4 ms·
Nitpick: the infinite countable union of countable well-ordered sets is countable. This statement is immune to the failure of Choice.
by Smaug123 10y ago
Nitpick: the infinite countable union of countable well-ordered sets is countable. This statement is immune to the failure of Choice.
- enugu 10y agoAs Lang used to say the Axiom Of Choice is obvious, 'I just pick the elements'. Just kidding, thanks for this interesting logical point.
- gjm11 10y agoThe 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.)
- Smaug123 10y agoIn my mind, "well-ordered" is fully distinct from "well-orderable" :) but "counted" is unambiguous, you're right.