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?