Ok, point, I was being too clever. Can I save face by claiming that the number of games is countably infinite as long as you require that all the games be of finite length?
I concede that if you allow for games which never terminate, then my method will not enumerate them. I should probably stop now before I annoy the Real Mathematicians among us. --gvc Eugene Deon wrote: > *That's like saying you can enumrate the real numbers by starting with > the first decimal digit and building a tree with 10 children per node > off into infinity. * > > *I'm quite sure I can find a unique game of scrabble for every real > number. And the decision tree is trivial. Both players exchange 7 > every turn. One decision. It's what's on the rack that makes it > different.* > > > Even allowing a game with an unbounded number of consecutive exchanges, > it's > still countably infinite, because you can enumerate not just all the > games, but > all the infinite moves in these possibly-infinitely -long games by a > breadth-first scan of the decision tree starting with turn one. > > Now, if you want to figure in the length of time taken for each move, > you're > getting somewhere, maybe, until you run into the quantization of time (the > nature of which has not yet been plausibly postulated to date). > --gvc > > > >
