Talk:Zorn's lemma - Knowledge

1570:"If it was correct then why it was replaced?" If you bother to read before writing, you might look at the discussion above involving myself and another editor on re-writing and clarifying, not out of concern that it was incorrect, but just to make it better. As to your failure of being familiar with the alternative phrasing, you will note that I provided three citations which use that phrasing (specifying that the both the set and the chains must be non-empty). Did I invent them? No. They happen to be the three that I easily located in my bookshelf after five minutes of searching. If you have not seen it elsewhere, then that speaks to your lack of familiarity with the literature and little else. 3517:

that chains must have upper bounds. That's why the two current versions are stated as they are: because that is how they are often found in the literature. (As an aside, the page uses "(1)" and "(2)" to refer to itemized sentences, not "1." in the middle of a sentence, since the period can be parsed as an end-of-sentence marker by renderers and by readers; also "pedantic" often has a negative connotation, which I am not entirely sure you are intending...) Again: the reason we state both versions is because both versions are often found in the literature, and stating just one led in the past to edits "correcting" it. Historically, having just one version

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theorem", which is an instance of the very confusion that you falsely tried to attribute to me. And your "rutabaga" example is wrong because "rutabaga" does not belong to the domain of discourse of the language of set theory in which the Zorn's Lemma is formulated. Also, the fact that the article in question is an encyclopedic entry is no excuse for being sloppy in details that some casual readers may not be willing to dwell upon. I suggest you try to write down your claims using the actual language of Set Theory before posting them in public as not doing so may cost you some credibility in this subject.

2740:. The rutabaga example is what is called a hyperbole, a way to exemplify by point via exaggeration to make it clearer. At this point, you have failed to produce any valid complaints, or for that matter suggestions for improvement of the article, which is what this page is for. You did succeed, through your errors, in prompting others to improve the article, so thank you for that. Otherwise, I have nothing else to add, and no fear about my credibility in this subject (and certainly no interest in what your opinion of that may be, so don't worry about sharing it with me or with the world). 3027:

repeating. Despite your protestations of doing things without bothering to try to figure out what they are (so much for cooperation) and objecting to language, you use fuzzy phrases like "invalidates it" (what is "it"? The original theorem? No. The new assertion? Maybe.) In the months between your initial false claim of a fallacy and now, nobody has objected to its current formulation. I agree that people don't care about the argument, but rather about the text. Right now, you seem to be the only one who objects to its current form.

95: 85: 64: 31: 535:; link is in postscript). Perhaps I wasn't clear, but my experience is rather that people who learned the "non-empty" version are taken aback when they see the other version and their initial reaction is to insist that there are hypothesis missing and hence the statement must be wrong. Given the prevalence of the "non-empty" version, I think it should be addressed. Whether on the text or as a footnote... I have no preference. The title of the section was a bit technical, though. 174: 3577:

nonempty. Yes, when we ask a chain (empty or not) to have a upper bound, that non-empty requirement is redundant, but we can just say so after the statement. I am not saying we don't need a discussion of whether the nonempty requirement is needed or not, but I am just saying that can be done in a much more efficient manner than currently done (and Trovatore seems to agree). That is,

1179:" is superfluous (and in fact, the linked reference to chain reads "While chain is sometimes merely a synonym for totally ordered set, it can also refer to a totally ordered subset of some partially ordered set. The latter definition has a crucial role in Zorn's lemma.") Yes, we are aware that an implication with a false premise is true, which means Zorn's Lemma holds for 22: 1400:

indicates that you smuggle some informal assumptions to what is supposed to be a, however sketchy, study of an aspect of ZFC. Chain is a linearly ordered set, and if in renowned Set Theory texts it means something narrower than that, then its is the requirement (in Kuratowski's verion) that the ordering relation in question is the set-theoretic inclusion

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The new paragraph gives the alternative version with a weaker hypothesis. In order to show that the two formulations are equivalent, it is enough to show that a set satisfies the hypothesis of one version if and only if it satisfies the hypothesis of the other version. In other words, what is being established is that a partially ordered set

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Please do not modify comments to which extensive replies have been posted; you are changing the terms of the discussion ex post facto, which may make replies seem incoherent or irrelevant. If you wish to modify your position, make a new comment. I will note that the changes you attempted to make seem

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being the empty set (in the sense that the empty set fails to satisfy the hypothesis of the Lemma; just like a rutabaga satisfies the Rank-Nullity Theorem, since it is not a linear transformation). As to your final statement, this is again incorrect, and again born of not bothering to read carefully.

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Not much a fan, given the history of this particular piece of text in the page. The most common ways in which the lemma is found in the literature either asks for "nonempty" in both clauses (that the set be nonempty, and that the chains which must have an upper bound be nonempty), or only specifying

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Interestingly, Zorn's Lemma is useful in considerations of non-classical approaches to the notion of truth and the Liar's paradox in Philosophy of Language; specifically, Kripke uses it to show that any fixed point can be extended to a maximal fixed point. I'd be up for writing up on this under this

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The problem here is that what you write is not always the same as what you mean. Mathematics does not tolerate qualifiers of the kind "that's not what I meant". I read what you wrote and I don't feel like trying to figure out what you meant. It is not my lack of care but your imprecise writing. Your

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I suggest you either learn the meaning of the phrase "you contradicted yourself", or else that you try to be more careful in reading. So far, most of your claims of a contradiction seem to stem from lack of comprehension of what you are reading, both now and in your original claim of a "fallacy" in

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The article currents states two versions, the one that explicitly requires the set to which the applies is nonempty and the other doesn’t and we even give a proof of the equivalence of two versions. I *fully* understand the pedantic reason for this sort of matter, but I think we shouldn’t make the

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does not satisfy the Theorem" to mean that it does not satisfy the hypothesis of the theorem; in my experience, the vast majority of mathematical readers understand from context what is meant; you did not: your complaint, however, was to claim earlier that a statement in the article was a fallacy;

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You just contradicted yourself. First, you claimed that I confused "satisfying the lemma" with "satisfying the hypothesis of the lemma" and then, in the very next para, you went on to saying that it is "quite common" that "does not satisfy the theorem" means "does not satisfy the hypothesis of the

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we need to state all the superficially different versions. As far as I know, almost no reliable source gives two versions of Zorn's lemma (maybe except Lang?). Since it is a common practice to check the nonemptyness of the set, I think it makes sense to state the version that asks the poset to be

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Meanwhile the entire Section was replaced with a different text, while I was still commenting on the original one. (If it was correct then why it was replaced?) I also wonder how could a rational reader of the original Section conclude that the purpose of that Section was not a "claim that Zorn's

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Maybe it would be better to say that the formulation of ZL containing the word "nonempty" appears formally weaker, but is actually equivalent, and here's why. Or maybe the section is more trouble than it's really worth, given that at the end of the day, no one is confused about whether the empty

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I have little doubt that this is a fine article. If any part of this article were comprehensible in plain English, I'd be tempted to read it. The convenience of mathematical language/jargon should maybe not prelude an explanation in English. Just a thought. IP address, because I can't be bothered

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You seem to be getting confused about the difference between "satisfying the lemma" and "satisfying the hypothesis of the lemma". You claimed Zorn's lemma applies to the empty set, or that "the empty set satisfies Zorn's lemma", which in a sense is true in that the premise of Zorn's lemma is not

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The lack of a quantifier was not your objection. In any case, what exactly is your point here? Your second variant above is the variant in question, and it appears in Lang, Dummit and Foote, Bergman's book (and handout mentioned above), and other locations (with minor variations); you know, that

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I hope we all agree that the empty partial order does not have a maximal element. Going into detail as to why that doesn't contradict the article's formulation of Zorn's lemma is tangential and unnecessary in this article. My opinion is that the new section should be removed. --Trovatore (talk)

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Thanks for your suggestion. I agree that it is not perfect as it is. I personally don't know how to improve it at the moment, but I think such an informal explanation of the motivation is better than no motivation section, because in my opinion it gives at least a feeling for what the intuition

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And the snark continues. Again, you seem to get hung up on irrelevant and persnickety details. Here, "some partially ordered set" means "some partially ordered ." With a narrowmindedness like that, how can you expect to ever understand a proof or be satisfied by anything, especially given the

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You are not only stubborn but also hopeless. I wrote a clear section with simple argument that you deleted while keeping your sloppy one in place. (It would have embarrassing errors if it weren't for my criticism last year.) I guess, it better fits your intellectual limitations. Or is it your

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phrasing of Zorn's lemma that explicitly assumes that the partially ordered set is non-empty, and an explanation of why that alternative version is equivalent to the one given, even though the alternative seems to be formally weaker."? I read many Set Theory textbooks, most of them written by

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Also, your statement "While chain is sometimes merely a synonym for totally ordered set, it can also refer to a totally ordered subset of some partially ordered set" is either a tautology, since any linearly ordered set is a subset of itself and linearly ordered implies partially ordered, or

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Eight months and a bit later you jump back in, begin with a few empty insults, and add a section that repeats information already in the article and does not provide anything new, while instructing the reader to go do exercises to justify the assertions given. You aren't clarifying, you are

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has been that the inference in the quoted-above original phrase (note the use of "thus" there) was a fallacy. Was it because of ambiguous/imprecise language? Perhaps. For clarity, a fallacy is an inference that has this property that, under some interpretation, its premises are true but its

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My feeling is that this is the sort of thing that is best addressed in an explanatory footnote. An inline section, I think, is distracting and confusing to readers who are thinking about the lemma in its ordinary application, which I would argue does not include the empty partial order.

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conclusion is false. So, if you say, that an inference X is not fallacy because such an interpretation is not what the authors mean then you come up with a very strong argument: you can defend that way all the fallacies that at least sometimes derive true conclusions from true premises.

2736:"the theorem is false when you substitute the empty set as a specific instance", when in fact it meant "the hypothesis of the theorem are not satisfied by the empty set." That is, you confused the two statements. So, no; I did not contradict myself. I described of what the confusion 3149:

In most cases, we don't go into details of proofs in Knowledge articles. One possibility would be to remove the proof sketch altogether. I don't think making it more detailed is really on the table, but we could certainly think about how to make it clearer at the current level of

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lemma and theorem is about proving a general situation so that one does not have to prove it by hand each time. There is nothing special about Zorn's Lemma in that respect to warrant highlighting it like that, in my opinion. This section should be tidied up considerably.

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you described that fallacy as a claim that the theorem did not hold, even though it did because substituting the empty set as a specific instance yielded a true implication because the antecedent was false. In short, you took the phrase "does not satisfy the theorem" to

1936:: "the empty subset viewed as a chain", particularly, the "viewed as" part of it, must puzzle those who passed a rigorous course that covers ZFC. In ZF, KM, ZFC,..., the empty set is the empty set, and from set-theoretic perspective it does not matter how one views it. 1125:

So, Zorn's Lemma doesn't need any "alternative formulation" as it is perfectly correct in its classic form. You may include an explanation why it apples to the empty set, too, if you would like to make the Lemma a bit easier to understand for some uninitiated readers.

1145:" and pretending that it was meant to be a formal statement, which is rather persnickety. Especially as that could not possibly be your original objection, given that the formulation as it appeared when you first asserted your claim included the words "upper bound in 3540:

May I gently remind the discussants that this whole controversy is completely trivial? The only difference between saying "any P" and "any nonempty P" is whether P can be empty. So it all comes down to asking whether the empty partial order has a maximal

1269:"In the formulation of Zorn's lemma above, the partially ordered set P is not explicitly required to be non-empty. However, the empty subset of P is a chain (trivially), hence is required to have an upper bound, thus exhibiting at least one element of P." 3571:

Sometimes (or often?) in math, there can be several ways to state a theorem. For example, when stating an inequality, we may explicitly exclude the case when the right-hand side is infinite or we don't (in the cast, the inequality holds trivially). That

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So I take it the concern is for the reader who thinks, "Aha! I've found a counterexample. The empty partial order appears to satisfy the hypothesis but not the conclusion." And maybe especially for the reader of that sort who then tries to edit the

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is true). In this sense, the empty set does satisfy the Zorn's Lemma. Those who claim otherwise, as one person implied in his or her message to me (a quote from that message is below), are urged to write the Lemma formally in the language of ZF.

3581:. By the way, the negative connotation of "pedantic" is somehow intended since the whole matter is somehow distracting (yes, we have to be careful about the way we state the lemma but that's all; no need to go into a lengthy discussion.) -- 2785:

I suggest that you try to focus on improving the article, as opposed to defending it in its current form. Most of readers don't care who won an argument - they want to read language that is correct and easy to understand. I added a section

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don't like that wording. Also, transfinite induction is not the only way to prove maximal elements exist, so the implication that Zorn's Lemma is used so as not to have to do "transfinite induction" each time is overbroad. And finally,

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What would it take to make the article more comprehensible? I'm not trying to be difficult, but I struggle to think of ways to describe Zorn's lemma without lots of mathematical language. After all, one has to understand the terms

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matter look more complicated than it actually is. It is true that in practice, one typically verifies the set is nonempty (so the version without "nonempty" looks superficially wrong.) So, how about the following formulation?

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A new motivation section has been added. It is probably a good idea to have one, but I'm not sure the current one does the job. "Sometimes, one wishes" is okay during a mathematical talk, but in Knowledge we need to provide

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The most relevant-looking line seems to be "As a consequence, maximal principles were rediscovered independently several times, the most important instances occurring in articles by Kazimierz Kuratowski and Max Zorn ."

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is false.] So you are incorrect in claiming that the Section implies that the empty set satisfies Zorn's Lemma. The section makes no such claim, it is making the claim whose contrapositive you are arguing for."

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That does not seem like a serious excuse as the empty chain has an upper bound in every non-empty set and it has no upper bound in the empty set. If, for some reasons, one doesn't want to deal with the empty set

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This, by the way, is an encyclopedia page; those who "have passed a rigorous course that covers ZFC" will not be confused by colloquial phrasing placed in a colloquial setting. On the other hand, those how have

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As the sketch states: since you can define one such element for each ordinal, you end up with a proper class of elements of P, which is impossible since P is a set. The contradiction is already stated there.

1359:". When writing out a proof, one does not usually write out in excruciating detail each and every part, especially when there is an explicit reference to the hypothesis in question. Here, since the statement 2659:, this is different from saying that the empty set satisfies the hypothesis of Zorn's lemma. It does not (because in the empty set, the empty chain does not have an upper bound). It is quite common to say " 1375:. As to your initial snark, no. Both statements you objected to were in point of fact correct and your objections were erroneous. There were no "corrections" as such, but rather a more explicit write-up. 2335: 2241: 479:

I think the word "required" is problematic. It may not be clear to the reader whether the "requirement" applies to the hypothesis or to the conclusion. I think that may be what tripped up the original

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variant that you claimed above you have never seen before. In your first variant, the hypothesis "non-empty" in "non-empty partially ordered set" is superfluous, because the condition "every chain in

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I'll take another pass at it; but I do think something is needed, since the "nonempty" formulation is common, and I've encountered many people who seem quite adamant that omitting it is a problem.

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It's possible that it could be worded more clearly; it might be that the "hence is required to" language is confusing. That might be what tripped you up. Can you suggest alternative wording? --

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at the end of the article. The new (corrected) formulation seems clearer than the original one. You may wish to read some comments about the new Section in the section that comes after this one.

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The proof seems incomplete to me? What do you do with the a_i's once you define them? I assume you show they are a chain with no upper bound? It needs an extra paragraph justifying this. --

1200:." At this point, you are not applying Zorn's lemma at all (just verifying that the two hypotheses are equivalent), so the comment "there is nothing in entire Zorn's Lemma that would make 1871:

in question is non-empty without ever worrying about the non-emptiness of the chains, as this variant of Zorn's (paraphrased from Yech's formulation in the Handbook of Logic) allows for:

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to still be born from missing the point of that section. It's not a claim that Zorn's lemma does or does not apply to the empty partially ordered set. It's a section about an alternative

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The above form is equivalent (in Zermelo - Fraenkel set theory without the Axiom of Choice) to the version given at the beginning of this section simply because for every non-empty set

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I've given it another try, and changed the title of the section along the way. (There is a whole paragraph discussing the two versions in Bergman's Universal Algebra book, and also on

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behind Zorn's lemma is (this is especially important here because Zorn's lemma can be quite overwhelming if one only reads the formal statement and asks oneself what the point is).

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Your original claim of a fallacy was, quite simply, incorrect and based on an incorrect reading. Your first objection here is based on taking a gloss that omitted the clause "in

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You are, apparently, arguing vehemently that the correct statement of Zorn's Lemma is... the statement given in the article. Which makes me wonder what, exactly, is your point.

1204:"exhibit at least one element"" is misguided. Zorn's lemma is not being applied. The hypothesis of Zorn's lemma is such that a set that satisfies it must in fact be non-empty. 1046: 891: 719: 2616: 2587: 2509: 2270: 1418: 1177: 3060:

The section you added is still here for others to discuss, if they so wish. Until then, perhaps you can refrain from adding nothing but insults that say more about you?

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accomplished set theorists, and I haven't noticed your "common" phrasing of the Lemma, which - in my opinion - just makes it more obscure, never mind unnecessary longer.

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It is a bit discouraging that the authors of this entry (or whoever watches that page) are resisting fixing it. Perhaps, they should consult this matter with an expert.

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It is an easy exercise to show that the above form is equivalent (in Zermelo - Fraenkel set theory without the Axiom of Choice) to the version given in the Section

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Stating there are two versions of the lemma, while pedantically true, is confusing and can be avoided by a formulation like the above. What does everyone think?

2187:. Of course, it's nothing wrong with proving a stronger fact then what's needed, but, taking into account that the Lemma was not formulated precisely in a form 3275:

If Zorn's lemma is ultimately the same as axiom of choice, what does it mean to "prove" it: "Proved by Kuratowski in 1922 and independently by Zorn in 1935"

35: 3640: 3229:. The issue of transfinite induction is actually complicated, and the current paragraph is glossing it a bit too much. "Tidies up the conditions"... no, I 141: 3610: 1429:

With a language like that and implicit assumptions that are not all standard in thee Set Theory, how can you expect a reader to deduce what you mean?

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As it happens, I am a professional, published mathematician, with experience as writer, referee, and editor of mathematical texts. How about you?

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Well, if you mean that after all the necessary corrections the said section is or will be free of fallacies then you are tautologically correct.

367:." is a fallacious inference. Thus the premise of Zorn's Lemma is not satisfied, contrary to what this Section implies, and the conclusion that 117: 3635: 267: 1116:

However, the empty subset of P is a chain (trivially), hence is required to have an upper bound, thus exhibiting at least one element of P

249: 235: 3047: 3012: 2761: 2711: 2626: 2338: 1943: 1915: 1779: 1506: 1450: 1430: 1127: 389: 375: 330: 241: 2675:. Just like Zorn's lemma is not about rutabagas. The "contradiction" you see comes from your failure to understand what is being said. 2755:

I suggest you either learn the meaning of the phrase "you contradicted yourself", or else that you try to be more careful in reading.

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is empty, then it is not the case that every chain has an upper bound. Or contrapositively, if every chain has an upper bound, then

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was proposing your third version, which leads me to wonder about why you bothered to present it, unless it's part of a strawman.

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May I gently remind the discussants that this whole controversy is completely trivial? The only difference between saying "any

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I've tried to clarify what this section is trying to say: "In the formulation of Zorn's lemma above, the partially ordered set

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However, assuming the non-emptiness of chain in the formulation of Zorn's lemma without assumption of the non-emptiness of

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A QUOTE from a message to me. "This is logically equivalent to saying that the empty set does not satisfy the hypothesis [

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It probably means proved using the axiom of choice. The axiom of choice is intuitively clear; Zorn's lemma is not. --

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satisfied by the empty set, and as such the implication of Zorn's lemma holds for the empty set being substituted for

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the article (which did not exist, despite your protestations). What I point out is that one often uses the phrase "

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is false. And the reason for it that you used a phrase "has an upper bound" rather than "has an upper bound in

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satisfies Zorn's lemma; it does, since the premise of Zorn's lemma is false when we substitute a rutabaga for

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What exactly is your experience writing and reading proofs? The formal statement specified "an upper bound in

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Note the condition 1. is actually superficial since 2. implies the empty set must have an upper bound and so

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The assertion is not true. Only connected graphs have spanning trees. The mistake in the proof happens here:

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Well, they do have spanning forests (a forest is a set of trees), and that does follow from Zorn's Lemma.

1878: 1729: 1669: 1621: 1311:"... However, the empty subset of P is a chain (trivially), hence is required to have an upper bound in 1300:" while the Section uses "has an upper bound". Do you claim that the correct reading was supposed to be: 284: 50: 2844:, but no upper bound of 0 (any element of any set is an upper bound of the empty set) is an element of 94: 3046:

defensiveness that overrides your weak commitment to truth? I guess, you are good at it for a reason.

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can be empty. So it all comes down to asking whether the empty partial order has a maximal element.

589:". While it is true that the empty set is a chain and, trivially, has an upper bound (for instance, 21: 3560: 3312: 3162: 3099: 522: 489: 428: 116:

on Knowledge. If you would like to participate, please visit the project page, where you can join

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article, but it's unclear under which seciton organisationally it would be worth mentioning ...

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This sequence is really long: the indices are not just the natural numbers, but all ordinals.

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claim that I need to learn is a projection: you project on me your own flaws that you deny.

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Your arguments exactly prove why the text in the article is correct, not why it's incorrect

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The following two variants are equivalent (are provable from one another in ZF) to Zorn's:

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I haven't read the new discussion in detail, but I think maybe that's a "support"? --

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lemma does or does not apply to the empty partially ordered set" but an "alternative

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Note: The above referenced Section has been removed and replaced with another Section

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Similarly, any chain in the formulation of Zorn's lemma may be assumed non-empty if

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is not explicitly required to be non-empty. However, in this formulation, if a set

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a deep understanding of set theory and familiarity with formal proofs to follow.

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is the point. And the paragraph is explicit as to what it is trying to establish.

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is still a fallacy since there is nothing in entire Zorn's Lemma that would make

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Reread the passage. Your arguments exactly prove why the text in the article is

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https://math.stackexchange.com/questions/1777443/original-proof-of-zorns-lemma

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does not satisfy the Theorem" to mean not that the theorem is false, but that

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There is a difference between "has an upper bound" and "has an upper bound in

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what has led to confusion by readers, not having both and the explanation.

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is non-empty then its every element is an upper bound of its empty chain.

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Zorn's lemma says that a maximal tree must exist, which is a spanning tree.

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To actually define the function b, we need to employ the axiom of choice.

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is non-empty; that was the whole point of the section you objected to.

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Zorn's lemma CAN'T be used to show that every graph has a spanning tree

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The bottom line is that the Zorn's Lemma applies to the case of empty

3153:(If you're interested for your personal understanding, please ask at 2589:, I suppose] of Zorn's Lemma, which is what you are saying. [Indeed, 445:

satisfies the given hypothesis ("..every chain has an upper bound in

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Here is what it probably (I am speculating) it was supposed to mean:

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is empty). In particular, the statement (quotation from the Section

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in the formulation of Zorn's lemma may be assumed non-empty, as in

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of the theorem and as such that the theorem is not "really" about

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without any "modifications" or "corrections" because the premise

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I am not proposing to state one version without any explanation

1858:

rather than deal with the empty chain in the general argument."

1854:"So many authors prefer to verify the non-emptiness of the set 561:

Your statement (quoted from your reply in Section "Fallacy in

453:

must be non-empty. To see this, note that the empty subset of

226:

Maximal trees in disconnected grpahs aren't spanning trees. --

203: 167: 15: 1713:

while the following variant is NOT equivalent to any of them:

457:

is a chain, and hence is required to have an upper bound in

1192:

is nonempty and every nonempty chain has an upper bound in

3489:

is necessary nonempty. However, in practice, one often ...

2395:

in the form mentioned above, assuming that the quantifier

2330:{\displaystyle \forall C(\xi (P,C)\Rightarrow \zeta (P,C)} 2236:{\displaystyle \forall P(\varphi (P)\Rightarrow \psi (P))} 2683:

will not appreciate a hyper-formalized presentation that

2100:{\displaystyle \Phi (\forall xA(x))=\Phi (\forall xB(x))} 2107:, while in the said Section, it implies more than that: 1778:" and use of "suppose" and "property" may confuse some. 1111:

if false in such a case, and the entire Lemma is true.

652:

Zorn's Lemma, as stated in the article, has this form:

609:), it may happen to have no upper bound in a given set 3278:

Apparently this question was asked and answered here:

2180:{\displaystyle \Phi (\forall x(P(x)\equiv Q(x)))=true} 3475: 3451: 3428: 3408: 3385: 3363: 2595: 2566: 2517: 2488: 2468: 2424: 2401: 2357: 2278: 2249: 2193: 2113: 2039: 2019: 1963: 1957:

Also, "equivalent" needs an explanation. In genreal,

1406: 1317: 1286: 1244: 1159: 1094: 1074: 1054: 1028: 1008: 988: 965: 939: 919: 899: 873: 850: 830: 810: 787: 767: 747: 727: 701: 681: 661: 635: 615: 595: 575: 2959:

satisfies the premiss but not the conclusion of it.

112:, a collaborative effort to improve the coverage of 2418:is implicit in it (which seems quite common), then 3481: 3457: 3434: 3414: 3391: 3369: 2610: 2581: 2547: 2503: 2474: 2454: 2410: 2387: 2329: 2264: 2235: 2179: 2099: 2025: 2005: 1412: 1323: 1292: 1250: 1171: 1100: 1080: 1060: 1040: 1014: 994: 971: 945: 925: 905: 885: 856: 836: 816: 793: 773: 753: 733: 713: 687: 667: 641: 621: 601: 581: 351:is empty then no chain can have an upper bound in 3536:I would just repeat what I said seven years ago: 3113:The proof sketch section looks really weak here. 2006:{\displaystyle \forall xA(x)\equiv \forall xB(x)} 3626:Knowledge level-5 vital articles in Mathematics 2848:. Therefore, the lemma is (vacuously) true for 2548:{\displaystyle \varphi (0)\Rightarrow \psi (0)} 2455:{\displaystyle \varphi (0)\Rightarrow \psi (0)} 2388:{\displaystyle \varphi (P)\Rightarrow \psi (P)} 1449:One question: are you, gentlemen, philosophers? 3292:of Zorn's lemma were proved in 1922 and 1935? 2852:= 0. The above formulation has been used in . 1363:specified that the desired upper bound was in 1774:A lack of explicit quantifier "For every set 1331:, thus exhibiting at least one element of P." 8: 2788:Notes on the assumption of the non-emptiness 1153:that is totally ordered, so that including " 3351: 3323:Application in Philosophy of Logic/Language 2620: 1937: 1909: 416:is nonempty. And that's what the article 383: 261: 58: 3474: 3450: 3427: 3407: 3384: 3362: 2594: 2565: 2516: 2487: 2467: 2423: 2400: 2356: 2277: 2248: 2192: 2112: 2038: 2018: 2013:iff under all admissible interpretations 1962: 1867:then they may just assume that their set 1474:evidence of your prior misunderstandings? 1405: 1316: 1285: 1243: 1158: 1093: 1073: 1053: 1027: 1007: 987: 964: 938: 918: 898: 872: 849: 829: 809: 786: 766: 746: 726: 700: 680: 660: 634: 614: 594: 574: 2645:. But this is as empty as saying that a 371:has a maximal element does not follow. 3616:Knowledge vital articles in Mathematics 2967: 2951:contains at least one maximal element. 2887:contains at least one maximal element. 2824:contains at least one maximal element. 2272:was not precisely formulated in a form 484:partial order has a maximal element. -- 60: 19: 3377:be a partially ordered set satisfying 2482:is the empty set, is a theorem of ZF ( 1735:has the property that every non-empty 1675:has the property that every non-empty 1280:The Lemma uses "has an upper bound in 864:in the above statement and concluded: 3631:B-Class vital articles in Mathematics 2812:has the property that every chain in 1843:Unconvincing purpose for the Section 461:, exhibiting at least one element of 299:just to be able to state the lemma. — 7: 804:In your discussion of the case when 106:This article is within the scope of 1932:Also, this phrase from the Section 1651:(I saw this version in Yech's text) 49:It is of interest to the following 3641:High-priority mathematics articles 3288:Is what we're saying that certain 2402: 2279: 2194: 2120: 2114: 2076: 2070: 2046: 2040: 2020: 1985: 1964: 533:his handout on AC and Zorn's Lemma 14: 2895:, the conditions "every chain in 2351:If one refers to Zorn's Lemma as 382:) 17:04, 23 September 2016 (UTC) 126:Knowledge:WikiProject Mathematics 3611:Knowledge level-5 vital articles 3130:Where do these steps come from? 2931:Suppose a partially ordered set 2337:), it is likely to mislead some. 1122:"exhibit at least one element". 172: 129:Template:WikiProject Mathematics 93: 83: 62: 29: 20: 146:This article has been rated as 3621:B-Class level-5 vital articles 2955:invalidates it; the empty set 2750:23:19, 29 September 2016 (UTC) 2720:22:23, 29 September 2016 (UTC) 2697:04:08, 24 September 2016 (UTC) 2635:01:17, 24 September 2016 (UTC) 2605: 2599: 2576: 2570: 2542: 2536: 2530: 2527: 2521: 2498: 2492: 2449: 2443: 2437: 2434: 2428: 2382: 2376: 2370: 2367: 2361: 2347:00:03, 24 September 2016 (UTC) 2324: 2312: 2306: 2303: 2291: 2285: 2259: 2253: 2230: 2227: 2221: 2215: 2212: 2206: 2200: 2159: 2156: 2153: 2147: 2138: 2132: 2126: 2117: 2094: 2091: 2085: 2073: 2064: 2061: 2055: 2043: 2000: 1994: 1979: 1973: 1952:23:40, 23 September 2016 (UTC) 1924:23:07, 23 September 2016 (UTC) 1831:04:00, 24 September 2016 (UTC) 1788:22:37, 23 September 2016 (UTC) 1580:04:00, 24 September 2016 (UTC) 1539:21:03, 23 September 2016 (UTC) 1515:00:43, 24 September 2016 (UTC) 1487:04:00, 24 September 2016 (UTC) 1459:00:37, 24 September 2016 (UTC) 1439:00:37, 24 September 2016 (UTC) 1385:04:00, 24 September 2016 (UTC) 1221:20:50, 23 September 2016 (UTC) 1136:20:17, 23 September 2016 (UTC) 545:19:12, 23 September 2016 (UTC) 527:18:38, 23 September 2016 (UTC) 508:18:31, 23 September 2016 (UTC) 494:18:27, 23 September 2016 (UTC) 475:18:07, 23 September 2016 (UTC) 433:17:10, 23 September 2016 (UTC) 408:, not why it's incorrect. If 398:17:00, 23 September 2016 (UTC) 339:23:26, 23 September 2016 (UTC) 250:16:55, 23 September 2016 (UTC) 1: 3317:22:57, 21 December 2021 (UTC) 3302:22:50, 21 December 2021 (UTC) 2915:" are equivalent. Indeed, if 953:contains a maximal element), 933:has an upper bound) implies ( 120:and see a list of open tasks. 3636:B-Class mathematics articles 2935:has the property that every 2871:has the property that every 1884:has the property that every 1627:has the property that every 1041:{\displaystyle C\subseteq P} 886:{\displaystyle 0\subseteq P} 714:{\displaystyle C\subseteq P} 563:Empty chain as boundary case 361:Empty chain as boundary case 345:Empty chain as boundary case 317:Empty chain as boundary case 3591:07:26, 31 August 2024 (UTC) 3565:18:00, 30 August 2024 (UTC) 3531:17:14, 30 August 2024 (UTC) 3511:09:05, 30 August 2024 (UTC) 2611:{\displaystyle \varphi (0)} 2582:{\displaystyle \varphi (P)} 2504:{\displaystyle \varphi (0)} 2265:{\displaystyle \varphi (P)} 1369:guaranteed by the statement 236:20:41, 5 January 2015 (UTC) 3657: 3265:16:56, 2 August 2019 (UTC) 3249:15:56, 2 August 2019 (UTC) 3204:15:08, 15 April 2021 (UTC) 3189:02:09, 15 April 2021 (UTC) 1413:{\displaystyle \subseteq } 1172:{\displaystyle C\subset P} 824:is empty, you substituted 309:09:53, 25 April 2016 (UTC) 276:20:38, 23 April 2016 (UTC) 3343:The non-empty requirement 3167:23:37, 8 March 2018 (UTC) 3142:23:17, 8 March 2018 (UTC) 2980:About the axiom of choice 2832:. Indeed, in the case of 2411:{\displaystyle \forall P} 1371:would necessarily lie in 629:(it happens exactly when 145: 78: 57: 3549:07:43, 5 June 2017 (UTC) 3338:10:52, 16 May 2024 (UTC) 3104:07:43, 5 June 2017 (UTC) 3070:01:28, 5 June 2017 (UTC) 3056:23:04, 4 June 2017 (UTC) 3037:22:20, 4 June 2017 (UTC) 3021:18:02, 4 June 2017 (UTC) 2770:16:06, 4 June 2017 (UTC) 2243:(and, to make it worse, 326:Alternative formulation 152:project's priority scale 3465:has a maximal element. 3181:Jordan Mitchell Barrett 3009:Alternative formulation 1934:Alternative formulation 1845:Alternative formulation 801:has a maximal element. 347:contains a fallacy. If 109:WikiProject Mathematics 3606:B-Class vital articles 3483: 3459: 3436: 3422:has an upper bound in 3416: 3393: 3371: 2953: 2943:has an upper bound in 2911:has an upper bound in 2899:has an upper bound in 2889: 2879:has an upper bound in 2867:partially ordered set 2830:Statement of the Lemma 2826: 2816:has an upper bound in 2808:partially ordered set 2612: 2583: 2549: 2505: 2476: 2456: 2412: 2389: 2331: 2266: 2237: 2181: 2101: 2027: 2007: 1904:contains at least one 1809:has an upper bound in 1755:contains at least one 1695:contains at least one 1647:contains at least one 1414: 1325: 1294: 1252: 1173: 1114:Your new formulation: 1102: 1088:has an upper bound in 1082: 1062: 1042: 1016: 996: 973: 947: 927: 907: 887: 858: 838: 818: 795: 775: 761:has an upper bound in 755: 735: 715: 689: 669: 643: 623: 603: 583: 3484: 3460: 3437: 3417: 3394: 3372: 2929: 2861: 2802: 2667:does not satisfy the 2613: 2584: 2550: 2506: 2477: 2457: 2413: 2390: 2332: 2267: 2238: 2182: 2102: 2028: 2026:{\displaystyle \Phi } 2008: 1879:partially ordered set 1730:partially ordered set 1670:partially ordered set 1622:partially ordered set 1415: 1326: 1295: 1253: 1174: 1103: 1083: 1063: 1043: 1017: 997: 974: 948: 928: 908: 888: 859: 839: 819: 796: 776: 756: 736: 716: 690: 670: 644: 624: 604: 584: 285:partially ordered set 36:level-5 vital article 3473: 3449: 3426: 3406: 3383: 3361: 3225:or risk engaging in 3109:Proof not convincing 3083:" and "any nonempty 3007:It subsumes section 2859:is non-empty, as in 2790:for clarification: 2593: 2564: 2515: 2486: 2466: 2422: 2399: 2355: 2276: 2247: 2191: 2111: 2037: 2017: 1961: 1877:Suppose a non-empty 1668:Suppose a non-empty 1620:Suppose a non-empty 1404: 1315: 1284: 1242: 1157: 1092: 1072: 1052: 1026: 1006: 986: 963: 956:which is a fallacy. 937: 917: 897: 871: 848: 844:(the empty set) for 828: 808: 785: 765: 745: 725: 699: 679: 659: 633: 613: 593: 573: 132:mathematics articles 3479: 3455: 3432: 3412: 3389: 3367: 3353: 3216:Motivation section 2840:, 0 is a chain in 2608: 2579: 2545: 2501: 2472: 2452: 2408: 2385: 2327: 2262: 2233: 2177: 2097: 2023: 2003: 1410: 1321: 1290: 1248: 1169: 1098: 1078: 1058: 1038: 1012: 992: 969: 943: 923: 903: 883: 854: 834: 814: 791: 771: 751: 731: 711: 685: 665: 639: 619: 599: 579: 101:Mathematics portal 45:content assessment 3550: 3545: 3542: 3482:{\displaystyle P} 3458:{\displaystyle P} 3435:{\displaystyle P} 3415:{\displaystyle P} 3392:{\displaystyle P} 3370:{\displaystyle P} 3227:original research 2984:Handbook of Logic 2793:BEGIN of section 2637: 2625:comment added by 2511:is false so that 2475:{\displaystyle 0} 1954: 1942:comment added by 1926: 1914:comment added by 1324:{\displaystyle P} 1293:{\displaystyle P} 1251:{\displaystyle P} 1101:{\displaystyle P} 1081:{\displaystyle C} 1068:is a chain) then 1061:{\displaystyle C} 1015:{\displaystyle C} 995:{\displaystyle C} 972:{\displaystyle P} 946:{\displaystyle P} 926:{\displaystyle 0} 906:{\displaystyle 0} 857:{\displaystyle C} 837:{\displaystyle 0} 817:{\displaystyle P} 794:{\displaystyle P} 774:{\displaystyle P} 754:{\displaystyle C} 741:is a chain) then 734:{\displaystyle C} 688:{\displaystyle C} 668:{\displaystyle C} 642:{\displaystyle P} 622:{\displaystyle P} 602:{\displaystyle 0} 582:{\displaystyle P} 400: 388:comment added by 278: 266:comment added by 209: 208: 166: 165: 162: 161: 158: 157: 3648: 3547: 3544: 3539: 3500: 3488: 3486: 3485: 3480: 3464: 3462: 3461: 3456: 3441: 3439: 3438: 3433: 3421: 3419: 3418: 3413: 3398: 3396: 3395: 3390: 3376: 3374: 3373: 3368: 3356: 3271:What was proved? 2997: 2996: 2972: 2617: 2615: 2614: 2609: 2588: 2586: 2585: 2580: 2554: 2552: 2551: 2546: 2510: 2508: 2507: 2502: 2481: 2479: 2478: 2473: 2461: 2459: 2458: 2453: 2417: 2415: 2414: 2409: 2394: 2392: 2391: 2386: 2336: 2334: 2333: 2328: 2271: 2269: 2268: 2263: 2242: 2240: 2239: 2234: 2186: 2184: 2183: 2178: 2106: 2104: 2103: 2098: 2032: 2030: 2029: 2024: 2012: 2010: 2009: 2004: 1419: 1417: 1416: 1411: 1330: 1328: 1327: 1322: 1299: 1297: 1296: 1291: 1257: 1255: 1254: 1249: 1178: 1176: 1175: 1170: 1107: 1105: 1104: 1099: 1087: 1085: 1084: 1079: 1067: 1065: 1064: 1059: 1047: 1045: 1044: 1039: 1021: 1019: 1018: 1013: 1001: 999: 998: 993: 978: 976: 975: 970: 952: 950: 949: 944: 932: 930: 929: 924: 912: 910: 909: 904: 892: 890: 889: 884: 863: 861: 860: 855: 843: 841: 840: 835: 823: 821: 820: 815: 800: 798: 797: 792: 780: 778: 777: 772: 760: 758: 757: 752: 740: 738: 737: 732: 720: 718: 717: 712: 694: 692: 691: 686: 674: 672: 671: 666: 648: 646: 645: 640: 628: 626: 625: 620: 608: 606: 605: 600: 588: 586: 585: 580: 565:Section" above) 301:Tobias Bergemann 204: 176: 168: 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 3656: 3655: 3651: 3650: 3649: 3647: 3646: 3645: 3596: 3595: 3494: 3471: 3470: 3466: 3447: 3446: 3424: 3423: 3404: 3403: 3381: 3380: 3359: 3358: 3354: 3345: 3325: 3273: 3223:reliable source 3218: 3111: 3004:END of section 3002: 3001: 3000: 2993: 2974: 2973: 2969: 2947:. Then the set 2883:. Then the set 2820:. Then the set 2591: 2590: 2562: 2561: 2513: 2512: 2484: 2483: 2464: 2463: 2420: 2419: 2397: 2396: 2353: 2352: 2274: 2273: 2245: 2244: 2189: 2188: 2109: 2108: 2035: 2034: 2015: 2014: 1959: 1958: 1906:maximal element 1900:. Then the set 1848: 1813:" implies that 1757:maximal element 1751:. Then the set 1697:maximal element 1691:. Then the set 1649:maximal element 1643:. Then the set 1402: 1401: 1313: 1312: 1282: 1281: 1240: 1239: 1155: 1154: 1090: 1089: 1070: 1069: 1050: 1049: 1024: 1023: 1004: 1003: 984: 983: 961: 960: 935: 934: 915: 914: 913:is a chain and 895: 894: 869: 868: 846: 845: 826: 825: 806: 805: 783: 782: 763: 762: 743: 742: 723: 722: 697: 696: 677: 676: 657: 656: 655:If (for every 631: 630: 611: 610: 591: 590: 571: 570: 321: 297:maximal element 257: 214: 205: 199: 181: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 3654: 3652: 3644: 3643: 3638: 3633: 3628: 3623: 3618: 3613: 3608: 3598: 3597: 3594: 3593: 3568: 3567: 3553: 3552: 3551: 3543: 3535: 3533: 3491: 3490: 3478: 3454: 3444: 3443: 3431: 3411: 3402:Each chain in 3400: 3388: 3366: 3350: 3344: 3341: 3324: 3321: 3320: 3319: 3272: 3269: 3268: 3267: 3217: 3214: 3213: 3212: 3211: 3210: 3209: 3208: 3207: 3206: 3172: 3171: 3170: 3169: 3151: 3128: 3127: 3121: 3120: 3110: 3107: 3092: 3077: 3076: 3075: 3074: 3073: 3072: 3040: 3039: 2999: 2998: 2991: 2966: 2965: 2961: 2922: 2783: 2782: 2781: 2780: 2779: 2778: 2777: 2776: 2775: 2774: 2773: 2772: 2757: 2702: 2701: 2700: 2699: 2676: 2654: 2607: 2604: 2601: 2598: 2578: 2575: 2572: 2569: 2544: 2541: 2538: 2535: 2532: 2529: 2526: 2523: 2520: 2500: 2497: 2494: 2491: 2471: 2451: 2448: 2445: 2442: 2439: 2436: 2433: 2430: 2427: 2407: 2404: 2384: 2381: 2378: 2375: 2372: 2369: 2366: 2363: 2360: 2326: 2323: 2320: 2317: 2314: 2311: 2308: 2305: 2302: 2299: 2296: 2293: 2290: 2287: 2284: 2281: 2261: 2258: 2255: 2252: 2232: 2229: 2226: 2223: 2220: 2217: 2214: 2211: 2208: 2205: 2202: 2199: 2196: 2176: 2173: 2170: 2167: 2164: 2161: 2158: 2155: 2152: 2149: 2146: 2143: 2140: 2137: 2134: 2131: 2128: 2125: 2122: 2119: 2116: 2096: 2093: 2090: 2087: 2084: 2081: 2078: 2075: 2072: 2069: 2066: 2063: 2060: 2057: 2054: 2051: 2048: 2045: 2042: 2022: 2002: 1999: 1996: 1993: 1990: 1987: 1984: 1981: 1978: 1975: 1972: 1969: 1966: 1930: 1929: 1928: 1927: 1860: 1859: 1847: 1841: 1840: 1839: 1838: 1837: 1836: 1835: 1834: 1833: 1795: 1794: 1793: 1792: 1791: 1790: 1767: 1766: 1765: 1764: 1763: 1762: 1761: 1760: 1719: 1718: 1717: 1716: 1715: 1714: 1706: 1705: 1704: 1703: 1702: 1701: 1700: 1699: 1659: 1658: 1657: 1656: 1655: 1654: 1653: 1652: 1611: 1610: 1609: 1608: 1607: 1606: 1598: 1597: 1596: 1595: 1589: 1588: 1587: 1586: 1585: 1584: 1583: 1582: 1561: 1560: 1559: 1558: 1557: 1556: 1542: 1541: 1520: 1519: 1518: 1517: 1496: 1495: 1494: 1493: 1492: 1491: 1490: 1489: 1475: 1464: 1463: 1462: 1461: 1444: 1443: 1442: 1441: 1424: 1423: 1422: 1421: 1409: 1394: 1393: 1392: 1391: 1390: 1389: 1388: 1387: 1346: 1345: 1344: 1343: 1337: 1336: 1335: 1334: 1333: 1332: 1320: 1304: 1303: 1302: 1301: 1289: 1275: 1274: 1273: 1272: 1271: 1270: 1262: 1261: 1260: 1259: 1247: 1233: 1232: 1231: 1230: 1224: 1223: 1209: 1168: 1165: 1162: 1097: 1077: 1057: 1037: 1034: 1031: 1011: 991: 968: 942: 922: 902: 882: 879: 876: 853: 833: 813: 790: 770: 750: 730: 710: 707: 704: 684: 664: 638: 618: 598: 578: 560: 558: 557: 556: 555: 554: 553: 552: 551: 550: 549: 548: 547: 514: 481: 421: 355:(just because 320: 313: 312: 311: 268:122.59.202.135 256: 253: 224: 223: 213: 210: 207: 206: 201: 197: 195: 192: 191: 183: 182: 177: 171: 164: 163: 160: 159: 156: 155: 144: 138: 137: 135: 118:the discussion 105: 104: 88: 76: 75: 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 3653: 3642: 3639: 3637: 3634: 3632: 3629: 3627: 3624: 3622: 3619: 3617: 3614: 3612: 3609: 3607: 3604: 3603: 3601: 3592: 3588: 3584: 3580: 3575: 3574:does not mean 3570: 3569: 3566: 3562: 3558: 3554: 3546: 3538: 3537: 3534: 3532: 3528: 3524: 3520: 3515: 3514: 3513: 3512: 3508: 3504: 3498: 3476: 3468: 3467: 3452: 3429: 3409: 3401: 3386: 3379: 3378: 3364: 3349: 3342: 3340: 3339: 3335: 3331: 3330:HerbertDibDab 3322: 3318: 3314: 3310: 3306: 3305: 3304: 3303: 3299: 3295: 3291: 3286: 3282: 3281: 3276: 3270: 3266: 3262: 3258: 3253: 3252: 3251: 3250: 3246: 3242: 3237: 3232: 3228: 3224: 3215: 3205: 3201: 3197: 3192: 3191: 3190: 3186: 3182: 3178: 3177: 3176: 3175: 3174: 3173: 3168: 3164: 3160: 3156: 3152: 3148: 3147: 3146: 3145: 3144: 3143: 3139: 3135: 3131: 3126: 3123: 3122: 3119: 3116: 3115: 3114: 3108: 3106: 3105: 3101: 3097: 3090: 3087:" is whether 3086: 3082: 3071: 3067: 3063: 3059: 3058: 3057: 3053: 3049: 3048:172.88.206.28 3044: 3043: 3042: 3041: 3038: 3034: 3030: 3025: 3024: 3023: 3022: 3018: 3014: 3013:172.88.206.28 3010: 3005: 2994: 2992:0-7204-2285-X 2989: 2985: 2981: 2977: 2971: 2968: 2964: 2960: 2958: 2952: 2950: 2946: 2942: 2938: 2934: 2928: 2926: 2920: 2918: 2914: 2910: 2906: 2903:" and "every 2902: 2898: 2894: 2888: 2886: 2882: 2878: 2874: 2870: 2866: 2860: 2858: 2853: 2851: 2847: 2843: 2839: 2835: 2831: 2825: 2823: 2819: 2815: 2811: 2807: 2801: 2799: 2794: 2791: 2789: 2771: 2767: 2763: 2762:172.88.206.28 2758: 2756: 2753: 2752: 2751: 2747: 2743: 2739: 2735: 2730: 2725: 2724: 2723: 2722: 2721: 2717: 2713: 2712:172.88.206.28 2708: 2707: 2706: 2705: 2704: 2703: 2698: 2694: 2690: 2686: 2682: 2677: 2674: 2670: 2666: 2662: 2658: 2655: 2652: 2648: 2644: 2639: 2638: 2636: 2632: 2628: 2627:172.88.206.28 2624: 2602: 2596: 2573: 2567: 2559: 2558: 2557: 2539: 2533: 2524: 2518: 2495: 2489: 2469: 2446: 2440: 2431: 2425: 2405: 2379: 2373: 2364: 2358: 2349: 2348: 2344: 2340: 2339:172.88.206.28 2321: 2318: 2315: 2309: 2300: 2297: 2294: 2288: 2282: 2256: 2250: 2224: 2218: 2209: 2203: 2197: 2174: 2171: 2168: 2165: 2162: 2150: 2144: 2141: 2135: 2129: 2123: 2088: 2082: 2079: 2067: 2058: 2052: 2049: 1997: 1991: 1988: 1982: 1976: 1970: 1967: 1955: 1953: 1949: 1945: 1944:172.88.206.28 1941: 1935: 1925: 1921: 1917: 1916:172.88.206.28 1913: 1907: 1903: 1899: 1895: 1891: 1887: 1883: 1880: 1876: 1875: 1874: 1873: 1872: 1870: 1866: 1857: 1853: 1852: 1851: 1846: 1842: 1832: 1828: 1824: 1820: 1816: 1812: 1808: 1803: 1802: 1801: 1800: 1799: 1798: 1797: 1796: 1789: 1785: 1781: 1780:172.88.206.28 1777: 1773: 1772: 1771: 1770: 1769: 1768: 1758: 1754: 1750: 1746: 1742: 1738: 1734: 1731: 1727: 1726: 1725: 1724: 1723: 1722: 1721: 1720: 1712: 1711: 1710: 1709: 1708: 1707: 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