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  • $\begingroup$ This argument is essentially the same as the argument that Andres gave in the comments above, but it seems to answer the whole question, and not just the issue about the reals being a countable union of countable sets. The proof shows more generally that the continuum, if well-orderable, cannot have countable cofinality, even in ZF. $\endgroup$
    – Joel David Hamkins
    Commented Oct 20, 2011 at 3:41
  • $\begingroup$ Thanks a lot, Joel. I swear I read somewhere that in the Feferman-Levy model $2^\omega=\omega_1$. I can't find it, and the proofs you and Andres gave convince me that it was probably in a dream. $\endgroup$
    – Asaf Karagila ♦
    Commented Oct 20, 2011 at 8:33
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    $\begingroup$ Just for the record, the standard proof of the more general Konig's theorem ($\kappa^{cof(\kappa)}>\kappa$ for $\kappa$ any aleph) doesn´t use AC (it is essentially the same proof that Joel already gave). $\endgroup$
    – Ramiro de la Vega
    Commented Oct 20, 2011 at 13:00