Axiom of Choice

Context

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.

Problems

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.

Footnote

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.