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Request for comment on small set / large set

I would like to request comment on the matter of the articles small set and large set. Another editor determined that "large set" and "small set" cannot share a disambiguation (I don't see why not), and split them up into two articles (I don't see why). I thought this was terribly redundant (just look at the two pages) and completely unnecessary. Maybe I'm wrong, but I propose that one ought merge them back together, so that people looking for large/small sets will find what they're looking for regardless. The two can (and have) coexisted in a disambig together for quite some time, and there's no reason they can't continue to do so. Do others have input on this? It would be greatly appreciated Thanks in advance. --Cheeser1 06:04, 24 August 2007 (UTC)

The problem is not as much with sharing a disambiguation as it is with listing Large set (Ramsey theory) as a possible meaning of small set. I suggest merging the two pages into Small and large sets with content along the lines of:
In mathematics, the term small set is used to refer to any set that is small in some sense. A large set is one that is not small. The terms have specialized meanings in the following contexts:
-- Meni Rosenfeld (talk) 06:45, 24 August 2007 (UTC)
See, but if large set redirects to small set, then large set (Ramsey theory) is not a specialized meaning of "small set," it's a specialized meaning of "large set," because the article (at that point) would cover both. That might require new wording, but I don't know if the title has to be changed to accommodate that. We have plenty of articles with multiple titles, and the content is allowed to reflect that multiplicity of titles. --Cheeser1 16:44, 24 August 2007 (UTC)
I had put together a page for this purpose User:CRGreathouse/Large and small sets, although as I recall Trove didn't like it much. CRGreathouse (t | c) 17:26, 24 August 2007 (UTC)
Seems OK to me. Such a thing could become over-extended, but that's a bridge to cross when we come to it. Charles Matthews 19:42, 24 August 2007 (UTC)
I don't see any problem with having separate disambiguation pages for "small set" and "large set" disambiguation pages, like redirects, are cheap. Paul August ☎ 19:49, 24 August 2007 (UTC)
That's the way I see it. I don't want to see an OR-ish exposition of all the various ways "small set" or "large set" might be used as nonce terms, under these titles. Our articles on ideals and filters and measures are the proper place for most of that content, and the disambig pages can have see-alsos to those articles. --Trovatore 03:31, 25 August 2007 (UTC)
But that requires double-duty for maintenance/upkeep of varying definitions of small/large. The "exposition" and see-alsos are not the matter in question, in any way (although "OR-ish" is a bit of a stretch). Small and large set often are the same thing (or rather, they describe the same notion). Combining the pages, as was the case previously, saves people trouble of looking for one or the other and getting different results. The exposition is unnecessary in the disambiguation, yes, and it could easily be removed. But that doesn't change the fact that the two topics can easily and sensibly be in the same disambig together. (This excludes the proposal by CRGreatehouse that we make it into a page with content, where exposition becomes necessary, of course.) --Cheeser1 08:09, 25 August 2007 (UTC)
The OR part is the idea of abstracting a commonality from the disparate ways in which the terms "small set" and "large set" are used. This we should not do. And that includes abstracting the notion that a small set is a set that isn't large, or is the complement of a large set, or anything of the sort. Your own objection to the term "small set" in Ramsey theory reinforces this point (quite possibly, that line item should be removed from the small set disambig page).
The upkeep issue is not terribly convincing -- all we need to do is make sure that the articles remain disambiguation pages in the strict sense, with zero exposition. That's fairly minimal upkeep, well worth it to avoid any suggestion that "small set" or "large set" has some general mathematical meaning apart from their particular meanings. The closest thing to such a commonality is the concepts described at ideal (set theory) and filter (set theory), but these are not standardly described using the terms "small" and "large" in any technical sense, so they are properly dealt with by listing them in the "see also" section without further comment. --Trovatore 19:26, 25 August 2007 (UTC)
All of that is about the exposition you're concerned about, not about whether or not the articles should be split up. Like I sad a week ago on the article's talk page, if you had an issue with the exposition, you should have removed the exposition, not split up the articles. The fact is, it's perfectly sensible to combine the two and link to the appropriate uses of the term (large or small). When the terms have specific meanings, they often mean opposite things. This is mentioned in this devilish exposition, when it appears in the content of each article. Nothing prevents us from disambiguating to them from one page, instead of two, by combining them as they had been combined up until a week ago. --Cheeser1 19:43, 25 August 2007 (UTC)
There is no sigificant cost to having the two articles disambig pages, and it's cleaner that way. Should have been done that way from the start; all I did was correct the original mistake. As I say, assuming you're right about the Ramsey theory issue, the small set (Ramsey theory) entry should probably be removed from small set. --Trovatore 20:17, 25 August 2007 (UTC)
There's also no reason to have two articles either (your points about exposition aside - exposition is an unrelated concern to be resolved at another time). As such, there was no reason to change it, and it seems that on Misplaced Pages, if there is no reason to go one way or the other, you leave it as it was first done (e.g. BC vs BCE). You keep referring to it as a "mistake" or "problem" that you need to "fix" but you've never presented an honest-to-goodness policy or reason why we can't have two things in one disambig. "Small set or large set may refer to: ...(list)" In fact, I'm relatively sure we can. I'm not convinced that having two pages wouldn't require extra upkeep and wouldn't be more difficult to navigate - even if we assume you're right and they wouldn't make things worse, the only reason you've given in their favor is the one about exposition - a point that isn't related. --Cheeser1 20:45, 25 August 2007 (UTC)
And the exposition would be simpler with one article; unless someone can give an example of a "small set" which is not a "not-large set". Septentrionalis PMAnderson 21:32, 25 August 2007 (UTC)
There is no such example, as far as we know. There is a "large set" here where "small set" is not used, but the only problem that could arise would be if there were a "small set" of this type (Ramsey theory) that is different in its definition from "not a large set." This is highly unlikely (I'd wager impossible), since I can't imagine terminology being accepted where "large set" and "small set" in a particular context are both defined but completely unrelated. --Cheeser1 22:16, 25 August 2007 (UTC)

Let's clarify our terms here. Disambig pages are not "articles". They are navigational tools, intended to get the reader to the article he/she is looking for. We should not have any article whatsoever about "small set" in general or "large set" in general, because there is no such mathematical concept. To the extent there's a mathematical concept to be extracted, it is that of an ideal or a filter (though I don't know whether "large sets" in the Ramsey-theory sense form a filter; I'd be interested in seeing that clarified, but just as a point of curiosity irrelevant to the discussion at hand).

Since the dab pages are purely tools to map search terms to real articles, and not real articles themselves, it is appropriate to have separate ones for separate search terms, even if they wind up mapping to the same collection of articles. --Trovatore 07:28, 26 August 2007 (UTC)

First of all, drop the filter stuff. It's irrelevant, and is obfuscating the fact that you have yet to address the question at hand. You continue to make points about how this is a disambig. Therefore it can't be an "article" or have too much "content." That's fine. That is not what we are discussing. It's perfectly reasonable to have two disambigs. It's also perfectly reasonable to have one. The two terms are often indistinguishable, and as a "navigational tool" either configuration works just as well as the other. You felt the need to throw your weight around by asserting that you're "fixing" the "problem" but you aren't. There is no guideline, policy, or reason (that you've given or that I've ever seen anyway) that tells us we can't have left it the way it was. If changes were unnecessary, as I believe they were, you should have left things as they were. So, tell me we can't have exposition or an "article" or how filters are really great, but I still feel like there's a perfectly reasonable solution: until there's a good reason to split the articles (a problem that doesn't just require cutting down on the exposition or whatever), don't split them. --Cheeser1 04:06, 27 August 2007 (UTC)
I find the analogy with filters, and the question of how sharp it is, to be very relevant here. Among regulars of this page, discussing the actual mathematical content while deciding how to arrange our description of it is common practice. There's no reason I can see that the discussion about filters obfuscates anything. If anything, it helps to clarify what's going on. — Carl (CBM · talk) 14:07, 28 August 2007 (UTC)
Cheeser, you started the whole thing by asserting -- quite incorrectly -- that redirects should be replaced by direct links. You were just wrong about that, but you wouldn't give up on uglifying the small set dab page by putting in a "large set" line item. It's reasonable to allow small variations on the basic search term as line items in a dab page, but antonyms, IMHO, are going too far.
I cut the Gordian knot by making the second dab page, which is an inherently better solution anyway, since it lets you (you personally, if you like, or you generically, otherwise) take out the "small set (Ramsey theory)" link, which we have your word is an unused term, and leave the "large set (Ramsey theory)" link in a dab page where it would be naturally found. --Trovatore 05:49, 27 August 2007 (UTC)
First of all, accusing me of anything, based on I "started the whole thing," is completely irrelevant, inflammatory, and a ridiculous side-point. You're twisting my words, taking things out of context, and trying to ignore my points, which are about the current discussion, not the one from three weeks ago. I inserted "large set" because there is no such "small" analog in use. That is the reason it was there. That is all that's relevant to this discussion, and I'll thank you not to dredge up old nonsense to try to obfuscate this issue.
I'm glad you think you've solved everything by taking bold decisive action. However, the only reason you've come up with is that it's "ugly" and that you seem to think that antonyms are not allowed on disambiguations. Now, I think it makes more sense there, and that's how it was, and until you find me a policy for this "it's ugly to have an antonym" opinion of yours, you've got neither policy nor consensus on your side. And yet you think your edits should stand because, what, you've decided you're the fixer and I'm the bad guy? Sorry, saying that you're "fixing" the "ugly" "problems" I add to pages doesn't make it so. --Cheeser1 06:41, 27 August 2007 (UTC)


Because there is no compelling policy-reason either way, and because there is consensus both for and against a merge, I suggest that we follow Misplaced Pages's general policy whenever two things are both perfectly good - keep it the way it was. In light of this, I suggest we merge them, like they used to be (although perhaps with slightly different content), since that's how they were in the first place. Does anyone object to following this widespread precedent for disputes without consensus? --Cheeser1 06:47, 27 August 2007 (UTC)

Cheeser, "the way it was" was with the link to small set (Ramsey theory) in small set, not to large set (Ramsey theory). There's no policy either way on that either (though there definitely is a guideline against replacing redirects with direct links). If we go back to the status quo ante, it should be back to that version. But this way makes more sense altogether. --Trovatore 16:30, 27 August 2007 (UTC)
"Small set" is not used in that sense. There is a policy on that, it's called WP:V. There's no such thing as "small set (Ramsey theory)" in this context. If you want to violate WP:V to prove some petty point about how you want to split it and I don't, be my guest. Otherwise, we will continue to discuss the issue at hand - whether or not to split the articles. There is no policy or consensus either way. I would like to proceed in a manner consistent with widespread precedent and collapse them back into two articles. If you'd like to stick to your guns, that's fine, but unless you have a policy you'd like to cite, no version is better than the other and you changing things just because you feel like it isn't a good way to work in a community of editors. --Cheeser1 18:55, 27 August 2007 (UTC)
So start a new section and make a straw poll. I'd like to see just what the consensus is -- or if, more likely, few really care. CRGreathouse (t | c) 20:35, 27 August 2007 (UTC)
That's what I just tried to do when Trovatore jumped back in to pick up the discussion (notice my: "does anyone object ...?"). It's already pretty clear, as far as I can tell, that consensus (among those who care to comment) is mixed. Unless an overwhelming consensus pipes up one way or the other, or unless some policy is dug up that clears any of this up, I'm inclined to believe that the do-it-the-way-it-was-done method is the way to go. I'll try again, with a new heading. --Cheeser1 07:13, 28 August 2007 (UTC)

The status quo ante had all the line items in small set as some small set (foo), which obviously makes sense for someone looking for the search term "small set". Cheeser1 wants to consider these issues separately, but they are in fact linked, and if it's to go back "the way it's always been" then the link should go back to small set (Ramsey theory). If as Cheeser1 claims, that's an unused term, then fine, let people looking for the search term "large set" see a disambig page where the articles are called "large set", and contrariwise. --Trovatore 07:26, 28 August 2007 (UTC)

The status quo had large set redirecting to small set, and thus this page served as disambiguation for both terms. The fact that someone inserted the unsourced and irrelevant commentary/exposition and formatting (use of "small set (subject)" throughout) based on the false assumption that all "large sets" are defined as "not small sets" is irrelevant. This is plainly obvious. You yourself argued that such exposition needed to be removed. --Cheeser1 08:36, 28 August 2007 (UTC)

"Straw poll"

In order to assess more succinctly if there is a consensus in this matter, please comment below with "split" "merge" or "no opinion" and, if you'd like, a sentence explaining your position (what policy, guideline, or personal opinion might be motivating your "vote"). Please keep discussion above, where it belongs. Thanks. --Cheeser1 07:13, 28 August 2007 (UTC)

  • Merge - no policy, guideline, or (apparent) consensus either way, so keep them together like they always have been. --Cheeser1 07:13, 28 August 2007 (UTC)
  • Leave as is. The status quo ante had all the line items in small set as some small set (foo), which obviously makes sense for someone looking for the search term "small set". Cheeser1 wants to consider these issues separately, but they are in fact linked, and if it's to go back "the way it's always been" then the link should go back to small set (Ramsey theory). If as Cheeser1 claims, that's an unused term, then fine, let people looking for the search term "large set" see a disambig page where the articles are called "large set", and contrariwise. --Trovatore 07:26, 28 August 2007 (UTC)
    • Further philosophical comment. Disambig pages are kind of like redirects to multiple pages, with one big exception: Whereas redirects are semantical, based on the expected meaning, dab pages are inherently syntactical -- you're looking for a particular search term. The least surprise principle suggests that people looking for the search term "large" should not be dumped into a disambig page titled "small". Then when they actually follow the links, that's different; it's normal to be redirected to an article and have to find the search term there; that's semantical. --Trovatore 07:26, 28 August 2007 (UTC)
  • No opinion between merging and keeping split. I had proposed exposition (see User:CRGreathouse/Large and small sets for an example) but Trovatore doesn't want exposition here, and I'm unwilling to oppose his wishes without some serious consensus. CRGreathouse (t | c) 13:36, 28 August 2007 (UTC)

Why are we having a poll here?

Why are we having a poll here? There aren't nearly enough people involved to expect to get a resolution from it. — Carl (CBM · talk) 14:07, 28 August 2007 (UTC)

In order to conclude more concretely the fact that there is no consensus on this matter. I suppose a poll isn't necessary, but someone suggested it, and I would feel remiss if I did not heed such a suggestion and instead acted in a way that might be interpreted as unilateral. I'm certainly of the opinion that the lack of the consensus is fairly evident, but I would not want to presume that it is until is more explicitly so. After a day or two more, I believe consensus or lack thereof will be determined clearly enough. --Cheeser1 17:25, 28 August 2007 (UTC)
That would not, of course, be sufficient for you to make your preferred change; at best your wikilawyering would get you back to the status quo ante (with all line items being "small set"). There's no consensus for your change either. --Trovatore 01:07, 29 August 2007 (UTC)
Sure. Unfortunately, I have WP:V on my side. "Large set" is the appropriate term, and if lack of consensus (which we obviously have) leads us to revert to the "status quo" we will have a disambiguation page for both terms together. WP:V leads us to change "Small set (Ramsey theory)" to "Large set (Ramsey theory)" to avoid original research. Stop mincing words. Why are you so obsessed with splitting these pages? There is no policy or consensus in favor, just your personal preference, albeit shared with some people. Without consensus, things should stay the way they are. The fact that you want to revert my unrelated contribution which has to do with WP:OR violations is irrelevant. Stop making a fuss. I know you want to "fix" the "ugly" "problems" so that Misplaced Pages is exactly the way you want, but you don't own Misplaced Pages. --Cheeser1 01:48, 29 August 2007 (UTC)
It really isn't appropriate to make any further changes to the pages' names until we find some consensus to go forward. Pushing towards "no consensus" isn't a productive strategy. Many editors here have long experience with WP and are not particularly vulnerable to wikilawyering. I recommend that you work towards finding some compromise. — Carl (CBM · talk) 01:54, 29 August 2007 (UTC)
First of all - are we changing pages' names? I don't recall that being proposed by anyone (maybe I misunderstood or missed something). Regardless, when somebody swoops in and makes a huge unilateral change to an article, I expect a reason besides "I like it better this way." And I'm not pushing for no consensus - there's already no consensus, this should be evident. The fact that I'm trying to move on from there is apparently counting against me. Either the article is split, or it isn't. There was no (objective) reason to split it, and no consensus to do so. So we should un-split it. I'm perfectly happy compromising by cutting the exposition and so forth - nowhere did I expect that to go back in, and I'm just as opposed to it as anyone else. What more compromise could I give, but to concede entirely to Trovatore just because he insists that his version is "better" because it "fixes" "ugly" "problems" in the article. This is the most compromising version one could use, without conceding entirely to Trovatore (hardly a compromise) or violating WP:OR by abusing/inventing terminology. --Cheeser1 02:47, 29 August 2007 (UTC)
Not a chance, not gonna fly. The only thing you can get with the "no consensus" version is back to the status quo ante. I will admit that I wouldn't have my back up about this if it weren't for your seriously substandard manners and comportment in the original discussion at talk:small set combined with the colossal gall then to lecture me about mine. But that's the way we stand; if you're going to play wikilawyer, the best you can get is back to where we were before your change. --Trovatore 03:27, 29 August 2007 (UTC)
The no consensus version should not violate WP:V, and as such, should include only verifiable mathematical terminology. If you want to give me the "pipe down whippersnapper" lecture, you can hold your proverbial tongue because I'm not interested. You accuse me of Wikilawyering - at least what I'm trying to do (include "Large set") is based on policy (and consensus, I might add), instead of an irrelevant personal preference that requires sweeping and unsupported changes to the article structure. I've given you everything except exactly what you want - I've agreed to reduce exposition, agreed to link only to valid mathematical terminology, agreed to avoid expanding it into any sort of article, all of which we've come to terms with - but when I still don't give you the split that you want (for no supportable reason), what? You refuse, and insist that we revert to a version that violates policy?? This is crazy, and if you're going to use my "manners" as motivation for refusing to compromise, I'd say you're the one who's out of line. I'm merging the articles, per the status quo, based on lack of consensus, policy, or other means to determine whether or not to split them. I will not include content that violates WP:V. If you want to reintroduce that content without splitting up the articles, feel free, but I will not be the one to reintroduce the WP:OR term "Small set (Ramsey theory)." How's that for compromise? --Cheeser1 05:22, 29 August 2007 (UTC)
There was no consensus to expand the small set dab page to include the search term large set. You work out how readers are going to find large set (Ramsey theory), without the large set dab page. --Trovatore 05:27, 29 August 2007 (UTC)
I'm afraid you'll find that before you split them up, this page functioned as the disambig for both. Hence it was the disambig for both terms in per the status quo. We've been over this 100 times. I don't want to have to repeat such a plain and obvious thing again. --Cheeser1 05:31, 29 August 2007 (UTC)
This "I don't want to repeat myself" when you're wrong is the single most offensive aspect of your comportment and manners. It was not the disambig page for both terms; it didn't mention "large". "Large" just redirected there. Redirects are semantical; disambigs are syntactical. For obvious reasons. --Trovatore 05:33, 29 August 2007 (UTC)
Fine. Since you've stooped to continually attacking my "conduct" (asking you not to make me repeat myself), why don't you take your snooty self-righteous attitude out of my face? "Redirects are semantical; disambigs are syntactical." Tthat's all well and good, and I'm glad you remember your SAT words, but this mythic "status quo ante" you keep referring to made explicit, unverifiable claims that "large" means "not small". The only reason it was not explicitly the disambig for both search terms is because of this completely false presumption. It was still implicitly the disambig for both. If not for those misguided assumptions, it would been (correctly) the disambig for both search terms. You were far more adamant than I about removing this "exposition," as you called it. Why can't you cope with the consequences of removing it?? --Cheeser1 05:39, 29 August 2007 (UTC)
The natural consequence is the split. Apparently they never should have been combined in the first place, since there's no uniform relationship between "small set" and "large set" in different contexts. --Trovatore 05:44, 29 August 2007 (UTC)
Ah, so apparently you're in charge of natural consequences. Good to hear that you've appointed yourself mother nature. There is an obvious uniform relationship: when the two exist, they are opposites. This clearly allows for them to remain together (in a natural way). But you'd rather presume that your idea of "natural" is best. Because it's what you prefer, even if it requires sweeping unilateral changes. If you think your idea of best is actually best, then I'm done talking to you. If there's one thing I've learned on Misplaced Pages, it's this: when someone doesn't know the difference between their completely subjective (in no way superior) preference and "the best possible way to do it," there's really nothing to discuss. --Cheeser1 05:48, 29 August 2007 (UTC)

Cheeser has promised not to respond, but I submit these Facts to a candid world: "small" and "large" are opposites, but "opposite" can mean a number of things. In the current case there are, a priori at least, two obvious possibilities: A large set might be simply not a small set (that is, a positive set in the relevant ideal), or it might be the complement of a small set (that is, a set in the ideal's dual filter). So we can't confidently predict any uniform relationship between the two terms that holds necessarily across all possible contexts ("intensionally" as opposed to "extensionally"). Not that that matters too much anyway, since disambig pages are based on finding a particular search term, and "small" is not the same sequence of letters as "large", nor even any similar sequence of letters. --Trovatore 06:11, 29 August 2007 (UTC)

And while I refuse to respond to Trovatore, I submit these facts (without being pompous or arrogant enough to capitalize "facts"): both articles on small sets explicitly define large sets as sets that are not small. Just like I said. I would ask that other editors disregard his absurd and inappropriate obfuscation of these issues and the related minutia he's using to distract us from the main issue here: There was a perfectly good "status quo" version with one very minor WP:V problem. Trovatore decided to make sweeping changes despite lacking support of consensus or policy, because he likes to have things his way. Since not everyone has rolled over and let him do whatever he wants, he has actually stooped to making the article worse, rather than have a version that he does not like. And keep in mind, I'm only ignoring him because he's explicitly stated that his refusal to compromise is based on some personal issue he's decided to have with me. I'd be happy to continue to discuss this with anyone else. --Cheeser1 06:24, 29 August 2007 (UTC)

Just for the record: I have not seen a single instance yet where Trovatore has made anything on wikipedia worse, and I have seen innumerous places where he has made great contributions. I did witness, on the other hand, basic lack of understanding of the meaning of consensus, cockish and inflammatory edit summaries, edit warring, Neanderthal manners in general, and bizarre insistence on "an administrator thinks that I am right, hence I AM RIGHT!", all of the above by Cheeser1 at Calculus, not even three months ago. Arcfrk 06:13, 31 August 2007 (UTC)
Yes, let's make this an issue about my "manners," especially pertaining to events from months ago. God forbid we discuss the issue at hand. If this is what I get for coming to WPM, I'd rather not come back here. Trovatore admitted to refusing to compromise because he didn't like my attitude, and I'm getting the same crap from someone else to? Forget this. --Cheeser1 11:58, 31 August 2007 (UTC)

"such that"

Such that.

Sigh.

Well, there's at least some truth in this article. It needs to be brought into conformance with the usual Misplaced Pages conventions. But if there's a reason to keep it, it needs a lot more than that. Michael Hardy 22:44, 24 August 2007 (UTC)

Banish it to wikitionary. Arcfrk 03:42, 25 August 2007 (UTC)
Along with "therefore" and "if ... then ..." ? — Carl (CBM · talk) 04:26, 25 August 2007 (UTC)
I am not positive, but I do imagine that wiktionary would want an article on this (as it's a 'set phrase', or whatever they call it). However, it might be appropriate to include it in mathematical jargon and table of mathematical symbols. Actually, I think that mathematical terminology and notation might make a good wikibook, but that is perhaps a bit outside the scope of WPM. --Sopoforic 07:33, 25 August 2007 (UTC)
"Such that" is hardly mathematical jargon, since it is used in precisely the same way in both mathematics and in common language. Mathematicians do apply it to certain properties that tend not to arise in everyday settings, of course. It may indeed belong in the table of symbols, though, given what's currently in such that. Ryan Reich 20:45, 25 August 2007 (UTC)
The article has a unique interpretation of the set-builder notation in which the central symbol is an independent notation. I don't think there is support for that interpretation in any printed reference. — Carl (CBM · talk) 21:08, 25 August 2007 (UTC)
Well, not jargon in the sense of slang, but certainly it is used mostly as a technical term. I don't ordinarily hear it used in casual speech: "J. Random Celebrity has recently purchased several cars such that each car has pink spots on its roof" doesn't quite sound right. However, looking at mathematical jargon, I agree that that isn't really the place for it. It already has a description in table of mathematical symbols and in Set-builder notation, so that's probably sufficient. I support redirecting it to Set-builder notation. --Sopoforic 21:41, 25 August 2007 (UTC)
A redirect sounds like a good solution to me. "Such that" is a term of art deserving some sort of explanation, but not really a whole article, and the proper context for the explanation is set-builder notation. —David Eppstein 07:39, 26 August 2007 (UTC)
I would point out in passing that set-builder notation is not the only context in which the term "such that" arises. However I don't have any strong objection to the redirect; the alternative seems to be deletion and as a practical matter it may not be worth the headache. --Trovatore 08:13, 26 August 2007 (UTC)
It does arise in other situations, of course, but I don't know where else to redirect it to. The article such that is at present totally about the colon or vertical bar used in set-builder notation to represent 'such that'. If you think it should redirect somewhere else, please recommend somewhere. --Sopoforic 19:23, 26 August 2007 (UTC)
To be honest, I think it should be deleted. I'm just not sure the difference between the redirect and getting rid of it entirely, is worth going through AfD. But I'll prod it and see what happens. --Trovatore 19:27, 26 August 2007 (UTC)
Oh, I guess it's been tried. OK, AfD it is, grumble grumble. --Trovatore 19:34, 26 August 2007 (UTC)

"Set-builder notation" (I've always thought that was a childish-sounding phrase) is not the only place where this phrase is used essentially as mathematical jargon. Just look at the usual epsilon-delta stuff. And many many similar cases. Michael Hardy 13:49, 26 August 2007 (UTC)

(AFAIAC, it is a childish phrase: the only place it's ever been uttered to me was in ninth grade.) I still argue that its status as jargon is low, and that it is, at worst, a piece of mathematical language, a manner of speaking like the manner of writing for a particular literary school. That language is de rigeur rigorous, so it tends to be a little more formal, and of course it includes many jargonish phrases, but this one is just a rather stiff alternative to the usual "who/which/that". For example:
"The numbers which are greater than zero..." versus
"The set of all x such that x > 0..."
mean the same thing in math and in English and the only reason we as mathematicians prefer the latter is that it is formulaic and, therefore, precise. This works for "the usual epsilon-delta stuff" too, though you won't believe me because no one names their variables in English. But here are two analogous sentences:
"For every opportunity missed, there's another right around the corner which is bigger and better." (A plausible maxim)
"For every ε > 0, there is an N such that for x > N, 0 < f(x) < ε." (Functions asympotic to zero; I was trying to keep the actual math short)
I assert that the only jargon in the second sentence is the mathematical content itself. I wouldn't advocate rewriting the casually upbeat motto in the same style, but swap out the appropriate blocks and tell me it doesn't sound but too formal. Ryan Reich 16:44, 26 August 2007 (UTC)
Well, no....the reason for preferring the latter is that sometimes you want to talk about sets; e.g. you want to say the cardinality of the set of all real numbers is the same as that of the powerset of the set of all integers, etc. etc. Michael Hardy 17:51, 27 August 2007 (UTC)
The use of "such that" in mathematical writing is jargon in the sense that it is shorthand for "which is such that", a meaning confined to mathematical texts. In non-mathematical use (and occasionally in mathematical texts), the usual meaning is "in such a way that", "so that".  --Lambiam 17:56, 26 August 2007 (UTC)

Some people use a backwards epsilon (with the epsilon stylized for set membership) to represent the phrase "such that" when introducing a variable (though this isn't mentioned in Quantification#Notation_for_quantifiers). I had a professor once that was vehemently against this practice, and called the symbol "meaningless" and "non-mathematical." nadav (talk) 19:54, 26 August 2007 (UTC)

The AfD entry is here. --Sopoforic 23:19, 26 August 2007 (UTC)

Result was delete. --KSmrq 10:52, 1 September 2007 (UTC)

Negation of definitions

I noticed that the following articles have been created:

In general, it seems strange to me to have an article that only serves to negate the definition of another article. — Carl (CBM · talk) 16:51, 28 August 2007 (UTC)

They are there now for no other reason than their absence previously. Certainly feel free to merge or otherwise place these. They do not seem to correspond exactly to the others in Mathematical relations. I do not know the reason for the discrepancy. I don't think these are different "in logic" and "in mathematics." I'm pretty sure they are the same concept. I have a high degree of confidence in their accuracy because I looked into it extensively at the time I took notes. The one I am NOT sure of is Antitransitive. If anyone can enlighten me on that one I would be grateful. It was in searching for that one I learned about the others. Gregbard 17:06, 28 August 2007 (UTC)
BTW, they are not strictly the negation of their base word as the use of a-, non-, anti-, and counter-, etc. are possible. Have a wonderful day. Gregbard 17:09, 28 August 2007 (UTC)
Do you know where you got these definitions from? The definitions given on these "non" articles are not the negations of the corresponding terms. Typically "non" is used to indicate a negation, while "anti" is used to indicated something stronger than negation. For example a nonsymmetric relation just isn't symmetric, while an antisymmetric relation fails to be symmetric to the greatest extent possible. — Carl (CBM · talk) 17:12, 28 August 2007 (UTC)
I didn't write down the sources at the time. I have an extensive collection of index cards with all kinds of logic (over 3000) on them. One that is different is "reflexive" vs. "totally reflexive." In my notes Reflexive is:
(x)(( ∃ {\displaystyle \exists } y)(Rxy ∨ {\displaystyle \lor } Ryx) → {\displaystyle \to } Rxx)
Totally reflexive is:
(x)Rxx
Gregbard 17:27, 28 August 2007 (UTC)


(edit conflict)These articles (which apart from anything are too short and technical to be decent articles) seem slightly confused. Why is it "the nontransitive relation", rather than "a nontransitive relation" (implying that "nontranstivity" defines scissors paper rock)? I have never heard "nonsymmetric" mean anything other than "not symmetric", yet the article equates it with "partimsymmetric", which apparently means "neither symmetric nor asymmetric", where asymmetric means antisymmetric and irreflexive. The nonreflexive article equates nonreflexive not with "not reflexive", but with "partimreflexive" (ie neither reflexive not irreflexive), giving partimreflexive examples but actually giving a definition for irreflexivity. At any rate, simple negations would not deserve an article of their own. Stronger concepts may, but even irreflexive relation is a redirect to reflexive relation. JPD (talk) 17:42, 28 August 2007 (UTC)
(I also echo the queries about syntax at Talk:Nontransitive relation JPD (talk) 17:52, 28 August 2007 (UTC))
We need reliable sources for these defnitions. Unfortunately "an extensive collection of index cards" does not constitute a reliable source. Until we can find such, I think the best thing would be for Greg to ask that they deleted. Is that ok with you Greg? Paul August ☎ 17:59, 28 August 2007 (UTC)
I think that just redirecting them to the definition articles, and adding the bolded negated term to the definition articles, would be enough. So for example into Reflexive relation we would add "A relation that is not reflexive is nonreflexive." What would need a source, in my opinion, is a claim that nonreflexive means something else than "not reflexive". — Carl (CBM · talk) 18:02, 28 August 2007 (UTC)
I agree. Paul August ☎ 18:32, 28 August 2007 (UTC)
Mergers/redirects appear to be in order. Notice, though, that a phrase like "not transitive" is ambiguous. The current article intransitivity covers this thoroughly. If there are no transitive triplets, the article says the relation is antitransitive (and also not transitive). OTOH, if there are some transitive triplets a>b>c, but there are also some exceptions to this rule, the relation does not define a partial order, or, as Gregbard would have it, the relation is nontransitive. DavidCBryant 23:28, 28 August 2007 (UTC)

I'm trying to talk some sense into this incredibly confused reader who's creating a rather large number of articles on these topics. See User talk:Gregbard. Can anyone else help bring him to his senses? He doesn't have a clue about the correct use of quantifiers. Quite possibly he understands the correct definitions of all these types of relations, but his ways of attempting to express them are horribly incorrect. Michael Hardy 17:56, 29 August 2007 (UTC)

Michael, on the nontransitive relation talk page you state
"If by "nontransitive" you mean simply "not transitive" then the way you've written it is utterly wrong. When you write ( ∃ {\displaystyle \exists } x)( ∃ {\displaystyle \exists } z)¬Rxz) you're necessarily referring to an "x" and a "z" that are not the same as what you called "x" and "z" earlier; you're just saying there exist two elements such that one is not related to the other. You're NOT saying there exist elements x, y, and z such that x is related to y and y to z but x is not related to z. In the notation your using, that would say \exists x\exists y\exists z(Rxy & Ryz & ¬Rxz. Those preceeding (x), (y), and (z), meaning for all z, etc...., should not be there at all."
No they are NOT "not the same." It says for all xs, ys, and zs there exists a particular x, and a particular z which don't have the relation. You are incorrect in thinking that they are not the same. The x, y, and z quantifier binds the whole expression in parentheses. The ∃ {\displaystyle \exists } x, and ∃ {\displaystyle \exists } z are particular members of those sets x, and z respectively. So I understand the use of quantifiers perfectly, and you already admit that I understand the relations (thank you for the concilliatory). Be well, Gregbard 23:15, 29 August 2007 (UTC)
You've quantified x twice, in two different ways, within the same sentence above: "for all xs" and "there exists a particular x" — although you wrote it in English rather than formal logical notation, those still look like quantifiers, but the double quantification makes it impossible to make sense of your statement. That does not suggest to me that you are skilled in the correct handling of quantifiers. —David Eppstein 23:21, 29 August 2007 (UTC)
I just want to make sure I understand perfectly clearly: YOU find it impossible to make sense of a precisely crafted expression, but that suggests that I don't know what I'm doing. Well, um, not really. Gregbard 00:15, 30 August 2007 (UTC)
You know, most reasonable people, when confronted with an error in their mathematics, would apologize and correct it. It's not a subject in which the greater ability to bluster will win the day. —David Eppstein 00:50, 30 August 2007 (UTC)
Do you think perhaps he's trying to say something like:
( ∀ y ) ( ( ∃ x ) ( ∃ z ) ( R x y ∧ R y z ) ) → ( ( ∃ x ) ( ∃ z ) ( R x y ∧ R y z ∧ ¬ R x z ) ) ) ? {\displaystyle (\forall y)\left(\left(\exists x)(\exists z)\left(Rxy\land Ryz\right)\right)\rightarrow \left((\exists x)(\exists z)\left(Rxy\land Ryz\land \lnot Rxz\right)\right)\right)?}
In any case, all clearly not-standard (yes, I meant "not standard", rather than "non-standard") terms (i.e., I don't recognize them ) should be proposed for deletion unless a reference can be provided. — Arthur Rubin | (talk) 23:37, 29 August 2007 (UTC)
Well I'm not going to put them up for deletion, for one; I assume that Gregbard put them up in good faith. But I'm having as much trouble as the rest of you making sense of the expressions. Gregbard, would you be so kind as to convert each of these (here on Talk) into prenex normal form? I think then we'd be better able to discuss the terms. Right now we all seem to be talking past each other.
CRGreathouse (t | c) 01:16, 30 August 2007 (UTC)
The disposition of an article should depend on whether its inclusion, after such improvements as may be needed and can reasonably be expected to be executed with our limited means, increases the value of the encyclopedia – a consideration that is independent of the assumption it was created in good faith. The article Such that was undoubtedly created in good faith, and yet you (CRGreathouse) recommended its deletion, I suppose.  --Lambiam 07:16, 30 August 2007 (UTC)
I do draw distinction between putting an article up for deletion, as Trovatore did for that one, and participating in the discussion on the AfD. Even on these articles, I don't think I'd 'vote' to keep them, depending on what else was suggested. But I'd rather wait until at least the discussion settles before voting them down.
Perhaps it was too fine a distinction, and perhaps I was inspired, at least in part, by a desire to keep the peace with Gregbard: he seems to have had a rather rough time with our project.
CRGreathouse (t | c) 12:02, 30 August 2007 (UTC)

relations redux

What have we learned? CRGreathouse was the first to raise the question about the definite article, and he was absolutely correct. The relation I stated was incorrect in that regard for certain. The negation of transitivity is the one stated by Michael Hardy and I think Dr. Rubin:

∃ x ∃ y ∃ z ( x R y  and  y R z  and not  x R z ) . {\displaystyle \exists x\,\exists y\,\exists z\,(xRy{\text{ and }}yRz{\text{ and not }}xRz).\,}

I never denied this fact onced faced with it. My further concern was that there may be degrees of nontransitivity, and that the statement I had origianlly thought was the nontransitive relation was still more general than this one stated above. That isn't so wild a thought. After all, the most general form will "screw up" the transitive relation as little as possible and that would seem to only require the participation of two variables. I was shown to be incorrect in this belief by Carl directly. Now I understand. Think about it: there is no such game as rock paper! It requires three. However "grossly incorrect" would be a dramatization.

Carl provided a counterexample that shows that my stated formulation permits for the existence of at least one (and I think we can see intuitively an infinite number) of transitive relations. However the relation I stated may still be some form of weakened nontransitivity, with a name begining with some strange prefix appended to -transitive. I do not know what.

I agree that everything in Misplaced Pages should have a reference if possible. However I do not agree that it should be deleted altogether, but rather improved and merged appropriately. My intention from the outset was that this would be a noncontroversial relation, and therefore survive to see other editors' contributions.

So everyone is more esteemed in my eyes because I learned from it. Everyone was fairly kind and I was a bit exasperating on the points I was sticking to. However, the low point was when David Eppstien said it looked like I didn't know what I was doing, while stating "the double quantification makes it impossible to make sense of your statement." That statement looks as if he doesn't know that one may put more than one form of quantification on a variable in a formula (that isn't prenex normal form, but still a formula). One can do that, and I know that, but it LOOKS as if he doesn't know that. So I think I played it pretty cool for everyone telling me how much I don't know.

He also said "You know, most reasonable people, when confronted with an error in their mathematics, would apologize and correct it. It's not a subject in which the greater ability to bluster will win the day." This I agree with 100%. So this is the conciliatory note you were waiting for. It hadn't been shown by Carl at that point that I was incorrect, David. Furthermore you hadn't said anything to disprove my claim AT ALL. Maybe someday I'll have a peanut gallery cheering me on instead of sandbagging me. It's okay David, like I said, I see that I was exasperating.

I'm not too worried about nontransitivity. Eventually, there will be plenty of information on it, even though right now there is a redirect to an article with NO information on it at all. I don't know how we are better off. What were you saying about increasing the value of the encyclopedia Lambiam? I agree with that too. Be well, Gregbard 10:09, 30 August 2007 (UTC)

see you in the talk pages...

Hi, Gregbard. I was surprised to see Weak order missing, and for the moment I made it a redirect to a section in an existing article which is devoted entirely to weak orders (there called total preorders). I think there's reason enough to make this its own article; it's quite important in social choice theory and more broadly in economics for the study of preferences.
If anyone's around, your new article Homogeneous relation could use another pair of eyes.
CRGreathouse (t | c) 12:36, 30 August 2007 (UTC)
Well I appreciate that these topics are getting some consideration. I think that's wonderful. However, I don't see that it is a lot of progress to just put a redirect without actually giving the reader looking for that topic something to read about it. I guess it's a little progress. I think I am going to put some material in my userspace and take it from there. You guys are too hypercritical for me. For all that discussion no one has anything useful to contribute to the entry on nontransitive? I really am not here for the debate club aspect. Gregbard 01:04, 31 August 2007 (UTC)


biorder, interval order, semiorder, and almost connected order can all also be defined axiomatically as relations. E.g. biorder: for all x,y,z,w, (xRy and zRy and zRw) imply xRw. Interval order: irreflexive biorder. I added them to the list.—David Eppstein 14:23, 30 August 2007 (UTC)

Am I missing something? Let R be an irreflexive biorder, and assume aRb and bRc. Irreflexivity tells us that ¬bRb. Applying the above defn of biorder with the substitution (w,x,y,z) := (b,b,c,a) tells us that (bRc and aRc and aRb) imply bRb, which can be simplified to ¬aRc. So R is totally antitransitive, and therefore not a partial order; on the other hand, interval orders are supposed to be partial orders.  --Lambiam 16:11, 30 August 2007 (UTC)
Sorry, there was a little bar in the definition of biorder that was too small to see clearly in the font size I was using to read the reference, but makes a big difference in the definition. The proper definition of biorder is: for all x,y,z,w, (xRy and ~zRy and zRw) imply xRw. The reference I was reading is unpublished, but if you want a published one, I think this material is in J.-P. Doignon and J.-Cl. Falmagne. Well-graded families of relations. Discrete Mathematics, 173:35–44, 1997, and C.W. Doble, J.-P. Doignon, J.-Cl. Falmagne, and P.C. Fishburn. Almost connected orders. Order, 18(4):295–311, 2001. —David Eppstein 16:18, 30 August 2007 (UTC)

Greg, you still seem slightly confused. "The nontransitive relation" implies that there is only one relation that is nontransitive, which you have conceded is wrong, yet you still talk about "the relation I stated" as though the property which makes relations nontransitive is itself a relation. As for generality, yes, negation of a condition X definitely gives the most general form of being non-X. It gives everything that is not X. Anything more general must include relations that are X and so could hardly be called "non-X" In my experience, "non-X" is used to mean "not X", the general lack of X-ness, and things like "anti-", etc. are used for stricter conditions, but with the way terminology works, I wouldn't be surprised to find a counterexample.

You also insist that your double quantification was ok. However you understand your formula, it doesn't mean what you said it means. At the very least using the same variable with two different thigns in mind is likely to cause confusion, so the simpler formulations would be better for an article even if you were right. As for the nontransitive article, you say it is now a redirect to an article with no information at all. What do you mean by this? The article intransitivity covers the correct version of everything that was in the nontransitive relation article, in a broader context. The only thing that could possibly be added is the word "nontransitive" as an alternative to "intransitive". JPD (talk) 18:18, 30 August 2007 (UTC)

No, intransitivity is a different thing than nontransitivity. I thought we all agreed on that. The entry was to be redirected to intransitivity and that article would touch on the difference. Think of intransitive as the ongoing pervasive condition of always avoiding transitivity among x, y, and z. Nontransitive is merely the existence of just one counterexample. The article is redirected but has no information that reflects this. Gregbard 01:19, 31 August 2007 (UTC)
the formula I designated uniquely describes a statement of predicate logic. So it wasn't ambiguous. If anyone else didn't understand it then we at least have to SHARE the blame (what do you say) Gregbard 01:19, 31 August 2007 (UTC)
Yes, the formula you designated uniquely describes a statement of predicate logic, assuming we defined the double-bond quantifiers correctly. However, it seems to mean the relation is not universal. — Arthur Rubin | (talk) 02:16, 31 August 2007 (UTC)
It's universal among all x, y, and zs if it begins with (x)(y)(z) Gregbard 05:50, 31 August 2007 (UTC)
For all xs, ys,and zs, it is true that(blah blah blah). If among the "blah blah blah" we have "there exists an x such that..." that's a particular x from among the set that x denotes. It's like saying for all cities, counties, and states there is a city which is its seat of government. (In this case it is trivially true that every city has itself as its seat of government). So like I said not in Prenex normal form, but an unambiguous formula nonetheless. (In these particular cases it's easier to understand and see directly when it is not in PNF). Gregbard 05:50, 31 August 2007 (UTC)

Is a separate article for each of the many red links above contemplated? Maybe some of them warrant individual articles, but if the articles are to be little more than dictionary definitions, perhaps it is better to put those that don't warrant more than a short paragraph all into one long article. Michael Hardy 02:45, 31 August 2007 (UTC)

I think they are called redirects with possibilities. Gregbard 05:50, 31 August 2007 (UTC)

Greg, please read the article intransitivity before claiming that the article has no information reflecting either version of nontransitivity/intransitivity. It starts by defining and describing "intransitivity" as the simple negation of transitivity (i.e. the existence of a single counterexample to transitivity), and then goes on to say that "a more common mathematical definition" is the anti-transitivity notion, where every chain of length 3 is a counterexample to transitivity. In other words, the article clearly states that "intransitivity" is used with more than one meaning, one of which is the most general possible.

As for your formula with double quantifiers, I didn't say it was ambiguous. I said it doesn't mean what you said it did (as Arthur says, it is equivalent to something without any universal quantifers), and that it would be easier to understand if it were written without double quantification. The very fact that you have misunderstood the formula suggests that in this case it is not "easier to understand and see directly when it is not in PNF". I am not sure what to make of your attempt at explanation by example. (x)(y)(z) means "for all choices of x, y and z", not "for any choice from any of three sets denoted by x, y and z". The x doesn't denote a set.

More importantly, what about these red links/articles. I agree with Michael Hardy that many would be better combined in one article. The resulting redirects may have possibilities, but I (and others) don't see what the possibilities could be. JPD (talk) 10:52, 31 August 2007 (UTC)

The article is saying that we use the same name for both, and I am saying that they are different. If there is a strong culture out there that uses the terms that way fine. However, it should still be noted that they are different and that there is a different name for it out there.
The example of the city which is the seat of government is an excellent example of the mixed use of quantifiers. I don't know off hand what the PNF form of that would look like, but I doubt it would be clear immediately that it was that simple a connection. I could be wrong.
( ∀ {\displaystyle \forall } city)( ∀ {\displaystyle \forall } county)( ∀ {\displaystyle \forall } state)( ∃ {\displaystyle \exists } city)
However I likewize cannot make anything of your statement "(x)(y)(z) means "for all choices of x, y and z", not "for any choice from any of three sets denoted by x, y and z". The x doesn't denote a set."
(x)(y)(z)(blah blah blah) means "for all xs, ys, and zs 'blah blah blah' is the case." This is more specifically understood as that idea that you can choose a member from each of the three sets and "blah blah blah" will be true of them. x, y, and z are particular members of sets. "(x)" corresponds to some set of values for x which is a set.
Be well, Gregbard 11:18, 31 August 2007 (UTC)
The article says the terms are used with different meanings in different places, including the meaning of intransitive that you advocate. As I said earlier, the only thing that is possibly missing is the "different name" "nontransitive". The fact that the meanings are different is already noted. You may be right to question whether the word "intransitive" is actually used in the way the article describes, but the article does cover both notions, and point out how they are different.
No, "( ∀ {\displaystyle \forall } city)( ∀ {\displaystyle \forall } county)( ∀ {\displaystyle \forall } state)(blah)" means "For any choice of city, county and state, blah holds" (that is, any choice of a triple (city, county, state)). For the seat of government example, you seem to be saying it means "For any choice of (a) city, (a) county or (a) state.", which is wrong. I am not familiar with this idea of using variable naming only to tell us which sets are being considered, but it is not relevant in the original case of hommogenous relations, and seems to only make more room for confusion, as demonstrated by your incorrect formula. For your example, we can simply write " ( ∀ x ∈ A ) ( ∃ y ∈ B ) G x y {\displaystyle (\forall x\in A)(\exists y\in B)Gxy} ", or more formally, " ( x ) ( ∃ y ) ( A x → ( B y ∧ G x y ) ) {\displaystyle (x)(\exists y)(Ax\rightarrow (By\wedge Gxy))} ", where A is the set of cities, counties and states, B the set of cities and G the relation "x is the seat of government of y". I fail to see how your style helps understanding at all, especially since you haven't actually produced a full formula for your statement. Restricting ourselves to cities, we may say that all cities have a seat of government, " ( x ) ( ∃ y ) G x y {\displaystyle (x)(\exists y)Gxy} ". However, if we wish to say all cities are their own seat of government, no existential qualifier is needed " ( x ) G x x {\displaystyle (x)Gxx} ". JPD (talk) 12:29, 31 August 2007 (UTC)
(edit conflict) I read the city/county/state example as
∀ x ∈ X , y ∈ Y , z ∈ Z ∃ w ∈ X ( G w x ∨ G w y ∨ G w z ) {\displaystyle \forall x\in X,y\in Y,z\in Z\exists w\in X(Gwx\vee Gwy\vee Gwz)} (X the set of cities, Y the set of counties, Z the set of states, Gab the binary relation "a is the seat of government of b")
or more simply
∀ x ∈ X ∪ Y ∪ Z ∃ w ∈ X ( G w x ) {\displaystyle \forall x\in X\cup Y\cup Z\exists w\in X(Gwx)}
While I agree that PNF isn't ideal for everything, and in fact I don't generally use it myself (I just wanted something unambiguous), I don't think it's hard to do for any of these transitivity relations or this example. In fact it's already in sloppy PNF, where set memberships for the quantified variables is allowed; to move those out, you'd just have
∀ x ∃ w ( x ∈ X ∪ Y ∪ Z ∧ w ∈ X ) → ( G w x ) {\displaystyle \forall x\exists w(x\in X\cup Y\cup Z\wedge w\in X)\rightarrow (Gwx)}
As for the comment you didn't get anything from, Greg: ∀ x y z {\displaystyle \forall xyz} (as I've been writing it), ∀ x , ∀ y , ∀ z {\displaystyle \forall x,\forall y,\forall z} (as a formal system might have it), or (x)(y)(z) (as you write), the x, y, and z are variables, not sets. If one writes ∀ x ∈ X {\displaystyle \forall x\in X} x is the variable and X is the set. You can say "for all choices of x from X" or "for all choices from X", but JPD doesn't want you to say "for all choices for x" because X, not x, is the set.
CRGreathouse (t | c) 12:44, 31 August 2007 (UTC)
At the risk of beating a dead horse, I'm going to throw a few statistics into this discussion. The statistics are just g-hits, from Google Scholar.
Google Scholar lets one choose from "Engineering, Computer Science, and Mathematics" (where most mathematical logic articles are found), or from "Social Sciences, Arts, and Humanities" (encompassing philosophy, plus disciplines like economics and behavioral psychology, etc.). After running a couple of searches, I noticed that the phrase "non transitive" or "non-transitive" is more common than "nontransitive" – I have lumped all three spellings together in the statistics below.
Searching on the terms "nontransitive" and "intransitive" I obtained the following results: Math – 1,941 instances of "non", 3,820 instances of "in"; Humanities – 976 instances of "non", 14,800 instances of "in". So "intransitive" occurs much more frequently than "nontransitive" in both subject areas. Notice, however, that most references to "intransitive" in the Humanities appear to be usages such as "intransitive verb", which really have nothing to do with this discussion.
Searching on the terms "nontransitive relation" and "intransitive relation" I obtained the following results: Math – 67 instances of "non", 40 instances of "in"; Humanities – 69 instances of "non", 63 instances of "in".
So on the preponderance of the evidence presented here it appears that the phrase "nontransitive relation" (or "non transitive relation", or "non-transitive relation") is slightly more common than the phrase "intransitive relation". And the perception most of us have (me, too!) – that "intransitive relation" is commonly used in mathematics, and "nontransitive relation" is not so common – is actually incorrect, and is probably related to the fact that the word "intransitive" is widely used, but is not often found in the phrase "intransitive relation". DavidCBryant 16:17, 31 August 2007 (UTC)
This is a tangent, but I'm not sure that the distinction between mathematics and social sciences is meaningful here. E.g., I tried searching the publications of Doignon (a Belgian mathematician who works in this area) and found slightly more under social sciences than math, because he's published in journals like J. Mathematical Psych., British J. Mathematical and Statistical Psych., etc. On the other hand, Int. J. Man–Machine Studies is listed under mathematics despite the similarity in content with those other journals. It's one of those borderline areas that simplistic classification doesn't work well for, I think. —David Eppstein 17:16, 31 August 2007 (UTC)

Proposed deletions (WP:PROD)

Paradoxes of set theory, redux

I know this was discussed in a previous invocation of this page, but I have to say that after multiple attempts to fix the article by myself (under ip address 71.198.111.245) and others such as User:Mathemaduenn and User:CRGreathouse, the article stands virtually unchanged since April 2007, with the exception that the completely bogus WM invention of "the binary tree" appears to have, finally, been deleted. I have no objection to an article describing the counter-intuitive aspects of infinite cardinality; but this article does more to encourage misunderstanding than it does to bring anything positive to the subject. Sets that have more "reality" than other sets? A cardinality for a "set of gaps", without a definition of "gaps"? "The most important theorem of set theory proves that there are uncountable sets"? The familiar WM usenet argument that omega is a member of the union of all finite sets of naturals? This really is pseudo-math, and shouldn't be on Misplaced Pages. On the other hand, after putting time into this page without any remnant remaining, I hesitate to enter the fray again. Suggestions on how to proceed are welcome! Chas zzz brown 02:08, 31 August 2007 (UTC)

I'll work on it some; I would appreciate it if some other editors would at least watch the article to give feedback, if not help with the editing. — Carl (CBM · talk) 13:17, 31 August 2007 (UTC)

the parable of the nontransitive relation

Everybody gather 'round as I tell the parable of the nontransitive relation. I may need some help telling it, so sit close...

I live on a planet with a very similar language to English, and you have come to visit...

Most of the words are the same, but some are a little different. For instance, you visit with a group of delegates in a conference room, and it comes up in conversation about how "Xtall" a delegate is. You inquire into the meaning, and the lead delegate explains to you (without indicating anything about height) that delegate A is "Xtaller" than delegate B, and delegate B is "Xtaller" than delegate C. He does this several times, and the meaning becomes clear to you that it just means the same thing as "tall." However, a new person walks into the room, and the the lead delegate introduces the president of the planet as the "Xtallest of us all." At this point you are confused because the person is obviously not the "tallest" person in the room. On this planet, you conclude, the relation "Xtaller than" is a nontransitive relation and that it may or may not have any connection at all to "tallness" as you know it.

Now it happens to be a strange planet and all that. However, it would have been equally strange if the meaning of "Xtall" had meant that the lead delegate said the same of any person other than the tallest person in the room. That means that there are number of ways in which the nontransitive relation could have occured. All of those ways have the nontransitive relation in common. These are the same relation as far as we are concerned in logic.

If, on the other hand you later find that the vice-president walks into the room and is also introduced as one of the "Xtallest" people on the planet, and is about as tall as the president, who herself is obviously not "tall." You still conclude that "Xtall" is nontransitive, because although all that is required for a relation to be nontransitive, is ONE counterexample, it still is called nontransitive if there in fact exist TWO. All of those ways have the nontransitive relation in common. They are still the same relation as far as we are concerned in logic.

Much later, (after several intergalactic plenary sessions, etc) you have learned a little more about this culture and "Xtallness." It turns out that the planet has a caste system, and every person of caste1 is always "Xtaller" than every person of caste2 and etc. Furthermore, you remember that your original meeting was with a room entirely filled with the diplomatic core. All of those people are members of the same caste AND at least one caste number higher (sociologically lower, that is, you suppose) than the president and the vice president. You also read in a political analysis report that the president only appoints people such as her veep to her most highest caste if they are shorter than she is. This explains a lot.

Now remember the language is close to english, but there are a few new words that are similar. Due to terrain, certain cultural factors, and city planning developments, etc; on this planet there is a word "quintasubproxihoodamate" for "lives five neighborhoods south of" (At least that is how your abridged translation dictionary defines it.) However, you learn from your cultural experiences that there is a lot more to being "quintasubproxihoodamate" ...

It turns out that in the political culture of this planet some neighborhoods are more quintasubproxihoodamate than others --if you know what I mean! You say you don't? How naive brother! I mean that even though everybody knows that two neighborhoods are on the same parallel, the one whose neighborhood association president is a close friend of the director of Geological Surveys will always be considered more southerly. This, it turns out is favorable in applying for government grants on this planet. The planet is just corrupt enough to allow an administration to say what is and is not "south of" but not corrupt enough to influence the grant application process once a neighborhood is labeled as one of those undeserving northerly neighborhoods!

Ordinarily, the quintasubproxihoodamate relation is an intransitive relation. If A has a quintasubproxihoodamate relation B and B has a quintasubproxihoodamate relation to C, then according to the abridged dictionary definition of quintasubproxihoodamate, A will NEVER have a quintasubproxihoodamate relation to C. But it isn't that simple here.

So can anyone tell me what happens when the Director of Geologic Surveys designates the neighborhood at the north pole of the planet as quintasubproxihoodamate to some other existing neighborhood on the planet?

What kind of relation would you call quintasubproxihoodamate then if not merely nontransitive? It sure isn't intransitive anymore. Intransitive is when the relation of the first to the third NEVER holds. In this case it sometimes does. This is still the nontransitive relation because all of the formulae which contain the quantifier ∃ {\displaystyle \exists } are satisfied by the existence of one, two or even all of the particular members of the set as designated by the formula. The nontransitive relation includes the intransitive, not necessarily the other way around.

Anyway, quintasubproxihoodamate isn't what it used to be anymore. Not under this administration.

Gregbard 07:33, 31 August 2007 (UTC)

Sorry, I got completely lost from the fourth paragraph on. Natural language has its limitations. Could you rephrase this parable using logical connectives and quantifiers?  --Lambiam 08:00, 31 August 2007 (UTC)

parable redux

Such is Transitive: (x)(y)(z)((Rxy ∧ {\displaystyle \wedge } Ryz) → {\displaystyle \to } Rxz)

One counter example screws up transitivity and makes the relation a nontransitive relation: Nontransitive: ( ∃ {\displaystyle \exists } x)( ∃ {\displaystyle \exists } y)( ∃ {\displaystyle \exists } z)((Rxy ∧ {\displaystyle \wedge } Ryz) ∧ {\displaystyle \wedge } ¬ {\displaystyle \neg } Rxz)

Such is Intransitive: (x)(y)(z)((Rxy ∧ {\displaystyle \wedge } Ryz) → {\displaystyle \to } ¬ {\displaystyle \neg } Rxz)

One counter example screws up intransitivity and makes the relation a what? It's only transitive for one set of triplets. All the rest are intransitive

i.e. ( ∃ {\displaystyle \exists } x)( ∃ {\displaystyle \exists } y)( ∃ {\displaystyle \exists } z)((Rxy ∧ {\displaystyle \wedge } Ryz) ∧ {\displaystyle \wedge } Rxz) however (u)(v)(w)((Ruv ∧ {\displaystyle \wedge } Rvw) → {\displaystyle \to } ¬ {\displaystyle \neg } Ruw) holds for all the other values of those sets.

I say this is still called nontransitive because of the meaning of ∃ {\displaystyle \exists } . Gregbard 10:41, 31 August 2007 (UTC)

Greg, you are still using the word "relation" to sometimes mean "property of a relation". You said: "That means that there are number of ways in which the nontransitive relation could have occured. All of those ways have the nontransitive relation in common. These are the same relation as far as we are concerned in logic." I would say "There are a number of ways in which the relation could have been nontransitive. All of these would still mean the relation is nontransitive. They all have the same property - nontransitivity" I don't understand the claim that this is the only property we would care about. (Of course, in your parable, the appearance of the Xtallest person in itself doesn't tell us that Xtall is nontransitive, as it is not a counterexample to transitivity - it simply tells us that Xtall is not the same as tall.)
Then, you abuse the terminology further (that is not always a bad thing, but let us look at it in this case). You seem to apply a universal property (transitivity) to individual triplets. By "It's only transitive for one set of triplets." do you mean "There is only one triplet for which the condition for intransitivity fails."? Yes, in that case, it is nontransitive but not intransitive (using the stricter definition of intransitive). This tells us, not that "the nontransitive relation includes the intransitive", but that the set of nontransitive relations includes the set of intransitive relations.
Your very last line also seems confused. What do you mean by "other values of those sets", when using universal quantifiers. I think what you should have said is ( ∃ {\displaystyle \exists } x)( ∃ {\displaystyle \exists } y)( ∃ {\displaystyle \exists } z)((Rxy ∧ {\displaystyle \wedge } Ryz) ∧ {\displaystyle \wedge } Rxz) however ((Ruv ∧ {\displaystyle \wedge } Rvw) → {\displaystyle \to } ¬ {\displaystyle \neg } Ruw) holds for all the other values. JPD (talk) 11:16, 31 August 2007 (UTC)
Thank you for your correction of my language. I appreciate the fine points you bring up. You are correct on all of them as nearly as I can tell.
I had to use u,v and w at that point because I was talking about a different set. It's the set x is in, the set y is in, and the set z is in minus the one counterexample that makes the intransitive property leave the scene.
I found your observation that the existence of the president is "not a counterexample to transitivity" quite a brilliant one that I missed. I'm not sure how to rephrase such a parable however. In fact the whole caste system I described is merely a very complex relation that still has the transitive property. One can't tell who is Xtaller just by looking, but once it is figured out we find it all quite transitive! My goodness.
Okay so I need a better example. However, I still think there is a better fate for some of these articles than redirects to related material. No one seems too excited to see them at all. I'm quite surprised. Be well, Gregbard 12:14, 31 August 2007 (UTC)
Transitive: ∀ x y z ( R x y ∧ R y z → R x z ) {\displaystyle \forall xyz(Rxy\wedge Ryz\rightarrow Rxz)}
Nontransitive: ∃ x y z ( R x y ∧ R y z ∧ ¬ R x z ) {\displaystyle \exists xyz(Rxy\wedge Ryz\wedge \neg Rxz)}
Intransitive: ∀ x y z ( R x y ∧ R y z → ¬ R x z ) {\displaystyle \forall xyz(Rxy\wedge Ryz\rightarrow \neg Rxz)}
Your "almost-intransitive" example (Q) is not transitive, nontransitive, and not intransitive. Relations are (1) transitive, but not the others; (2) nontransitive, but not the others; (3) nontransitive and intransitive, but not transitive; or (4) empty, and thus transitive and intransitive but not nontransitive. This covers all relations.
I'm not sure what the story illustrates, though. "Most" relations (viewed as random binary square matrices of arbitrary large size) are of my category (2) like the one in your example. Being transitive or intransitive is 'hard'.
CRGreathouse (t | c) 12:16, 31 August 2007 (UTC)

May I suggest that you continue your talk at a more specific talk page, e.g. Talk:Relation (mathematics)? Jakob.scholbach 12:21, 31 August 2007 (UTC)

Another math article on AfD

Uncertainties of the limits, a recently created math article (without categories, so not found by the bot — I just added one) has been listed for deletion. —David Eppstein 15:48, 3 September 2007 (UTC)

I've redirected it. It was a badly written article on a topic worthy of inclusion by long-established consensus, and there was an already existing article. Michael Hardy 18:38, 3 September 2007 (UTC)

Is fuzzy logic math?

I am wondering whether Category:Fuzzy logic should be added to the list of mathematics categories. This would have the effect that all articles in this category would be listed in the list of mathematics articles by the bot.

So the question is, is fuzzy logic math? The answer could be a bit fuzzy, I guess. :) At the core it may be math, but it has a lot of applications outside math, and for example, Fuzzy electronics could not be considered math. Comments? Oleg Alexandrov (talk) 20:19, 4 September 2007 (UTC)

There is a recent proposal that WP:LGC keep two worklists, one philosophy, and one mathematics. The fuzzy logic category is proposed to be among the mathematical logic categories. Gregbard 22:26, 4 September 2007 (UTC)
Fuzzy is not logic. Fuzzy is not mathematics. (Although it might be a technique in control theory.) To the extent that it pretends to be logic or mathematics, it is a scam, a fraud. It would be more accurate to call it an attempt to destroy logic rather than logic. JRSpriggs 04:18, 5 September 2007 (UTC)
Huh ? I am not an expert, but I have always had the impression that fuzzy logic is either a sub-topic or possibly a generalisation of the perfectly respectable mathematical subject of modal logic. Indeed the term "fuzzy modal logic" throws up a lot of academic-looking hits on Google. Am I mistaken ? Could you expand on your comments, JRSpriggs ? Gandalf61 10:48, 5 September 2007 (UTC)
It seems that fuzzy logic has multiple meanings. I think of it mostly as a topic in control theory. In that guise it's an important tool in engineering but has had little to no impact on mathematical logic. The study of modal logic in mathematics is also quite small, although present; modal logic is usually studied by philosophers. — Carl (CBM · talk) 12:50, 5 September 2007 (UTC)
That seems a bit harsh, JRSpriggs. A lot of the probabilistic variables used in models in eg cluster analysis, or data compression and transmission, strictly speaking are actually fuzzy variables -- ie Bayesian variables whose "true" value would still not be definitively known, even given a complete physical description of the state of the universe. And recognising this can sometimes be useful. Where "fuzzy logic" becomes much more questionable is in its many ad-hoc prescriptions and rough-and-ready shortcuts for manipulating such variables, which to Bayesian eyes generally look chancy at best, and often significantly wrong-headed. Nevertheless, fuzzy approaches are widely used in model making, and calling them "not mathematics" seems, shall we say, a bit strong! -- Jheald 13:31, 5 September 2007 (UTC)

More possible vandalism in mathematics articles.

User:WAREL is adding seriously wrong information to Riemann hypothesis again.
Waxex (talk · contribs) is making almost sensible, although clearly incorrect, edits to Function (mathematics) and Grammatical function.
I've reverted 3 times already today in each. (45 in Function (mathematics), but the last two waswere clear vandalism.) — Arthur Rubin | (talk) 00:46, 5 September 2007 (UTC)
I blocked User:218.133.184.93 for 24 hours. Arthur, I understand your hesitation in blocking the guy (as a person involved in the edit conflict at that article), but next time if you (or anybody else) sees him doing the thing he's been doing for the last year or so, he should just be blocked on sight. I think there will be full support from the community here. Oleg Alexandrov (talk) 01:04, 5 September 2007 (UTC)
How funny, no real number ε is known to make that hold? You'd think ε = ½ would be easy enough to prove, both functions being bounded by 0 and x. Yeah, I think a block has to happen. CRGreathouse (t | c) 14:57, 5 September 2007 (UTC)

Mathematics Wikia

One of the annoying things I've found here at Misplaced Pages is the tendency for mathematical lists and tables to be nominated for deletion as "indiscriminate lists of information" (a recent example). Such information has been transwikied in the past to Wikisource, only to be deleted from there, as well. I suppose the best hope for that kind of stuff now is Wikibooks, but I haven't checked the state of their math section lately... Anyway, the Mathematics Wikia (link is to English version, but other languages exist) hasn't seen a lot of activity since it was created few years ago, and I was hoping we could use it for — at the very least — transwikiing useful mathematical content deleted from the various Wikimedia projects. If anyone has old versions of deleted math-related stuff (that's actually worth keeping) — or wants to expound on mathematical topics more than would be appropriate in a Misplaced Pages article — please consider putting it up at the Math Wikia. Obviously, not every deleted math article will be worth saving, but if it's legitimate, correct mathematical information, it probably should go somewhere. Before diving in over there, please see wikia:math:Mathematics:Guidelines (and its talk page!) for some ideas about what we should be doing with the wiki(a). - dcljr (talk) 08:13, 5 September 2007 (UTC)

Article protection at the beginning of the school year

The academic year is beginning in the United States, which will probably lead to an increase in IP vandalism from students at computer labs during the school day. There are only three math articles currently semiprotected from editing by IP editors and new editors: Actuary, Mathematics, and Space. We have had problems in the past with vandalism of other articles such as Geometry and Randomness. If you notice an article is getting enough vandalism to warrant semiprotection, contact me or another admin to ask about it. — Carl (CBM · talk) 13:13, 5 September 2007 (UTC)

And just to repeat myself, one can track the changes to the math articles from the list of mathematics articles (now with a link to all the changes combined in one page). Oleg Alexandrov (talk) 14:59, 5 September 2007 (UTC)

FA reconsidered

I requested a copyedit for an FA article a while back; I am somewhat pleased to report it has been promoted. I have commented on the process on the FAC talk page; I would like to thank Jim and Ksmrq particularly.

I am only somewhat pleased because FA seems to have worsened significantly in the last few months; my evidence is also in the link above. For here, some regulars insist on enforcing every tittle of MOS, even when it is merely stating general good advice, not literally applicable to every article. (They do not seem to understand that it's a {{guideline}}.) Other articles pass with no substantive attention at all.

What should we do?

  • Withdraw from FA and GA and ignore them to death; deprecate efforts to promote math articles, as giving recognition to a broken process?
  • Try to fix it, as Geometry Guy is trying to fix GA?
  • Institute dispute resolution against the worst reviewers? (While personally satisfying, this will have limited long-term effect; FA and GA have a fatal attraction to the sort of mind who would like to contribute to Misplaced Pages by demanding changes in other people's work, without doing anything themselves.
  • Other. What?

Regards. Septentrionalis PMAnderson 20:51, 6 September 2007 (UTC)

I had to click on the link you gave above to be reminded what the abbreviation "FA" stands for, and if anyone's not a newbie here, it's certainly me. Michael Hardy 21:38, 6 September 2007 (UTC)