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Define a Russell socks set as a countable set of (pairwise disjoint) pairs such that no infinite subset has a choice function. Of course, if ZFC is consistent then it proves that no such set exists (as the axiom of choice is precisely the statement that every set has a choice function). On the other hand, it is known to be consistent with ZF that such a set exists.

Many wonderful and entertaining consequences of such a set existing in a model of ZF can be found in papers such as 'On the number of Russell's socks [...]' by Herrlich and Tachtsis or in Ethan Thomas' undergraduate thesis on the subject.

Neither of these papers explicitly construct a model of ZF containing a Russell socks set. Are any of the more common models such as Cohen's known to contain such a set? Is it easy to construct a model containing one?

Any reference would be much appreciated!

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  • $\begingroup$ Ethan Thomas's thesis. $\endgroup$ – Andrés E. Caicedo Oct 30 '16 at 17:41
  • $\begingroup$ In his thesis he constructs models containing so called psuedo-Russel socks sets. These are models containing an infinite set of pairs such that any subset contained in a given non-principal ultrafilter has no choice function. This is slightly weaker than the definition of a Russel socks set above I believe. $\endgroup$ – Guy Paterson-Jones Oct 30 '16 at 17:44
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    $\begingroup$ (The paper by Brunner mentioned in the references of the Herrlich-Tachtsis paper exhibits a model with Russell cardinals where every infinite set admits a Hausdorff topology with infinitely many nonisolated points, and the power set of the reals is well-orderable. The paper is available here.) $\endgroup$ – Andrés E. Caicedo Oct 30 '16 at 17:50
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    $\begingroup$ Take a look at MR3411399. Eleftherios Tachtsis. On the existence of free ultrafilters on $\omega$ and on Russell-sets in $\mathsf{ZF}$, Bull. Pol. Acad. Sci. Math. 63 (2015), no. 1, 1–10. In that paper, Tachtsis shows that in Blass's model where every ultrafilter is principal there are Russell sets. $\endgroup$ – Andrés E. Caicedo Oct 30 '16 at 17:54
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    $\begingroup$ MR0476510. Andreas Blass. A model without ultrafilters, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 4, 329–331. $\endgroup$ – Andrés E. Caicedo Oct 30 '16 at 17:57
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Yes. Cohen's second model of $\lnot\sf AC$ is a model in which there is a Russell set.

The proof can be found in Jech, "The Axiom of Choice" in Chapter 5, section 4. While Jech does not include the statement that the resulting set is a Russell set, it is implicit in the proof of Lemma 5.19.

Additionally, Fraenkel's second model of $\sf ZFA$ has a Russell set, and in the same book by Jech, he provides "transfer theorems" for transferring some results from models with atoms to models of $\sf ZF$ (without atoms). These include the existence of a Russell set as well. Other transfer theorems (Pincus, Hall) are equally suitable for the job also.

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