The road to epsilon-zero: Shortlex order also orders sequences of numbers

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 Coin-moving games with no coins Productive programs and well-founded orders In part 4, Infinite Nim as a coin-moving game, we saw how to reinterpret infinite Nim into an equivalent game: instead of heaps of beans and various special tokens, we interpreted it as a game about moving coins first on a track, then on a grid, and then in a rather difficult-to-visualize infinite-dimensional space. In part 5, Coin-moving games with no coins, we stopped thinking about the coins and their locations in space, and just wrote down the coordinates of each coin. Each coin has an infinite sequence of coordinates, each of which is a non-negative number. But crucially, only a finite number of the coordinates are greater than zero, so every coin's list of coordinates can be written down as a finite sequence, with the infinite tail of zeroes left implicit.…