Talk:Equivalent Conditions for Dedekind-Infinite Set

There seems to be two things that are being stated here: one that Dedekind-infinite is equivalent to having the same cardinality as a proper subset.

The second (and more profound) thing is that you need ACC to prove that a set which has no bijection to a finite subset is necessarily Dedekind-infinite. That is, if ACC does not hold then there exist infinite sets which are not Dedekind-infinite.

So I suggest the two proofs be separated out into two pages. --prime mover 02:18, 5 April 2012 (EDT)


 * Good idea. --Lord_Farin 03:26, 5 April 2012 (EDT)


 * On this page, I didn't mean to include a proof that if the axiom of countable choice does not hold then there could exist infinite sets that are not Dedekind-infinite. That, I think, will definitely deserve its own page (but I don't have enough knowledge to create it). On this page, I only mean to prove logical equivalences. --abcxyz 09:48, 5 April 2012 (EDT)


 * I still contend that the case surrounding point 3 deserves its own page, as the initial assumptions are different. This is another instance of keeping a page as minimal as possible, not trying to prove too much at once. --prime mover 09:52, 5 April 2012 (EDT)


 * I probably should have mentioned this earlier, but I planned to just have a set of links like in Sequence of Implications of Metric Space Compactness Properties. Yes, I meant separate pages for proving separate statements, just as suggested. --abcxyz 10:01, 5 April 2012 (EDT)