This is just a question about terminology. I had thought that "enumerable" is a synonym for "countable," and you could call a set "enumerated" to mean it comes with some specific ordering of type $\omega$ (or an initial segment, if finite). Is that standard or is there another concise terminology for the distinction?

The point is I want a concise way to express the fact that, while ZF does not prove every countable union of countable sets is countable, Zermelo set theory is already more than you need to prove countability of every countable union of sets where each set is given with a specified listing as a sequence.

countedsets is countable", is what I've seen. – David Roberts Oct 24 '12 at 4:48