text:
Talk:Zorn's lemma

Zorn's lemma CAN'T be used to show that every graph has a spanning tree
The assertion is not true. Only connected graphs have spanning trees. The mistake in the proof happens here:
 * Zorn's lemma says that a maximal tree must exist, which is a spanning tree.

Maximal trees in disconnected grpahs aren't spanning trees. --Jobu0101 (talk) 20:41, 5 January 2015 (UTC)

Well, they do have spanning forests (a forest is a set of trees), and that does follow from Zorn's Lemma.172.88.206.28 (talk) 16:55, 23 September 2016 (UTC)

Too technical
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 logging in. — Preceding unsigned comment added by 122.59.202.135 (talk) 20:38, 23 April 2016 (UTC)

 * 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 partially ordered set, chain, upper bound and maximal element just to be able to state the lemma. —Tobias Bergemann (talk) 09:53, 25 April 2016 (UTC)

Fallacy in Empty chain as boundary case Section
Note: The above referenced Section has been removed and replaced with another Section Alternative formulation ''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.''172.88.206.28 (talk) 23:26, 23 September 2016 (UTC)

The Section Empty chain as boundary case contains a fallacy. If P is empty then no chain can have an upper bound in P (just because P is empty). In particular, the statement (quotation from the Section Empty chain as boundary case): "However, the empty set is a chain (trivially), hence is required to have an upper bound, thus exhibiting at least one element of P." is a fallacious inference. Thus the premise of Zorn's Lemma is not satisfied, contrary to what this Section implies, and the conclusion that P has a maximal element does not follow.

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.172.88.206.28 (talk) 17:04, 23 September 2016 (UTC) — Preceding unsigned comment added by 172.88.206.28 (talk) 17:00, 23 September 2016 (UTC)
 * Reread the passage. Your arguments exactly prove why the text in the article is correct, not why it's incorrect.  If P 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 P is nonempty.  And that's what the article says.
 * 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? --Trovatore (talk) 17:10, 23 September 2016 (UTC)
 * I've tried to clarify what this section is trying to say: "In the formulation of Zorn's lemma above, the partially ordered set P is not explicitly required to be non-empty. However, in this formulation, if a set P satisfies the given hypothesis ("..every chain has an upper bound in P..."), then P must be non-empty. To see this, note that the empty subset of P is a chain, and hence is required to have an upper bound in P, exhibiting at least one element of P."  Magidin (talk) 18:07, 23 September 2016 (UTC)
 * 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 poster.
 * 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 partial order has a maximal element. --Trovatore (talk) 18:27, 23 September 2016 (UTC)
 * 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. Magidin (talk) 18:31, 23 September 2016 (UTC)
 * 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 article.
 * 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. --Trovatore (talk) 18:38, 23 September 2016 (UTC)
 * 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 his handout on AC and Zorn's Lemma; 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. Magidin (talk) 19:12, 23 September 2016 (UTC)

Your statement (quoted from your reply in Section "Fallacy in Empty chain as boundary case Section" above) Your arguments exactly prove why the text in the article is correct, not why it's incorrect is false. And the reason for it that you used a phrase "has an upper bound" rather than "has an upper bound in $$P$$". While it is true that the empty set is a chain and, trivially, has an upper bound (for instance, $$0$$), it may happen to have no upper bound in a given set $$P$$ (it happens exactly when $$P$$ is empty).

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

If (for every $$C$$, if ($$C$$ is a set and $$C \subseteq P$$ and $$C$$ is a chain) then $$C$$ has an upper bound in $$P$$) then $$P$$ has a maximal element.

In your discussion of the case when $$P$$ is empty, you substituted $$0$$ (the empty set) for $$C$$ in the above statement and concluded:

($$0 \subseteq P$$ and $$0$$ is a chain and $$0$$ has an upper bound) implies ($$P$$ contains a maximal element),

which is a fallacy.

The bottom line is that the Zorn's Lemma applies to the case of empty $$P$$ without any "modifications" or "corrections" because the premise

(for every $$C$$, if ($$C$$ is a set and $$C \subseteq P$$ and $$C$$ is a chain) then $$C$$ has an upper bound in $$P$$)

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

Your new formulation: 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 is still a fallacy since there is nothing in entire Zorn's Lemma that would make P "exhibit at least one element".

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. 172.88.206.28 (talk) 20:17, 23 September 2016 (UTC)

 * 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 P" 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 P". Note also that "chain" is (as is often the case) assumed to mean subset that is totally ordered, so that including "$$C\subset P$$" 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 P 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. 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 P satisfies "P is nonempty and every nonempty chain has an upper bound in P" if and only if it satisfies "every chain has an upper bound in P." 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 P "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. That is the point. And the paragraph is explicit as to what it is trying to establish.
 * 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. Magidin (talk) 20:50, 23 September 2016 (UTC)

 * 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.

 * There is a difference between "has an upper bound" and "has an upper bound in $$P$$". Here is a verbatim quotation from the original version of the Section:

 * "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."

 * The Lemma uses "has an upper bound in $$P$$" while the Section uses "has an upper bound". Do you claim that the correct reading was supposed to be:

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

 * What exactly is your experience writing and reading proofs? The formal statement specified "an upper bound in P". 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 already specified that the desired upper bound was in P, it was unnecessary to repeat this yet again when saying that the upper bound guaranteed by the statement would necessarily lie in P. 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. Magidin (talk) 04:00, 24 September 2016 (UTC)

 * 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 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 $$\subseteq$$.

 * 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?172.88.206.28 (talk) 00:37, 24 September 2016 (UTC)

 * One question: are you, gentlemen, philosophers?172.88.206.28 (talk) 00:37, 24 September 2016 (UTC)

 * And the snark continues. Again, you seem to get hung up on irrelevant and persnickety details. Here, "some partially ordered set" means "some [specific] partially ordered [that is understood from context]." With a narrowmindedness like that, how can you expect to ever understand a proof or be satisfied by anything, especially given the evidence of your prior misunderstandings?
 * As it happens, I am a professional, published mathematician, with experience as writer, referee, and editor of mathematical texts. How about you? Magidin (talk) 04:00, 24 September 2016 (UTC)

 * My main point 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 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.172.88.206.28 (talk) 00:43, 24 September 2016 (UTC)

 * 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 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 common 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. Magidin (talk) 21:03, 23 September 2016 (UTC)

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

 * "If it was correct then why it was replaced?[sic]" 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. Magidin (talk) 04:00, 24 September 2016 (UTC)

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

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

 * Suppose a non-empty partially ordered set P has the property that every chain in P has an upper bound in P. Then the set P contains at least one maximal element (I saw this version in Yech's text)

 * Suppose a non-empty partially ordered set P has the property that every non-empty chain in P has an upper bound in P. Then the set P contains at least one maximal element

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

 * Suppose a partially ordered set P has the property that every non-empty chain in P has an upper bound in P. Then the set P contains at least one maximal element.

 * A lack of explicit quantifier "For every set P" and use of "suppose" and "property" may confuse some. 172.88.206.28 (talk) 22:37, 23 September 2016 (UTC)

 * 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 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 P has an upper bound in P" implies that P is non-empty; that was the whole point of the section you objected to. Nobody was proposing your third version, which leads me to wonder about why you bothered to present it, unless it's part of a strawman. Magidin (talk) 04:00, 24 September 2016 (UTC)

Unconvincing purpose for the Section Alternative formulation
A quote:

 * "So many authors prefer to verify the non-emptiness of the set P rather than deal with the empty chain in the general argument."

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 P then they may just assume that their set P 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:

 * Suppose a non-empty partially ordered set P has the property that every chain in P has an upper bound in P. Then the set P contains at least one maximal element. — Preceding unsigned comment added by 172.88.206.28 (talk) 23:07, 23 September 2016 (UTC)

Also, this phrase from the Section Alternative formulation: "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. — Preceding unsigned comment added by 172.88.206.28 (talk) 23:40, 23 September 2016 (UTC)

Also, "equivalent" needs an explanation. In genreal, $$\forall x A(x) \equiv \forall x B(x)$$ iff under all admissible interpretations $$\Phi$$, $$\Phi(\forall x A(x)) = \Phi(\forall x B(x))$$, while in the said Section, it implies more than that: $$\Phi(\forall x (P(x) \equiv Q(x))) = true$$. 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 $$\forall P(\varphi(P) \Rightarrow \psi(P))$$ (and, to make it worse, $$\varphi(P)$$ was not precisely formulated in a form $$\forall C(\xi(P,C) \Rightarrow \zeta(P,C)$$), it is likely to mislead some.172.88.206.28 (talk) 00:03, 24 September 2016 (UTC)

If one refers to Zorn's Lemma as $$\varphi(P) \Rightarrow \psi(P)$$ in the form mentioned above, assuming that the quantifier $$\forall P$$ is implicit in it (which seems quite common), then $$\varphi(0) \Rightarrow \psi(0)$$, where $$0$$ is the empty set, is a theorem of ZF ($$\varphi(0)$$ is false so that $$\varphi(0) \Rightarrow \psi(0)$$ 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.

 * A QUOTE from a message to me. "This is logically equivalent to saying that the empty set does not satisfy the hypothesis [$$\varphi(P)$$, I suppose] of Zorn's Lemma, which is what you are saying. [Indeed, $$\varphi(0)$$ is false.] So you are incorrect in claiming that the Section implies that the empty set satisfies Zorn's Lemma. [Now, you contradicted yourself; if the Zorn's Lemma is correctly formulated then 0 satisfies it; if you claim 0 does not then of the two of us you are the one who is wrong.] The section makes no such claim, it is making the claim whose contrapositive you are arguing for." [You are wrong, again, period.] — Preceding unsigned comment added by 172.88.206.28 (talk) 01:17, 24 September 2016 (UTC)
 * 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 satisfied by the empty set, and as such the implication of Zorn's lemma holds for the empty set being substituted for P. But this is as empty as saying that a rutabaga satisfies Zorn's lemma; it does, since the premise of Zorn's lemma is false when we substitute a rutabaga for P.
 * However, 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 "X does not satisfy the Theorem" to mean not that the theorem is false, but that X does not satisfy the hypothesis of the theorem and as such that the theorem is not "really" about X. Just like Zorn's lemma is not about rutabagas. The "contradiction" you see comes from your failure to understand what is being said.
 * 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 not will not appreciate a hyper-formalized presentation that requires a deep understanding of set theory and familiarity with formal proofs to follow. Magidin (talk) 04:08, 24 September 2016 (UTC)

 * 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 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.172.88.206.28 (talk) 22:23, 29 September 2016 (UTC)
 * 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 the article (which did not exist, despite your protestations). What I point out is that one often uses the phrase "X 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; 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 mean "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 was. 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). Magidin (talk) 23:19, 29 September 2016 (UTC)
 * I suggest you either learn the meaning of the phrase "you contradicted yourself", or else that you try to be more careful in reading.
 * 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 claim that I need to learn is a projection: you project on me your own flaws that you deny. 172.88.206.28 (talk) 16:06, 4 June 2017 (UTC)

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 Notes on the assumption of the non-emptiness for clarification:

BEGIN of section

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

Suppose a non-empty partially ordered set P has the property that every chain in P has an upper bound in P. Then the set P contains at least one maximal element.

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 Statement of the Lemma. Indeed, in the case of P = 0, the assumption of the Zorn's lemma is not satisfied because the empty set 0 is a subset of P, 0 is a chain in P, but no upper bound of 0 (any element of any set is an upper bound of the empty set) is an element of P. Therefore, the lemma is (vacuously) true for P = 0. The above formulation has been used in.

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

Suppose a non-empty partially ordered set P has the property that every non-empty chain in P has an upper bound in P. Then the set P contains at least one maximal element.

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 P, the conditions "every chain in P has an upper bound in P" and "every non-empty chain in P has an upper bound in P" are equivalent. Indeed, if P is non-empty then its every element is an upper bound of its empty chain.

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

Suppose a partially ordered set P has the property that every non-empty chain in P has an upper bound in P. Then the set P contains at least one maximal element.

invalidates it; the empty set P satisfies the premiss but not the conclusion of it.

END of section

It subsumes section Alternative formulation. 172.88.206.28 (talk) 18:02, 4 June 2017 (UTC)

 * 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 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. Magidin (talk) 22:20, 4 June 2017 (UTC)

 * 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 defensiveness that overrides your weak commitment to truth? I guess, you are good at it for a reason. 172.88.206.28 (talk) 23:04, 4 June 2017 (UTC)
 * 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? Magidin (talk) 01:28, 5 June 2017 (UTC)

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 element. 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) 07:43, 5 June 2017 (UTC)

Proof not convincing
The proof sketch section looks really weak here.

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

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

Where do these steps come from?

Vishal0123 (talk) 23:17, 8 March 2018 (UTC)
 * In most cases, we don't go into details of proofs in Wikipedia 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 detail.
 * (If you're interested for your personal understanding, please ask at WP:RD/Math.) --Trovatore (talk) 23:37, 8 March 2018 (UTC)

 * 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. --Jordan Mitchell Barrett (talk) 02:09, 15 April 2021 (UTC)
 * 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. Magidin (talk) 15:08, 15 April 2021 (UTC)

Motivation section
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 Wikipedia we need to provide reliable source or risk engaging in original research. 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 really 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, every 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. Magidin (talk) 15:56, 2 August 2019 (UTC)
 * 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 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).Zaunlen (talk) 16:56, 2 August 2019 (UTC)

What was proved?
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"

Apparently this question was asked and answered here: https://math.stackexchange.com/questions/1777443/original-proof-of-zorns-lemma

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 [1922] and Max Zorn [1935]."

Is what we're saying that certain implications of Zorn's lemma were proved in 1922 and 1935? KVenzke (talk) 22:50, 21 December 2021 (UTC)
 * It probably means proved using the axiom of choice. The axiom of choice is intuitively clear; Zorn's lemma is not. --Trovatore (talk) 22:57, 21 December 2021 (UTC)

Application in Philosophy of Logic/Language
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 article, but it's unclear under which seciton organisationally it would be worth mentioning ... HerbertDibDab (talk) 10:52, 16 May 2024 (UTC)

title: Talk:Zorn's lemma


text:
Talk:Fermat's Last Theorem

Renewed interest from serious mathematicians
This section 2.5 is only reported by the NY Times and Wiles has had no comment on it. Saying there was renewed interest from serious mathematicians due to Vaughn seems inconsistent with the sections immediately above where there is clearly work being performed on it all through the centuries including just before Vaughn's money became involved. — Preceding unsigned comment added by 75.80.196.159 (talk) 22:56, 16 February 2022 (UTC)

Section about Harvey Friedman's grand conjecture
There is a short section in the article beginning with While Harvey Friedman's grand conjecture implies ... It strikes me as irrelevant, and honestly, rather stupid. It contrasts with the rest of the article which is so well written and sourced. I propose to delete the section. Can I get some support from mathematicians? --Herbmuell (talk) 21:11, 14 May 2022 (UTC)
 * It was added in 2010 by R.e.b. (Special:Diff/373497345), whose taste in mathematical interestingness I tend to trust more than those of most other editors, even the ones who frequently edit mathematics articles here. The issue of whether FLT has an elementary proof is still as far as I know unresolved and directly relevant to the question of whether Fermat himself could have had a proof (the previous sentence in the article). See also McLarty 2010, Grosholz 2017 , and McLarty 2020 on exactly this issue, and especially the final section "The Wide-Open Question" of McLarty 2020.  —David Eppstein (talk) 21:59, 14 May 2022 (UTC)
 * Okay, David, thanks for your reply. There is no information about R.e.b. on his user page, contrary to your page. This grand conjecture is very broad, it's not in formal mathematical language, so it's impossible to prove or disprove. It smacks of talk of mathematicians over a cup of coffee. But no problem, we disagree on this, and you have done much more reading than I, so we let the section stand as it is. --Herbmuell (talk) 09:34, 15 May 2022 (UTC)
 * I believe that the program of reverse mathematics makes this quite a precise and formal conjecture, not vague. Unfortunately Wikipedia rules prevent me from saying more about why I trust R.e.b.'s judgement on this sort of thing. —David Eppstein (talk) 18:22, 15 May 2022 (UTC)

Popular culture: verbiage
There is far too much verbiage about minor Star Trek episodes. I'm removing the obvious excess. People who want to save it should move it to Fermat's Last Theorem in fiction. Here is the deleted material.

 * Also in a Simpsons episode, "Treehouse of Horror VI", in the "Homer³" segment, an equation appears on the blackboard that reads 178212 + 184112=1922.12. This is another equation that appears to be a counterexample to Fermat's Last Theorem, but is also wrong. This equation does, however, similar to the one found in "The Wizard of Evergreen Terrace", appear correct if entered in a calculator with 10 significant figures. David Cohen is the Simpsons writer responsible for these appearances of false counterexamples to Fermat's Last Theorem.

Wiles' proof was referenced in the Star Trek: Deep Space Nine season three episode "Facets", where Jadzia Dax says to Tobin Dax that his proof of the theorem was "the most original approach to the proof since Wiles over three hundred years ago". )
 * In "The Royale", a 1989 episode of the 24th-century-set TV series Star Trek: The Next Generation, Picard tells Commander Riker about his attempts to solve the theorem, still unsolved after 800 years. He concludes, "In our arrogance, we feel we are so advanced. And yet we cannot unravel a simple knot tied by a part-time French mathematician working alone without a computer." (Andrew Wiles's insight leading to his breakthrough proof happened four months after the series ended.)

Zaslav (talk) 22:30, 21 May 2022 (UTC)

 * @Zaslav I concur; the trivia referenced directly to a ST episode was the worst. See also the ongoing Piotr Konieczny aka Prokonsul Piotrus| reply here  07:30, 22 May 2022 (UTC)
 * Despite my keep !vote on the AfD, I also strongly agree with removing self-sourced trivia such as this. Either here or (if kept) at the in fiction article, we should only include material covered in secondary sources that clearly state the significance of the theorem to the work in question. —David Eppstein (talk) 07:46, 22 May 2022 (UTC)
 * I agree, except for the classic "Simon Flagg" story (!) -- it should be easy for anyone who is motivated to find secondary sources for that story. (By the way, the significance of Fermat's theorem to that work is the most obvious possible.)
 * I note that many articles have sections of arbitrary trivia of the same nature. It's a bit silly. Zaslav (talk) 22:00, 22 May 2022 (UTC)
 * In the "Simon Flagg" story it's central to the plot. In the Star Trek one it appears to be just a throwaway remark. I think if we're going to have a section like this here at all, we should swap the Star Trek bit out for The Last Theorem and Arcadia (play) where the connection is again more central. —David Eppstein (talk) 23:33, 22 May 2022 (UTC)

I am disappointed that the example of Wile's proof mentioned in Star Trek Deep Space Nine was removed as it was aired on 12 June 1995, when Wile's proof had only been published in May 1995. This surely shows that the proof of Fermat's Last Theorem was incredibly important to the writers of Star trek who felt it should receive a special mention. Star trek has long been pointed to as an inspiration for young scientists and mathematicians, indeed, Stephen Hawking even appeared in an episode as he was a huge fan. I very much doubt that many young scientists would have read the Annals of mathematics at the time but being informed that the theorum had been proved through Star trek would likely have piqued their interest and led them to find out more about what the characters were talking about. But hey, I guess Star trek isn't for everyone CiaranMacD (talk) 16:07, 13 October 2022 (UTC)

 * All additions of such material, even in Fermat's Last Theorem in fiction, require in-depth published sources (not merely TV-guide like episode listings) attesting to the significance of the subject to the episode in question, and preferably also explaining why it could not have been possible to substitute some random other unsolved mathematical problem with no change in meaning to the rest of the episode. The section of this article that summarizes Fermat's Last Theorem in fiction should be even more selective. Your addition did none of that. If you want to list TV episodes that mention stuff, maybe TV Tropes would be a better site to edit than Wikipedia. —David Eppstein (talk) 17:06, 13 October 2022 (UTC)

Please delete all of my contributions and my account. CiaranMacD (talk) 04:24, 16 October 2022 (UTC)
 * I would agree that the Star Trek:Deep Space Nine mention should be returned. As mentioned, it was made only weeks after Wiley's discovery and was significantly pointed out in the episode. Let's return this one. Randy Kryn (talk) 04:30, 16 October 2022 (UTC)
 * I don't see the point of having a list of some random media mentions here and a list of more random media mentions in the Fermat's Last Theorem in fiction article, without any clear criterion for which ones are important enough to mention here. The paragraph here should be a WP:SUMMARY of the child article, not content fork filtered only by which fans whine the most when their fandom is snubbed. It should certainly not be the case that equivalent content here is both more detailed and less well-sourced than in the "in fiction" article, which was the case in the recent go-round. And judging from our one-sentence description of the episode in the "in fiction" article, it was a throwaway line, not anything intrinsic to the plotline or historically significant as a particularly early piece of Fermat-fiction (as the Simon Flagg piece is), or having any nontrivial mathematical content (as maybe the Simpson's reference does). —David Eppstein (talk) 06:11, 16 October 2022 (UTC)
 * The relevance: being mentioned on a science fiction show set far in the future just weeks after the discovery occurred, and being added as an iconic event to the already completed script (or possibly the completed show), indicates its importance within society and the structure of mathematics. Randy Kryn (talk) 11:54, 16 October 2022 (UTC)
 * That gets over the edge into drawing our own new conclusions. I don't think we can make that judgment call. XOR'easter (talk) 14:46, 16 October 2022 (UTC)
 * You have a point if the conclusion was used in the page text, which it wouldn't be, but in discussion for or against inclusion the point of view seems fine (although I'm not a wikilawyer). Randy Kryn (talk) 14:58, 16 October 2022 (UTC)
 * If the only grounds for inclusion is our speculation about its possible cultural significance based on guesses about the production schedule of a TV show, then we don't really have grounds for inclusion. XOR'easter (talk) 20:45, 16 October 2022 (UTC)

A Commons file used on this page or its Wikidata item has been nominated for deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion: Participate in the deletion discussion at the. —Community Tech bot (talk) 02:53, 12 March 2023 (UTC)
 * Diophantus-II-8.jpg

"It is impossible to separate a cube into two cubes" versus Banach-Tarski paradox?
> "It is impossible to separate a cube into two cubes - Fermat"

The article should clarify for layman readers how that 1670 statement by Fermat can co-exist / reconcile with the Banach-Tarski paradox from 1924? I mean, one may use B-T-P as a workaround: transform the cube into a ball, split that ball up into as few as 5 pieces and re-combine them in a different way, which yields TWO identical copies of the original ball (or abritrarily many copies, per Churkin 2010). Transform those balls into cubes and finished, or not if the logic is broken somewhere? 94.21.229.113 (talk) 23:12, 22 June 2024 (UTC)

 * One is about integers and the other is about continuous shapes (or, if you prefer, real numbers). They are not very closely related. —David Eppstein (talk) 23:24, 22 June 2024 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Fermat's Last Theorem


text:
Singular function

In mathematics, a real-valued function f on the interval [a, b] is said to be singular if it has the following properties:

 * f is continuous on [a, b]. (**)
 * there exists a set N of measure 0 such that for all x outside of N the derivative f (x) exists and is zero, that is, the derivative of f vanishes almost everywhere.
 * f is non-constant on [a, b].

A standard example of a singular function is the Cantor function, which is sometimes called the devil's staircase (a term also used for singular functions in general). There are, however, other functions that have been given that name. One is defined in terms of the circle map.

If f(x) = 0 for all x ≤ a and f(x) = 1 for all x ≥ b, then the function can be taken to represent a cumulative distribution function for a random variable which is neither a discrete random variable (since the probability is zero for each point) nor an absolutely continuous random variable (since the probability density is zero everywhere it exists).

Singular functions occur, for instance, as sequences of spatially modulated phases or structures in solids and magnets, described in a prototypical fashion by the Frenkel–Kontorova model and by the ANNNI model, as well as in some dynamical systems. Most famously, perhaps, they lie at the center of the fractional quantum Hall effect.

When referring to functions with a singularity
When discussing mathematical analysis in general, or more specifically real analysis or complex analysis or differential equations, it is common for a function which contains a mathematical singularity to be referred to as a 'singular function'. This is especially true when referring to functions which diverge to infinity at a point or on a boundary. For example, one might say, "1/x becomes singular at the origin, so 1/x is a singular function."

Advanced techniques for working with functions that contain singularities have been developed in the subject called distributional or generalized function analysis. A weak derivative is defined that allows singular functions to be used in partial differential equations, etc.

title: Singular function


text:
Mohammad Daoud Khan

Mohammad Daoud Khan (also romanized as Daud Khan or Dawood Khan; 18July 1909 – 28April 1978) was an Afghan military officer and politician who served as prime minister of Afghanistan from 1953 to 1963 and, as leader of the 1973 Afghan coup d'état which overthrew the monarchy, served as the first president of Afghanistan from 1973 until his assassination in the Saur Revolution.

Born into the Afghan royal family and addressed by the prefix "Sardar", Khan started as a provincial governor and later a military officer before being appointed as Prime Minister by his cousin, King Mohammad Zahir Shah, serving for a decade. Having failed to persuade the King to implement a one-party system, Khan overthrew the monarchy in a virtually bloodless coup with the backing of Afghan Army officers, and proclaimed himself the first President of the Republic of Afghanistan, establishing an autocratic one-party system under his National Revolutionary Party.

Khan was known for his autocratic rule, and for his educational and progressive social reforms. Under his regime, he headed a purge of communists in the government, and many of his policies also displeased religious conservatives and liberals who were in favor of restoring the multiparty system that existed under the monarchy. Social and economic reforms implemented under his ruling were successful, but his foreign policy led to tense relations with neighboring countries. In 1978, he was deposed and assassinated during the 1978 Afghan coup d'état, led by the Afghan military and the communist People's Democratic Party of Afghanistan (PDPA). His body was discovered 30 years later and was identified by a small golden Quran gifted by King Khalid of Saudi Arabia he always carried. He received a state funeral.

Early life
Khan was born in Kabul, Emirate of Afghanistan, into a Barakzai Pashtun family and was the eldest son of the diplomat Prince Mohammad Aziz Khan (1877–1933; an older half-brother of King Mohammad Nadir Shah) and his wife, Khurshid Begum. His father was assassinated in 1933 in Berlin, while serving as the Afghan Ambassador to Germany. He and his brother Prince Naim Khan (1911–1978) then came under the tutelage of their uncle Mohammad Hashim Khan (1884–1953). Daoud proved to be an apt student of politics.

Educated in France, he served as a senior administrator in the Kingdom of Afghanistan, serving as Governor of the Eastern Province in 1934–35 and in 1938–39, and was Governor of Kandahar Province from 1935 to 1938.

In 1939, Khan was promoted to Commander of the Central Forces. As commander, he led Afghan forces against the Safi during the Afghan tribal revolts of 1944–1947. From 1946 to 1948, he served as Defense Minister, then Interior Minister from 1949 to 1951. In 1948, he served as Afghan Ambassador to France.

In 1951, he was promoted to General and served in that capacity as Commander of the Central Corps of the Afghan Armed Forces in Kabul from 1951 to 1953.

Royal Prime Minister
Khan was appointed prime minister in September 1953 through an intra-family transfer of power, replacing Shah Mahmud Khan. His ten-year tenure was noted for his foreign policy turn to the Soviet Union, the completion of the Helmand Valley project, which dramatically improved living conditions in southwestern Afghanistan, as well as tentative steps towards the emancipation of women, giving women a higher public presence, which led to significant amounts of freedom and educational opportunities for them.

With the creation of an independent Pakistan in August 1947, Prime Minister Daoud Khan had rejected the Durand Line, which had been accepted as international border by successive Afghan governments for over a half a century. Khan supported a nationalistic reunification of the Pakistani Pashtun people with Afghanistan, but this would have involved taking a considerable amount of territory from the new nation of Pakistan and was in direct opposition to an older plan of the 1940s whereby a confederation between the two countries was proposed. The move further worried the non-Pashtun populations of Afghanistan such as the minority Hazara, Tajik, and Uzbek, who suspected his intention was to increase the Pashtuns' disproportionate hold on political power.

Abdul Ghaffar Khan (founder of Khudai Khidmatgar movement), stated "that Daoud Khan only exploited the idea of reunification of Pashtun people to meet his own political ends. The idea of reunification of Pashtun people never helped Pashtuns and it only caused trouble for Pakistan. In fact it was never a reality". Moreover, Daoud Khan's project for the reunification of the Pashtun people failed to gain support from the majority of Pashtuns in Pakistan.

In 1960, Khan sent troops across the poorly-marked Durand Line into the Bajaur Agency of Pakistan in an attempt to manipulate events in that area and to press the Pashtunistan issue, but the Afghan forces were defeated by the Pashtun Tribal militias. During this period, the propaganda war from Afghanistan, carried on by radio, was relentless. In 1961, Daoud Khan made another attempt to invade Bajaur with larger Afghan army this time. However, Pakistan employed F-86 Sabres jets which inflicted heavy casualties against the Afghan army unit and the tribesmen from Kunar province who were supporting the Afghan army. Several Afghan soldiers were also captured and were paraded in front of international media, which in turn caused embarrassment for Daoud Khan.

In 1961, as a result of his policies and support to militias in areas along the Durand Line, Pakistan closed its borders with Afghanistan and the latter severed ties, causing an economic crisis and greater dependence on the USSR. The USSR became Afghanistan's principal trading partner. Within a few months, the USSR sent jet airplanes, tanks, heavy and light artillery, for a heavily discounted price tag of $25 million, to Afghanistan. That same year he attended the 1st Summit of the Non-Aligned Movement in Belgrade making Afghanistan one of the founding members of the Non-Aligned Movement.

As a result of continued resentment against Daoud's autocratic rule, close ties with the USSR and economic downturn because of the blockade imposed by Pakistan, Daoud Khan was asked to resign. Instead of resigning, Daoud Khan requested King Zahir Shah to approve new 'one-party constitution' proposed by him which would in turn increase Daoud Khan's already considerable power. Upon rejection, Daoud Khan angrily resigned.

The crisis was finally resolved with his forced resignation in March 1963 and the re-opening of the border in May. Pakistan continued to remain suspicious of Afghan intentions and Daoud's policy left a negative impression in the eyes of many Tajiks who felt they were being disenfranchised for the sake of Pashtun nationalism. He was succeeded by Mohammad Yusuf.

In 1964, King Zahir Shah introduced a new constitution, for the first time excluding all members of the royal family from the Council of Ministers. Khan had already stepped down. In addition to having been Prime Minister, he had also held the portfolios of Minister of Defense and Minister of Planning until 1963.

President of the Republic
Khan was unsatisfied with King Zahir Shah's constitutional parliamentary system and lack of progress. He planned rebellion for more than a year before he seized power from the King on 17 July 1973. The coup was bloodless, and backed by a large number of army officers who were loyal to him, facing no resistance. Departing from tradition, and for the first time in Afghan history, he did not proclaim himself Shah, establishing instead a republic with himself as President. The role of pro-communist Parchamite officers in the coup led to him receiving the nickname "Red Prince" by some.

King Zahir Shah's constitution establishing a parliament with elected members and the separation of powers was replaced by a now largely nominated loya jirga (meaning "grand assembly"). The parliament was disbanded.

Although he was close to the Soviet Union during his prime ministership, Khan continued the Afghan policy of non-alignment with the Cold War superpowers. Nor did he bring drastic pro-Soviet change to the economic system.

In Khan's new cabinet, many ministers were fresh faced politicians, and only Dr Abdul Majid was a ministerial carryover from Khan's Prime Minister era (1953–1963); Majid was Minister of Education from 1953 to 1957, and from 1973 was appointed Minister of Justice until 1977. Initially about half of the new cabinet were either current members, former members or sympathizers of the PDPA, but over time their influence would be eradicated by Khan.

A coup against Khan, which may have been planned before he took power, was repressed shortly after his seizure of power. In October 1973, Mohammad Hashim Maiwandwal, a former Prime Minister and a highly respected former diplomat, was arrested in a coup plot and died in prison before his trial set for December 1973. This was at a time when Parchamites controlled the Ministry of Interior under circumstances corroborating the widespread belief that he had been tortured to death by the leftists. According to one account, Daoud Khan planned to appoint Maiwandwal as prime minister, leading to the Parchamite Minister of Interior, Faiz Mohammad, along with fellow communists, framing Maiwandwal in a coup plot, then torturing him to death without Daoud Khan's knowledge. Louis Dupree wrote that Maiwandwal, one of few Afghan politicians with an international reputation, could have been a leader in a democratic process and therefore a target for communists. One of the army generals arrested under suspicion of this plot with Maiwandwal was Mohammed Asif Safi, who was later released. Khan personally apologized to him for the arrest.

In 1974, he signed one of two economic packages that aimed to greatly increase the capability of the Afghan military. At this time, there were increasing concerns that Afghanistan lacked a modern army comparable to the militaries of Iran and Pakistan.

In 1975, his government nationalized all banks in Afghanistan, including Da Afghanistan Bank, Afghanistan's central bank.

Khan wanted to lessen the country's dependence on the Soviet Union and attempted to promote a new foreign policy. In 1975 he visited some countries in the Middle East, including Egypt, Turkey, Saudi Arabia, and Iran, all of which were anti-Soviet states, to ask for aid, He also visited India. Regarding the Non-Aligned Movement summit in Havana, Khan said that Cuba "only pretends to be non-aligned." Surprisingly, he did not renew the Pashtunistan agitation; relations with Pakistan improved thanks to interventions from the US and the Shah of Iran. These moves alerted the Soviets.

Constitution of 1977
In 1977, he established his own political party, the National Revolutionary Party, which became the focus of all political activity. In January 1977, a loya jirga approved a new constitution. It wrote in several new articles and amended others - one of these was the creation of a presidential one-party system of government.

He also began to moderate his socialist policies, although the 1977 constitution had a nationalist bend in addition to previous socialism and Islam. In 1978, there was a rift with the PDPA. Internally he attempted to distance himself from the communist elements within the coup. He was concerned about the tenor of many communists in his government and Afghanistan's growing dependency on the Soviet Union. These moves were highly criticized by Moscow, which feared that Afghanistan would soon become closer to the West, especially the United States; the Soviets had always feared that the United States could find a way to influence the government in Kabul.

During his latter years in charge, his purge of communists in his government strained his relations with them, while his desire for one person rule created conflicts with the liberals who had been in charge during the monarchy. At the same time, his persecution of religious conservatives engendered enmity with them and their followers as well.

Relations with Pakistan
As during his time as Prime Minister, Daoud Khan again pressed on the question of Pashtunistan, again leading to sometimes tense relations with Pakistan.

Daoud hosted General Secretary of the National Awami Party Khan Abdul Wali Khan, Ajmal Khattak, Juma Khan Sufi, Baluch guerrillas, and others. Khan's government and forces also commenced training of anti-Pakistani groups to conduct militant action and sabotage in Pakistan. The campaign was significant enough that even one of Bhutto's senior colleagues, minister of interior and head of the provincial branch of Bhutto's party of/in the then-North-West Frontier Province (renamed Khyber Pakhtunkhwa in 2010), Hayat Sherpao, was killed, ostensibly on the orders of the later-acquitted Awami Party. As a result, Afghanistan's already strained relationship with Pakistan further dipped and Pakistan likewise started similar kinds of cross-border interference. By 1975, Pakistani Prime Minister Zulfiqar Ali Bhutto, through its Inter-Services Intelligence (ISI), had begun to engage in promoting a proxy war in Afghanistan.

Since coming to power, under pressure from the PDPA and to increase domestic Pashtun support, Khan took a stronger line on the Pashtunistan issue and promoted a proxy war in Pakistan. Trade and transit agreements with Pakistan were subsequently severely affected.

The year 1975 was a watershed in Afghan-Pakistan relations. Pakistan blamed Afghanistan for unrest in Bajaur agency and the bombing of a PIA B707 at Islamabad airport in 1975. The 130 passengers of PIA B707 had deplaned before the explosion took place and thus no one was harmed in the explosion inside the aircraft.

At the same time, Afghanistan also faced several short lived uprisings in retaliation in eastern Afghanistan and in Panjshir valley, which Afghanistan blamed on Pakistan. There was also deployment of additional troops by both the countries along the Afghanistan-Pakistan border.

The same year Kabul was also quietly negotiating with Islamabad to defuse the tensions between the two countries. In early 1976, relations between the two countries improved and the leaders of the two countries, Zulfiqar Ali Bhutto and Douad Khan, exchanged visits. Daoud Khan was also worried about the growing power of communists within his government so he started improving his relations with Pakistan and Iran. The same year Pakistan also provided aid to Afghanistan to help alleviate the suffering caused by earthquake and floods in northern Afghanistan. This gesture by Pakistan had helped mollify Afghan public opinion about Pakistan.

By October 1976, the head of Pakistan intelligence agency, Jilani was informing a US diplomat that Afghanistan was no longer creating troubles for Pakistan. By August 1976 relations with Pakistan had improved to a high degree. Later on, while promoting his new foreign policy doctrine, Daoud Khan came to a tentative agreement on a solution to the Pashtunistan problem with Ali Bhutto.

Diplomatic relations with the Soviet Union
Khan met Leonid Brezhnev on a state visit to Moscow from 12 to 15 April 1977. He had asked for a private meeting with the Soviet leader to discuss with him the increased pattern of Soviet actions in Afghanistan. In particular, he discussed the intensified Soviet attempt to unite the two factions of the Afghan communist parties, Parcham and Khalq. Brezhnev described Afghanistan's non-alignment as important to the USSR and essential to the promotion of peace in Asia, and warned him about the presence of experts from NATO countries stationed in the northern parts of Afghanistan. Daoud bluntly replied:"'we will never allow you to dictate to us how to run our country and whom to employ in Afghanistan. How and where we employ the foreign experts will remain the exclusive prerogative of the Afghan state. Afghanistan shall remain poor, if necessary, but free in its acts and decisions'"

After returning to Afghanistan, he made plans that his government would downscale its relationship with the Soviet Union, and instead forge closer contacts with the West as well as the oil-rich Saudi Arabia and Iran. Afghanistan signed a co-operative military treaty with Egypt and by 1977, the Afghan military and police force were being trained by Egyptian Armed Forces. This angered the Soviet Union because Egypt took the same route in 1974 in distancing itself from the Soviet Union.

Communist coup and assassination

After the murder of Mir Akbar Khyber, the prominent Parchamite ideologue, his funeral on 19 April 1978 served as a rallying point for the Afghan communists. An estimated 1,000 to 3,000 people gathered to hear speeches by PDPA leaders such as Nur Muhammad Taraki, Hafizullah Amin and Babrak Karmal.

Shocked by this demonstration of communist unity, Khan ordered the arrest of the PDPA leaders, but he acted too slowly. It took him a week to arrest Taraki, Karmal managed to escape to the USSR, and Amin was merely placed under house arrest. Khan had misjudged the situation and believed that Karmal's Parcham faction was the main communist threat. In fact, according to PDPA documents, Amin's Khalq faction had extensively infiltrated the military and they outnumbered Parcham cells by a factor of 2 to 3. Amin sent complete orders for the coup from his home while it was under armed guard, using his family as messengers.

The army had been put on alert on 26 April because of a presumed coup. On 27 April 1978, a coup d'état, beginning with troop movements at the military base at Kabul International Airport, gained ground slowly over the next twenty-four hours as rebels battled units loyal to Daoud Khan in and around the capital.

Khan and most of his family were assassinated during the coup by members of the People's Democratic Party of Afghanistan (PDPA). The coup climaxed in the Arg, the former chief royal palace, during the early hours of 28 April 1978, involving heavy fighting and many deaths. Shortly afterwards, the new military leaders announced that Khan had been killed for refusing to pledge allegiance to the new regime by Lieutenant Imamuddin of the 444th Commando Battalion. Upon Daoud's assassination, Afghan singer Fazal Ghani wrote the song “Khalqi Nizam” which mocked the former president for his baldness and for being “finished with one strike”, which was aired on Radio Television Afghanistan.

Body and state funeral
On 28 June 2008, his body and those of his family were found in two separate mass graves outside the walls of Pul-e-Charkhi prison, District 12 of Kabul city. Initial reports indicate that sixteen corpses were in one grave and twelve in the other. On 4 December 2008, the Afghan Health Ministry announced that Daoud's body had been identified on the basis of teeth molds and a small golden Quran, a present he had received from the King of Saudi Arabia, found near the body.

On 17 March 2009, General Daoud was given a state funeral. His only surviving child, Dorkhanai, attended the funeral. Daoud and following family members that were killed (resembling the Murder of the Romanov family by the Bolsheviks) on the same day on 28 April 1978 are buried at an incomplete hilltop tomb located 2.5 km west of Darul Aman Palace, Kabul:

 * Brother: Sardar Muhammad Naim Khan (b. 1910, age 68)
 * Daughter: Bibi Zarmina Naim Ghazi (unmarked grave)
 * Wife: Bibi Zainab Daoud (b. 1917, age 61)
 * Sister: Bibi Aisha Daoud (b. 1920, age 58)
 * Son-in-law: Sardar Niezamudeen Ghazi (b. 1934, age 44)
 * Daughter: Bibi Safora Abdulaziz Ghazi (b. 1965, age 13)
 * Son: Sardar Muhammad Umar Daoud (b. 1934, age 44)
 * Daughter: Bibi Hila Umar (b. 1961, age 17)
 * Daughter: Bibi Ghizal Umar (b. 1964, age 14)
 * Daughter: Bibi Shinkay Daoud (b. 1940, age 38)
 * Son: Sardar Wais Daoud (b. 1947, age 31)
 * Wife: Bibi Shaima Asefi Daoud (b. 1946, age 32)
 * Mother: Bibi Belqis Asefi (b. 1920, age 58)
 * Son: Sardar Ares Daoud (b. 1973, age 5)
 * Son: Sardar Waygal Wais Daoud (b. 1976, age 2)
 * Son: Sardar Khalid Daoud (b. 1947, age 31)
 * Daughter: Bibi Zarlasht Daoud (b. 1953, age 25)

Public image
News sources in the 1970s claimed that General Daoud Khan said he was happiest when he could "light his American cigarettes with Soviet matches."

Mohammad Daoud Khan was retrospectively described as an "old-fashioned statesman, compassionate yet reserved and authoritarian" by The Guardian's Nushin Arbabzadeh. Then-President Hamid Karzai hailed Khan's courage and patriotism in comments after his 2009 state funeral, saying he was "always thinking of the advancement and prosperity of the country." Some Afghans fondly consider him to be the best leader their country has had in modern times. During his time as prime minister and president, Khan was highly unpopular among the non-Pashtun minorities in Afghanistan because of his alleged Pashtun favouritism. During his regime, all significant positions in the government, army and educational institutions were held by Pashtuns. His attempt at the Pashtunisation of Afghanistan reached such an extent that the word 'Afghan' started being used to refer only to Pashtuns and not to the other minority groups who collectively formed a majority in Afghanistan.

The Afghan Armed Forces were allied with Daoud Khan and supported his goal of promoting Pashtuns to higher posts in the Afghan Armed Forces. In 1963, Afghan Uzbeks were barred from becoming high-ranking officers in the Afghan armed forces. Similarly only a few Tajiks were allowed to hold the position of officer in the Afghan army, while other ethnicities were excluded from those positions.

Daoud Khan viewed the Afghan armed forces as a crucial vector in the Pashtunisation of Afghan state. The Panjshir uprising in 1975 is also believed to be result of anti-Pashtun frustration which had been building up in Panjshir valley as result of Daoud Khan's policies.

Personal life
In September 1934, Daoud Khan married his cousin, Princess Zamina Begum (11 January 1917 – 28 April 1978), sister of King Zahir (15 October 1914 – 23 July 2007). The couple had four sons and four daughters:
 * 1. Zarlasht Daoud Khan (1953 – k. 1978)
 * 2. Khalid Daoud Khan (1947 – k. 1978). Had a son:
 * Tariq Daoud Khan
 * 3. Wais Daoud Khan (1947 – k. 1978). Had four children:
 * Turan Daoud Khan (1972–)
 * Ares Daoud Khan (1973 – k. 1978)
 * Waygal Daoud Khan (1976 – k. 1978)
 * Zahra Khanum (1970–)
 * 4. Muhammad Umar Daoud Khan (1934 – k. 1978). Had two daughters:
 * Hila Khanum (1961 – k. 1978)
 * Ghazala Khanum (1964 – k. 1978)
 * 5. Dorkhanai Begum
 * 6. Shinkay Begum (1940 – k. 1978). Had two daughters:
 * Ariane Heila Khanum Ghazi (1961–)
 * Hawa Khanum Ghazi (1963–)
 * 7. Torpekay Begum. Had three children:
 * Shah Mahmud Khan Ghazi
 * Daud Khan Ghazi
 * Zahra Khanum Ghazi

title: Mohammad Daoud Khan


text:
AutoLISP

AutoLISP is a dialect of the programming language Lisp built specifically for use with the full version of AutoCAD and its derivatives, which include AutoCAD Civil 3D, AutoCAD Map 3D, AutoCAD Architecture and AutoCAD Mechanical. Neither the application programming interface (API) nor the interpreter to execute AutoLISP code is included in the AutoCAD LT product line (up to Release 2023, AutoCAD LT 2024 includes AutoLISP). A subset of AutoLISP functions is included in the browser-based AutoCAD web app.

Features
AutoLISP is a small, dynamically scoped, dynamically typed Lisp language dialect with garbage collection, immutable list structure, and settable symbols, lacking in such regular Lisp features as macro system, records definition facilities, arrays, functions with variable number of arguments or let bindings. Aside from the core language, most of the primitive functions are for geometry, accessing AutoCAD's internal DWG database, or manipulation of graphical entities in AutoCAD. The properties of these graphical entities are revealed to AutoLISP as association lists in which values are paired with AutoCAD group codes that indicate properties such as definitional points, radii, colors, layers, linetypes, etc. AutoCAD loads AutoLISP code from .LSP files.

AutoLISP code can interact with the user through AutoCAD's graphical editor by use of primitive functions that allow the user to pick points, choose objects on screen, and input numbers and other data. AutoLisp also has a built-in graphical user interface (GUI) mini- or domain-specific language (DSL), the Dialog Control Language, for creating modal dialog boxes with automated layout, within AutoCAD.

History
AutoLISP was derived from an early version of XLISP, which was created by David Betz. The language was introduced in AutoCAD Version 2.18 in January 1986, and continued to be enhanced in successive releases up to release 13 in February 1995. After that, its development was neglected by Autodesk in favor of more fashionable development environments like Visual Basic for Applications (VBA), .NET Framework, and ObjectARX. However, it has remained AutoCAD's main user customizing language.

Vital-LISP, a considerably enhanced version of AutoLISP including an integrated development environment (IDE), debugger, compiler, and ActiveX support, was developed and sold by third-party developer Basis Software. Vital LISP was a superset of the existing AutoLISP language that added VBA-like access to the AutoCAD object model, reactors (event handling for AutoCAD objects), general ActiveX support, and some other general Lisp functions. Autodesk purchased this, renamed it Visual LISP, and briefly sold it as an add-on to AutoCAD release 14 released in May 1997. It was incorporated into AutoCAD 2000 released in March 1999, as a replacement for AutoLISP. Since then, Autodesk has ceased major enhancements to Visual LISP and focused more effort on VBA and .NET, and C++. , Autodesk ended support for VBA versions before 7.1, as part of a long-term process of changing from VBA to .NET for user customizing.

AutoLISP has such a strong following that other computer-aided design (CAD) application vendors add it to their products. Bricscad, IntelliCAD, DraftSight and others have AutoLISP functionality, so that AutoLISP users can consider using them as an alternative to AutoCAD. Most development involving AutoLISP since AutoCAD 2000 is performed within Visual LISP since the original AutoLISP engine was replaced with the Visual LISP engine. There are thousands of utilities and applications that have been developed using AutoLISP or Visual LISP (distributed as LSP, FAS and VLX files).

Examples
A simple Hello world program in AutoLISP would be: Note the final line inside the function definition: when evaluated with no arguments, the  function returns a null symbol, which is not displayed by the AutoCAD command-line interface. As the AutoCAD command line functions as a read–eval–print loop (REPL), this would normally print "Hello World!" to the command line, followed immediately by the return value of the call to. Therefore, without the final call to the  function, the result of this would be:

The  function may also be used to achieve the same result.

A more complex example is: The above code defines a new function which generates an AutoCAD point object at a given point, with a one-line text object displaying the X and Y coordinates beside it. The name of the function includes a special prefix 'c:', which causes AutoCAD to recognize the function as a regular command. The user, upon typing 'pointlabel' at the AutoCAD command line, would be prompted to pick a point, either by typing the X and Y coordinates, or clicking a location in the drawing. The function would then place a marker at that point, and create a one-line text object next to it, containing the X and Y coordinates of the point expressed relative to the active User Coordinate System (UCS). The function requires no parameters, and contains one local variable ('pnt').

The above example could also be written using built-in AutoCAD commands to achieve the same result, however this approach is susceptible to changes to the command prompts between AutoCAD releases.

title: AutoLISP


text:
South Platte River

The South Platte River is one of the two principal tributaries of the Platte River. Flowing through the U.S. states of Colorado and Nebraska, it is itself a major river of the American Midwest and the American Southwest/Mountain West. Its drainage basin includes much of the eastern flank of the Rocky Mountains in Colorado, much of the populated region known as the Colorado Front Range and Eastern Plains, and a portion of southeastern Wyoming in the vicinity of the city of Cheyenne. It joins the North Platte River in western Nebraska to form the Platte, which then flows across Nebraska to the Missouri. The river serves as the principal source of water for eastern Colorado. In its valley along the foothills in Colorado, it has permitted agriculture in an area of the Colorado Piedmont and Great Plains that is otherwise arid.

Description
The river is formed in Park County, Colorado, southwest of Denver in the South Park grassland basin by the confluence of the South Fork and Middle Fork, about 15 mi southeast of Fairplay. Both forks rise along the eastern flank of the Mosquito Range, on the western side of South Park, which is drained by the tributaries at the headwaters of the river. From South Park, it passes through 50 mi of the Platte Canyon and its lower section, Waterton Canyon. Here, it is joined by the North Fork before emerging from the foothills southwest of the Denver suburb of Littleton. At Littleton, the river is impounded to form Chatfield Reservoir, a flood-control basin for the Denver metropolitan area.

The river flows north through central Denver, which was founded along its banks at its confluence with Cherry Creek. The valley through Denver is highly industrialized, serving generally as the route for both the railroad lines, as well as Interstate 25. On the north side of Denver, it is joined somewhat inconspicuously by Clear Creek, which descends from the mountains to the west in a canyon that was the cradle of the Pike's Peak Gold Rush. North of Denver, it flows through the agricultural heartland of the Piedmont (a shale region that was formed through erosion by the ancestor of the river following the creation of the Rockies). It flows directly past the communities of Brighton and Fort Lupton, and is joined in succession by Saint Vrain Creek, the Little Thompson River, the Big Thompson River, and the Cache la Poudre River, which it receives just east of Greeley.

East of Greeley, it turns eastward, flowing across the Colorado Eastern Plains, past Fort Morgan and Brush, where it turns northeastward. It continues past Sterling, and runs into Nebraska between Julesburg, Colorado, and Big Springs, Nebraska. In Nebraska, it passes south of Ogallala and joins the North Platte River near the city of North Platte.

The South Platte River through Denver is on the U.S. Environmental Protection Agency's (US EPA) list of impaired waterbodies for pathogen impairment, with E. coli as the representative pathogen species. Other water issues involve the appearance of the New Zealand mud snail and zebra mussel.

The flows of the South Platte have been greatly modified by human activity. Originally the river was seasonal north of Fort Morgan to its confluence with the North Platte. In the 1800s the river would disappear in July, August, and September due to low flows and a sandy river bottom. By 1910 the increase in agriculture caused the river to reach the border of Colorado and Wyoming due to return flows and dams holding back water later in the year. Increasing diversions from the upper Colorado River in the 20th century mean that flows reach the North Platte year round.

History
The South Platte was originally called Niinéniiniicíihéhe by the native Arapaho people who lived on its banks. The early Spanish explorers called it the Rio Chato (calm river). In 1702, it was named the Rio Jesus Maria by Captain Jose Naranjo, the Indigenous-Spanish scout and captain of war of the New Mexico Indian Auxiliaries, who was ordered by the viceroy of New Spain to search the Tierra Incognita for a French incursion into New Mexico. The South Platte also served as a vital water source in Colorado. Long before the city of Denver was created, many travelers came to the South Platte River to escape the arid Great Plains. These people could survive the heat, but not without the vital water source that the South Platte gave them. Buckets and wells sufficed as a water system for a while, but eventually, the Denver Water System was created.

Dams
In an arid region of the United States, the South Platte is marked with several dams. The first notable water impoundment on the South Platte is Antero Reservoir. "Antero" is derived from the Spanish word "delantero", which means "foremost" or "head", as the reservoir was the first dam on the South Platte River near the river's origin.

The next dam is Spinney Mountain Reservoir. At capacity, Spinney Mountain covers 2500 acre. A bottom-release dam, Spinney releases to the east of the inlet.

Two miles below Spinney Mountain Reservoir, the river enters Eleven Mile Reservoir, with a capacity of 97000 acre.ft. The Eleven Mile Reservoir Dam drains into Eleven Mile Canyon, which runs through Forest Service land. Three former Colorado towns, Howbert, Idlewild, and Freshwater Station, were submerged under the reservoir when it was built.

From Eleven Mile Canyon, the South Platte runs northeast to Cheesman Reservoir, named for Denver water pioneer Walter S. Cheesman. At its completion in 1905, the dam was the world's tallest gravity dam, at 221 ft above the streambed. The reservoir and related facilities were purchased in November 1918 by the Denver Water Board. Cheesman was the first reservoir of Denver's mountain storage facilities, and has been designated a National Historic Civil Engineering Landmark. Cheesman Reservoir feeds Cheesman Canyon. Six miles below Cheesman Reservoir is the town of Deckers; there, the river bends north for about 17 mi to the confluence with the North Fork of the South Platte.

In the late 1980s, a proposal was put forth for the Two Forks Dam, which would have created a reservoir flooding the entire section from the North Fork confluence to the town of Deckers. In 1990, the US EPA vetoed the permit, calling the project an "environmental catastrophe."

From the confluence, the river flows towards Denver and enters Strontia Springs Reservoir.

Below Strontia Springs, the South Platte runs through Waterton Canyon before entering Chatfield Reservoir. Chatfield marks the seventh and final dam on the South Platte until it merges with the North Platte.

Fly fishing overview
The South Platte River is a gold medal western trout river on the eastern slope of Colorado, meaning it produces at least 12 trout of greater than 14 inches in length or greater per acre in areas accessible to the public. The river is well known for its wild trophy population of brown and rainbow trout. As a result of its close proximity to Denver, the river has thousands of fly fishing enthusiasts visit each year. With seven dams on the river, the South Platte is considered a tailwater fishery. Most of these dams are bottom released, which allows for both stable water temperatures and year-round fly fishing. Popular fly-fishing stretches of the river include Waterton Canyon, Cheesman Canyon, and the Dream Stream.

title: South Platte River


text:
User:Andrew Szanton

Andrew Szanton is a memoir collaborator who has assisted with their memoirs Nobel Laureate Eugene Wigner; NAACP Man of the Year Charles Evers; U.S. Senator Edward Brooke and founding director George Pake of Xerox PARC.

After taking a degree in American History at Princeton University in 1985, he began work with the Smithsonian on an oral history program interviewing surviving participants in the Manhattan Project -- the building of the world's first atomic bomb.

This oral history project led to his first memoir collaboration, with the Nobel laureate in physics, Eugene Wigner, of Princeton, New Jersey, an important member of the Manhattan Project.

He followed that memoir with a memoir collaboration with the civil rights leader Charles Evers, of Mississippi. That memoir is called "Have No Fear" and describes how Evers ran the NAACP in Mississippi, and managed to get elected mayor of the formerly white-run town of Fayette, Mississippi.

Szanton has written the Humphrey Bogart profile for Wikipedia.

He helped George Pake, of Palo Alto and Tucson, with a memoir describing his years as founding director of the seminal research firm Xerox PARC.

Other memoir clients he has assisted include: John Whitehead, former co-chairman of Goldman Sachs investment bank in New York City; Peter Raven, celebrated director of the Missouri Botanical Garden; Charles Epps, a distinguished surgeon in Washington, D.C.; and Marvin Gilmore, a prominent developer, banker and civil rights activist in Boston.

Andrew Szanton is also a veteran freelance editor and writing instructor.

In memoir collaboration, he works by assembling a narrative, usually chronological, and then by focusing on three key issues: character, setting and motive.

title: User:Andrew Szanton