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)


 * I don't think you understood my intention. I fully recognize that we are not going to accomplish anything by trying to change traditional names, and I don't see any point in trying. The main point I was trying to make at the time will be reflected in my next edit to this article, which you can accept or revert. The secondary matter of how to deal with the fuzzy nature of axioms vs. theorems is up to you. Do you want to choose a specific collection of axioms and rules of logic to serve as the foundation for the site? Do you want to allow different articles to choose their own in some fashion?


 * Interesting point. The aim of this site is (purportedly) to document all possible approaches that aren't too far out; for example, we thus aim for different systems of propositional logic (Hilbert style, natural deduction, Gentzen sequents). In the light of the recently established consensus that to possible extent, definitions should be treated with equal merit, it seems only natural to extend this principle to the realm of the axioms.
 * However, before we embark on such a journey, it seems to me that we need a (much) better coverage of the axioms being used with reasonable frequency, and their interdependencies. I hope that you do not oppose the simplification of e.g. saying that AoC is required when we in fact use Zorn. OTOH, we have had an improvement a few months back by abcxyz, who introduced a distinction between AoC and ACC. Such is valuable. I have no idea where the Ultrafilter Lemma fits in this partial ordering; if it is in an equivalence class not covered yet, it seems reasonable that it gains equal status to AoC and ACC.
 * If we are to adopt this new convention, there are several consequences. In my opinion, it will lead to a substantial growth of the axiom namespace (not a bad thing); it will also naturally require the pages describing the axioms to take a more encyclopaedic nature, because we'll have to document interdependences exhaustively. At this point, I consider the issue minor, and suggest that we first cover more ground before abandoning our island again, setting sail to what appears to be even more fertile soil (i.e., go through a major refactoring operation again).
 * TL; DR / conclusion: I'm undecided as of yet. --Lord_Farin (talk) 16:13, 11 December 2012 (UTC)


 * From seeing your edit, I now see where you're coming from. The short answer is no. The long answer is not yet. Although yes, the Ultrafilter Lemma has as great a claim to be axiomatic as does AoC, as does Zorn's Lemma and as does the HMP and Uncle Tom Cobley's Supposition for all I care, the fact is that it makes no sense to increase the number of axioms beyond the minimum number necessary. The fact that any one of these statements can be used instead of the AoC in ZFC, the fact is that in the "conventional" approach they generally aren't.


 * We've had this discussion before on ProofWiki at mind-numbingly violence-inducing nauseatingly-tedious length and the conclusion which seems to have been arrived at is to set up a structure in which the various logical equivalents to AoC are gathered and grouped and explained. The interested reader can then see all this stuff at a glance and have it all neatly explained in one place, rather than have to suffer the confusion of several disparate slightly and trivially different axiom schemas to choose from with no indication as to why.


 * TL;DR: I can't be bothered to do all the work necessary to put all this trivial detail into place and the various exponents of this approach failed to apply finger to keyboard in that direction after all. So I assumed that the matter was clearly not important enough to anyone for them to actually do any of the work. --prime mover (talk) 16:30, 11 December 2012 (UTC)

I believe that our point of view can be accurately summarized by expressing that this may be a direction we could go, but the point is sufficiently minor that we do not want to put so much effort in - at the moment, at least. Maybe it can be taken into account when someone starts to put effort into the logic/foundations area of the site. That won't be me, or it will be some months at the least. --Lord_Farin (talk) 17:29, 11 December 2012 (UTC)


 * Lord Farin, since you asked, based on what I've found online the ultrafilter lemma is equivalent to the boolean prime ideal theorem, and to the fact that a product of compact Hausdorff spaces is compact. It is strictly weaker than the axiom of choice. It is strong enough to prove the Ordering Principle (which is itself sufficient to prove the axiom of finite choice), the Dimension Theorem for Vector Spaces, and the existence of non-measurable sets. I know no more at the moment. --Dfeuer (talk) 02:39, 12 December 2012 (UTC)


 * prime.mover, it is true that dealing with multiple non-equivalent, overlapping sets of axioms would require a lot of work. It's not just about cataloging alternatives to AoC and weaker forms of choice, since some people are interested in discarding or weakening other ZFC axioms, some people are interested in completely different ways of building set theory (e.g., NF), some prefer different rules of logic, etc... I personally have only a fraction of a clue about how one could structure a wiki around that mess. --Dfeuer (talk) 02:39, 12 December 2012 (UTC)


 * The problem we have is that, as a mentor at my place of work so colourfully puts it, you can only shit with the arse you've got. Placing requirements onto talk pages detailing what the site ought to provide and pointing out areas where it is in need of improvement doesn't actively do anything to actually put those improvements in place. I understand it must be frustrating coming to a website expecting to learn stuff you don't know, but the fact is, this is where we are. The actual contribution towards the expansion of this site into new work has (apart from a few exceptions) come to a complete halt, and all the changes I can see consist of minor changes and trivial hair-splitting amendments to (and comments about) existing pages, and areas which are already fairly well covered. (My own contributions towards the site is, at the moment, little more than rework, and has been for the last 18 months or so, but my excuse is that quite a few the pages I'm working on were contributed by me in the first place.) --prime mover (talk) 06:29, 12 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)

Using Zorn's Lemma
The proof is being flagged because its nontrivial that the arbitrary union of filters over a set is again a filter. In general this is not true. Take the power set of $\{\varnothing, \varnothing^+\}$ ordered by inclusion. Then $\uparrow\{\varnothing\} \cup \uparrow\{\varnothing^+\}$ is not a filter. But these two filters aren't comparable when considered as elements of the poset $P(P\{\varnothing,\varnothing^+\})$ ordered by inclusion. The chain restriction is what makes the arbitrary union of filters also a filter. Namely, the chain restriction is required to prove that the union is closed under finite intersections. The other three properties of a filter (empty set is not a member, underlying set is a member, upper set) are satisfied even when the filters of the union aren't comparable.

I don't know how to create pages, nor would know what to even title that proof. But the use of Zorn's Lemma is valid, and a reference to a page that proved that the arbitrary union of comparable filters is a filter would alleviate the need to flag the proof. --Robertbiggs34 (talk) 03:13, 21 July 2013 (UTC)