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I wanted to verify the following considerations on the context of Russell's infinite sock pair conundrum. The conundrum pointed out that a rule for choosing from pairs of shoes is possible a-priori. For indistinguishable socks, such rule is not possible a-priori, and has to be assumed.

Thus, in Set Theory there are two major avenues for context of Russell's Sock Pair example for the role of Infinite Choice (choice from an infinite number of unordered pairs).

In the first situation the pairs arise from the set {1,2} acting on some set in repeated fashion. In this case, there is an ordering of the pairs, and choice is not needed. Alternatively, the set of pairs may arise from an linearly ordered set (e.g. set of pairs of elements of $\Bbb R$). In such case a well order can be defined on each pair a-priori. In such situations Russell's conundrum does not apply (the pair of shoes situation).

For Russell's sock pair conundrum to arise one needs a set of pairs drawn by comprehension from the powerset of an not totally ordered (t.o.) set. Use of replacement schema to create the pairs (but not to change an already existing collection to something else) would lead to the first situation (existence of an ordering approach).

Is this correct? is there any other (non-equivalent) path to get to the infinity of sock pairs beyond comprehension acting upon a non-t.o. set?

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There's no definite way to create a set of sock pairs that don't admit a choice function, using the usual constructions allowed in set theory. After all, it is known that the Axiom of Choice is consistent with the usual axioms of set theory, meaning that it cannot be proved that such at sock set exists at all -- and an explicit construction of one would certainly count as an existence proof.

It is also consistent with ZFC that there is a single formula that always picks out one sock from each pair, and which works no matter which set you apply it to. (For example, if $V=L$ then there's a particular formula that well-orders the entire universe).

So the best you can hope for is a particular description of a definite set of pairs, such that no formula can be proved in ZFC to pick out one sock from each pair. I don't know for sure whether that is possible, though I would intuitively expect $$ \{\{(x,y),(y,x)\} \mid x\subseteq \mathbb R, y\subseteq \mathbb R, x\ne y \}$$ to be an example.

So a more precise question would be:

Is there a formula $\phi(x)$ such that

  1. ZFC proves that exactly one $x$ satisfies $\phi$,
  2. ZFC proves that every element of $x$ has cardinality $2$, but
  3. ZFC does not prove for any $\psi(y,z)$ that $\{ (y,z) \mid y\in x, z\in y, \psi(y,z) \}$ is a choice function on $x$?
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  • $\begingroup$ How about {$x|x\in P(P(\Bbb R)),|x|=2$}? That seems to provide infinite collections of pairs that do not have an apriori choice function attached? $\endgroup$ – Dacian Bonta Nov 23 '15 at 16:10
  • $\begingroup$ @DacianBonta: Yes, that is essentially the same as my conjectured example, except that I went to some extra trouble to make the pairs disjoint (it being somewhat untidy to have a sock that is in several different pairs). $\endgroup$ – hmakholm left over Monica Nov 23 '15 at 16:17
  • $\begingroup$ Sorry. Yes, that was my question in some way. Is there any other intrinsically/structurally different way of describing such set outside comprehension + powerset. $\endgroup$ – Dacian Bonta Nov 23 '15 at 16:34
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Existence is not a predicated property. It is a semantic property.

What does it mean? It means that after you've fixed a universe of set theory, there are things which exists there, and that's that. They don't have to be defined a-priori in order to exist, and they don't have to have a definition either in order to exist. They just do.

It is consistent with the failure of the axiom of choice that there is a family of pairs without a choice function. That is a mathematical theorem which tells you that it is possible to have a universe of set theory in which there is a set which can be partitioned into countably many pairs without a choice function.

We don't create anything. If anything is "created" it means, generally, that we proved its existence from the axioms, which implies that we can always find that set which was created (or its likened image). And of course we cannot prove from $\sf ZF$ that there is always such set. In fact, even if we assume that there is a set which cannot be linearly ordered we still cannot prove that there is a countably family of pairs without a choice function.

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  • $\begingroup$ Thank you for the clarification. It is however useful to understand if/how/why an object of one type leads organically to the existence of one of another type within the theory. The furniture is there, to use A. Wiles' analogy, sure; it gets trying to move one thing and seeing that it bumps against another to understand how they're all positioned. $\endgroup$ – Dacian Bonta Nov 23 '15 at 16:16
  • $\begingroup$ I'm not entirely sure what you're trying to say. $\endgroup$ – Asaf Karagila Nov 23 '15 at 16:22
  • $\begingroup$ Confusion arose from language barrier. H. Makholm's answer and subsequent comments were illuminating. My inquiry is on the existence of different descriptions of a Russellian (sock) set by means of set theory axioms (in particular ZFC). $\endgroup$ – Dacian Bonta Nov 23 '15 at 16:42
  • $\begingroup$ Descriptions and axioms have nothing to do with one another; and the axioms of ZFC include AC so they prove there are absolutely no Russell sets like that. $\endgroup$ – Asaf Karagila Nov 23 '15 at 16:43
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    $\begingroup$ And the point in my answer is that comprehension is not a method of "displaying such set" to begin with. If it were, you could have done that in $\sf ZFC$ as well, and we know that you can't quite do that. I think that there is a language barrier here, and you're using "displaying such set" in the wrong sense of the term. Because to me it would mean that you can give a definition that you can prove (from ZF) to be a Russell set (of socks). $\endgroup$ – Asaf Karagila Nov 24 '15 at 10:28

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