Review: Is classical set theory compatible with quantum experiments?

Yesterday, I attended a talk at the Quantum Foundations seminar at the beautiful Perimeter Institute for Theoretical Physics (Waterloo, Ontario).
The (somewhat provocative) title of the talk was “Is Classical Set Theory Compatible with Quantum Experiments?”, and the speaker was Radu Ionicioiu. Here are the links to the slides and videotape.

To make a long story short, the speaker addresses the Sakurai’s Bell inequality that can be thought of yielding a decomposition of a certain set $S(a_x,b_y)$ into two parts $S(a_x,b_y,c_+)$ and $S(a_x,b_y,c_{-})$, each of which being unknown/non-understood/non-definite, while each of these sets can be described as the set of all elements of the original set obeying a certain property. Thus:

“Recalling that ZFC‘s axiom of separation asserts that if $S$ is a set, and $\mathcal P$ is a property, then $\{ x\in S\mid \mathcal P(x)\text{ holds}\}$ is a set, we conclude that the results of the Sakurai-Bell experiment are incompatible with classical set theory (or incompatible with even more foundational logical rules, such as the law of excluded middle that implies that the union of the two sets would resurrect $S(a_x,b_y)$).”

Now, here are briefly my thoughts on this argument:

  • The axiom of separation does not apply to arbitrary properties $\mathcal P(x)$. It only applies to formulas $\mathcal P(x)$ in the language of set theory! (Recall the paradox of the heap.)
  • The discussed set (that admits an unclear decomposition) is finite – one of the participants of the seminar mentioned that it contains around 20k many elements. Partitioning a 20k-sized-set got nothing to do with ZFC!
  • The superadditivity idea on slide 43 is a consequence of a confusion between undefined sets and the empty set. (some sets are undefined, some undefined objects are simply not sets, and in any case, undefined object is not necessarily the empty set).
  • On slide 48, the speaker suggests that just as that the elimination of the fifth postulate from Euclidean geometry give rise to fertile (“non-Euclidean”) theories, the elimination of the separation scheme from ZFC could give rise to valuable theories. While this may sounds plausible, one should take into account that the fifth postulate is a restrictive axiom (hence, removing it, gives more freedom), while the separation axiom is responsible for the “existence” of sets (hence, removing it, kills many arguably-reasonable sets). To be even more picky, the axiom of separation anyway follows from the axiom of replacement, provided that an empty set exists.
  • Nine years ago, I attended a talk by Guy Gildor, on his master thesis concerning a development of Set Theory within Fuzzy Logic. His work is perfectly rigorous, and may be found more convenient as foundations for quantum physics. I wonder if anyone ever considered that..?

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7 Responses to Review: Is classical set theory compatible with quantum experiments?

  1. Good job writing a well-reasoned criticism, carfefully pointing out each fallacy. I probably wouldn’t have been able to contain my frustration with this kind of thing. (I really don’t see any mathematical, philosophical, or physical merit in it… although I’m rather unqualified to judge on the second and third points.)

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    • saf says:

      Thanks! The patience here came from my impression that the speaker is trying to do an honest job, and that the source of confusions is simply the lack of firm background in set theory.
      I contacted Arnon Avron, asking for an electronic copy of Guy Gildor’s thesis. While he does not have one, he pointed out that Zuzana Hanikova and Petr Hájek have some joint works on the subject, so my tip for the speaker would be: talk to these people!

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  2. Ari B. says:

    I haven’t looked at the slides so I’m just commenting based on the quote in your description. I wonder whether it’s ever possible to “conclude that the results of [any physical] experiment are incompatible with classical set theory”. I would say that, at worst, the results could be incompatible with the way someone thought classical set theory should be applied to the experiment.

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  3. Assaf sent me an email asking me what purpose math is supposed to serve in the context of physics. So here’s my very personal opinion.
    Math is a fascinating science in its own. It’s not an experimental one, but rather based only on logic. Math gets a lot of intuition from physics, but it obviously does not restricts itself to what we “observe” in nature. Actually, a large part of physics today is analyzing toy models which obviously are not met in nature, in the hope that after understanding them, we may be able to build a model which does describe some natural phenomena. Personally what draws me to math is the pathologies and wackiness. Investigating PDES/ODES are interesting only when it is attached to some physical problem. We attempt to describe nature in terms of equations. That’s the physics part, but from there on the mathematical rules of solving the equations apply. It is true that we gloss over some mathematical rigour. For example people used Dirac delta function long before it was finally “mathematically” defined. Similarly people use the path integral for decades, even though its measure is still ill defined. However, I can partially defend both uses, by saying that the measure is defined on a lattice, so it’s not completely out of the blue. As far as I know in all occasions mathematicians eventually managed to construct a rigorous notion. The reason we gloss over, is because the main interest is predicting an observable phenomena, or explaining a phenomena which makes sense in terms of physical principles. Therefore since the path integral has successfully predicted endless number of experiments, it is a useful physical tool and the mathematicians are the ones who should work on it now. What physicists might expect from math is: 1) Use mathematical concepts to describe physics. Whenever anew mathematical concept was introduced into physics, it was revolutionary, from Newton through Einstein, and in the last 50+ years groups and topology. 2) Physics is limited by the mathematical knowledge, so if there’s new math being done, which could be applied to physics that would again be huge. I’m an old fashioned guy, but the amount of data being accumulated now in astrophysics and particle physics is becoming difficult to manage with current knowledge. So any mathematical concept that will enable different understanding of the data, not just a more efficient way of processing it, will again be a revolution. Last but not least, to the best of my knowledge the 20K is just a random number, he could’ve said any number larger than 20. Don’t take that part too literally.

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    • saf says:

      Thanks for your comment, Ido!
      The more I come to think of it, the more it seems like Set Theory in Fuzzy Logic is indeed an example of framework that is worth testing with respect to such QM experiments (assuming this wasn’t done before). Just like non-standard analysis is sometime the right angle to see and prove theorems, Boolean-Valued models of set theory may be found to be convenient here.

      As for the conrente/random number: $20k$. Don’t worry! $20k$ is as small as $10^{20k}$ from the partitioning point of view.

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      • Ido Ben-Dayan says:

        Perhaps we should really look this up. Though the number of new things I’m thinking of is really becoming huge. Any chance you can learn epsilon analysis. At least I’m pretty sure that both me and Misha will be interested if we can say something about the path integral measure.

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