Terminology for sequences and countability

Question

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.

Answer 1

Might as well add it:

A countable set is one for which there exists a bijection with $\omega$, and a counted set is one equipped with a bijection with $\omega$. Thus the statement true in ZF is

A countable union of counted sets is countable.

Answer 2

I would say that the terms enumerate, enumerable,enumerated etc. are flexible enough to be used for a variety of related purposes. You just need to make sure that your readers/listners share your understanding of how it is being used. I've never done it, but I could imagine enumeration by ordinals which are infinite but larger than $\omega.$ Hiwever enumerable is useful as a synonym for "countable or finite." If $S$ is a set of finite cardinality $n$ (maybe {fred, bill,joe}={the lawyer, the doctor, the CPA}={Mr. Jones, Mr. Smith, Mr. Collins} although I don't know which is which), then I can enumerate $n!$ permutations without having a standard ordering. Likewise the algebraic integers are enumerable and one can in principle declare an ordering but in practice it would be hopeless to say what position is occupied by $\sqrt[7]{13+\sqrt[11]{73}}.$