Talk:Infinite Set has Countably Infinite Subset/Proof 3

Can it be investigated whether ACC can be used to produce this proof? I wonder whether perhaps no, because the power set of a countable set is uncountable, which rules out ACC at practically the first line. --prime mover 15:35, 5 April 2012 (EDT)


 * The statement can be adapted to use a choice function for $\mathcal C$ union with the cofinite subsets (I think). But that is countable precisely when $S$ is, so it doesn't help. I suspect it can't be modified to use ACC. $\mathcal C$ may simply be too big. --Lord_Farin 16:38, 5 April 2012 (EDT)


 * The more I read this proof, the more I think that it is pretty much the same argument as Proof 2, except that Proof 2 seems to leave the existence of $f$ (as defined there) unjustified. I could make out a version of Proof 2 that uses (at the very weakest) ADC. So far, I can't figure out how to use only ACC. --abcxyz 16:44, 5 April 2012 (EDT)


 * If it can genuinely be established that some of these proofs are genuinely too unwieldy to be sensible, or that two proofs are similar enough except for a minor detail (sorry but I'm not getting my head round this, the more I try the more confused I get) then feel free to consolidate it down a bit. Ignore all the impassioned pleading I made around new year - but I was worried that something important was going to get lost in the cutting-down.


 * In particular, though, it would be good to keep any proof which has a citation at the bottom from a published work - although it might be an idea to comment on the possibility that the proof itself "may not be very good" or whatever, because it uses AoC rather than the more logically economical ACC. Because if the result relies on ACC, then in particular the proof should use ACC - and unless we specifically raise pages with "ACC implies that ..." and "AoC implies that ..." then again we might start to lose track of what the "ultimate" truth really is.


 * Am I making sense, or am I sounding like David Cameron lecturing on religion, or Silvio Berlusconi on the subject of financial probity? --prime mover 17:13, 5 April 2012 (EDT)

So what you want/suggest is a note expressing 'Although this proof uses AoC, "different proof" shows that only the weaker ACC/ADC is needed. Therefore, one may consider this proof unsatisfactory.'. If that is correct, then this particular dissertation does not (yet) qualify you as a crackpot.

In expressing this, you also argued against deleting such 'unsatisfactory' proofs and against removing (approximate) duplicate proofs without reasonable discussion. Again, I can only agree. --Lord_Farin 17:27, 5 April 2012 (EDT)