Talk:Set theory/Draft

Nice work! Boris Tsirelson 18:22, 16 September 2009 (UTC)


 * Thanks! I only joined CZ because I thought the world needed a decent introductory account of set theory :-) Mark Wainwright 04:22, 9 May 2010 (UTC)


 * Are you sure the world needs nothing else? :-) Boris Tsirelson 07:57, 9 May 2010 (UTC)


 * It may well do, but perhaps nothing that I am so easily able to supply. Mark Wainwright 10:44, 20 May 2010 (UTC)

Probably this article is approvable. For now I am not ready to approve it, because some finer points are beyond my competence. I understand the given text about that, but I have no other sources to be sure. Namely: Boris Tsirelson 08:19, 18 May 2010 (UTC)
 * "An ingenious axiom of Goedel's, Limitation of Size";
 * "Montague proved in 1961 that ZF cannot be finitely axiomatised";
 * "NF ... is finitely axiomatisable";
 * "NFU, whose consistency is implied by that of simple type theory".


 * That would be great. A web search suggests I was completely wrong about Limitation of Size and it was von Neumann. Also if http://planetmath.org/encyclopedia/Class.html is correct I shouldn't have claimed it as subsuming Powerset. I've amended the article accordingly. Thanks for the glitches below, which I see someone has fixed now. I can't provide references but I'll e-mail you if I can work out how. Mark Wainwright 10:44, 20 May 2010 (UTC)

Also I observe some minor errors: Boris Tsirelson 08:34, 18 May 2010 (UTC)
 * "difference X-Y or X\Y contains of all those" — either "consists of all" or "contains all", I guess;
 * "X∪Y ={x|P and Q}, X∩Y={x|P or Q}" — swap them.

--

Being advised by Mark (the author) I have found a source for "NF ... is finitely axiomatisable" here: "Stratified comprehension is an axiom scheme, which can be replaced with finitely many of its instances (a result of Hailperin). Using the finite axiomatization removes the necessity of referring to types at all in the definition of this theory."
 * Hailperin, T. [1944] A set of axioms for logic. Journal of Symbolic Logic 9, pp. 1-19.

Boris Tsirelson 18:16, 20 May 2010 (UTC)

I also see on the same page (by Holmes) the phrase "NFU: New Foundations with urelements. This system is consistent, ..." which I fail to understand (surely because I do not work in logic). Consistent relative to what?? How is it related to the phrase "NFU, whose consistency is implied by that of simple type theory"? Boris Tsirelson 18:33, 20 May 2010 (UTC)

Yes, ZF cannot be finitely axiomatised:
 * Montague. Semantic closure and non-finite axiomatizability. In Infinitistic Methods, pages 45–69. Pergamon, 1961.
 * K. Kunen. Set Theory: An Introduction to Independence Proofs. North-Holland, 1980.

Boris Tsirelson 18:46, 20 May 2010 (UTC)

Yes, consistency of NFU is implied by that of simple type theory:
 * R. Jensen. On the consistency of a slight(?) modification of Quine's New Foundations. Synthese, vol. 19 (1968/69), pp. 250/263.
 * M. Boffa, The consistency problem for NF. The Journal of Symbolic Logic, vol. 42, no. 2, 1977, pp. 215–220.

Boris Tsirelson 17:40, 22 May 2010 (UTC)

Thus, now I am ready to approve it. Boris Tsirelson 17:45, 22 May 2010 (UTC)

Power set and Cantor
I have replaced the statement:
 * "This was the basis of his demonstration that different sizes of infinity must exist."

It is historically incorrect. Cantor's first proof (1973) that the real numbers are uncountable did not use the power set idea. --Peter Schmitt 13:37, 25 May 2010 (UTC)
 * I see. (But "1973" means "1873", I guess.) Boris Tsirelson 13:52, 25 May 2010 (UTC)


 * Oops, of course.
 * G. Cantor: Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen. (23.December 1873)
 * Journal für die reine und angewandte Mathematik. Band 77 (1874) 258-262 (pdf)
 * --Peter Schmitt 15:20, 25 May 2010 (UTC)


 * Thanks, and likewise for being much more scrupulous than I was about links and wiki formatting. I notice you've also made some minor stylistic changes, which I don't particularly prefer but don't propose to argue about except one which I have changed back faute de mieux: I think 'The difference X-Y' was more correct than 'The difference, written X-Y', since in the latter the noun phrase 'The difference' does not mention which difference. One could I suppose write 'The difference of X and Y, written ...' but the given form is conciser (and grammatically parallel to 'my cousin John'). Mark Wainwright 15:28, 27 May 2010 (UTC)


 * Then let us use the same style for all three operations. It less "concise", but it makes a difference between the operation and the notation. This is, I think, particularly important when there are two notations given. --Peter Schmitt 17:05, 27 May 2010 (UTC)

Notation
The bar may be the commonest notation, as stated in the article (though I don't know how you'd tell), but shouldn't the colon notation be mentioned too, as it's pretty common? And French mathematicians seem to use semicolons. Peter Jackson 14:48, 25 May 2010 (UTC)


 * Sure, but this page is a summary of set theory (with a personal touch) and to discuss all variations of notation does not fit into it. This can be done elsewhere, where more details are discussed. (There are variants for complement, as well, and for negation.) (Moreover, the variants are intuitive and easy to understand.) --Peter Schmitt 15:05, 25 May 2010 (UTC)


 * I took the same view as Peter; the aim is to present the ideas and just enough notation for the reader to understand the rest of the article. As to common, you're right, I didn't do a survey, I just stuck my finger in the air and thought I'd seen more bars. (I have no preference; indeed the history shows I originally put a colon in one place and corrected it for consistency later.) To avoid questions about the 'commonest' I've reworded to 'A common notation uses ...' which as a bonus is slightly conciser too. Mark Wainwright 15:28, 27 May 2010 (UTC)

Approval date
As far as I'm concerned, if an Editor on the 22nd, say, sets an Approval date for, say, the 25th, and then someone comes along in the meantime and makes a number of changes, as Peter has done here, the Approval date should then be delayed *another* three days, not just having a change make in the date of *when* the Editor approved things. The whole point of a delay between a proposed approval and the date of the approval is to give other people time to look at things. Or such is my understanding of things. If you disagree with this, please feel free to contact Matt Innes or Joe Quick about it. Thanks. Hayford Peirce 17:09, 26 May 2010 (UTC)


 * Let me first say that a few days earlier or later do not matter. So it is not a problem. And, of course, it is the purpose of the period before approval to allow some input and corrections and improvements. But I cannont follow your logic, Hayford: If, during this period, some changes are made that are considered an improvement and lead to an update of the version -- why should this delay approval? It is the result of the review period. (A delay may be useful if substantial changes happen. But in the case of only minor edits?) --Peter Schmitt 19:30, 26 May 2010 (UTC)


 * Hopefully, some day we'll have a charter that defines all that unambiguously, no one will violate it, and no one will demand more than that. Boris Tsirelson 19:46, 26 May 2010 (UTC)


 * This article is looking very good. I do suggest adding at least one reference before approval, if nothing more than an in-depth book that would cover the topic in greater detail.  Fine job!Pat Palmer 21:44, 26 May 2010 (UTC)


 * I have Hausdorff's book purely on set theory, and a couple of graduate-level discrete mathematics texts with considerable set theory. How much citing is needed, as opposed to putting them in the bibliography? (The Lange review book is actually decent). --Howard C. Berkowitz 21:55, 26 May 2010 (UTC)


 * My answer to Peter is simply the comment by Pat. Any time changes are made, enough to have the original nominating Editor change the date of his Nomination, then, at least in MY opinion, more time ought to be given for the other Citizens to vet the changes. Doesn't that make sense?  In times past I *think* that when we were waiting for an Approval to be carried out, a *couple* of times an Editor asked a *Constable* to go into the text and correct a typo or misspelling. Am I wrong about this?  Could be.  But my own feeling remains that before an article is Approved, every Citizen should have at least 3 days to review the changes. And, of course, I *do* agree with Boris, that this is something that eventually should be very, very clearly spelled out somewhere. Hayford Peirce 22:03, 26 May 2010 (UTC)


 * Pat's comment has nothing to do with the version update. I still believe that there is no reason that mere copy edits (corrections of typos, formatting, etc.) should delay the approval date. (The current CZ:Approval Process does not even mention a date change -- only the updating of "now", moreover, the minimum approval period is 24 hours. This may be objectionable, of course, but it is current.) --Peter Schmitt 22:35, 26 May 2010 (UTC)


 * I was using Pat's comment as a *general* example of what I mean when I say that I think there ought to be a little more time allowed in these cases. He wrote, "I do suggest adding at least one reference before approval" -- that's the sort of update I think every Citizen should have to the time to furnish.  I frankly can't see why Boris is in such an absolute rush to get these articles approved within three days of his first nominating them without giving time to other people to look at them.  Many of these articles have been around for *years* -- is another two or three days waiting for Approval going to sink the Citizendium project? Hayford Peirce 15:42, 27 May 2010 (UTC)


 * I see that I missed the presence of references on the Bibliography page. The 3 that were there were all mooshed up and I just didn't see them what with all the "nominated for approval" and other notices.  So that is good.  Incidentally, guys, I am a she.  Just to be honest, not only do I feel there should be no hurry to approve any article, I don't even like the entire idea of approval, because I feel that people should always be free to improve any article and also, the knowledge available about many topics continues to evolve.  So I wish there were not any such thing as Approval, as I think it tends to make things competitive--once someone is seeking approval, they tend to push for it, which is their right the way the rules now work.  So I am not going to block any specific article based on my dislike of approvals, but I don't feel that hurrying is necessary either.  The article should be made as good as possible if it is has to get frozen in place.Pat Palmer 04:29, 29 May 2010 (UTC)


 * Sorry, Pat, about the he/she thing -- I've also wondered about this for a long time about Sandy Harris. But regarding your comments about the Approval stuff, then you are basically against the whole point of CZ -- that we *do* have Approved articles that *cannot* be touched. Sure, work on the draft version, and eventually it will become Approved. But if we don't have blocked Approved articles, then what's our difference from WP -- except politeness, which, of course, *does* count for a lot? Hayford Peirce 05:00, 29 May 2010 (UTC)


 * Pat, I had to revert your edits to the Bibliography: It is only one book (with two editions cited).
 * As for Approvals: They are a defining feature of CZ. Their purpose is to "guarantee" that an article meets some quality requirements, is reliable and stable, and thus citable, not to stop improvements (or complete rewrites). I did not much care about approval because of several problems. Making even minor corrections was/is very difficult, and editing an acceptable article before nominating it for approval is very restricted (unless there are at least two active Editors who cooperate). Under these conditions I preferred to repair problems I happened to notice. But when Boris took the initiative things became easier, and I am happy to cooperate. (I do not mind a delay of a few days, though in some cases I wonder about the reasons given.) --Peter Schmitt 09:29, 29 May 2010 (UTC)

Could do with some references
I'm not sure where I'd start, but it would be useful if there were some references. Not because any of what is said doesn't pass the smell test (I'm not a mathematician though...) but actually so people can follow up and read further. As a non-editor, it's not really my decision, but I would support approval iff some references are added. –Tom Morris 14:36, 29 May 2010 (UTC)


 * Ooops, thank you, Tom. why did I not notice that the headings were wrong? Probably, because I was not aware that =...= works at all...
 * As I said above: Some good sources should (and will) be added to the bibliography, but there is no need for references(=footnotes) in the article.
 * I do not like the links you added. They have nothing to do with set theory. Not even the link to "number" is needed here.
 * I know about "naive" (and noticed it) -- but isn't "naïve" thought to be manieristic by some (most?) and therefore only a matter of taste?
 * I am not sure that your remarks on the empty set fits in here. This is a brief outline of the mathematical development only. There are more philosophical implications and questions related to set theory, but they are also not inlcuded here. Would deserve its own article, of course.
 * --Peter Schmitt 15:16, 29 May 2010 (UTC)


 * Sorry, it was my fault about the headings; I hadn't realised they start at the second level for some reason.
 * I agree with Peter about the links and philosophical material - I don't think they have anything to do with set theory, so interesting though they may be, they should not be here. Readers are quite capable of looking up terms that interest them in the encyclopedia; a link should be a cue that the linked page will shed more light on the page currently being read, i.e. set theory. It's clear that 'colour', for instance, won't do that. I think there is an explicit CZ policy on this.
 * Why have you rewritten the 'notation' section to use two technical terms, 'comprehension' and 'extension', instead of a plain English account? The term 'comprehension' is not concealed from the reader: it is defined later, in the section of the same name, by reference to the earlier example. This is an article that belongs relatively near the top of a notional 'mathematics' tree, and I designed the article on the principle that any technical terms used could be defined clearly first.
 * I'm also not sure that 'unordered' adds much. One could equally say that sets are unpointed, unequipped with a topology, etc. But this isn't an article about those subjects. Mark Wainwright 16:43, 29 May 2010 (UTC)


 * "When an article is nominated for approval ... important role for non-experts is to make recommendations and criticisms on the talk page." (CZ:Approval Process) I do not understand why Tom did otherwise. Definitely I'll not approve his changes. Changes by Hayford could be approved, but unfortunately they are made after Tom's changes. This is exactly an example of "fuss and bustle" that I prefer to avoid. Boris Tsirelson 16:49, 29 May 2010 (UTC)


 * Might I suggest that a good deal of this discussion could go into a draft CZ: Mathematics Workgroup Style Guide, as Russell started for History?


 * Hayford, at times, does come across as a Monty Python variant crying "burn the which"! ;-) --Howard C. Berkowitz 18:12, 29 May 2010 (UTC)


 * Naw, living in Tucson, as I do, I'm just the Wicked Which of the West. Hayford Peirce 18:32, 29 May 2010 (UTC)


 * Okay, why did I edit a version that is going up for approval in contravention of the oh-so-sacred approval process policy? Firstly, because as a member of the great unwashed Author underclass, I am totally unfamiliar with the intricacies of the approval process. I've written hundreds of blimmin' articles, but thanks to the absence of philosophy editors, I can't get no approval. Which is kind of dispiriting in the same way that shouting into the abyss is.


 * My justification for my mortal sin of editing an article? I was only following the overriding commandment of Citizendium, namely Be Bold! I don't apologise for that. I don't apologise for my edit sucking either. It was made in good faith, and if you consider it "fuss and bustle", then you have the option to change it or revert it. I do apologise for editing a wiki. Actually, I don't. I've edited other articles that have been put forward to approval, namely cryptography. If the only way non-editors can edit articles in the pre-approval stage is by proxy on the talk page, then that is a really broken policy and needs to change, "fuss and bustle" be damned!


 * The approval process ought not stop any good faith contributions. It is there to be a useful way for readers to have stable articles. But if the article hasn't been approved yet, stability is not expected. When an article is put forward for approval, the author of the article ought to expect that there will be an increase in the number of edits from us non-expert authors. The whole reason we put articles awaiting approval on the Recent Changes page now seems to be so that people can take a look at them, and see if we can make them better. Otherwise, why advertise them on Recent Changes if you don't want people to actually edit the article? Someone sound the alarm, I have detected potential absurdity! If that is really the policy, then someone hasn't quite understood the point of it being a wiki.


 * Having to edit via the talk page seems totally silly - why hamper the process by requiring people to do the wiki equivalent of back-seat driving? And the editorial process should be able to filter good and bad edits. I want to help produce good articles, and I certainly will monitor articles that are awaiting approval and help improve them - policy be damned! If the policy is getting in the way of good-faith editorial changes, the policy is wrong.


 * I do try and structure my edits to keep editorial and structural changes separate so that the edits can be applied selectively and atomically - I got a little bit carried away this afternoon though, and I apologise for that. (It'd be nice if we could do actually have atomic changes - if we're gonna do this approval thing, then it seems a fairly essential thing to be able to selectively pull some changes and not others into the stable version. As Joel Spolsky says "Subversion = Leeches. Mercurial and Git = Antibiotics." - MediaWiki is definitely in the leeches/Subversion category when it comes to branching, which is essentially what the approval process is.)


 * I strongly disagree about the links - I included them because part of the point of a wiki is that it has hyperlinks between entries. Hypertext gives you serendipity and requires only a little bit of added colour in the text! The links don't have to be relevant to set theory to be included - if someone is reading a traditional encyclopedia or other reference work, it is a pretty standard stylistic feature to mark out works where another article is available - in a comprehensive encyclopedia and compendium of information (as Citizendium aims to be), this may mean a lot of links. Links unequivocally add value in my opinion. If there is an explicit policy on not linking, then we need to have a debate about that, because that's a pretty obviously broken policy! A wiki without links is a very sad beast indeed! Why not have links? Because the text goes blue - or red? That's not a good reason at all. Because people might leave the article? I just don't understand why one ought not have links. It's a wiki. Standard operating procedure.


 * I attempted to rewrite the notation section because I felt it was slightly unclear: "a set can also be written by naming a property common to its elements". I was rather clumsily trying to rewrite it because, well, you aren't "writing" a set, you are building a set or maybe describing a set (depends on your metaphysics, I guess). And you aren't "naming" a common property, you are describing a property or selecting or specifying a property which constructs the set. I'm not sure what the best way to describe this in simple terms is. Adding more jargon doesn't help though - but I do think the description of set-building by comprehension needs a bit of a brush up.


 * I do think the article ought to be more than just set theory for beginners. Consider the Stanford Encyclopedia of Philosophy article on set theory - it contains both a main article and a primer. Set theory has an important role in the problem of universals and that should be described in this article. Perhaps instead of my poor attempt to weave it into the 'primer' section, we ought to have a section at the end - something like "applications of set theory" or "role of set theory" or something - can't quite put my finger on it. It wouldn't just be stuff in philosophical logic and metaphysics, but also parallels in computer science like list comprehension (or, as Scala calls it, sequence comprehension) which uses set-builder notation-derived syntax in Python, Haskell, Scala, Lisp, Erlang, OCaml, JS 1.8 and plenty of other languages to construct lists or sequences.


 * If we don't have this kind of material - and references, as I'll come on to - I think that approval may be premature. It's an encylopedia, not just a primer.


 * As for references, I do think that the article ought to cite sources as proper footnotes as well as on the Bibliography subpage. Firstly, because most other articles on CZ do so - and that includes articles that have more of a primer style. Secondly, because we have them for a reason. Consider: if someone is reading the section on, say, Russell and type theory, it tells them that Russell and Whitehead attempted to resolve difficulties by introducing the theory of types. Where did he do that? In Principia Mathematica, of course. Okay, I should be able to click on the footnote and see that. Now, without footnotes, I need to go to the Bibliography subpage, look for Russell (or Whitehead?) and then there might be five publications. What about if I want secondary literature? No, there needs to be a direct link between the text and the citations. And there needs to be citations. It is an essential tool for readers, especially school or university level readers who may be using this as their first introduction to set theory - and not the mathematicians: the philosophy students, the computer science students and so on. –Tom Morris 22:00, 29 May 2010 (UTC)


 * Hi, Tom, I've certainly disagreed with you many times in the past over various (totally forgotten) matters, but I gotta say that I basically agree with everything you've written above. As you say, you shouldn't have to apologize for writing something on a wiki.  That's what it's there for! Hayford Peirce 23:11, 29 May 2010 (UTC)


 * Tom, though I do not like all of your changes I never meant to imply that you should not have made them. No need to apologise. There is now a slight complication: How to include only some of your edits? Boris was impatient because he wanted the page to be approved for some days now and did not like the delay.
 * As for the linke: Our views differ -- links are means to guide the reader, not to colour some words. Encyclopedias do not put arrows before every word for that they have an entry. (And this is contained in -- current -- guidelines, marking a difference to WP.)
 * As for the references: Practice is different in science and philosophy. Footnotes are used very rarely. The bibliography may be annotated, and the Principia mathematica are not discussed here.
 * Of course, this article is not a sufficient treatment of set theory. Hopefully, much more will be written in fhe future. But it is correct, and gives a (brief) survey for those wanting basic information.
 * --Peter Schmitt 23:42, 29 May 2010 (UTC)


 * Yes, I wasn't clear: I certainly think there should be all kinds of other stuff in an ideal article, which may well include a section on philosophical applications, which may well include Tom's remarks on the empty set or something like them. (I don't know, it's not my area. I'd have guessed that discussion would have called for mentioning the intensional/extensional distinction.) But if they're to be included, my preference would be to create that section, even if it's embryonic and needs to be filled out, rather than interrupt the flow of an expository section.


 * I really did mean that a set could be *written*, not merely described: this was a section on notation and in mathematics one does talk of 'writing' an object, meaning having a notation to denote it.


 * The policy on links is in CZ:Article_Mechanics, linked from every page. "If our target audience would find that the linked article illuminates the present article, then we should link to it." I understand you don't agree with this policy but in my view (and seemingly that of others), being selective about links in this way is one of the things that can *add* value to CZ, and corresponds with practice in other good reference works. Readers are not stupid - they know they can look up any term they like. What they don't know unless they're clairvoyant is which articles have some bearing on the subject they're reading about - which it is reasonable to assume they have, for the moment, a particular interest in learning more about.


 * I've retired to the talk page in case I offend anyone's sensibilities by marching in and reverting changes; I'm new round here and don't know the etiquette. But I think I've made my views clear. Mark Wainwright 03:01, 30 May 2010 (UTC)

No more changes for this approval
Dear all, you are welcome to improve this article further, of course; but not toward this approval. I shall not update the approved version anymore, anyway. Very probably, my approval will be withdrawn. In that case, Mark and Peter, please accept my apologies. I did my best, as I understand it. Boris Tsirelson 19:36, 29 May 2010 (UTC)


 * Please be aware that the following is my own personal interpretation of the Approval Process. Others may disagree, and I may be wrong. However, I believe that withdrawing your support for an article is not necessary. Quoting from the Approval Process (bold highlighting is my own):
 * If the article undergoes revision during the days following the initial filling of the "ToApprove" metadata (and this is common), and the nominating editor is still willing to approve the article in its revised state, he or she should update the Article URL and the Now field in the metadata section to point at the most recent satisfactory version of the article found under the "history" tab. At any given moment it might or might not be the most recent version.
 * My understanding of this is that if, as the approving Editor, you do not wish to approve any changes that have occured since your original approval then this is your right. The version of the article that should be approved is the version specified in the metadata. Therefore, you should update the metadata to match the most recent version that you are happy to approve. Any changes past this point will be included in the new Draft version, but NOT in the approved version. Of course, if another Editor from a workgroup relating to the article objects to the approval they may cancel the nomination altogether. From a Constables perspective, his duty is to (again a quote)
 * Ensure that at least one editor currently supports the approval of the article version mentioned in the approval metadata. If there is any reason to doubt editor support, request clarification before doing the honors. Reasons to doubt editor support include the nominated version not being the latest one and/or the existence of unanswered author critiques.
 * As long as you, as an Editor, clarify to a Constable that you support approval of the specific version (and not the updated version) then there should be no problems having it approved. Of course, if a specific Constable is unhappy for any reason he or she may choose to not approve it. There is nothing to stop you approaching another Constable in this case.
 * Anyway, again I'd like to point out that this is my personal interpretation. I am not a Constable and I did not help write the policy (nor was I here when it was written). I may be wrong. --Chris Key 20:50, 29 May 2010 (UTC)


 * You've got everything correct, Chris, and, now that I see that you've made your very practical suggestion in the Forums, you've apparently taken care of the major issue that I've had recently with Approvals. Hayford Peirce 21:56, 29 May 2010 (UTC)


 * I have made a version on which we (hopefully) can all agree on (and updated Metadata), and the put back Tom's paragraph on the empty set. --Peter Schmitt 00:09, 30 May 2010 (UTC)


 * Indeed. Thank you. Boris Tsirelson 06:57, 30 May 2010 (UTC)

Toward Approval
I'm glad to see robust conversation concerning the best way to improve an article as it reaches approved status. I am in no way concerned with obstructing the process of making improvements to this article or to the approval process. However, any discussions about the approval process should be elsewhere.

From what I can tell, it appears that we have a single editor approval on a date prior to the last date edited as the editor does not agree with edits made after that date. Before performing the mechanics of approval, a constable is asked to look at the talk page and make sure that the approving editor has at least addressed concerns on the talk page. I do see that there are some concerns, but I also see that Boris has at least read the changes and still prefers the version that he placed in the metadata. It is up to the author to satisfy the editor's concerns. No-one can force him to endorse a version that he disagrees with. Peter has made some changes and added his name to the metadata, but this would require that a third editor sign on (assuming that Boris remains on) because Peter has contributed significant content to the article.

Citizendium gives editors the option to lock a version and make decisions about content. My interpretation of the current status is that the version that Boris has approved be locked as version 1.0 and any subsequent changes can be re-approved using the same provedures as the first approval - either single or multiple editor.

I also agree with Tom that he is not only able, but encouraged and welcome to make whatever changes he likes to the article, unless specifically asked by an editor not to touch a specific area of content. I haven't seen that is the case.

So, unless I hear otherwise, I'll stop by tomorrow to see if there is anything that would stop Hayford or I from performing the approval and locking the version that Boris last approved. D. Matt Innis 02:30, 30 May 2010 (UTC)


 * Many thanks, Matt, you are our last and only resort.
 * "...unless specifically asked by an editor not to touch..." — I did not know it is my duty to ask this; from now on I'll bother to do so. However, I believe, it should be written in our policy. (Or is it written already?)


 * It is documented. See here where it says "When an editor has made a certain edit, and has specifically requested that some limited portion of text should not be changed (or that it must not be changed in certain limited ways), then authors should respect the request. (Editors may not request that articles be simply left alone.)". --Chris Key 09:05, 30 May 2010 (UTC)


 * Just now the article starts with a wrong statement: "Boris Tsirelson has nominated the version dated 00:00, 30 May 2010 for approval". :-) And by the way, the last version of Peter is dated 00:07, not 00:00 (or is it by intention?). More seriously, here is a real problem. I was able to learn four points of the set theory that I did not know before, in order to be able to approve this article. However, in order to approve a philosophical section (and I agree with Mark that it should be a separate section) we need a more widely educated editor than me. Boris Tsirelson 06:28, 30 May 2010 (UTC)


 * As for the philosophical material, we could poke Larry to approve a philosophy section. Or I could go and poke some professors who aren't CZ members - on a Bank Holiday to come and approve the little scrap of philosophical material that I've added. Or I can provide some pretty comprehensive references as to the philosophical stuff I've added and the current editor in mathematics can take an educated guess as to whether it is right or not. It is such a minimal amount and so utterly non-controversial, it seems silly to go to too much effort at this stage to approve that section - obviously, in the future, it would be sensible to get proper approval from one of the many non-existent editors in the philosophy workgroup! –Tom Morris 10:27, 30 May 2010 (UTC)


 * Ah, now I understand that Peter (being much more patient than me) has made a great work preparing the version dated 00:00, to which I can agree with pleasure. Thus, be it needed or not, I did some void edit of the template in order to formally agree with that version (in spite of my menacing declaration). However, I feel uncomfortable to remove Peter's support, as Matt demands. This matter is left to Peter himself. Boris Tsirelson 06:40, 30 May 2010 (UTC)


 * And by the way, here is an interesting set-theoretic paradox local to CZ. :-) The two-element set {Boris,Peter} contains the one-element set {Boris}. Nevertheless the former set of editors is not enough to approval, while the latter one is enough. What could it mean? How can it be that more is less? Can anyone say a word in defense of this rule (if it is really our rule)? See also User talk:Approvals Manager. Boris Tsirelson 08:18, 30 May 2010 (UTC)


 * I said in a similar discussion recently (on Cryptography) that I think it does not matter whether there is a second approving Editor (involved in the drafting or not), as long as we have one non-involved approving Editor and the approving editors agree on a version to approve. --Daniel Mietchen 13:13, 30 May 2010 (UTC)