I was reading the book "Howard and Rubin, “Consequences of the Axiom of Choice”.

For example, why do not exists sets that are neither countable nor uncountable?

It is interesting to point out that there are some statements on countable and uncountable sets that depend on the presence of weak choice principles (which are weaker versions of the Axiom of Choice).

For instance, there are models of ZF on which the statements “A countable union of countable sets is countable”, or even “A countable union of finite sets is countable”, do not hold. In fact, there is a model of ZF with a countable family of pairs whose union is uncountable.

One could intuitively think that this union of a countable family of pairs, which lives in a model where the Axiom of Choice is not available and because of that enjoys its existence as an awkward uncountable set, would become a countable set as soon as the Axiom of Choice would enter the room, but this is just a tale we tell to our students and has no formal meaning.