Role of Equality Axiom in Russell Sock Axiom Of Choice Example

Question

Apologies for asking such a basic question, but I am trying to understand how Set Theory represents identical sets (in Quantum Physics sets of identical particles can behave oddly) and the question arose when I read about Russell's countably infinite set of pairs of Socks in various Axiom of Choice questions.

Instead of socks turn it into something mathematical, so represent a sock as the set for the number 1. So a pair of socks is represented as the set {1,1} and call this set P. This means that the countably infinite set of pairs of socks becomes {P,P,....}.

The axiom of extensionality is in Set Theory $\forall a \forall b (a=b \leftrightarrow \forall z (z\in a \leftrightarrow z \in b))$

This presumably means {1,1} = {1} and also {P,P,...}={P}.

Also presumably the cardinal number of any set of identical sets is always 1.

So it appears that Set Theory can't actually distinguish a pair of socks from a single sock when considered as sets, as Set Theory would always think each pair of socks is actually only one sock. So how can the Russell question about selecting one sock from each of an infinite pair of socks (either as a sequence of pairs or as a set of pairs) be formally represented in Set Theory in a form that is relevant to discussions about the axiom of choice ?

Answer 1

Russell's example is divulgative, in order to clarify the need of the Axiom of Choice when dealing with infinite sets :

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.

Socks are individual objects and not sets. Thus, the Axiom of Extensionality will not apply to them.

If they were sets, having no elements they will be all the same set, and specifically the empty set.

Thus, a pair of socks is a set with two "distinct" elements.

The issue is : from a mathematical point of view there is no "uniform" property that can identify in one fell swoop one of the two socks of each pair from the other one, in the case of an infinite collection.

This is not the case with the infinite collection of pairs of shoes, where we can simply say : "choose the right shoe".

Answer 2

So... you're living in a "one sock universe", where all the socks are the same? Or a "one ant universe", where all the ants are really just the same ant, moving really fast?

In mathematics two things are equal if they are the same. Two socks are not one sock. Two ants, even if you cannot distinguish which one is which, are not the same ant. Usually.

So you would represent one sock as $1$, but another sock as $2$ and another sock as $3$, and so on and so forth. But even that's wrong. Because that makes Russell's analogy odd, why can't you choose from pairs of socks? If the socks are labelled $1,2,3,4,5$ and so on, just choose the one with the smaller label from each pair.

The point is that Russell's analogy is based on the idea that you cannot label all the socks simultaneously. You can label all the pairs at the same time, yes. And you can distinguish between two given socks in the same pairs, but there is no feature that you can use that works to distinguish the socks in all the pairs at the same time.

Answer 3

The real problem here is that the sock analogy is a bad analogy that misses important features of how set theory works.

Yes, whenever you have two sets (in ordinary well-founded set theory), the axiom of extensionality tells you that they are distinguishable -- there will be something that is an element of one of them and not an element of the other one.

A better analogy would be, instead of speaking of infinitely many pairs of indistinguishable socks, you have infinitely many pairs of various woolen objects, and you want to instruct a stupid assistant to pick one object from each pair.

Whenever you look at one of the pairs, it is easy to see a difference between them. Some are red, some are blue. Some are big, others are small. Some are knit, others are crotcheted. Many are just random snarls of yarn.

However, whenever you ask your assistant to pick one from each pair, he will go working for some time, and then come back: "Boss, which of these two ones do you want? You haven't told me which one to pick." Each time you can find something to do with that pair, but the questions keep coming. Take the heaviest? Somewhere there's a pair of woolen things that weigh exactly the same. Take the one with the fewest stiches in total? After you've defined what counts as a stitch for a random snarl of yarn, you'll find that one of the pairs end up with the same count under that procedure. (Oh, but they are different colors! How did you forget to instruct him to pick the red one when that happens?)

The axiom of choice states that you have an assistant with sufficient initiative to just pick one of them already each time he comes to a pair that you haven't left instructions for distinguishing between.