I’m fairly new to set theory and I think my book (Kolmogorov, /Introductory Real Analysis/) uses old terminology, but I followed that as ultimately making X a countable union of countable subsets, one for each “description format”.

As described in my counterexample, it occurs to me that the description formats themselves might have a finite description (eg I could define the nth format as “an n-tuple of real numbers”), and then *that* becomes the finite description of the set elements.

]]>Clearly the elements don’t all need to have the same method of finite description either—we can partition the set according to which description is used.

That leads to another question though: for what I’ve just said to work, does the number of description formats need to be finite? My feeling is that it does, but I’ve not tried to prove it yet. ]]>