Talk:Union of Countable Sets of Sets

Countable Union of Countable Sets is Countable depends on AC, shouldn't one rather put up an argument like for the proof that $\Q$ is countable? It may at least be worthwhile to put that up as a second proof. --Lord_Farin 18:01, 14 February 2012 (EST)

If you like. --prime mover 01:07, 15 February 2012 (EST)

Proof up. I would argue that it deserves the status of Proof 1 as it doesn't use AC. Also, funny how I didn't need those induced orderings I thought of after all. --Lord_Farin 17:58, 15 February 2012 (EST)

... then again I'm not fully convinced that Countable Union of Countable Sets is Countable actually depends on AC either. The book I'm currently making heavy work of (Gaal) doesn't raise AC until well after these results have been established. So this also needs a rethink.

I think we also need to separate these proofs out into their own pages (even if only to establish exactly what the citation refers to, i.e. what is currently in as Proof 1). Not tonight, it's late and I have an early start tomorrow. --prime mover 18:04, 15 February 2012 (EST)

... Done. --prime mover 01:31, 16 February 2012 (EST)

My lecture notes cite an authoritative source, but it is in German; I won't be able to understand. So for now, suffice with

http://math.stackexchange.com/questions/55181/countably-infinite-union-of-countably-infinite-sets-is-countable --Lord_Farin 18:42, 15 February 2012 (EST)