User talk:Grambottle
2 questions which we need to sort out ...
1. Can an uncountable group be cyclic? Is $\mathbb{R}$ a cyclic group? If so, what's its generator?
2. "Countable" has two alternative definitions, according to "taste": some authors exclude finite sets from being countable, others include finite sets in the definition. I prefer the former as it makes certain theorems more easily worded, and if your set is finite, you just need to say "finite".
I just googled for it and Wikipedia suggests a way of weasel-wording our way out of this dilemma.
Suggestion: differences of opinion in terminology are probably best thrashed out in the discussion page of an entry before editing the actual page, because you never know when a proof may depends on a definition being "just so", causing confusion if the definition were to change. If there is a true alternative definition, then that is best indicated by indicating this fact by means of "alternative definition" and saying something like "Some sources define it differently ..." or whatever. --prime mover (talk) 06:31, 17 December 2008 (UTC)
Answers:
1. Good point. Any cyclic group can be written as a Z-module with generating set $\{1\}$. Hence it must be countable. The first thing I think when I see something that can pertain to an infinite case is "works for countable AND uncountable?". I jumped the gun on that one.
2. I've ALWAYS seen countable defined that way; by different authors and different profs at different universities. If a set is finite, of course you can just say it is finite. It seems very silly, however, to say something like:
"A cyclic group is either finite or countable"
since you can count a finite number of elements (hence making a finite set countable). As far as I know, it's a much, much more common convention and shouldn't be strayed from. It also makes no sense (as stated above) to exclude a finite set from being countable.
I'll keep the discussion thing in mind. I contributed a little bit to ProofWiki back when it fired up (creators = my friends) but haven't been on it much in quite some time. Guess I'm not used to it being a full-fledged wiki :P.
P.S. Nice Barnstar.
--Grambottle 07:19, 17 December 2008 (UTC)
Okay then, I've amended all the pages that assume "countable" to mean "countably infinite" to accord with the new definition, and added a comment on your observations about the actual definition of "countable" to the "Countable" page. Thanks for your input btw, I should have been more alert with this one in the first place! --prime mover (talk) 07:30, 17 December 2008 (UTC)
ALL the pages?! In little over an hour? HOW THE.... ya know what, just not going to ask :P. Good job. Thanks a lot! --Grambottle 16:03, 17 December 2008 (UTC)
there weren't many. I did a "what links here" and there were about 10 and all most of them needed was "countable" to be changed to "countably infinite". No problem for a complete saddo who has nothing better to do with his life ... ;-) --prime mover (talk) 20:40, 17 December 2008 (UTC)