*Consider a deterministic, perfect information, abstract strategy, finite game , with absurdly large state space, say ...chess*


**Q1** Is the game translatable to an axiomatic system?


**Q2** Can all statements in the game be proven or dis proven, are they decidable? Do we have an 'Incompleteness" at play here? (Assuming that the game has not/cannot be solved given our computation resources.)



> 
> **Examples : Statements in Chess** : 
> 
> 
> 


\*"There is a set of moves (starting from initial position) which lead white to win regardless  

 of black's moves" (Perfect strategy)


"There is a set of moves which lead white not to loose regardless of black's moves" (Perfect 
 strategy)


"All games are decidable (White wins looses or stalemates)after n-moves into the game" (Forcing a 
 win)


"Given a position, exists a function which tells if a game is decidable from that point on"\* 


**Q3** Is there an 'intuitive' truth in statements of chess, say as seen by grandmasters which can not be proven?



> 
> Aim of the questions is to check behavior for infinitely large state spaces with rule based transitions.
> 
> 
> 



Game of Chess and axiomatic systems
Steven Landsburg has now answered the question in the case of ordinary finite chess, which because it is finite has no undecidability or independence phenomenon to speak of. 


Meanwhile, the kind of phenomenon you seek in Q1, Q2 and Q3 does seem to occur in the context of infinite chess, where one plays from a finite position on an infinite board. This would be a somewhat vaster state space than you had suggested, since it is infinite, but the argument that Steven mentions shows that an infinite state space is a necessary condition for undecidability.


Specifically, as I explain in my answer to Richard Stanley's question on the [Decidability of chess on an infinite board](https://mathoverflow.net/questions/27967/decidability-of-chess-on-an-infinite-board/86755#86755) (see also [my blog posts on infinite chess](http://jdh.hamkins.org/tag/chess/)), my co-authors and I prove the decidability of the mate-in-$n$ problem of infinite chess by introducing what we call the first-order *structure of chess* $\frak{Ch}$, with the associated formal language of chess, in which various chess concepts are expressible. Our proof proceeds by showing that this structure is an automatic structure in the sense of finite automata theory, and its theory is consequently decidable. Thus, any infinite chess concept expressible in this formal language of chess will be decidable.


Meanwhile, not all chess concepts seem to be expressible in this particular formal language, and in particular it remains open whether the won-position problem is decidable. If it isn't, then like all undecidable problems, it will involve an independence phenomenon as in Q3, for there will be specific finite positions in infinite chess, such that the question of whether or not they are won for white or not will be independent of your favorite axiomatization. 


Thus, your desired independence phenomenon seems intimately connected with the decidability problem in this infinitary context.


