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 the 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? 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-constructible 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.
Rigorous arguments in functional analysis 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 integrals 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 subtleties that make a difference only when one begins to care about precise definitions and arguments in the presence of infinity.
The following paper may be of interest:
Norbert Brunner, Karl Svozil, Matthias Baaz, "The Axiom of Choice in Quantum Theory". Mathematical Logic Quarterly, vol. 42 (1) pp. 319-340 (1996).
The abstract is as follows:
We construct peculiar Hilbert spaces from counterexamples to the axiom of choice. We identify the intrinsically effective Hamiltonians with those observables of quantum theory which may coexist with such spaces. Here a self adjoint operator is intrinsically effective if and only if the Schrödinger equation of its generated semigroup is soluble by means of eigenfunction series expansions.
Also relevant is the fact that classical analysis doesn't require much more than dependent choice, which is consistent with "All sets of reals are Lebesgue measurable". However the combination of the two statements requires a stronger assumption as a theory (inaccessible cardinals).
What does baffle me, however, with physicists that have strong objections to the Banach-Tarski paradox, that it makes much less sense that a set can be partitioned into strictly more [non-empty] parts than elements. And that is a consequence of having all sets Lebesgue measurable.
So while you may sleep quietly knowing that you cannot partition an orange into five parts and combining the parts into two oranges (thus solving world hunger), you have an equally disturbing problem. You can cut out a line [read: the real numbers] into more parts than points.
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.
The axiom of choice can be applied in all mathematical problems concerning physics. However there it is not required because there are at most potentially infinite sets and every element can be chosen without an axiom.
The axiom of choice has gotten its bad reputation because it leads to contradictions with uncountable sets. But it is a very natural axiom and is frequently applied without notice in mathematics as Zermelo has correctly pointed out when defending his invention against objections of Borel, Peano, Poincaré, and others. [E. Zermelo: "Neuer Beweis für die Möglichkeit einer Wohlordnung", Math. Ann. 65 (1908) pp. 107-128]
The problem is only, as mentioned above, the application of the axiom to uncountable sets. In the history of mathematics, it was usual to fix factual conventions by an axiom, like "it is possible to draw a straight line from any point to any point" or "if n is a natural number, then n+1 is a natural number". The application of the axiom of choice claims for the first time a counterfactual convention, namely to choose an element without knowing what is chosen.
At least in 1904 it was clear that there are only countably many finite strings of letters, including strings defining mathematical objects. Cantor knew this theorem, as he wrote in a letter to Hilbert in 1906, although he did not believe that it is true. "If König's theorem was true, according to which all 'finitely definable' real numbers form a set of cardinality aleph_0, this would imply that the whole continuum was countable, and that is certainly false." [G. Cantor, letter to D. Hilbert (8 Aug 1906)]
Today there is no doubt that König's theorem is true. In order to maintain transfinite set theory, it is necessary to have the (in this realm) counterfactual axiom of choice for proving the basic theorem of set theory: Every set can be well-ordered. Otherwise a lot of ordinal theory would be unprovable. Therefore set theorists have agreed that the axiom is "not constructive", i.e., we can prove that we can choose every element, but we cannot choose every element. Although Zermelo used the axiom to prove that every set can be well-ordered, i.e., he thought it could be done and not only be proven that it could be done, knowing that in fact it cannot be done. [E. Zermelo: "Beweis, daß jede Menge wohlgeordnet werden kann", Math. Ann. 59 (1904)]
But what is the value of a counterfactual axiom? We could state many other axioms of same value like:
Axiom of three points on a line: Every triple of points belongs to a straight line. (But in most cases provably no geometrical construction can be given.)
Axiom of ten even primes: There are 10 even prime numbers. (But provably no arithmetical method to find them is available.)
Axiom of prime number triples: There is a second triple of prime numbers, besides (3, 5, 7). (But provably this second triple is not arithmetically definable.)
Axiom of meagre sum: There is a set of n different positive natural numbers with sum n*n/2. (This axiom is not constructive. Provably no such set can be constructed.)
All theories based upon such axioms would have the same value as transfinite set theory, namely none.
Keeping this in mind and ignoring absurd attempts to apply uncountable alphabets or infinite definitions to define uncountably many elements, we can be sure that the axiom of choice is true in every world with correct mathematics and therefore without uncountable sets.
Countable and uncountable sets require completed, finished, or actual infinity like the complete set of all natural numbers to quantify over. But what means quantifying over all natural numbers? Does it mean to take only those natural numbers which have the characteristic property of every natural number, i.e., to be followed by infinitely many natural numbers? Then you don't get all of them because always infinitely many are remaining. Or do you take all natural numbers with no exception? Then you include some which are not followed by infinitely many and hence are not natural numbers because they are lacking the characteristic property of every natural number. That means you get more than all natural numbers.
To make a long story short: The set of all natural numbers is a set that cannot exist because its elements cannot be in a set together. Therefore we have at most potential infinity. Actual infinity with countable and uncountable sets is a contradictio in adjecto.
Although it was Cantor's aim to apply set theory to physical, chemical, and even psychological and political problems
"The third part contains the applications of set theory to the natural sciences: physics, chemistry, mineralogy, botany, zoology, anthropology, biology, physiology, medicine etc. It is what Englishmen call 'natural philosophy'. In addition we have the so-called 'humanities', which, in my opinion, have to be called natural sciences too, because also the 'mind' belongs to nature." [G. Cantor, letter to D. Hilbert (20 Sep 1912)]
his ideas have turned out to be inapplicable everywhere and in particular to physics.