Talk:Infinite Set has Countably Infinite Subset

Discussion originally in here moved to Talk:Infinite Set has Countably Infinite Subset/Proof 2.

We have a perfectly adequate and technically simple proof in here now as Infinite Set has Countably Infinite Subset/Proof 1 which has a demonstrable reliance upon AoC.

To my mind the Infinite Set has Countably Infinite Subset/Intuitive Proof and Infinite Set has Countably Infinite Subset/Proof 2 are all just so much irritating noise - they're confusing and (despite protestations) consist of handwavery.

Anyone else support the view that they are either deleted completely, or relegated to links from this page, and surrounded by caveats and warnings?

I also don't believe the paragraph in "Note" really belongs here. General discussion around the use and implications of a definition (in this case AoC) would generally belong on the page on which that definition is raised, but even then I'm not sure. --prime mover 06:42, 4 December 2011 (CST)


 * I propose to remove everything before Infinite Set has Countably Infinite Subset/Proof 2, as well as the remark about the "intuitive proof" in Proof 2. Infinite Set has Countably Infinite Subset/Proof 1 is obviously nonsensical, and Infinite Set has Countably Infinite Subset/Proof 2 makes Infinite Set has Countably Infinite Subset/Intuitive Proof superfluous. Some renumbering and merging would then be in order. Also, I reiterate my claim that Template:AoC is opinionated and inaccurate and ought to be changed or removed. -- ratboy 19:09, 18 February 2012 (MST)


 * How would you write the AoC template?--GFauxPas 23:04, 18 February 2012 (EST)


 * A few points to address here:
 * a) UPDATE! I've taken the time to study this proof and conclude you may be right. Let me go and ponder it some more.
 * Having looked at it again, I can see it does make sense, but requires one more result in order to be tight. Work in progress on that result. --prime mover 07:10, 19 February 2012 (EST)
 * ... and having now done that, I think I've cleared any remaining problems with that result. I believe this is now clear. --prime mover 07:25, 19 February 2012 (EST)
 * b) An "intuitive" interpretation of Infinite Set has Countably Infinite Subset/Proof 2 is fully welcome on this site. (Perhaps intuitive is a bad word as intuition frequently leads one astray. The idea is that it's a wordy, expository proof that tries to explain in "layman's language".) Such sections are used elsewhere and (according to early students to whom the entirety of mathematics is not yet quite so blindingly obvious) they serve a useful purpose. But, as you suggest, we may well merge the "intuitive" proof into the same page as Proof 2 because they are tantamount to "the same thing".
 * c) Exactly what is "opinionated" about the AoC discussion? I appreciate that you are a full-on believer in AoC, and that you also seem to believe that everybody else is likewise nowadays convinced of it. I contend that this latter statement is not true - I have seen contemporary discussions on the web as recently as a couple of years ago (although I confess I've had other things to do than search for them). The real point is that there has been a recent mathematical endeavour that has been involved in determining which statements require AoC and which don't. It would be a mistake to ignore such an important stage in mathematical history just because it is considered by some to be no longer relevant.
 * Still and all, I have made an attempt at adding some words to the AoC template so that it expresses a fact and not an opinion. --prime mover 05:21, 19 February 2012 (EST)

axiom of countable choice
(1) Should this page be called "Infinite Set has Countably Infinite Subset" instead?

(2) If I'm not mistaken, this result follows from the axiom of countable choice, but it appears as if all the proofs use the full axiom of choice. I'm not sure how to fix up the proofs, without fundamentally altering them, as to use only the axiom of countable choice, or if that is even possible.

--abcxyz 01:49, 5 April 2012 (EDT)


 * Proofs which use AoC but only really need ACC should be amended so as to use ACC instead. Then Template:AoC needs to be copied and amended for Template:ACC (and also Template:ADC if that proves necessary). I'm doing other things at the moment. --prime mover 02:02, 5 April 2012 (EDT)


 * "Proofs which use AoC but only really need ACC should be amended so as to use ACC instead." Exactly, but I just said that I don't know how to do that, or if that is even possible without fundamentally altering the proof. I'll just add a proof that uses only ACC. --abcxyz 09:52, 5 April 2012 (EDT)


 * Unfortunately, nor do I ... most expositions on this subject don't go anywhere down the route of specialising between different choice axioms. Suggest we leave these proofs up, but what I think I'll do is move the "AoC" template into the actual proofs themselves rather than leave them on the main page. --prime mover 15:28, 5 April 2012 (EDT)


 * I think that's a good idea. --abcxyz 15:32, 5 April 2012 (EDT)