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I am trying to explain to someone (and also to understand better) Godel's Theorems through a chess example.

Indeed I found some examples of True but Not Provable math statements, but they are honestly difficult to visualize.

So I am looking for a chess example to visualize better this concept of Truth/Unprovability, where: the way the pieces move are the Axioms, the Theorems are board configurations and a Proof would be to show a configuration is reacheable by a certain order of moves.

I understand that chess, as far as I know, is not exactly a math system capable of expressing basic arithmetics, but it would be nice to visualize at least this concept of Truth/Unprovability. As I was unable to found this connection but feel that there is one, any insight would be valuable.

Alexandre Tourinho
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  • It may be possible to simulate a Turing machine with a chess board, but you would for sure need infinite board (and arbitrary large number pf pieces). – Momo Jul 22 '21 at 19:10
  • Of interest: https://arxiv.org/abs/1201.5597 [The mate-in-n problem of infinite chess is decidable] https://math.stackexchange.com/questions/71760/is-chess-turing-complete – TomKern Jul 22 '21 at 22:57

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I would argue that in fact it is provable (in theory) whether any position is reachable from the starting position. My argument goes like this:

  • There are a finite number of squares and a finite number of pieces, so there are a a finite number of positions. Board states aren't quite the same as positions, but you can add in a few variables to keep track of whether en passant and castling are legal, how close you are to three-move repetition, etc.

  • It's easy to calculate which positions are reachable from a given position.

  • Therefore, one could generate a finite directed graph where each node corresponds to a position and each edge means that there's a play that will take you from the first node to the second.

  • Then, it's quite easy in theory to write down an algorithm that will search a finite directed graph for all nodes that are reachable from a given starting node, in this case the starting position of a chess game.

Therefore, despite the fact that (except for three-move repetition and length limits or whatever) chess games can take an infinite number of moves, they're played on a finite number of positions, so they're more like modular arithmetic than infinite arithmetic and it's always possible (in theory) to calculate whether a position is attainable.

David
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  • Even more simply, there's only finitely many reachable configurations in the first place - just write down a list of them! – Noah Schweber Jul 22 '21 at 19:23
  • @NoahSchweber That's certainly simpler, but it's not as easy for me to see why it works. Once you have your finite number of configurations, or even just one configuration, how do you know whether it's reachable? A priori, you could have an infinite number of potential processes to check. The only way I can see to rule that out is to note that each potentially useful process should involve visiting some of those finite states no more than once each. But by that point, the two arguments seem pretty equivalent. – David Jul 22 '21 at 19:29
  • No, you write down the finite list of all reachable configurations. Don't even bother thinking about ways of reaching them, just jump right to the end goal. Your argument provides a specific algorithm which works and a proof that it works, but if we merely want existence of such an algorithm we can just get it for free from finiteness. – Noah Schweber Jul 22 '21 at 19:29