Does any physical process of finite duration embody the axiom of choice?

Does any physical process of finite duration embody the axiom of choice? No. But then no physical process of finite duration embodies an irrational number either. Nor a square or a cube. Nor any mathematical object, even the (finite) counting numbers...I think you missed the memo about Mathematics being an abstraction. It is used from time to time to model physical reality but it is not reality itself.The primary purpose of axioms is to provide a basis for reasoning. We choose axioms that simplify reasoning and lead to interesting results. In the "good ol' days" we used to strive for axioms that were "self evidently true", but we gave that up when we realised that mathematical models and physical reality were not as intimately connected as we thought. For example https://en.m.wikipedia.org/wiki/Euclidean_geometry used to be "the way the world was", but then we discovered https://en.m.wikipedia.org/wiki/Non-Euclidean_geometry, and then we found that it was a better model for how we now think the world "really is" in General Relativity.It turns out that certain mathematical abstractions, like Real numbers, are a lot simpler if you include transfinite operations. Almost all Real numbers are not computable and yet they make practical computation and analysis much easier than doing everything with only Rational numbers.Similarly using the https://en.m.wikipedia.org/wiki/Axiom_of_choice (AC) makes things much simpler. You can do lots of mathematics without it, but it is the proverbial pain in the you-know-where to do so. Interesting to examine, but largely fruitless.I would have more objection to AC if you could demonstrate some real physical process where it clearly did not apply, rather than trying to find one where it clearly does apply. In practice that is not going to happen: AC works every time in any finite model, and we can (in principle) approximate reality as closely as we like with mere finite models even if (in practice) it is very hard to do.

One form of the axiom of choice says that for any binary relation PPP, that is, subset PPP of X×YX×YX\times Y, (a)  if for each x∈Xx∈Xx\in X there is exists at least one y∈Yy∈Yy\in Y such that P(x,y)P(x,y)P(x,y) holds, then (b) you can specify such a yyy for each xxx, more precisely, there is a function f:X→Yf:X→Yf:X\to Y such that for each xxx, P(x,f(x))P(x,f(x))P(x,f(x)). Now, suppose you have a mathematical model of some aspect of physical reality in which (a) holds. Is it reasonable to conclude (b) also holds, or is there a good reason to deny (b)?An example. Every person has at least one cell in their body. Is there a way to specify a particular cell for each person? The physical process to make the choice is allowed to be as extensive as you like. How about taking that cell which is furthest north on the person's body? If there's more than one, take the one of those which is furthest east, and if more than one of those, take the the highest one.If you can come up with an example in which (a) holds but it's not possible to do (b), that would be a good reason to deny the applicability of the axiom of choice when modeling physical reality. ——————— The second paragraph in the details includes the statement "no one can point to an observable irrational number". In the same sense, on one can point to a decimal number with more than 100 significant digits. Does that matter? Isn't more useful to allow decimals without limiting their significant digits than putting a limit on the number of significant digits? Likewise allowing square roots to exist for all positive numbers is more useful than denying the square root of two. Mathematics is used to model physical reality. It isn't physical reality itself. A simpler model that allows arbitrarily many significant digits, irrational numbers, and the axiom of choice is preferable to one that's more complex. On the other hand, if you can find a simpler model that works just as well and turns out not to involve those things, use it if you like.