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I was wondering if any of the finite predictions of Quantum Mechanics depend on what set theoretic axioms are used.

We will say that Quantum Mechanics makes a finite prediction about an experiment if, when you do the experiment, you can determine whether or not the prediction is correct after a finite amount of time. (One caveat is that many predictions of QM are probabilistic. In this case, predictions of the form "E will occur during experiment X with probability p" is permissible as finite prediction (as long as we can determine if E happened after a finite amount of time, and ZF can compare p to any ratio of numerals).)

For example, deciding whether a material is gapped or gapless does not count, due to a limit approaching infinity (i.e. it is an unbounded question).

My question is if there are any finite predictions in QM that depend on the set theoretic axioms used? That is, is there some finite prediction such that the statement "QM makes that prediction" is independent of ZF set theory?

EDIT: We can ask a weaker question that is interesting as well if we replace ZF with second-order arithmetic (under Henkin semantics) or one of its prominent subsystems.

EDIT: There's an interesting paper that talks about how QM works in different models of ZF. The statement about QM that they prove is independent of ZF is not a finite prediction, but it may be helpful since it deals directly with ZF, instead of using a more general computability based approach.


First of all the difference between Does the Axiom of Choice (or any other "optional" set theory axiom) have real-world consequences? and this question is that mine focuses solely on Quantum Mechanics, making it answerable. Additionally, it is my opinion at least that if some prediction of QM depends on set theory, that would reveal a flaw in QM, not an empirical fact about set theory.

The reason for limiting this to finite predictions is that it is easy to see that QM has predictions without the finite restriction that are dependent on the set theoretic axioms. For example if ZFC is consistent, then the QM predicts "a quantum computer will never find an inconsistency in ZFC", but if ZFC is inconsistent, it predicts "a quantum computer will find an inconsistency in ZFC". However, this is entirely ununique to QM, since the same thing happens with Newtonian Mechanics and regular computers, or even with Conway's Game of Life and Universal computers.

However, when we restrict to finite predictions, the situations changes. For any finite configuration in Conway's game of life, for example, PA can completely describe what the configuration will be in $n$ steps for any numeral $n$.

You might ask then "can any physical theory depend on the set theoretic axioms used"? The answer is technically yes. For example, consider PyRulez mechanics. It laws are:

  • The axiom of choice implies that the Earth is flat.
  • The negation of the axiom of choice implies the laws of Newtonian physics.

Seeing as the Earth isn't flat, this empirically proves the axiom of choice is false, of course. At least it would if there was not an equally (im)plausible theory that proves the opposite. :P

Of course, there is no law of Quantum Mechanics that outright states something like that. However, it does make extensive use of the real number system. Some properties of the real numbers are independent of ZF, so maybe one of them will translate into finite predictions in QM?

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  • $\begingroup$ I'm confused about your comment on "a limit approaching infinity." For example if someone were to prove "ZFC implies a mass gap, but ZF is agnostic about a gap" would that answer your question? We all know there's a mass gap, and if someone were to rely on ZFC to be able to prove that, then "yay ZFC?" $\endgroup$ – Mark S Feb 15 at 21:35
  • $\begingroup$ @MarkS (1/2) Disclaimer: I do not really know any research level physics, so I may have phrased that weirdly. My understanding of the spectral gap problem comes from Scott Aaronson's post. Anyways, the reason that problem would not count is that the definition talks about the behavior of a material as its number of particles $L^2$ approaches infinity. Therefore, to experimentally test it, you would need an infinite series of experiments for larger and larger $L$. As noted by Aaronson, the properties of a material for any fixed $L$ is decidable, so just using one really big $L$ would not work. $\endgroup$ – PyRulez Feb 15 at 23:17
  • $\begingroup$ @MarkS (2/2) As for the mass gap problem? Again, I do not know enough about the problem to tell for sure, but I guess the question is if a solution to the problem would imply some prediction that could be tested by a single experiment that takes finite time. $\endgroup$ – PyRulez Feb 15 at 23:20
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    $\begingroup$ I'm voting to close because this is not a well-defined mathematical question. It's not even clear to me that it's a well-defined physical question. Lurking in the background are some hidden and not-very-well-articulated philosophical assumptions about the relationship between physical theories, mathematical theories, and experimental observations. $\endgroup$ – Timothy Chow Feb 15 at 23:28
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    $\begingroup$ @PyRulez: It is hard to know how to start answering this question. First, QM is not a mathematical theory. It is a physical theory. ZFC is a well-defined mathematical object but QM is not. Second, what is a "prediction"? It's not just a mathematical theorem. It is something about the results of a thought experiment in the physical world. Again, experiments and the physical world are not mathematically precise entities. So your question is not a well-defined mathematical question. $\endgroup$ – Timothy Chow Feb 17 at 19:12
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The Pauli exclusion principle for identical particles (fermions) may be one of the "finite predictions" of quantum mechanics dependent on set theory. S. Tarzi argues in Exclusion Principles as Restricted Permutation Symmetries (2003) that the description of collections of sets of identical particles requires a failure of the Axiom of Choice and proposes models of set theory that allow for a negation of that axiom in order to fundamentally derive the Pauli exclusion principle.

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    $\begingroup$ There are no Hilbert spaces in that paper at all, as far as I can see, only sets. How can one obtain Bell inequality violation from such a model? Quantum particles are not elements of a set. It is not possible to answer this by looking at the paper because it doesn't give any physical theory that would use the facts about amorphous sets described there. $\endgroup$ – Robert Furber Feb 15 at 14:09
  • $\begingroup$ Although I like this answer, the axiom of choice is not actually changing the predictions; its just that a simpler formulation exists when the axiom of choice fails, if I am interpreting the paper correctly. (Unless using the axiom of choice actually makes predictions, but those predictions are different from the ones in the paper.) $\endgroup$ – PyRulez Feb 16 at 18:58
  • $\begingroup$ Also, something in the paper is weird: "This is because to mathematically model a set or class of identical objects, which a collection of electrons is usually taken to be, we require a set theory in which AC is false." Electrons aren't completely identical, they have different positions. It just that position is their only difference. Even if this was not the case, you could just use multisets instead. $\endgroup$ – PyRulez Feb 16 at 19:00

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