Is chess infinite?

Maybe just because within my memory people were better at chess than machines, period, and with that came an assumption (by some anyway) that human thought was qualitatively different than computer processing. Then computing machines got tiny and incredibly powerful and probably radically changed the way people trained for tournaments. I doubt we're too far from engineering either silicone-based "external" memory for our brains, or genetic innovations that will enhance biological CPUs. 

The infinity question is sort of all balled up in metaphysics (any maybe regular physics) because my imperfect understanding is that what humans perceive as "infinite" is perhaps just very, very large. Like "space," whatever that is.

BTW you were right about the chess strength of ordinary PC chess programs. My pre-packaged Mac software was not running up to speed. I was getting "better" suspiciously fast, but tried a properly functioning program today and was humbled.

A question, my new machine actually takes time (more than milliseconds, I mean) to make its moves. Is the program really "thinking" all this time, or are the pauses for effect? It is a Mac Book Pro running with a pre-installed program called "Chess." It's pretty basic, but "powerful" relative to early machines, I'm sure.

Well, even if chess is solved, even then it would be interesting as no one will be able to learn and remember about ALL the positions of the chess borad

I'm going to retract my statement that there are more Chess games than real numbers. I realized last night after posting that showing a one-to-one relationship with Chess games left over is not sufficient to prove that the cardinality is greater.

Even if particular openings were shown to be losing with best play, I expect people would still play them for that very reason - to show that they can still win in a handicapped line. A bit like the King's Gambit then (allegedly)!

I didn't notice that you said "greater cardinality" than real numbers, and I interpreted it as greater than the natural numbers or at least equal to the real numbers, which was what you were originally trying to prove. That proof still seems valid, so the number of chess games is at least uncountably infinite (cardinality equal to or greater than that of the real numbers).

You could argue that it is infinite, on the grounds that the 50-move rule and the 3-fold-repetition rule do not kick in automatically, they have to be claimed by one of the players.

sapientdust said: "That proof still seems valid, so the number of chess games is at least uncountably infinite (cardinality equal to or greater than that of the real numbers)."

Saying that a set is uncountable is not the same thing as saying its cardinality is equal to or greater than that of the real numbers.  The equivalence of these two properties (that there are no uncountable sets with cardinality strictly less than that of the real numbers) is precisely the continuum hypothesis.  So, saying that a set has at least continuumly many elements (cardinality at least that of the real numbers) is a stronger statement than saying it's uncountable, because every set with continuumly many elements is uncountable, regardless of whether the continuum hypothesis holds, but the converse is only true assuming the continuum hypothesis.

Also, saying there are precisely continuumly many games is stronger than saying there are at least continuumly many games.  The latter requires, as I showed above, proving you can inject the set of all games into a set of cardinality the continuum.

Even if two games would be identical with regards to the moves they could still be quite different, and even have different results. One player could lose on time after having made exactly the same moves as in another game where the players agreed to draw in the same position. Not that this ever will be particularly common, maybe it has never happened.

Then there are many variations that end with more or less forced repetitions in the opening. Two players happy with a draw could blitz it out and go home, while another player that needs to win blunders into the line without knowing about it, and sits for an hour trying to find something better before accepting the draw. In one way the games would be identical, but in another way they would be slightly different.

Wouldn't the threefold repetition rule ensure that the number of possible games is finite?  

Is is finite but computers will take years to "solve" chess.

        I once read somewhere that the number of different positions that could arise in a chess game up to the 25th move on either side is reckoned to be greater than the number of observable atoms in the universe. If this is so, it does not show that chess is "infinite", but that a certainly winning series of moves, which would have to be mind bogglingly huge in number, is a long way off being discovered and would no doubt take an altogether impractical period of time to make.

        Further to the above, this puts me in mind of those mathematicians using super computers to find the largest prime number yet discovered. The highest one so far, I believe, is thousands of digits if not millions long, when expressed fully.

What's an 'observable atom'? I Googled, but it talks about 'Observable Universe'. Not sure if they are the same.

Exactly - I think RG1951 means "greater than the number of atoms in the observable universe".

I see. Yeah, I'd have a hard time believing there are more possible moves to the 25th move in chess than there are atoms in the observable universe. I'm certain there are more atoms in my personal viewable space.

But, I suppose, true or not, it doesn't really matter. There is no denying; there are a hell of a lot of combinations in chess and attempting to 'solve' it is going to be a monumental task.

http://en.wikipedia.org/wiki/Shannon_number

Difficult to argue with numbers. 

At the end it talks about an 80 move game. What would the values be for the first 25 moves?

I thought the same thing - 80 moves seems a bit atypical.

Sorry, not a mathmatician. Which of these numbers is the largest:

4×10⁷⁹  10⁸¹ 10¹²³

10^123 is a one with 123 zeros following which is hugely larger than 10^81 (one with 81 zeros following) or 4x10^79 (4 with 79 zeros following).

 I'm with you, though - intuitively, I'd never have dreamed there were more chess positions than atoms in the universe! I think the 80 move assumption is big here - agree with your question, what would it be in just 25 moves; a lot less I bet.

-Mark R.