The formulation I prefer: a countable union of counted sets is countable. (Any countable set has a well-ordering, because a bijection with the natural number g

The formulation I prefer: a countable union of counted sets is countable. (Any countable set has a well-ordering, because a bijection with the natural number gives you one in an obvious way. The trouble is that you need well-orderings for all of them together. The unusual term "counted" emphasizes that we need the actual "countings" to do the job, whereas for me "well-ordered" is sufficiently commonplace that it doesn't shove in my face the requirement that each set come along with a specific choice of well-ordering.)