The road to epsilon-zero: Coin-moving games with no coins

Previously: Ordinal numbers and basic set theory Ordinals as nim-heaps Nim always ends, even with infinite ordinals Infinite Nim as a coin-moving game In the previous article we saw how to interpret Nim heaps of up to beans as coins on a quarter-infinite array: The coin here represents a heap of beans. The heap can be reduced to any smaller number of beans. In the coin version of the game, this corresponds to moving the coin to any square to the left in the same row, or to any square in any lower row. To extend this past , though, was a little clumsy. We had to pile up an infinite stack of these grids, and that got us only to . Then to go further we had to move into the fourth dimension, and to get all the way to we had to imagine a sort of discrete Hilbert space with an infinite number of dimensions, not easy. I personally have trouble imagining anything with more than about dimensions, and an infinite number of dimensions is a couple more than I can handle comfortably. We can do…