notes-math-transfiniteOrdinals

http://blog.sigfpe.com/2008/10/whats-use-of-transfinite-ordinal.html

here he's interpreting an ordinal as 'the player gets to choose any number below that ordinal for their next move'. So e.g. 3 is a choice of 0 or 1 or 2, w is a choice of any natural number. The ordinals-as-numbers represent the greatest possible number of moves that you could make. In this interpretation it's more clear that w is not 'infinity realized' but just a symbol that means 'arbitrarily large'.

" There are three kinds. There's

(1) the zero game 0=∅. (2) the games where there is one 'largest' next move in the game, the one you should take. These are games of the form a+1 and the next position in the game, if you're playing to survive, will be a. For example, the game 7 is 6+1. (3) the games where there is no single best choice of next move. These are games where there is no largest next move, but instead an ever increasing sequence of possible next moves and you have to pick one. The first example is ω where you're forced to pick an n. These are called limit ordinals.

We can use this to recursively define ab using the classification of b.

(1) If b=0 then by definition ab=1 and that's that. (2) if b=c+1, then ab=ac·a. (3) if b is a limit ordinal then a move in ab works like this: you pick a move in b, say c, and now you get to play ac (or, by transitive closure, any smaller game).

For example, you play ωω like this: in your first move you pick an n, and then you play ωn. Note how the state of play never becomes infinite, you're always playing with finite rules on a finite 'board'. This is an important point. Even though ωω3+22+ω+4 seems like it must be a very big kind of thing, it describes a finite process that always terminates. "

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