Your work is very interesting. Of course you are correct that pawns will be limited to a narrow band if they promote at ranks 1 and 8. But this is only fair, as a pawn is the only piece that can’t move backwards. (Without this rule, a pawn might be doomed to a condition of eternal diminished significance, as things become less interesting far from the normal playing area).

Thanks for your analysis of the decidability of infinite chess, and how it is affected by the huygens. This problem is indeed interesting, just as the knight’s tour problem was interesting in the 9th century. But today’s problems might not lend themselves to solving even with computers. I’ll continue to follow the available resources to see if other pawn promotion rules have been devised (besides the null rule, and the classical rule). I’ll also follow the work on Goldbach’s conjecture, and prime numbers in general, and your work to see if there are any rules devised on how the huygens can best be played in a chess game.

Best regards,

]]>Another question, in your notable paper “The mate-in-n problem of infinite chess is decidable” (2012), you concluded that chess (with certain restrictions) is decidable.

On page four, you define the scope of the study as “(the) argument relies are: (i) no new pieces enter the board during play, and (ii) the distance pieces—bishops, rooks and queens—move on straight lines whose equations can be expressed using only addition.”

So this bring me to two questions: (1) if pawns promote at ranks 1 and 8 as they do in chess normally, do we know anything about the decidability of infinite chess?

Secondly (2), (and separately), if the huygens is added to the set of chess pieces (a huygens is a chess pieces which jumps prime numbers of squares), again, do way know anything about the decidability of infinite chess?

In the case that the decidability of these versions of chess is unknown, is there any branch of mathematics that can be used to answer such questions, or would such an effort be a futile endeavor?

]]>But I the problem is that naive intuition is not something concrete. The Banach–Tarski Paradox is indeed paradoxical to the naive intuition, since the naive intuition certainly does not expect it. And only a handful of people who might disagree with that naively.

Russell’s paradox is also a bit paradoxical, but mostly because we usually present comprehension as “the naively correct intuition” (which may have been true back in the 19th century, but I argue that 120 years later, our intuitive sensitivities in logic may have already got to the place where this naive intuition is no longer that common).

But in this case? I just don’t know if I agree. Maybe because I’m a set theorist and I’m more used to the whole “move to a different universe” thing. Maybe it’s just my philosophical views that give me that point of view. But I am certainly someone not versed in game theory at all.

As far as CH goes, though, here I think that you’re just missing the point. CH is not about reals, it’s about sets of reals. Different models with the same reals which disagree about CH, will invariably disagree about the sets of reals as well.

]]>My perspective is that we use the word paradox only when a result goes against what might be naive intuitions on the topic. Most of the so-called paradoxes in set theory (Banach-Tarski, Russell, Burali-Forti) are not paradoxical or even confusing to someone with a more sophisticated understanding of the topic. The paradoxes evaporate with a more informed understanding. (One exception might be the liar paradox, whose confounding nature seems to be more enduring.) For this reason, I prefer a more relaxed use of the word ‘paradox’.

In the instance of my post, I find the result paradoxical in that someone with a naive understanding of game theory might not expect that it would matter for the purpose of having a winning strategy, whether one insisted on computable play or not, especially since the game tree is computable. Indeed, I find the fact that there is a computable tree with no computable branch to itself be similarly paradoxical.

Similarly, I do find it paradoxical that CH can be true in some models of set theory and false in others, especially when those models have the same real numbers.

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