I Axiom of Choice: Choosing Identical Objects

[Swamp Thing]

TL;DR Summary

Finite set of indistinguishables -- is axiom of choice required?

From Wikipedia entry on the Axiom of Choice:

Bertrand Russell coined an analogy: for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection; this makes it possible to directly define a choice function. For an infinite collection of pairs of socks (assumed to have no distinguishing features), there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice.

[1] What about a finite set of indistinguishable things (e.g. identical socks)? Do we need to invoke the axiom?

[2] Is there any physical consequence of this axiom, i.e. is there any physics experiment where the calculations to predict the result must incorporate the axiom of choice, even indirectly? Perhaps some experiment involving identical particles, say?

 

[fresh_42]

Summary:: Finite set of indistinguishables -- is axiom of choice required?

From Wikipedia entry on the Axiom of Choice:

[1] What about a finite set of indistinguishable things (e.g. identical socks)? Do we need to invoke the axiom?

No. We do not need AC in any case you can give an algorithm to make a choice. And if there are only finitely many things, just pick the first, e.g.

[2] Is there any physical consequence of this axiom, i.e. is there any physics experiment where the calculations to predict the result must incorporate the axiom of choice, even indirectly? Perhaps some experiment involving identical particles, say?

No. In the real world there are always only finitely many things involved in any experiment, even if you would count the particles. The infinities in models in field theory do not require AC either. It could only be important if we needed to construct a basis in an infinite dimensional space. I doubt that this is necessary and even if, then it probably hasn't any effects on real world measurements.

 

[Swamp Thing]

And if there are only finitely many things, just pick the first, e.g.

If the notion of "first" exists for a finite number of pairs of things, then it's not clear why it becomes invalid for an infinite number of such pairs. If the Nth pair supports the notion of "first" and "second", then the (N+1)th pair should support it as well. Ad infinitum?

 

[fresh_42]

If the notion of "first" exists for a finite number of pairs of things, then it's not clear why it becomes invalid for an infinite number of such pairs. If the Nth pair supports the notion of "first" and "second", then the (N+1)th pair should support it as well. Ad infinitum?

Yes, but what about uncountably many? What is the first complex number? The examples always involve a description of a set, so we can probably choose one. But what if we can only prove existence? How to select such a solution?

 

[Swamp Thing]

Ok, I think I see. The very notion of Nth and N+1th may not exist for an uncountable set of collections. Is that it?

 

[fresh_42]

Ok, I think I see. The very notion of Nth and N+1th may not exist for an uncountable set of collections. Is that it?

Yes, one part of it. E.g. consider the interval $[0,1)$. What is the last number in some ordering? Does it exist? AC has several equivalent formulations, one of it is that it does exist.

 

[Swamp Thing]

So in a sense, the axiom of choice implies that you can apply the principle of mathematical induction (arguing from Nth to N+1th) without being able to actually specify an N-like label for each collection?

 

[fresh_42]

So in a sense, the axiom of choice implies that you can apply the principle of mathematical induction (arguing from Nth to N+1th) without being able to actually specify an N-like label for each collection?

This is another concept, called transfinite induction. And yes, it requires the axiom of choice.