, consider a game that starts with nim-heaps of size 1, 3, 4, and 8 beans. Here the first player can win, if they might the right opening move. But there's only one…