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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?

Asaf Karagila
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Marcus
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  • First: what version of the axiom of choice do you know (there are many equivalent formulations)? Second: what does your question deal with the axiom of choice? – Crostul Jun 09 '15 at 12:57
  • What does that question have to do with the axiom of choice? – Thomas Andrews Jun 09 '15 at 12:57
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    It looks like the trouble you're having is not with understanding the actual axiom of choice, but with understanding someone's attempt to explain it in popular terms without being technical. Such attempts are often doomed; if you want to understand what the axiom says you need to go to source material that has formulas in it, such as a textbook in logic and axiomatic set theory. – hmakholm left over Monica Jun 09 '15 at 13:03
  • Thanks Henning. I will work to understand the technical definition - I expect that the confusion will disappear. – Marcus Jun 09 '15 at 13:20
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    @Marcus: Once you have sufficient familiarity with how axiomatic set theory works, this old answer of mine may be illuminating. – hmakholm left over Monica Jun 09 '15 at 13:27

2 Answers2

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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)

Hagen von Eitzen
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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.

Asaf Karagila
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  • But I would say that if you cannot distinguish between the socks, then you do not have two socks but only one sock... – Marcus Jul 09 '15 at 08:54
  • Suppose that I showed you two plain white socks (which pair with one another). Never worn, never washed. Brand new. You examine them with your eyes. Then you leave the room and return a minute later, and I tell you that I may or may not switched between the socks. Can you say if I did? – Asaf Karagila Jul 09 '15 at 10:55
  • My point is that this situation is impossible. The socks will be the same sock. It isn't possible to hold two identical socks in either hand, since identical socks are the same sock. If you are able to hold them apart from each other, they are distinguishable, they have different xyz coordinates. – Marcus Jul 16 '15 at 11:29
  • You haven't answered my question. I show you two socks. One in my left hand, one in my right. They look just like one another. Are they the same sock? I go outside for 30 seconds and return. I show you the socks again. Did I show you the same socks as before? Is the one I hold in my right hand the same one I held in my right hand before, or did I switch them? Forget mathematical properties or xyz coordinates or temporal interference from aliens. I'm showing you two socks. Can you tell me if I switched them when I was outside? – Asaf Karagila Jul 16 '15 at 11:33
  • Ok. No I cannot tell if you have switched them. – Marcus Jul 17 '15 at 13:55
  • And that is the whole point. While these are two socks, you can't really distinguish them by anything else than being able to say "Okay, I know that is not the same sock". Two objects can be indiscernible in the sense that you are able to say that they are not the same object, but they have no distinguishable properties that you can use to be able and recognize them. – Asaf Karagila Jul 17 '15 at 14:03
  • "Two objects can be indiscernible in the sense that you are able to say that they are not the same object, but they have no distinguishable properties that you can use to be able and recognize them." I'm sorry for me this fundamentally makes no sense. It is impossible to have two objects which have no features which allow you to recognise them and pick one from the other. The very definition of an object separate from another object is that it does have such a property. Two objects indistinguishable from the other are not two objects, but one object. – Marcus Jul 20 '15 at 09:20
  • @Marcus: If I show you two white socks. Just show you. You have no tools in your language to write down a property that discerns them. If I happened to have shown you that on the other side of the sock they have different symbols sewn into them it's fine. But currently, the language does not allow you to distinguish them, except that you know they are not the same sock. They have the same properties. When coming to choose from infinitely many sets, these are sets, they have no "immediate" distinguishable features, and their elements might also have no distinguishable features [...] – Asaf Karagila Jul 20 '15 at 10:03
  • [...] so you cannot "immediately" say anything more than "there are two elements in this set". Of course, the two objects are different, they have some distinguishing features, but at this point in time you have no access to this information. And since you cannot use this information, for all practical uses, it is irrelevant for the discussion. Therefore the two objects are indiscernible. I really don't know what else to tell you other than sit down and study logic and set theory for a while, and then it might make more sense. Good luck. – Asaf Karagila Jul 20 '15 at 10:04
  • Ok, thanks for your time. I'm not sure a standard text book on set theory is going to answer things for me, but I need to go and try.... – Marcus Jul 20 '15 at 11:39