The present post is intended to tackle the possible interactions of two bizarre realms of extremely large and extremely small creatures, namely large cardinals and quantum physics.
Maybe after all those paradoxes and uncertainty phenomena among weird tiny particles, which follow their own weird quantum logic, and after all those controversies surrounding the right interpretation of what is going on in the sub-atomic universe, the last straw that would break the camel's back could be the discovery of a series of statements in quantum theory which are independent or have large cardinal strength set theoretically. The fact that will send such physical statements beyond the realm in which the so-called usual mathematical tools can afford us a solution.
Not to mention that inspired by Hilbert's sixth problem and Godel's incompleteness theorems, some prominent physicists already brought up discussions concerning the possibility of obtaining independence results or existence of undecidable facts/theories in physics. In this direction see Stephen Hawking's lecture, Godel and the end of universe. The corresponding post on MSE might be of some interest as well.
Anyway the bad (good?) news is that the intersection of large cardinal theory and quantum physics is non-empty (if not potentially large). For example one may consider the following theorem of Farah and Magidor in the Independence of the existence of Pitowsky spin models which contains an assumption of consistency strength of measurable cardinals. [cf. R. Solovay, Real-valued measurable cardinals, Axiomatic Set Theory, 1971.]
Theorem (Farah - Magidor): If the continuum is real-valued measurable then Pitowsky's kind spin function does not exist. The same holds in the model one gets from any universe of $ZFC$ by adding $(2^{\aleph_0})^+$-many random reals.
See also some related philosophical discussions regarding this result in:
Menachem Magidor, Some set theories are more equal.
Jakob Kellner, Pitowsky's Kolmogorovian models and Super-Determinism. [Related: Super-determinism]
One also might be interested in taking a look at the following papers which shed some light on the the way forcing, Cohen reals, ultrafilters and various set theoretic concepts and tools play role in connection with some problems of quantum physics including hidden variables:
Jerzy Krol, Model and Set-Theoretic Aspects of Exotic Smoothness Structures on $\mathbb{R}^4$.
William Boos, Mathematical quantum theory I: Random ultrafilters as hidden variables.
Robert Van Wesep, Hidden variables in quantum mechanics: Generic models, set-theoretic forcing, and the emergence of probability.
Inspired by Farah-Magidor's theorem and the other mentioned papers, the following question arises:
Question: What are some other examples of the statements in (quantum) physics which are mathematically independent or have some large cardinal strength (or at least make use of large cardinal assumptions in their formulation)?
Please provide references if you are aware of any such result.