Talk:Ultrafilter Lemma

Axiom of Choice
The template claims that this theorem depends on the axiom of choice. In fact, this theorem (perhaps better called an axiom?) is strictly weaker than the axiom of choice, and is equivalent in ZF to the boolean prime ideal theorem. Therefore, I think the current proof should be introduced as a proof that AoC implies this axiom, and we need a proof that it is equivalent to BPIT. Dfeuer (talk) 07:19, 10 December 2012 (UTC)


 * Since we don't have the BPIT on this site yet, that will need to be done. I can't follow the direction of the rest of your comment. We have plenty of proofs of "A implies B" where A is stronger than B - it would be a daunting task to go into all of those proofs and indicate explicitly where, in each of them, the converse does not apply. In general it is sort of assumed that unless there's a page with a proof of the arrow pointing the other way, then the latter does not hold. --prime mover (talk) 07:40, 10 December 2012 (UTC)


 * The terminology gets confusing (to me at least), in large part because what one considers an axiom and what one considers a theorem depends on context/philosopy/personal preference. There's no particular reason for having the axiom of choice, Zorn's lemma, Tychonoff's theorem, and the Hausdorff maximality principle. In any case, there is a strong tradition of investigating the relative strength of various axioms that are implied by AoC, and just calling them all "theorems" and proving them using AoC doesn't seem quite fair. You wouldn't for example, be very likely to see a "proof" of countable choice that invokes AoC, but you might see a (trivial) proof that AoC implies countable choice. My concern is not with the mathematics here, but with how that is presented in English. --Dfeuer (talk) 08:03, 10 December 2012 (UTC)


 * The reason we call them "the axiom of choice, Zorn's lemma, Tychonoff's theorem, and the Hausdorff maximality principle" on this website is because that's what they're called in the outside world, so to speak. If this is a problem, then the solution is more wide-ranging than what we do on this website, which is more committed to providing a repository to document existing practices than it is to changing the existing paradigms. It would I believe be a sub-optimal approach to provide well-known concepts with a set of names which are not well-known throughout the mathematical community. "Where's Zorn's Lemma?" will come the anguished cries (and negative reports on peer websites).
 * While we don't provide a blanket ban, like Wikipedia does, stating "no original research", and while we are eager to document and utilise newly-evolved notation (as can be seen in the pages on real intervals, divisor and coprime), developing a completely new and (to a certain extent) arbitrary naming paradigm to well-known and (saving your presence) well-understood concepts seems to me contrary to the direction in which we might want this website to develop. --prime mover (talk) 09:04, 10 December 2012 (UTC)

Traditional name
This axiom is traditionally known as the "ultrafilter lemma". Should it be renamed to reflect that? If not, should there be a redirect page at Ultrafilter Lemma? Dfeuer (talk) 07:21, 10 December 2012 (UTC)


 * done --prime mover (talk) 07:40, 10 December 2012 (UTC)