Sorry, not an answer, but too long for a comment.

I am the author of the "Pitowsky's Kolmogorovian models and Super-Determinism" paper. I would reject the claim that this paper is a "philosophical discussion". Rather, I (try to) demonstrate (using simple physics and mathematics, not philosophy) that the whole Pitowsky model business is physically meaningless (and accordingly not a good example for a connection between set theory and physics).

Let me give a comparison: One can prove that no strategy will guarantees a steady income playing roulette, but of course for such a proof you have to assume your strategy is "measurable". One can now claim "aha! this indicates a deep connection between financial mathematics and set theory". Which is of course silly.

The Pitowsky construction is more complicated, and therefore it is harder to see, but basically he does something similar: he sabotages the simple, elegant proof of Bell's theorem by assuming that some stuff is not measurable. Of course such an assumption does not constitute a hidden variable theory (just as claiming "the winning strategy might not be measurable" only sabotages the roulette-proof, but doesn't give you a winning strategy). Pitowsky then goes on to actually "construct" such non-measurable hidden variables, by introducing a non-standard notion of probability. But, as I try to point out in my paper, in the end this notion just says: "We assign (nonstandard) probability 1 to exactly those results that will cone out of those of experiments that we will actually perform". This is logically consistent, but basically equivalent to super-determinism: Once we know exactly what will happen, in particular what will be measured, there is no locally problem with Bell or GHZ; this has been obvious from the the beginning of the investigations if no-go theorems.

To come back to the comparison: I can just as well claim to have a roulette winning strategy: I will always play the color that will then be picked. Again, this doesn't seem to me to indicate a connection between games of chance and the axiom of choice...

Sorry, not an answer, but too long for a comment.

I am the author of the "Pitowsky's Kolmogorovian models and Super-Determinism" paper. I would reject the claim that this paper is a "philosophical discussion". Rather, I (try to) demonstrate (using simple physics and mathematics, not philosophy) that the whole Pitowsky model business is physically meaningless (and accordingly not a good example for a connection between set theory and physics).

Let me give a comparison: One can prove that no strategy will guarantee a steady income playing roulette, but of course for such a proof you have to assume your potential strategy is "measurable". One can now claim "Aha! This indicates a deep connection between financial mathematics and set theory!" Which of course would be silly.

The Pitowsky construction is more complicated, and therefore it is harder to see, but basically he does something similar: He sabotages the simple, elegant proof of Bell's theorem by assuming that some stuff is not measurable. Of course such an assumption does not constitute a hidden variable theory (just as claiming "the winning strategy might not be measurable" only sabotages the roulette-proof, but doesn't give you a winning strategy). Pitowsky then goes on to actually "construct" such non-measurable hidden variables, by introducing a non-standard notion of probability. But, as I try to point out in my paper, in the end this notion just says: "We assign (nonstandard) probability 1 to exactly those results that will come out of those of experiments that will actually be performed". This is logically consistent, but basically equivalent to super-determinism: Once we know exactly what will happen (in particular: which measurement will be performed), there is no locality problem with Bell or GHZ. This has been obvious from the the beginning of the investigations of no-go theorems.

To come back to the comparison: I can just as well claim to have a roulette winning strategy: I will always play the color that will then be picked. Again, this doesn't seem to me to indicate a connection between financial mathematics and the axiom of choice...