Axiom of choice confusion: what does it mean an element to have no distinguishing features?

I’m trying to understand the axiom of choice, but am stuck on this point:

How can an element of a set ever have no distinguishing features?
Two things which are identical are the same thing – surely?

So why would you ever need to invoke the axiom of choice?

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I assume that your confusion is maybe related to the famous (distinuishable) shoes vs. (indistinguishable) socks example.

However, you should notie that even socks are distinguishable – but not in a way that allows us to reliably/predictably pick one of them: If you send two people around and tell them to bring you all left shoes then they will always agree which shoe to bring. But whatever attempt you make in describing to them which of two socks to bring (e.g. “pick the one closer to the door, or if they are at the same distance pick the southernmost, or if this still doesnnot help, …”) there may always be corner-cases where they need not agree (what if the socks are wrapped around one another or in floating in te air and rotating around another?) hence your minions must be intelligent enough to make a choice on their own – and possibly infinitely often.

That being said, I suggst to follow Henning Makholms advice to look up a more technical formulation (not necessarily a heap of formulas, but at least related to sets). Contemplate for example if you can explicitly describe a map $f\colon \mathcal P(\mathbb R)\setminus\{\emptyset\}\to \mathbb R$ such that always $f(x)\in x$. On the other hand you should have no problems with this task if we replace $\mathbb R$ with $\mathbb N$ and hardly any problem if we use $\mathbb Q$ (where there is always something like a left-most shoe)

Let me give an answer which is more intuitive than formal.

What does it mean to distinguish between two elements?

What does it mean to distinguish between two socks? If you hold one sock in your left hand, and one sock in your right, of course you can distinguish them! One of them is in your right hand, and the other is in your left hand.

But suppose that I have two plain white socks, and I tell you to turn around and maybe I switched the order they were presented and maybe I didn’t. You turn back, can you tell the difference? Can you tell me which one was in my right hand before and if it is the same as the one in my right hand now?

This means that you can’t distinguish the two socks without examining them concretely. There are no features of these two socks which tell you in advance that one sock will satisfy the one feature and the other one will not.

The same is true for elements of a set. Given a set, you can ask if there is one element of that set which you can distinguish from the others. If there is some property that you can express, that without referring to the actual element, you can say with certainty that there is only one element in your set which satisfy that property.

If your set is a finite set of real numbers, then of course you can say that. Finite sets of real numbers can be linearly ordered by the order of the real numbers and you can say with certainty that only one of these real numbers can be the smallest one. In fact, even if you take many finite sets of real numbers, none of which is empty, you can guarantee that the fact the real numbers are linearly ordered tells you that exactly one of the elements from each set is “the smallest one in that set”.

On the other hand, if I told you that your set is countably infinite and have the same ordering as the rational numbers. Can you pick an element then? Sure, you can pick some way to match it with the rationals and then pick the one corresponding to $0$ or something. But there are many ways to match this set with the rational numbers, and we can show that those ways themselves are (in general) indistinguishable (there are too many automorphisms of the rational numbers, order-wise). So you can’t, in the general case, distinguish between two real numbers in a set if all you know is that the set “looks like the rational numbers”.

The axiom of choice gives us a way to distinguish elements. It gives us a function which gives a particular, unique, element of each set. And that is a wonderful way to make that element distinguished. It is the unique element which is the output of the function, when given that set.