On 2 Jul., 04:03, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> My answer to your question above is similar. The question whether a countable > union of countably many sets is countable is implied by the axiom of countable > choice, which is not provable from ZF. So negating the axiom of countable > choice (which implies negating the axiom of choice) we can have situations > where the conclusion is false.
Feferman and Levy showed that the statement that the set of all real numbers is the union of a denumerable set of denumerable sets cannot be refuted.
So does it depend on the chosen axioms, whether R is countable?