Is there a chess example of True but Not Provable statement?

Question

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.

Answer 1

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.