I have been wondering about the axiom of choice and how it relates to physics. In particular, I was wondering how many (if any) experimentally-verified physical theories require axiom of choice (or well-ordering) and if any theories actually require constructability. As a math student, I have always been told the axiom of choice is invoked because of the beautiful results that transpire from its assumption. Do any mainstream physical theories require AoC or constructability, and if so, how do they require AoC or constructability?
No, nothing in physics depends on the validity of the axiom of choice because physics deals with the explanation of observable phenomena. Infinite collections of sets – and they're the issue of the axiom of choice – are obviously not observable (we only observe a finite number of objects), so experimental physics may say nothing about the validity of the axiom of choice. If it could say something, it would be very paradoxical because axiom of choice is about pure maths and moreover, maths may prove that both systems with AC or non-AC are equally consistent.
Theoretical physics is no different because it deals with various well-defined, "constructible" objects such as spaces of real or complex functions or functionals.
For a physicist, just like for an open-minded evidence-based mathematician, the axiom of choice is a matter of personal preferences and "beliefs". A physicist could say that any non-contractible object, like a particular selected "set of elements" postulated to exist by the axiom of choice, is "unphysical". In mathematics, the axiom of choice may simplify some proofs but if I were deciding, I would choose a stronger framework in which the axiom of choice is invalid. A particular advantage of this choice is that one can't prove the existence of unmeasurable sets in the Lebesgue theory of measure. Consequently, one may add a very convenient and elegant extra axiom that all subsets of real numbers are measurable – an advantage that physicists are more likely to appreciate because they use measures often, even if they don't speak about them.
The textbook formulation of functional analysis depends on the axiom of choice, eg via Hahn-Banach.
This means that discarding the axoim of choice will break the textbook formulation of quantum mechanics as well. However, as we're dealing with (separable) Hilbert spaces, there exists countable bases and we should be able to replace the axiom of choice with a less 'paradox' alternative like the Solovay model and still get the right physics.
The full Hahn-Banach theorem cannot be recovered, though, as it implies the existence of an unmeasurable set.
Rigorous arguments in functional analisis are made much simpler by employing the axiom of choice. As we are free to model our physics in any set theory we like, and any set theory containing ZF contains a model of ZFC, we are entitled to use this simplification without fear of inconsistency. Discarding the axiom of choice would only make concepts and proofs more tedious, without giving any higher degree of assurance of the results.
For example, the standard proof of the spectral theorem for self-adjoint operators depends on the axiom of choice, I believe, and much in mathematical physics depends on the spectral theorem.
On the other hand, already on the level of theoretical physics, one often replaces scrupulously integral by finite sums, takes limits irrespective of their mathematical existence, and employs lots of other mathematically dubious trickery to get quickly at the results.
So on this level of reasoning, nothing depends on subtilities that make a difference only when one begins to care about precise definitions and arguments in the presence of infinity.