I am thinking about the Axiom of Choice and I am trying to understand the Axiom with some but a little progress. Many questions are arising in my head. So, I know that there exists a model of ZF set theory in which the set of real numbers, which is provably uncountable, is a countable union of countable sets.


**Question:** does there exist a model of ZF set theory for which there exists a collection $A\_n$, $n\in\mathbb{N}$, of pairwise disjoint **two-element** sets such that their union is not countable?


Some thoughts. Let $A\_n$, $n\in\mathbb{N}$, be a collection of pairwise disjoint two-element sets.
Then for every $n\in\mathbb{N}$ there exists a bijection $f:\{1,2\}\to A\_n$.
But when we want to prove that $\bigcup\_{n\in\mathbb{N}}A\_n$ is countable, we have to choose a countable number of bijections $f\_n:\{1,2\}\to A\_n$, $n\in\mathbb{N}$, at once (simultaneously).
After this we plainly define the bijection $f:\mathbb{N}\to\bigcup\_{n\in\mathbb{N}}A\_n$
by $f(1):=f\_1(1)$, $f(2):=f\_1(2)$, $f(3):=f\_2(1)$, $f(4):= f\_2(2)$, and so on.
Rigorously, we write $f(k)=f\_l(1)$ if $k=2l-1$ and $f(k)=f\_l(2)$ if $k=2l$.
Clearly, $f$ is a bijection and we are done.
But without the Axiom of Countable Choice we can not choose $f\_n$, $n\in\mathbb{N}$, simultaneously and the argument does not work.


It is worth mentioning that if $A\_n$ are subsets of $\mathbb{R}$, then we **can** choose $f\_n$, $n\in\mathbb{N}$, simultaneously.
Indeed, we can define $f\_n(1):=\min A\_n$ and $f\_n(2):=\max A\_n$, $n\in\mathbb{N}$,
and the natural proof given above works.
So if a counterexample exists, the sets $A\_n$, $n\in\mathbb{N}$, have to be "abstract", say pairs of socks.



Can a countable union of two-element sets be uncountable?
Yes, it is possible. This phenomenon is sometimes called Russell's socks, named after an analogy due to Russell about how one can pick out a shoe from an infinite set of pairs of shoes, but not for socks since socks in a pair are indistinguishable.


[Horst Herrlich, Eleftherios Tachtsis, On the number of Russell’s socks or 2 + 2 + 2 + . . . = ?](https://www.emis.de/journals/CMUC/pdf/cmuc0604/herrtach.pdf) is a nice overview which proves some basic properties, including consistency of existence of Russell's socks.


