Talk:Main Page

We've gone social.
I've just setup a Twitter and FaceBook accounts for ProofWiki. If anyone is interesting in helping out with the facebook page let me know. --Joe (talk) 15:40, 4 October 2012 (UTC)


 * We've already had a comment on Twitter appreciating our Template:Handwaving ... Sheer class. --prime mover (talk) 16:53, 4 October 2012 (UTC)


 * Should anyone find or post locate something worthy to share with the world, don't hesitate to request a tweet. I for one have just posted about Group Example: inv x = 1 - x. --Lord_Farin (talk) 21:04, 10 October 2012 (UTC)

Spam

 * Changed to use reCAPTCHA. Hopefully this will solve some spam issues. --Joe (talk) 11:14, 10 October 2012 (UTC)


 * ... and instantly a spam posting appears.
 * Interestingly the precise same message from the precise same username (Jeremykippler728) has just been deleted by me from another website I assist in moderating. That website has a horrifically bad problem with spammers, and (mow I think about it) may be a major contributor towards my chronically bad temper. --prime mover (talk) 20:13, 11 October 2012 (UTC)

I'm not sure what else to be doing to stop the spam. --Joe (talk) 20:15, 11 October 2012 (UTC)


 * Can you go back to what we had before yesterday? It was considerably more effective than reCAPTCHA which is showing the same weaknesses as I'm having serious problems battling elsewhere. --prime mover (talk) 22:30, 11 October 2012 (UTC)

I think I've mitigated the spam problem for the time being. Changing the question we were asking seems to have done the trick. It's not a question anymore, just a copy the following "blah...." into the provided box. --Joe (talk) 01:37, 22 October 2012 (UTC)

Xymatrix
Allow me to draw your attention to this fantastic package again. Through the hard work of sonoisa, the xymatrix commutative diagram drawing package is now implemented.

A few months ago, the more powerful (but also more difficult and cumbersome) back-end language for xymatrix, xypic, was introduced, which I have employed several times.

But now, with xymatrix, I feel that most PW users should be able to draw the CDs they need. Documentation for xymatrix is easily located through a web search, but I find http://www.jmilne.org/not/Mxymatrix.pdf to be a good source.

The package can be used inside a xy environment, like so:

$\begin{xy}\xymatrix{

//xymatrix code here

}\end{xy}$

I have adopted the convention to precede such diagrams with two colons :: so as to place it a bit more centered in the flow of text.

In case you find functionality of xymatrix that is not yet supported by the implementation, notify me; I'll pass it through to sonoisa. That's all for now - happy editing. --Lord_Farin (talk) 19:52, 20 October 2012 (UTC)


 * I'm currently not in the area of mathematics which requires me to use this - but I will give it a workout once I need to. What I see of its results looks good. --prime mover (talk) 05:23, 22 October 2012 (UTC)


 * Should the basic reference above be insufficient for your purposes, then check out this, or as a last resort (very technical but complete) this. --Lord_Farin (talk) 17:35, 22 October 2012 (UTC)


 * I am still sorting these issues out for myself, but does this mean, in particular, that you implemented commutative diagram capabilities by installing XyJax</tt> 'extension' of MathJax</tt>. Or, instead, are you running xymatrix</tt> as a math system separate and apart from MathJax</tt>, such that the latter is used to do most math typography, but for commutative diagrams, a separate system is used ? Thanks and regards --Pickle (talk) 04:09, 7 January 2013 (UTC)


 * We have the XyJax extension installed - it supports the \xymatrix</tt> command now. How to implement the extension is explained at its web site. --Lord_Farin (talk) 08:42, 7 January 2013 (UTC)


 * Noted and many thanks. --Pickle (talk) 13:02, 7 January 2013 (UTC)


 * Followed the instructions and I have mathjax running locally on a local mediawiki (hard drive). Also, the XyJax sample runs fine when I click on it. I have placed the fp.js and xypic.js files in their places as described. However, I can not render XyJax (using the code for the same commutaive diagram) on the MediaWiki, even though the rest of my MathJax code renders just fine in the MediaWiki. I suspect (given very limited technical skills) that this is because I am not making any use of the script code as given in step 6 of the XyJax install instructions (https://github.com/sonoisa/XyJax). The reason for that is that I have no idea where to put it. Obviously, I'd like to place somewhere so that I need use only once as opposed to placing into every page, but unfortunately, I have no clue as to how or where to place this in the case of a mediawiki install. Would be grateful for any advice on this issue since I believe I am very close. Thanks very much! --Pickle (talk) 03:58, 15 January 2013 (UTC)


 * I'm afraid I don't know how to do such a thing with MediaWiki. Sorry, I don't think I can help you with this - you will probably find adequate assistance on some MediaWiki help thread, though I can't point you to one of those. --Lord_Farin (talk) 07:44, 15 January 2013 (UTC)


 * May be I am way off base in how I'm approaching the whole thing (I wouldn't be surprised), but how are you running XyJax thru MathJax for ProofWiki ? I followed the directions for installing XyJax with the exception of step 6. (Is MathJax running locally or remotely via the CDN. And, for XyJax, I thought you needed to move those two .js files mentioned above (per the XyJax directions) into the two subdir's of the MathJax distribution. Is there a different way to activate XyJax in MathJax. As I say, the test file for XyJax works just fine when I click on it. And, I have MathJax installed per the directions given on the MediaWiki discussion on the topic.) Thanks for any insight or thoughts about your experiences in XyJax up ! --Pickle (talk) 12:22, 15 January 2013 (UTC)

Our setup currently comprises a local MathJax copy, amended with the XyJax files in the locations as suggested by step 6. I think MathJax supports configuration files and extensions being located elsewhere; this should be documented in its manual. It may even be possible to use the CDN version of MJ while using local copies of the XyJax files. As for the functionality of the extension, I think it's simply amazing, especially considering it's one man's work. Whenever I encounter problems with it, I send an e-mail to the developer about it; usually it's then resolved within the next few weeks. I'd recommend it to anyone desiring vector graphics CD support in browsers. --Lord_Farin (talk) 16:09, 15 January 2013 (UTC)

Thanks. I finally got it to work. The trick was adding to the .js code of the MathJaxConfig file the elements described in step 6. The particulars came directly from the developer who was very helpful. Thanks again for your help. Best regards --Pickle (talk) 05:03, 17 January 2013 (UTC)

Equation template amendment
I have amended the eqn template to put the "ro" column to the left of the "r" column so as to be able to build multi-line expressions on the right with greater effectiveness. See this in effect on the $> 1$ proof in Minkowski's Inequality for Sums (incidentally up for a major refactor). --prime mover (talk) 11:23, 4 November 2012 (UTC)

Axiom namespace
It occurs to me that Definition:Group Axioms, Definition:Open Set Axioms and also potentially the axioms for a Definition:Synthetic Basis should perhaps all be somewhere in the Category:Axioms namespace.

Of course, that will mean the Sysyphean task of changing Definition:Group Axioms to Axiom:Group Axioms and all the appropriate pages (and also for Open Set Axioms but there are little links for that yet).

Anyone else think that's a good idea? Oh please say no, my fingers hurt. --prime mover (talk) 22:38, 5 November 2012 (UTC)


 * I'd say no. Almost every concept defined introduces its "axioms". You have probably noticed yourself referring to them as "conditions" as well. I can't craft a definitive argument, though. It seems so natural to have Peano's axioms in the Axiom namespace. It also feels natural (to me) to not include the "totally ordered field axioms" there. Maybe the benchmark is that a certain axiom system should be more-or-less universal, or at least universally accepted as a possible axiomatisation (like e.g. Hilbert style deduction axioms for PropLog). Abstract algebra and topology seem to be less suitable for such axiom systems. --Lord_Farin (talk) 22:55, 5 November 2012 (UTC)


 * As is apparent from my answer, there is a case for this change. However, I think that we're better off tidying proofs, refactoring pages and eviscerating stubs than letting our fingers bleed for such an ultimately aesthetic change. --Lord_Farin (talk) 22:58, 5 November 2012 (UTC)


 * I have an idea PM if you're quite serious about your fingers hurting. Perhaps make something that could take "[ [D:Set union] ]" and turn it into "[ [Definition:Set Union] ]" (that could of course be extremely difficult I don't know).
 * I am quite confused as to what an axiom is in any given context due to conflicting sources. Some sources call proof rules axioms, others make no distinction between an axiom and an axiom schema and others further still call the defining properties of a class the axioms of some theory about that class. I understand all these senses and that the essential quality they have in common is that they are starting points but I wish there was a bijection between words and concepts. I will have to leave it up to you. --Jshflynn (talk) 01:29, 6 November 2012 (UTC)


 * No I'm sorry, you've lost me again: why would I need to turn [ [D:Set union] ] into [ [Definition:Set Union] ]? It already does say [ [Definition:Set Union] ] thruout. --prime mover (talk) 06:16, 6 November 2012 (UTC)


 * As far as I'm aware when creating an article you have to type [ [Definition:Something] ] over and over when you want to refer to a definition page. It might be easier if upon saving a page some program could turn [ [D:Something] ] into [ [Definition:Something] ] (you know like \R instead of \mathbb{R}). The idea being that it would do the transformation once you press the save page button (so people in the future would still see it as [ [Definition:Something] ]). It is just an idea though and may save no time at all to someone such as yourself. --Jshflynn (talk) 09:46, 6 November 2012 (UTC)


 * Interesting that you say that. I suspect it not to be too hard to implement this; I'll discuss it with Joe. In fact, MediaWiki already supports a similar feature, which I happily use: if you type, say  Definition:Set Union </tt>, that becomes Set Union. Make use of it :). --Lord_Farin (talk) 13:12, 6 November 2012 (UTC)


 * On the note of making shortcuts, if it is simple to implement them, can we make one for \operatorname? --GFauxPas (talk) 14:26, 6 November 2012 (UTC)


 * I see that the shortcut LF mentioned works for other namespaces too (even nonexistent ones should you need to implement them in future). How delightfully efficient :)

Changing letters
I have recently seen that some editors prefer the use of the symbols $S$ or $A$ rather than $X$ to denote topological spaces and the like, because "$X$ tends to denote an unknown." I have no objection against this, but I cannot comprehend why this is an improvement. Would anyone mind to explain this (somewhat) more thoroughly? --abcxyz (talk) 00:06, 18 November 2012 (UTC)


 * Whim. --prime mover (talk) 08:45, 18 November 2012 (UTC)


 * It's good to have consistency of presentation in a wiki. When the topology work was started, a lot of the original work had $X$, a lot had $A$ and a lot had $S$, depending on who posted up what. Either we leave it like it is, with lots of different symbology, or we make an attempt to bring it into line. I've been working hard on generally trying to bring consistency to the site, and it cna be frustrating to find that what I'm doing has been undone. --prime mover (talk) 09:13, 18 November 2012 (UTC)


 * Fair enough. I haven't bothered to go through all the topology pages and see which symbols are more common than others.
 * Also, I have a question about linking to redirects (such as  Definition:Neighborhood of Point </tt> as opposed to  Definition:Neighborhood (Analysis)/Topology/Point </tt>). Would anyone mind to explain why this is done? I just don't understand it. --abcxyz (talk) 19:13, 20 November 2012 (UTC)


 * It is mainly for two reasons:


 * 1) Prettier text when hovering over a link
 * 2) In the case a page gets refactored in the future, only the redirect needs to be adapted; this saves mountains of work


 * I'm sure you understand now. As far as I'm concerned, the second argument is the more important of the two. --Lord_Farin (talk) 19:38, 20 November 2012 (UTC)


 * Thanks; I hadn't thought of that. It appears to be a good idea. --abcxyz (talk) 21:10, 20 November 2012 (UTC)

clock
Any chance of getting the system clock reset? It's 5 minutes or more fast. --prime mover (talk) 23:05, 19 November 2012 (UTC)
 * I shall see what I can do! --Joe (talk) 22:23, 20 November 2012 (UTC)
 * Fixed. --Joe (talk) 13:25, 21 November 2012 (UTC)
 * Much better. I no longer panic when I see the time on a page I've just edited. Thx. --prime mover (talk) 08:29, 22 November 2012 (UTC)

Of vs. for (moved from Talk:Equivalent Definitions of Direct Limit of Sequence of Groups)
We now have Equivalent Definitions for T5 Space and this page [ Equivalent Definitions of Direct Limit of Sequence of Groups, LF]. There are various other names for this kind of page (e.g. "Equivalence of Definitions of ..."). We need a consistent convention. What will it be? I like "Equivalent Definitions" better than "Equivalence of Definitions" (it's shorter). --Lord_Farin (talk) 18:22, 26 November 2012 (UTC)
 * Pedantically, "Equivalence of Definitions ..." is more accurate, as "Equivalent Definitions ..." suggests that the definitions themselves are being introduced on that page - whereas the definitions themselves are in fact elsewhere. I confess I have not thought it through; but were we to use the latter it ought to be "Equivalent Definitions of ..." not "... for ..." (when I was refactoring over the weekend, "for" felt wrong) but I'm not in the mood to change them all at the moment.


 * If someone else were to volunteer to do the work, then making everything either "Equivalence of Definitions of..." or "Equivalent Definitions of..." would work for me. --prime mover (talk) 18:50, 26 November 2012 (UTC)


 * The new structure with definitions on subpages makes it much easier to have these pages adhere to a common standard, and to ensure that updates on the def. pages get reflected properly on other pages. --Lord_Farin (talk) 13:55, 27 November 2012 (UTC)

What to do about bad work
Some of the work on this site is shockingly bad. This has got to stop.

I recommend that people who post up unacceptable pages get punished for it. I recommend a fine of up to and including 1000USD and a ban from posting for up to and including life.

That's the only way we can get the quality of submissions up to a decent level. --prime mover (talk) 07:09, 27 November 2012 (UTC)


 * I don't deem it commendable to expel new enthusiasts adding a proof for a previously stubbed article because their writing style is not as we demand it. I do however agree that users notorious for posting substandard material (not that I know many) should be warned and if necessary restricted to their own user page until they demonstrate to know how to make stylistically bearable contributions. I fear that a too repressive regime (and I'd say we're already pretty relentless, only not finding the time to display it) would scare off (highly desired) new contributors, thus limiting the influx of information unnecessarily. --Lord_Farin (talk) 13:55, 27 November 2012 (UTC)


 * Bad formatting and lack of attention to appropriate writing style is not a problem. It's bad mathematics that I was suggesting needs to be eliminated. --prime mover (talk) 19:41, 27 November 2012 (UTC)


 * Would you care to give a rather more objective description of what you consider 'bad'? Plainly false, or incoherent, or phrased in specialists' language? --Lord_Farin (talk) 19:47, 27 November 2012 (UTC)


 * In particular, invalid arguments like affirming the consequent. Incoherent as well. Specialists' language is fine, it just needs someone to come in and say "This means ..." and so on. --prime mover (talk) 20:17, 27 November 2012 (UTC)

Repeating the hypotheses
When I was at school and my perspiring perpetually-defeated teachers tried vainly to bludgeon the basics of mathematics into the solid lump of bone that passes for my skull, one of the rules actually caught hold. That one is: when you're answering a question, it makes sense to repeat the basic facts of the question at the top of your answer so you have a precise grasp of your hypotheses.

In this context, I believe it does no harm to repeat the hypotheses of the theorem within the body of the proof (at the top) in order to clarify the statements which hold at any particular time.

It does no harm to repeat the hypotheses in the proof. OTOH, I think it would be bad to insist that "All hypotheses need to be repeated at the beginning of all proofs and all new work is to be abandoned until every proof adheres to this commandment". But there's a case for saying: if a hypothesis has been repeated at the top of a proof, it does not necessarily improve the proof by removing it just because you can. --prime mover (talk) 19:48, 1 December 2012 (UTC)


 * Fair enough. I'll take this bullet. --Lord_Farin (talk) 20:10, 1 December 2012 (UTC)


 * OK, fair enough. I removed the phrase to make the proof easier to read; it was at the end of a rather lengthy sentence. (Not that you said I "removed it just because I could", but this is why.) Moving it to the beginning of the proof also does the job. --abcxyz (talk) 21:57, 1 December 2012 (UTC)


 * Same thing happened somewhere else I think, I misremember exactly where. No worries. --prime mover (talk) 22:30, 1 December 2012 (UTC)

Differentiation and integration
Is there already a proof that the derivative of the integral of $f$ is $f$, i.e., that differentiation and integration are inverses? — Timwi (talk) 12:29, 3 December 2012 (UTC)


 * Cf. Fundamental Theorem of Calculus (first part, I think). --Lord_Farin (talk) 12:31, 3 December 2012 (UTC)

On results for concepts that have multiple definitions
We have recently decided to try and treat all (sourced) definitions for a concept on equal footing. An example of how the site is progressing can be found at Definition:Connected (Topology)/Topological Space. In time, Definition:Exponential will go through a similar remodeling, and I hope to add other descriptions of a Definition:Boolean Algebra. This brings me to the following point, concerning how results on such "multi-defined" concepts should be structured. As it stands, we have pages like Exponential as Limit of Sequence; this will of course become irrelevant given that we may define the exponential in terms of this limit. Such pages therefore naturally move to the associated Equivalence of Definitions result.

Up to now, such has not posed a problem. However, in the case of Boolean algebras, we have results of the kind Meet is Commutative, or, more relevantly, Join Distributes over Meet. In order to prove that a BA may equivalently be defined as a complemented, bounded, distributive lattice, it is then required to effectively prove Join Distributes over Meet to establish the equivalence. OTOH, given that a BA is a distributive lattice, the result is not even a result, but plainly imposed. Because there are rather a lot of such results involved in proving equivalence of the definitions, there are two, both unappealing paths that I see:


 * Firstly, that the pages remain in place, and have a "this is by definition" one-line proof (which seems rather irrelevant).
 * Secondly, that the pages become lemmata on the Equivalence of Definitions page, thereby unnecessarily lengthening that page (and I can foresee it will already be quite comprehensive from itself).

Of these options, neither of which is satisfying my elegance standard, I'd prefer the first, if one were to be chosen. We are also approaching a similar issue with the natural numbers. There are a lot of results which either prove basic facts about certain paradigms, or prove partial equivalence of definitions, or prove things by means of one paradigm that are trivial in another (without mentioning the equivalence).

It will benefit the consistency and completeness of ProofWiki if we pursue a meta-structural approach that treats, to the extent possible, definitions on an equal footing. In the long run, this will:


 * Make proving stuff easier, because the paradigm(s) most suitable to a typical can be invoked to achieve results that are nearly infeasible to prove directly from other definitions.
 * Make ProofWiki more appealing to all mathematicians, since there is no need for abandoning preferred frameworks and mindsets.
 * Make the site easier to navigate, and make it easier to locate what one is looking for; to this end, the "Also known as" sections should be as comprehensive as possible.

While I am convinced (as is apparent) that this is the situation ProofWiki should pursue, I think it is of the utmost importance that all editors are aware of, and should be able to comment on, this new direction, before we start rolling back each other's edits and create a hostile environment where the covering of mathematical knowledge is no longer the paramount goal of every editor. Thus, please comment. --Lord_Farin (talk) 14:10, 4 December 2012 (UTC)


 * Completely agree.


 * There are two types of definiend here:


 * One, e.g. Connectedness, which is one particular concept which can be treated monolithically; equivalence is proved "in one step", so to speak.


 * The other type of definiend exemplified is that of natural numbers, which is defined by means of an axiom schema - and in this case there are several approaches. In this case, what is done is for separate threads to be established, each one aiming towards the "defining properties" of the natural numbers - and indeed, some of the proofs along the way may be trivial.


 * The transition from the definition-and-equivalences to the multiple-definitions paradigm can be fraught with trouble if other editors are simultaneously trying to "correct" stuff which is still in the process of being developed, but if we are rigorous about using (and respecting) templates indicating work in progress / refactoring / etc., and perhaps adding a field for a username to those templates which do not have them, then inter-editor friction should be kept to a minimum.


 * I would also exhort newer editors to learn the ropes before being too strident in their opinions of how they think things ought to be. Many of the structural conventions have evolved over time and (despite appearance to the contrary) do actually make sense. Some MediaWiki tools and MathJax constructs, for example, are specifically not utilised because they have been shown to cause problems (although with more recent releases of the s/w some of these may well have been resolved).


 * In short, if new editors find their work is being systematically altered, tagged, refactored etc. etc. then treat it as a learning experience as to how pages are to be styled rather than as a challenge to their creative abilities. They are invited to take note of the changes being made and adjust their style accordingly. There are reasons. --prime mover (talk) 15:03, 4 December 2012 (UTC)


 * As to that last point of PM, I'd like to stress that anyone who doesn't understand a certain edit is urged to ask the relevant editor for a justification. In this way, we are both ensured that people get the rationale behind paradigms such as house style, and that such rationale continues to exist in the first place. All editors are invited to put up examples of possible changes/advances in house style or otherwise on their user sandbox, and invite others to comment. This seems to me like the appropriate way to try and nudge the rules and regulations for editing in one's preferred direction (as opposed to simply posting pages in the newly desired format in the main wiki). --Lord_Farin (talk) 22:13, 4 December 2012 (UTC)


 * Sounds good. I have a question about proofs (e.g. Synthetic Sub-Basis and Analytic Sub-Basis are Compatible) that use multiple definitions of a concept (in that case, the generated topology). Evidently the equivalence proof has to be mentioned somewhere. I can't think of a way to deal effectively with pages such as those. Any ideas or comments?


 * Less importantly, does anyone find the phrase "the definition" strange, when one of the multiple definitions is being invoked? --abcxyz (talk) 16:52, 16 December 2012 (UTC)


 * Obviously what's happened is that the proof was written before the refactoring was done on the definition page to split it into multiple definitions. Equally obviously, the proof needs to be rewritten accordingly. --prime mover (talk) 18:22, 16 December 2012 (UTC)


 * Both definitions were still invoked (before the refactoring), so I don't see how the refactoring matters. (This was always a problem; I just didn't bring it up until now.) --abcxyz (talk) 18:37, 16 December 2012 (UTC)


 * So I haven't got round to sorting it out yet. My fault. I will try to do better. I understand my employment is up for review. Please feel free to vote on whether I get to continue to be allowed to contribute. --prime mover (talk) 21:33, 16 December 2012 (UTC)


 * No, it's not your fault one bit. I posted the proof and I want to know if anybody has an idea about how to make it work better. --abcxyz (talk) 21:38, 16 December 2012 (UTC)


 * Link to the specific subpage holding the definition you need in order to provide the required property that will move your proof along? --prime mover (talk) 21:54, 16 December 2012 (UTC)


 * Sorry, what required property? --abcxyz (talk) 23:35, 16 December 2012 (UTC)


 * Whatever property of whatever object you are using whatever definition of to prove whatever theorem you're writing a proof of. --prime mover (talk) 06:51, 17 December 2012 (UTC)


 * If I'm understanding you correctly, I think I did that already. The problem is that both definitions are used in the proof. --abcxyz (talk) 04:23, 18 December 2012 (UTC)


 * So link to one definition at the point at which you use that definition, and link to the other definition at the point at which you use that one. --prime mover (talk) 01:04, 26 December 2012 (UTC)


 * That's exactly what I did. So that's enough because the equivalence proof is in the "Also see" section of the definition page? --abcxyz (talk) 16:37, 8 January 2013 (UTC)

Enough for me, at least if you refer to "Definition X of ...". --Lord_Farin (talk) 17:53, 8 January 2013 (UTC)


 * Like this? --abcxyz (talk) 18:00, 8 January 2013 (UTC)


 * Yes, that's what I meant. In this way, it is apparent to the reader that there are multiple definitions, which may be overlooked otherwise. --Lord_Farin (talk) 18:02, 8 January 2013 (UTC)

Sine and Cosine of Sum
I split this (and all its subpages) into Sine of Sum and Cosine of Sum. Please let me know how I did! — Timwi (talk) 15:00, 13 December 2012 (UTC)

What should we do with the page Sine and Cosine of Sum now? Delete? Redirect to one of the two (which is what it does now)? Turn into a disambiguation page? The latter two seem silly, but the first one breaks existing links. — Timwi (talk) 15:00, 13 December 2012 (UTC)


 * From what I see, it looks good. Tasks still to be done are to make all links to Sine and Cosine of Sum point to the appropriate page and to turn Sine and Cosine of Sum into a transclusion amalgamate of Sine of Sum and Cosine of Sum. --Lord_Farin (talk) 15:19, 13 December 2012 (UTC)


 * Is that really necessary?... — Timwi (talk) 15:31, 13 December 2012 (UTC)


 * Such pages form convenient entry points to PW for wandering people doing web searches. While they are not strictly speaking necessary, they serve largely to disclose PW to incoming traffic. --Lord_Farin (talk) 15:33, 13 December 2012 (UTC)


 * OK well, I did the “simple” thing of just transcluding the two (see Sine and Cosine of Sum now), but it means it doesn’t have the proofs, and getting to the proofs is unobvious (you have to know that clicking the link in the header takes you to a proof...) — Timwi (talk) 17:38, 13 December 2012 (UTC)


 * If there's an existing page split into several, it is never a good idea to delete such a page. As you say, it breaks links. And even once the internal links are all amended, there may be external links which access this site from all over the place which would suddenly arbitrarily fail, particularly for old pages. It is also usually a bad idea to leave it as a redirect, because e.g. "blah blah sine and cosine blah" suddenly going to page containing "blah blah sine blah" etc., the reader is going to justifiably feel annoyed at not finding the cosine there. This contributes towards the generally negative view in which this site is held in the great wide world outside. --prime mover (talk) 17:41, 13 December 2012 (UTC)


 * We'll have to find a way to make clear that clicking links will take people to proofs, either now or some time in the future. As it stands, several people (rightfully IMHO) have indicated that the system isn't self-explanatory for non-veterans. Ideas are most welcome. --Lord_Farin (talk) 17:45, 13 December 2012 (UTC)

Separation of Proof and Theorem pages
Take for example Arctangent Logarithmic Formulation.

How necessary is it to put the proof on a separate page from the statement?

My idea was that we put the theorem and proof on the same page, except where we have multiple proofs in which case we separate out the proofs into different pages.

Have I been doing it wrong all this time? --prime mover (talk) 07:00, 14 December 2012 (UTC)


 * To the best of my knowledge, no. No need for duplication unless it allows for proper referral to knowledge. Your current approach to refactoring is reasonable and should be retained. --Lord_Farin (talk) 11:44, 14 December 2012 (UTC)

Template:Refactor
Personally, I’m quite confused by the fact that you seem to use the same template for two completely different meanings: I think we should split this into two to distinguish these two meanings. — Timwi (talk) 11:51, 15 December 2012 (UTC)
 * “Someone is currently working on this page, please don’t edit it”, and
 * “A change to this page is desirable, please edit it”.

And for the please edit it one, I think the template should mandate that whoever put the tag on the page should explain what it is that needs refactoring, otherwise the tag is not useful. — Timwi (talk) 11:51, 15 December 2012 (UTC)


 * My personal view is that until a person has a good overall feel for the structure behind the website, it would be preferable for that person to concentrate on adding new stuff rather than tidying up the existing structure. --prime mover (talk) 21:33, 15 December 2012 (UTC)


 * And how might one get such a “good overall feel”? By jumping in and doing it, not by staying away. — Timwi (talk) 22:37, 16 December 2012 (UTC)


 * The refactor template is to the best of my knowledge not used in the first sense. Template:Inuse is the thing supposed to do that, but it seems to have fallen into obscurity. As to the explanation: sometimes the work required is either blatantly obvious to its intended audience (site maintenance workers) or hasn't fully crystallised yet (in which case it functions like a stronger version of Template:Improve). Since until very recently, I and PM were the only ones in the "maintenance workers" group (he more so than I) there wasn't really a need. I'll try to explain things better in the future. --Lord_Farin (talk) 21:39, 15 December 2012 (UTC)


 * Perfect — in that case, I’ll be bold and remove the first sentence from the template. If you disagree, feel free to revert it and then perhaps we should move this discussion to Template talk:Refactor. — Timwi (talk) 22:37, 16 December 2012 (UTC)


 * Reverted it, and protected so that only editors with admin privileges can edit it. Sorry, but it was correct as it is: it has either been identified as a candidate for refactoring, or it is in the process of being refactored. That is how it is being used.


 * The bottom line is: if there is a maintenance template whose purpose you do not fully understand, then it would be better to leave it alone. Basically, such a tag was not meant as an invitation to all to contribute, but more as a flag to a particular few who have been involved in what is an ongoing major restructure of the fundamental architecture on which ProofWiki is based.


 * TL;DR: thanks for the offer but we have the refactoring under control because we know precisely where we are going with it. Please work on other stuff. --prime mover (talk) 22:55, 16 December 2012 (UTC)

FAQs
The same questions (perfectly legitimate and sensible) are being asked by different contributors concerning things like house style, philosophical approach, structural conventions and all sorts of non-obvious and seemingly arbitrary stuff which we impose on the entries on this site.

It might be a good idea to bag up some of these, e.g. "Do we really need to put \left and \right round every parenthesis and if so why?" and "Why did you delete my link to Wikipedia in the citations?" and "Why did you change my favourite notation?" and publish them on a FAQ page - so if they get asked again, we can point the asker to that page.

Thoughts? --prime mover (talk) 00:11, 16 December 2012 (UTC)


 * I think this is a good idea! --Joe (talk) 12:31, 16 December 2012 (UTC)


 * Okay - I don't feel like working on it immediately, but as and when questions crop up that have been asked before, I will probably start building such a page. --prime mover (talk) 13:19, 16 December 2012 (UTC)


 * I don't know why we didn't have this page earlier! --Joe (talk) 15:02, 20 December 2012 (UTC)


 * May I recommend implementing a system to uniquely identify the questions? This will lend itself to redirecting inquiries. --Jshflynn (talk) 18:36, 20 December 2012 (UTC)

Something wrong with the rendering of references
Does anyone know why we’re getting the rendering bug shown in the screenshot on the right? I looked at the page source, it looks fine, so the bug must be with the parser or the template system... — Timwi (talk) 14:05, 20 December 2012 (UTC)


 * It's the parser, I think. Something is wrong with how it handles nowiki tags (e.g. around square brackets for references). Various talk pages are littered with my remarks on the matter. --Lord_Farin (talk) 14:27, 20 December 2012 (UTC)


 * Once the templates have been cleared it should render okay. Anyway, house style discourages footnotes. --prime mover (talk) 17:32, 20 December 2012 (UTC)


 * ... oops: I notice that that was a page I wrote myself. Definitely needs a rethink ... I was infodumping a college course. --prime mover (talk) 18:19, 20 December 2012 (UTC)

Milestone approaching
We are 36 proofs away from the 7000th proof.

As this is a small number times a higher power of 10 (in this case a cube), this number would appear to have significance to a hypothetical race of intelligent beings which have a 10-based number system (probably because they have ten fingers or something silly and arbitrary like that).

On this website it is traditional to reserve such a number to be the ordinal of a particularly significant result. I therefore alert all active contributors to this site to keep an eye open for something significant which can be hooked onto this number, and make sure it is not something simple and/or trivial.

Newcomers: take a look in the "Landmark Pages" category for an idea of what (at least in recent times) sort of thing I mean. Note that at this stage it is not necessary to have a proof immediately available if you need some space to work up to it. --prime mover (talk) 23:03, 24 December 2012 (UTC)


 * My suggestion is Law of Large Numbers. It's the first significant result of probability calculus, and we don't have it yet. If we restrict ourselves to the Weak Law of Large Numbers, it should also be possible to prove the Lemmas that are required to prove the theorem. If you find that LLN is too trivial, then I'd suggest the Central Limit Theorem. --Anghel (talk) 18:44, 25 December 2012 (UTC)


 * If you happen to be sitting in front of your screen when proof 6999 appears, then you are more than welcome to post up LLN as the 7000th, or, if you prefer, CLT. :-) --prime mover (talk) 01:02, 26 December 2012 (UTC)


 * CLT was already up. I put up something quirky, although nowhere near being able to put a proof up at the moment. --prime mover (talk) 21:26, 31 December 2012 (UTC)

Duality
I have been thinking about a way to connect dual pairs of theorems like Infimum of Infima and Supremum of Suprema. What I've come up with most recently is something like Template:About, i.e. a sentence below the theorem title. It could be along the following lines:
 * The dual of this [theorem/definition] is Infimum of Infima. See Duality Theorem (Ordered Sets) for more information on duality.

where of course there are three parts ("[theorem/definition]" and the two links) that need to be changeable by the template governing this. Possibly we want to say "information on duality for [concept duality applies to]". Maybe the last two links may be governed by a standard setting like  or s.t. along those lines. Thoughts? --Lord_Farin (talk) 18:23, 3 January 2013 (UTC)


 * a) How well-defined a concept is "duality" apart from it being an intuitive thing?


 * It can be made quite formal, by e.g. interpreting a definition in the Definition:Opposite Ordered Set (defined along Inverse of Ordering is Ordering), and then translating that statement back to the original poset. This process is described in more generality on Duality Principle (Category Theory) (although stated there for theorems, it can easily be extended to cover definitions as well).


 * b) I'm not happy about it being right at the top of the page - too much of that and the statement becomes in danger of getting "lost" so to speak. In the "Also see" section sounds more appropriate, where most of these duals already exist - suppose it does no harm to emphasise their duality. --prime mover (talk) 19:40, 3 January 2013 (UTC)


 * Meh. The Also see section is an option, but I feel this detracts from the importance of duality. OTOH, a dedicated section titled "Dual something" could easily turn into overkill. If there were a nice option to stylistically emphasize the dual theorem, along with a few accompanying words, that'd be great. Maybe we have to settle with the dedicated section nonetheless. --Lord_Farin (talk) 20:46, 3 January 2013 (UTC)


 * So, I've decided to delve into the task of covering duality (for ordered sets only at this point). Would we like to anticipate on similarities coming up in the (hopefully not too distant future) concerning duality for ordered sets, groups, lattices, boolean algebras, categories (incidentally the last covers all the others) and call everything "Dual Blabla", i.e. "Dual Ordered Set", "Dual Ordering" (currently Definition:Opposite Ordering), "Dual Group" (Definition:Opposite Group), "Dual Boolean Algebra". Definition:Dual Category is already up. I have a feeling that this path will contribute to a more cohesive feeling for the whole site. --Lord_Farin (talk) 19:02, 4 January 2013 (UTC)


 * Can I just leave it up to you to see what it looks like? Maybe when I see it in action I'll like the look of it ... you know me, anti any idea that gets floated in a talk page, but once I see it working I usually quite like it. --prime mover (talk) 21:07, 4 January 2013 (UTC)

Question regarding use of copyrighted sources
I was wondering about sourcing proofs from copyrighted works, and what is permissible. Are there any guidelines on this from a legal perspective? To what extent can a proof be copyrighted? If a proof can not be copied in verbatim, to what extent must it differ from the original to avoid infringement? --Icarus (talk) 16:10, 5 January 2013 (UTC)


 * I'm afraid it's hard to put a definitive line in the inevitable greyscale area between "copied verbatim" and "rewritten in own words". However, if it's not self-evident that there are no legal issues, you're probably pushing it. That should be a safe guideline to work with. --Lord_Farin (talk) 20:12, 5 January 2013 (UTC)


 * I was curious, because I spent a significant amount of time working through Rudin's 'Principles of Mathematical Analysis', and writing out most of the proofs up through Chapter 9 with omitted steps and clarifying details filled in. A lot of this was done in LaTeX, and I would like to post it to this or a similar site, but I don't want to violate the copyright. When I asked the publisher about doing so, they denied me permission to use that much work. I guess one other way might be if they were willing to sell the copyright, and enough people were interested in contributing to purchase it in order to put it in the public domain. I don't know how likely either case would be though. --Icarus (talk) 21:57, 21 January 2013 (UTC)


 * The view I take is that the individual proofs, theorems and definitions which make up an elementary source work such as Rudin are not themselves subject to copyright. If one were to post them all up, one would note that practically all of it existed in other published works, in some form or other. You will probably find that the most of this work already exists on this site. All you therefore need to do (probably) is to add a citation in the "Sources" subsection to the specific paragraph in Rudin that contains the contents of the page. --prime mover (talk) 22:03, 21 January 2013 (UTC)


 * As a related side note, I wonder then if it would be useful, if it hasn't already been done, to have a page for each source that lists and links back to the proofs that reference it. --Icarus (talk) 22:15, 21 January 2013 (UTC)


 * The fundamental difference between what PM suggests and what you appear to have done is, that your work follows formulation and ordering of Rudin exactly (and subsequently expands upon it) and may thus be considered largely a derivative of orginal Rudin, while ProofWiki is an assembly of mathematical proofs formulated according to our own style premises - each on its own separate page. That such a work contains results coincident with Rudin is inevitable as they discuss the same matter (to some extent). This means it is permissible to add all the proofs you have explicated to ProofWiki and put them on the respective pages (which may or may not exist at this moment), while it won't be permissible to bundle them all and have the PDF distributed over the internet.


 * As for your note: Many of the sources that have already been covered to some extent at ProofWiki (and a lot of sources that haven't) are listed on Books. The "What links here" tool in the side bar provides a right inverse to the links in the Sources section. --Lord_Farin (talk) 22:19, 21 January 2013 (UTC)


 * If that is the case, I would be glad to publish it here as separate proofs. Do you know how I can legally verify that? Sorry, I'm just a little paranoid, especially after the publisher said that I didn't have permission to reproduce that much work.

The intent of Copyright Acts around the world is not to copyright ideas. It is to copyright particular implementations and renderings. Therefore, since we will undoubtedly fundamentally alter wording and presentation to match house style and our definitions, and hence not much but the idea of proofs will be retained, it is reasonable to conclude that doing so will not cause Copyright violation. Admittedly that's not a legal verification (for which I would need to be an expert in copyright law and jurisprudence, which I am not) but it's as close as I can get. I hope such suffices for you. --Lord_Farin (talk) 22:47, 21 January 2013 (UTC)


 * PS: I hit cancel on an edit and it seemed to post it anyway? --Icarus (talk) 22:39, 21 January 2013 (UTC)


 * Joe, can you look into this? --Lord_Farin (talk) 22:47, 21 January 2013 (UTC)
 * I'll dissable the fancy editor for now until I have time to do some more testing. --Joe (talk) 02:53, 22 January 2013 (UTC)

Oddness in Chrome
On my machine, when you try to edit a page in Google Chrome, it changes from (Courier New) font to some other font (some naff san-serif font that's all tiny and narrow and not equal-width, making it difficult to edit). It does this in all applications, not just ProofWiki. Even does it on Facebook.

Anyone experienced this? Only just started doing it. Version 23.0.1271.97 m.

I've sent a feedback note to Google in case it's something their end. Any advice? I lack the patience to dig around in the source code to debug it myself. --prime mover (talk) 20:02, 5 January 2013 (UTC)


 * I use the exact same version of Chrome, and I still get Courier New. --Lord_Farin (talk) 20:08, 5 January 2013 (UTC)
 * I think a solution is to add the code:

textarea{ font-family: monospace; }
 * to your user.css. This will force the textarea HTML environments (like the edit window is) to use a monospace font. --Lord_Farin (talk) 20:08, 5 January 2013 (UTC)


 * Doesn't work. As I say, it's not a ProofWiki issue, it does it all over, on all websites.--prime mover (talk) 20:20, 5 January 2013 (UTC)


 * Sorted it. In case anyone else has the same trouble, it turns out that I had Diigo bookmarked (it's an insidious little beast) which hijacks the stylesheet when in a textarea. Burn it with fire. --prime mover (talk) 22:44, 9 January 2013 (UTC)

Expressions involving restrictions
Just a quick note:

I noticed that the spacing of expressions involving restrictions can look somewhat awkward. For example:

is rendered as:
 * $f \restriction_A: A \to Y$

which doesn't look good. I think that a solution is to use braces, i.e.:

which is rendered as:
 * ${f \restriction_A}: A \to Y$

I have already done this in several places. --abcxyz (talk) 18:48, 8 January 2013 (UTC)


 * Well spotted. Maybe this technical note could be added to the Definition:Restriction page as well? --Lord_Farin (talk) 18:55, 8 January 2013 (UTC)


 * Sounds like a worthwhile structural paradigm: perhaps another standard section name below everything except "Also see" and "Sources" called "LaTeX technicalities" or some such. The exact wording of that subtitle may need to be discussed. --prime mover (talk) 19:07, 8 January 2013 (UTC)


 * Definition:Convergence in Measure uses "Technical Note" which we deemed to be pretty good back then. --Lord_Farin (talk) 19:10, 8 January 2013 (UTC)
 * I would say however that the section should be below Also see, because we don't want to burden the casual reader with LaTeX information they may well find obscure and irrelevant, aside of incomprehensible. --Lord_Farin (talk) 19:12, 8 January 2013 (UTC)


 * As we progress down the page we go to concepts less and less directly relevant to the item in question. As an "also see" is not directly about the subject, I would argue that the LaTeX for presenting the subject is more directly relevant, so therefore deserves to be above Also see.


 * Vote? --prime mover (talk) 19:43, 8 January 2013 (UTC)


 * The "Also See" section is about mathematics; the "Technical Note" is not. Thus, I would prefer having the "Also See" on top. --abcxyz (talk) 21:45, 8 January 2013 (UTC)


 * I completely agree with Abcxyz. The sources section is naturally at the bottom; this quite uniquely determines the position of the section. --Lord_Farin (talk) 21:51, 8 January 2013 (UTC)


 * Yep okay, no worries then. --prime mover (talk) 22:01, 8 January 2013 (UTC)


 * I'll just add this: the rendering of MathJax is different from how it renders in the MediaWiki LaTeX package, and so we had to go through and change a lot of the rendering. As MathJax evolves, presentational awkwardnesses are fixed and fiddly extras are no longer required. So it's possible that a recommended technical implementation on one version of MathJax may not be optimal on a later version (e.g. at the moment it is advisable to add a half-space before primes on certain letters, $T$ for example, otherwise the prime overwrites the letter making it obscured - it is possible that this may be fixed in future MathJax versions). So a technical recommendation for presentational optimality may change during the course of evolution of this site. --prime mover (talk) 22:07, 8 January 2013 (UTC)

I have added a "Technical Note" section to each of the subpages of Definition:Restriction.

I also noticed that, on this site, the restriction symbol (e.g. $\restriction_A$) is used as a unary operation on the relation/mapping. However, $\LaTeX$ apparently treats harpoons as binary relations, as it does for other arrows (e.g. $\rightarrow$). This causes an excessive (for my taste) space between the relation/mapping and restriction symbols. Thus, I suggest inserting braces around the restriction symbol instead, which results in:
 * $f {\restriction_A}: A \to Y$

In addition, this looks similar to $\preceq \restriction_A$.

Regarding the use of, its behavior seems to be the same as that of   (as in  , for example). That is, it treats its argument as a unary operation. When it is not intended as such, it can have an undesirable effect, e.g.:
 * $\operatorname{id}_X \mathop{\circ} \left({g \circ f}\right)$

which is produced by the $\LaTeX$ code:

One $\LaTeX$ command that treats its argument specifically as a binary operation is. I have used it in Definition:Restriction of Operation. It also can be applied to $\restriction$ as in $f \mathbin{\restriction} A$.

Correct me if I'm wrong on any of the above. Comments? --abcxyz (talk) 02:59, 10 January 2013 (UTC)


 * I'm (guardedly) in favour of replacing "mathop" with "mathbin" when the operator in question is binary, and using "mathop" instead for unary ones. As per current style, suggest the curly braces are used only when the operator itself is multipart, so e.g. you'd do  rather than  . This matches current usage of mathop and saves time when inserting new instances. --prime mover (talk) 06:13, 10 January 2013 (UTC)


 * Oh yeah, of course. Sorry, I just forgot. --abcxyz (talk) 16:51, 10 January 2013 (UTC)


 * Apparently the original suggestion to use "mathop" came from some $\TeX$ noob who encountered it on a quick web search. Braces around the harpoon seems a good idea because we want to treat it as a postfix operator. There is no dedicated $\TeX$ command for that, sadly. It may be that in certain rare cases the braces do not provide an adequate solution but the situations I can think of where this happens are very unlikely to be on PW, so we may decide to work with this (for the sake of brevity of source code). Thank you for your input on the matter, abcxyz. --Lord_Farin (talk) 09:03, 10 January 2013 (UTC)

Compare and contrast:
 * : $\displaystyle \sum_{i \mathop = 1}^n$
 * : $\displaystyle \sum_{i \mathbin = 1}^n$

On my setup  wins -- no contest. --prime mover (talk) 19:21, 10 January 2013 (UTC)


 * Yes, I know. But, as I said earlier,  can sometimes create problems when the right-hand side starts with parentheses. For example:
 * renders as:
 * $\displaystyle \sum_{i \mathop = \left({m + n}\right)/2}^n i$
 * We would need to add  after , which leads me to think that we might as well use   instead of.
 * By the way, I think that a $\LaTeX$ command for a relation is, which inserts more space than  . As one might expect, that doesn't do anything because   already acts as a relation. --abcxyz (talk) 22:36, 10 January 2013 (UTC)
 * By the way, I think that a $\LaTeX$ command for a relation is, which inserts more space than  . As one might expect, that doesn't do anything because   already acts as a relation. --abcxyz (talk) 22:36, 10 January 2013 (UTC)


 * No no and thrice no. We do not want to go adding arbitrary amounts of whitespace into the formulae just to make it look good on the browser you happen to be using.


 * : $\displaystyle \sum_{i \mathrel = 1}^n$
 * is indeed rubbish, but:
 * : $\displaystyle \sum_{i \mathop = \left({m + n}\right)/2}^n i$
 * works fine.


 * Bottom line: mathop as currently being used is fine. IMO all this farting about is a complete waste of time and effort. --prime mover (talk) 23:02, 10 January 2013 (UTC)


 * The point about restrictions was valid enough, but I am inclined to agree with PM that with  we have something that works, and therefore, since it's only a source code issue, stick with it; after all, it's not like we're adding lines of cumbersome code that could be replaced by a single character. --Lord_Farin (talk) 23:09, 10 January 2013 (UTC)


 * Okay, if you don't like inserting white space, fine. I'll just add that the unequal spacing before and after the  when preceded by   also occurs on my personal copy of $\LaTeX$, not just on my browser. I'm curious why you're so adamant about not inserting white space (in this case). I can't immediately see why.
 * By the way, I replaced  with   on Definition:Relation Compatible with Operation to make the spacing less lopsided (and a bit larger). Anything you'd like me to know about it? --abcxyz (talk) 04:31, 11 January 2013 (UTC)

How about we open a Technote namespace instead of putting them in definition space? --Dfeuer (talk) 04:44, 11 January 2013 (UTC)


 * How about we just do something useful instead? --prime mover (talk) 06:06, 11 January 2013 (UTC)

Parentheticals
The house style insists on $\left({blah}\right)$ in all cases rather than $(blah)$. There are three problems with this, ranked from most important to least important in my opinion:
 * It's much harder to read in source form, particularly when nested.
 * Try expanding the font size - that might help.


 * It takes longer to type.
 * Cut'n'paste works here.


 * It requires different structure for $\left\{{a} \,\middle|\, {b}\right\}$ than for $\left\{{a:b}\right\}$. --Dfeuer (talk) 22:15, 8 January 2013 (UTC)
 * The house style remains unchanged. --prime mover (talk) 22:22, 8 January 2013 (UTC)


 * Even if it takes longer to type, it's only by a couple of seconds (for me). As for your last point, I think  does the job. --abcxyz (talk) 22:24, 8 January 2013 (UTC)


 * does not automatically size:


 * $\left\{{\dfrac a b \mid \dfrac c d}\right\}$ (yes I know it's meaningless mathematics)


 * as against:


 * $\left\{{\dfrac a b} \,\middle|\, {\dfrac c d}\right\}$


 * One of the main reasons behind the rigorous enforcement of \left{ ... }\right is purely because of the resizing. Cut'n'paste on a given construct is more straightforward, you don't then have to put the left-right in. The irritation about adding the spaces either side of the \middle is a nuisance, I agree - but it's minor because that construct is not often used. --prime mover (talk) 22:34, 8 January 2013 (UTC)


 * Y'all do know of the mightily useful option of right-clicking a piece of rendered TeX, going to "Show Math As > TeX Commands" to obtain a small window with the TeX generating that construct, no? (Admittedly I only found out one minute ago, but that's not the point.) --Lord_Farin (talk) 22:37, 8 January 2013 (UTC)


 * Occasionally use it, but (a) muscle memory has developed facility typing \left etc., and (b) when I'm editing a page with a lot of f \left({x}\right) etc. cut'n'paste is easy. --prime mover (talk) 22:42, 8 January 2013 (UTC)


 * Muscle memory FTW :). --Lord_Farin (talk) 22:43, 8 January 2013 (UTC)


 * Let's not get hung up on my third point, which is relatively trivial. $\left({ \left({ f \left({x}\right) + g \left({y}\right) }\right) h \left({z}\right) }\right) ^ 2$ is just inherently harder to read in source form (and thus to keep track of while writing) than is $((f(x)+g(y))h(z))^2$, and they look the same once processed. As for cut and paste, unless you're really quick with your custom emacs/vi macros, that's going to be slow too. I understand the motivation to make cutting and pasting easier, but just how often does one replace a standard-height expression in parentheses with a tall one? --Dfeuer (talk) 22:45, 8 January 2013 (UTC)


 * Not so much less often than one needs more than two levels of nested bracing (in TeX, that is; regular contributors know I can get pretty deep in my prose (such as this illustrative yet meaningless phrase)), I would say. While I do admit the paradigm is awkward at first, it works quite well (says this one, having been fully internalised into PW). I'm still not inclined to change my view - sorry if that comes across as stubborn. --Lord_Farin (talk) 22:53, 8 January 2013 (UTC)


 * Let's suppose you're writing $((f(x)+g(y))h(z))^2$ and you make a mistake miscounting the brackets (easy to do). Using your preferred style:  you don't miss the missing close bracket in $((f(x)+g(y))h(z)^2$ but with this:   it doesn't render: $\left({ \left({ f \left({x}\right) + g \left({y}\right) }\right) h \left({z}\right) ^ 2$.
 * Case rests. --prime mover (talk) 23:05, 8 January 2013 (UTC)

I personally get confused/annoyed by quantified expressions, disjunctions, etc., that aren't sufficiently parenthesized, but these become relatively deeply nested once parenthesized enough, at which point I find them unreadable under PW style rules. Take, say $\forall x \in A:((3(x+a)<4) \lor (x>17))$. However: I would frankly be much, much happier if we even had a very slightly laxer rule, not requiring innermost parentheses to be fancified, while requiring it for all others. As for the automatic error detection, as you're clearly using a fancy editor with fancy custom macros I'm sure your editor is quite capable of checking for mismatched parentheses. --Dfeuer (talk) 23:19, 8 January 2013 (UTC)


 * I'm not using a fancy editor with fancy custom macros. Those are for girls. --prime mover (talk) 23:21, 8 January 2013 (UTC)


 * On-topic, I think there may be a case for Dfeuer's suggestion. It seemingly combines the best of the two approaches. I pose the slightly adapted suggestion that innermost parentheses with a content of just one entity (like e.g. $x$, $\overline \R$ or $\operatorname{id}_X$) needn't be endowed with lefts and rights. --Lord_Farin (talk) 23:25, 8 January 2013 (UTC)


 * I disagree because then somebody's going to go through all the existing pages and change them. I don't think that would be helpful. --prime mover (talk) 06:10, 9 January 2013 (UTC)

onlyinclude
Is there any way to have different includeonly "classes"? Sometimes it would be neat to be able to have, say,

Theorem:

(onlyinclude class = theorem)

Detailed setup

(onlyinclude class=theorem-summary)

The good parts

(/onlyinclude)

Some less-good parts

(/onlyinclude)

Proof:

(onlyinclude class=proof)

Good, solid proving.

(/onlyinclude)

Miscellanea. --Dfeuer (talk) 23:10, 10 January 2013 (UTC)


 * Have you read the MediaWiki documentation? --prime mover (talk) 23:12, 10 January 2013 (UTC)


 * This annoyance was one of the main reasons for me to start developing an extension tailored to ProofWiki. Among other things, it allows precisely what you want. Cf. User:Lord Farin/Long-Term Projects/Extension. --Lord_Farin (talk) 23:15, 10 January 2013 (UTC)


 * If I were convinced of a need for such complexity I might be persuaded to come on board with this, but I'm sceptical. --prime mover (talk) 06:19, 11 January 2013 (UTC)


 * The need is clear. Sometimes you want to put together a list of theorems. Sometimes you want to put together a list of proofs. Say you have three theorems, each of which has two proofs. You want on page that transcludes the three theorem statements. You want three pages that each transclude two proofs. BLAM. --Dfeuer (talk) 06:34, 11 January 2013 (UTC)

Lax definitions
A question about complex analysis: all sources that I have consulted speak about a connected domain which they define to be "a path-connected open subset of $\C$". Some texts even omit the word "connected" and simply call the thing a domain. This definition is not in line with the topological definition of connected, where a connected set need not be path-connected, and certainly not open. Nor does it follow our definition of a domain. Of course, we could simply write "open path-connected set", but that would sound differently from the book texts. My suggested solution is to create Definition:Connected Domain (Complex Analysis), and then explain the lax definition in a note --- similar to what we have done with Definition:Smooth Path (Complex Analysis), which is not smooth. Thoughts? --Anghel (talk) 14:28, 11 January 2013 (UTC)


 * Sounds like an adequate solution to me. Ideally we would also loosen the assumptions on each theorem as much as possible, and then e.g. put corollaries for common stronger assumptions. This path, however, lies relatively unexplored at the moment, so maybe we needn't make it a priority. Having the results in the first place is good enough for the moment I think. --Lord_Farin (talk) 14:38, 11 January 2013 (UTC)

A Fond Farewell
Hello all,

As has probably been apparent from my gradual decline in editing/general participation over the last year or so, I've been finding it progressively harder to keep up with the shear volume of content and changes that have been going on in ProofWiki. I've finally decided that it isn't really beneficial to the site or to me to try to flip through the hundreds of edits a day (and that's after filtering out the ones I don't think will be important). In light of that, I'm going to step back from ProofWiki and let it move on without me. This doesn't necessarily mean that I won't add a proof here or there if I run across something worthwhile, but I won't be here on a daily basis anymore. If you need to get in touch with me about something, I'll still get email and my talk page will notify me of changes (to address the only specific question I've thought of that might come up: I have no objection to future changes in the specifics of the copyright/licensing stuff, as long as ProofWiki stays available to those who want it).

I've had a lovely four and a half years as a regular part of the ProofWiki community, and I've really enjoyed seeing all of the changes the site has been through and getting to play a part in them. It's almost ridiculous to look back on what the site was like that long ago; I thought just for fun I'd link to what the main page looked like when I joined:. Back then we were excited about putting up proofs one through six, and I never would have thought I'd still be here at proof 7000. I was only the 22 user to join, and I guess of the semi-regular contributors, only Joe has been here longer, and he helped start this whole thing.

I think I've grown up mathematically along with ProofWiki. Back in the day when I first joined, I was still in high school, just starting to learn what math was really like. Now I'm looking towards the all too rapidly approaching end of my time in college, and while I know a lot more, perhaps even more importantly I realize how much more there is that I don't know yet. ProofWiki has helped teach me about the need for rigor and clarity in mathematics, and I'm sure I owe no small part of what academic success I've had to the practice I've gotten as a part of this community. Sorry for the nostalgic mini-essay here, but I wanted to say a last goodbye and thank you to all of you for helping make this site the incredible repository of knowledge that it has become.

If I have one piece of advice I'd like to leave you all with, it's to stay open to new participants and the ideas they bring. Of course they won't always be good ideas, and disagreements will inevitably come up, but if you approach them with an open mind and welcome into the community as warmly as I was welcomed over 1,700 days and 2,000 edits ago, they might become a valuable part of ProofWiki, they might learn something from all of you, and you might just learn something from them. I'll no doubt be stopping back in to ProofWiki occasionally to see how it's coming along, but I'm confident that the group of all of you that I'm leaving behind will keep up the good work for many years and many proofs to come.

--Alec (talk) 20:24, 12 January 2013 (UTC)


 * *Applause* You were an excellent editor Alec. You shall be missed :') --Jshflynn (talk) 22:35, 12 January 2013 (UTC)


 * It's been good. If only there were a simple way to ensure the structural integrity of the site continues to be maintained ... --prime mover (talk) 23:34, 12 January 2013 (UTC)


 * Although we haven't been involved as much as you and PM undoubtedly have been, I still consider you one of the most valued editors of this web resource. It would appear that at this point, we're hitting the inflationary stage in the growth of the site (considering that I joined only 15 months ago and witnessed the 5000th proof), so that it's understandable that you can't keep up with it as you have the last few years. I hope we all do our best to make you proud and honour your piece of advice. Hope to see you around sometime as well. --Lord_Farin (talk) 00:36, 13 January 2013 (UTC)


 * So long, and thanks for all the fish! --Joe (talk) 23:56, 17 January 2013 (UTC)

LEM
We seem to have a template to warn about LEM. That's great. However, we also have a slew of pages that rely on LEM without saying so, or that use it without necessarily needing it. If someone wants to try to actually keep track of what does and does not need LEM, may I suggest that we add a "no-LEM" template for theorems known NOT to rely on LEM? That way, we'll be able to tell the difference between "does not rely on LEM" and "no one's checked if this relies on LEM". Personally, I don't care about LEM (unless I'm dealing with a computer language whose type system doesn't support it), but clearly some people do. --Dfeuer (talk) 10:38, 24 January 2013 (UTC)


 * Far too much work. Results that use LEM are restricted to a) those in propositional logic, and b) those which are specifically dependent upon it like one of the proofs of Gelfond-Schneider. The reason why there are plenty of results which do use LEM that are not specifically flagged as using LEM is purely because the template was created like a couple of days ago, and it is taking me that length of time to go through the entire category and refactor it.


 * Incidentally I'm interested to know which pages specifically use LEM without actually needing it - I thought I'd made sure that wasn't the case. Please point out all such pages.


 * Is it just that you want me to do this work to add that template to every single page in PoofWiki that doesn't rely upon LEM (I mean FFS that's thousands) so that I've got plenty to do and it will stop me interfering with what anyone else is up to? If so, then I quite understand and will get on with it straight away.


 * Oh, and if you don't care, why are you raising this issue? --prime mover (talk) 11:02, 24 January 2013 (UTC)


 * My view on the point is that it should be taken up, if so at all, by people that *do* care about LEM. I have the impression this group of people is disjoint from the group of active PW contributors. It is thus not likely to happen in the near future. --Lord_Farin (talk) 11:10, 24 January 2013 (UTC)

Template:BookReference
As my current state of mind does not permit expanding the mathematics on ProofWiki, I have taken on the project of template documentation and general documentation guidelines (again?, some may say). I have written a comprehensive technical manual regarding the use of Template:BookReference and intend to start producing pages for other templates and support pages (such as Help:Editing/Creating Pages for Mathematicians or whatever I decide to call it). I hope this effort will lighten the ongoing burden on some of the editors pertaining to documenting sources, and referring to them, as well as allow editors so inclined to learn by themselves how to perform such tasks. I hope the style and presentation are comprehensible (and appreciated). --Lord_Farin (talk) 13:42, 24 January 2013 (UTC)

Incidentally, my work has brought to me the need for an option to indicate which edition of a particular source is used. This will reduce ambiguity and allow for accurate display of how recent a source is. I was thinking of something like replacing the year - currently e.g. "(1976)" - by "(2nd ed., 1976)" if the second edition is referred to. I think however that it is not constructive to cover multiple editions of a work; the main intent is to allow for this added detail that makes it more accurate to refer to year of publication. I envisage this new trinket to gradually be incorporated and do certainly not desire a marathon of changing all sources (that'd be insane). --Lord_Farin (talk) 13:47, 24 January 2013 (UTC)


 * We will need to work out exactly what to do with multiple editions in due course. For a work like  which has multiple versions written by several people, it is probably worth documenting the different editions, although the reading trail will probably be done (if at all) for only one of these. For most books it is "probably okay" just to include one edition (naturally, the one on the contributor's bookshelf) but cite it with the date as for the first edition. It's not ideal, but it should work well enough; IMO the initial publication date is significant. Why? Because, among other things, it is interesting to see which concepts come into and go out of fashion. Some concepts are not referred to in books after, e.g. year m, while others do not appear until year n. And so on. You can get a glimpse into which works may have been used as source works for others, by how the flow in one directly parallels that in another. And so on. There is fuel for scholarly study here which may not have been gathered anywhere else. --prime mover (talk) 14:10, 24 January 2013 (UTC)


 * Then I think it's best to try and adhere to your stated rule for the moment and put the suggestion into the fridge until I've come up (if I ever will) with a way of ensuring both accurate reference and availability of year of initial publication. --Lord_Farin (talk) 14:40, 24 January 2013 (UTC)


 * I have finished the important parts of the manual - that is, the parts we use frequently. Comments are appreciated, before I proceed with replacing the current material. --Lord_Farin (talk) 14:44, 29 January 2013 (UTC)


 * I've been following it on and off - I'll give it a good workout presently, but it looks good - far more detailed than I would have the patience for ... --prime mover (talk) 16:35, 29 January 2013 (UTC)


 * Which is why I took several days of reduced contribution to write it up; the expected benefit helped a lot in finding the patience. Also, it's admittedly incomplete at certain points, but as stated, it's complete regarding the features most used and most relevant to newcomers. --Lord_Farin (talk) 18:32, 29 January 2013 (UTC)

Symbolic Logic: paradigm shift
The Axioms of Natural Deduction as defined here are basically taken from (it was the first book that came to hand when I was first studying this subject). However, further books I have read, in particular and  use a slightly different set. Other books use a (subtly) different set again, and so on.

So specifying a particular statement of PropCalc as "absolutely" axiomatic seems a bit previous.

What I am considering doing is changing the namespaces of all those results whose names are e.g Axiom:Proof by Contradiction and Axiom:Modus Ponendo Ponens and so on, merely calling them Proof by Contradiction and so on.

Then, in order to bring us back to the concept of how Natural Deduction works, the various source works which do specify a set of axioms are used as bases for a set of axioms each, and the "Axiom" namespace will thereby contain a master page for each such set, each one containing / transcluding the specific results as defined as "axiomatic" in that source work.

This will inevitably require links to specific proofs (as opposed to specific results) in order to prevent circularity within a particular axiom space, even though e.g. LEM proves DNE and similarly DNE proves LEM - either one can be chosen as axiomatic, or neither (some sources using Modus Tollendo Tollens as an axiom instead).

So basically, this is a heads-up that the good old PropLog section is up for another seismic shift again. The plan is for the all the axiom schemata to be unified in a homogeneous manner.

If any person can show just cause why they may not be joined together – let them speak now or forever hold their peace. --prime mover (talk) 21:42, 28 January 2013 (UTC)


 * Since I support this kind of organization across the whole site, you won't get any objection from me. --Dfeuer (talk) 21:52, 28 January 2013 (UTC)


 * This will be a marvellous project once complete. It will be nice and modular (or so I hope) so that we can reasonably easily accommodate new approaches which will undoubtedly arise from covering more sources. Because the task is so monumental, we need to be very careful to ensure a smooth transition from the one state to the other. I would like to use this operation as an excuse to rigorously separate the proof theory and model theory branches of PropLog (the PredLog section will follow once we finish PropLog) because in the future we will not only want different approaches on the proof theory side (which is what you try to allow presently) but also on the model theory side (Kripke semantics, categorical semantics); the latter are important when considering nonstandard logics. I am willing to help out - I fear that we are up for a huge rewriting, almost as if we were to start from scratch. --Lord_Farin (talk) 21:56, 28 January 2013 (UTC)


 * The exercise to separate out every small result is progressing well. (This is in itself important as it makes sure that we can adequately isolate the specific threads of implication from others, despite the obvious-looking commonality between, say, $p \implies \neg p \vdash \neg p$ and $\left({p \implies \neg p}\right) \implies \neg p$.


 * I have started by moving "Axiom:Proof by Contradiction to Proof by Contradiction because that's probably the most work-heavy one, and I can iron out all the technical issues up front. Then the others will be more straightforward to process. --prime mover (talk) 22:49, 28 January 2013 (UTC)


 * Right, I've burned myself out by refactoring the entire PropLog space by replacing all lines in tableau proofs with template invocations. Those pesky abbreviations have been replaced with full names as appropriate, and a consistent approach has been taken with respect to sequent introduction. I may have missed some, but it's mostly complete.


 * For my next trick I will start construction of axiom schemata according to the various works I have available (there are a few) and make an attempt to match up where they match and where they differ, with a view to synthesising something out of it. But that may not be today, or even tomorrow. --prime mover (talk) 21:36, 31 January 2013 (UTC)


 * While at it, and I know I've asked this before but it's important to me for future plans for dashes into Model Theory, could you please see to a rewrite of Boolean Valuations and the whole mess around it? --Lord_Farin (talk) 22:11, 31 January 2013 (UTC)


 * Point me towards it. Can't find any pages referring to "Boolean Valuation" so it must be under some other name. --prime mover (talk) 23:56, 31 January 2013 (UTC)


 * In particular, Definition:Boolean Interpretation, Definition:Boolean-Valued Function, Definition:Propositional Function and whatever may be around those (results as well). --Lord_Farin (talk) 08:12, 1 February 2013 (UTC)

The problem here is in how to define such an object in terms which do not rely on the language of set theory to do so - as set theory itself requires the definition of a propositional function to get it off the ground in the first place (via Axiom of Replacement or whatever the axiom is). (Yes I know ZFC is not the only game in town, I just haven't had access to any such materials, particularly sources expounding paradigms which do not require this fundamental dependency.) --prime mover (talk) 09:02, 1 February 2013 (UTC)


 * It is impossible to circumvent the distinction between naive and axiomatic set theory. We simply cannot start with nothing. This does not pose a real problem, since it is intuitively clear that we can talk about "sets" (well at least as long as stuff is non-circular - Cantor's "set" simply doesn't exist because it is self-referent). We can suitably develop model theory in this naive setting, introduce ZFC (or any other approach to set theory) and then exhaustively formalise every construction used in naive set theory (which until now hasn't posed a problem to the mathematical community) by treating the concept "set" as "element of a model of ZFC". This leaves only the development of model theory as subject to discussion and possible confusion.
 * That said, my main problems with said stuff are due to the rather ad hoc presentational style which appears at best a bit dislodged from the rest of the site. A not worthless distinction would be that between set and meta-set (the latter corresponding to the naive notion), but since I haven't encountered that in any of the literature I've seen I don't consider that a good idea. Of course, once the whole thing is set up one can nest the construction, considering models of ZFC in a model of ZFC, rinse and repeat - this does however not add anything to the rigour IMO. The current Definition:Set is pretty good for a description of a "naive" set; since also Definition:Mapping is accurate, we should be fine. It's really only the presentation of said section that is up for improvement (the other pages could be amended with an elaboration on axiomatic set theory but that's for a later project). --Lord_Farin (talk) 09:51, 1 February 2013 (UTC)


 * I will make an attempt. Bear in mind that the PropLog section was among the work done on ProofWiki, and as such the style and direction had not completely crystallised. If the issues you have are primarily presentational, then feel free to experiment with those areas you have issues with. The Natural Deduction section will continue to be reskinned independently of the model-theoretic work, although it may yet be appropriate for the "Boolean Interpretation" and "Truth Function" sections of the e.g. Definition:Conjunction pages to be extracted into subpages, with a view to transcluding much as the Definition:Truth Table page has been presented.


 * I confess to being uncertain still as to exactly how to proceed in that area - so for now will just continue to work on the Natural Deduction stuff. --prime mover (talk) 10:09, 1 February 2013 (UTC)

Complex Analysis: assignment of responsibility
Anghel has now been put completely in charge of developing the Complex Analysis field on this site until further notice. Any amendments made by other collaborators are to be restricted to stylistic and cosmetic amendments only (of which few are expected to be necessary).

If you have any major questions regarding the direction of the documentation of this field of mathematics, particularly if you need the existence of a particular result or definition, place a comment on the relevant Talk page.

Anghel: please feel free to post your thoughts on this arrangement. --prime mover (talk) 14:57, 31 January 2013 (UTC)


 * I do not want that responsibility. I've stated my reservations here. Also, if I should drop dead tomorrow (or get a 9 to 7 job, which is equally likely), you would have a problem. Please, delete this notification from the talk page. This wiki should remain open. --Anghel (talk) 17:09, 31 January 2013 (UTC)


 * While I'm all for someone being " in charge" of a section, I think this should only be as a maintainer, not only contributor. Any user should be able to add anything they like. --Joe (talk) 17:14, 31 January 2013 (UTC)


 * I'm interpreting it like Joe. It is a good idea to let someone have the overview over an entire section so as to ensure consistent notation, terminology, etc. The site benefits from someone who knows the way in a field, what has and hasn't been covered, what inroads are up next and generally up to which level of complexity the whole coverage has advanced (which is a laughable level for most of the site - let's try and do something about that!).--Lord_Farin (talk) 18:34, 31 January 2013 (UTC)


 * The thing is this. When somebody is working methodically through a section of work, it is usually not a good idea for someone else at the same time to be looking at what is going up and deciding that the structure needs to be changed. There are always different ways of putting these things together, and if one person has a vision of something that works, then it is a bad move for anyone else to change things to suit the way they want it. As this potential danger appeared to exist at the time, I posted up what I did. --prime mover (talk) 20:46, 31 January 2013 (UTC)

Lord_Farin's backup, etc.
I just looked at the double redirects page, and I see that there are a number of redirects from Main into Lord_Farin's backup. Structurally, I don't think there should be links of any kind from Main into User. If a Backup area is necessary for pages moved from Main, I think it should go under Main. On the other hand, what are the chances of external links to material so incomplete or shaky that it had to be taken out of Main? --Dfeuer (talk) 19:58, 31 January 2013 (UTC)


 * Leave them, please. It's on my to-do list. --prime mover (talk) 20:41, 31 January 2013 (UTC)


 * Those moves were made because all the terminology and the whole approach to lattices and boolean algebras was being made rigorous. This obsoleted the results, but I wanted to keep them accessible in some way, hence the moves. --Lord_Farin (talk) 22:11, 31 January 2013 (UTC)

A proposed resolution of the naming problem
We are confronted time and time again with the problem that some perfectly valid results suffer from the lack of a properly descriptive name for them. There are usually two solutions practised to this issue: to post the result with a very inadequate or even misleading title; and to not post the result.

Both of these are really unacceptable and (while it be noted that I somehow feel a bit of credit to Jshflynn is due here - however, I couldn't locate the source of this thought) the solution that occurred to me just tonight is to effectively give them a number, something like "Lemma N". And while this is obviously not an elegant solution, it will allow us to post up, and refer to, results that haven't been assigned a proper name yet. Of course, as soon as a good and reasonably concise title is proposed, the page is to be redirected to that proper name.

Furthermore this will instigate the need of an overview page, listing all such "Lemmas" as well as good categorisation both on the Lemma pages and on the overview page (a bit like the Books section perhaps). I repeat: this isn't very elegant but it does take care of it, for now. More thought on how to keep the Lemmas accessible (so that there won't be five lemmas with the same content) would be appreciated. Finally, I'd appreciate any critique to be supported by arguments - rational ones, not purely emotional blubber. --Lord_Farin (talk) 19:38, 1 February 2013 (UTC)


 * I'm not sure what I think about the idea in general, but if we do it then the titles should have both numbers and a description, such as Lemma 5:Condition for Potato being Fried, even if the actual condition is too hard to describe. The current practice with such seems to be just to call it Condition for Potato being Fried, which is terribly arbitrary in that there might be various other conditions under which the potato is fried. --Dfeuer (talk) 19:46, 1 February 2013 (UTC)


 * I fail to see that it's a big problem in the first place - but as I've already been a proponent of using numbers in the recent work on PropLog I would be insane to say I objected to the numbering technique. As for providing them with consistent descriptive names I don't think that's important. Anyone who wishes to use a lemma that contributes towards the proof of such-and-such a proof will probably be reading that proof in the first place.


 * Certain people are rather more than normally concerned with naming, categorising, boxing, labelling and strangling the entities on this website (while being unconcerned about the presentational quality of such entities) and IMO such over-concern is not necessarily healthy for the site.


 * And if someone comes up with a better name, then good, we can rename. --prime mover (talk) 23:08, 1 February 2013 (UTC)


 * I have had some issues with results defying description to a point where I simply decided not to put them up. In many cases this applies to e.g. examples satisfying certain conditions subtly stronger than a certain definition or equivalent conditions for a def in some specific, hard to state context. I have generally decided not to cover those things in the past. Admittedly the first of the two current solutions isn't much of a problem. The second manifestly is.
 * Perhaps "Unnamed Result N" is better than "Lemma N" but that's up for discussion. It's not so much that I want to rename things, it's just that I want to have a means to let people post results they can't make up a title for. And with the lack of a title, the categorisation becomes quite important for already stated reasons. The "descriptive names" could instead be supplied by an explanatory line at the lemma collection page accompanying the link to said lemma. --Lord_Farin (talk) 23:36, 1 February 2013 (UTC)
 * In my suggestion, the descriptive name is optional but encouraged. Condition for Tent to be Leaky, Consequence of Tent being Leaky, Relationship between Two Notions of Leakyness, etc. --Dfeuer (talk) 23:40, 1 February 2013 (UTC)


 * But if you have a descriptive name for the page you don't need to number it. L_F: can you provide a solid concrete example so we know what we're talking about? If necessary post up one /some of those results and we can then discuss naming at that point. --prime mover (talk) 06:16, 2 February 2013 (UTC)

From Schilling, Exc. 6.11 (paraphrased):


 * Let $\Sigma$ be the Sigma-Algebra of Countable Sets on $\R$. Let $\mu$ be the co-countable measure. Let $\mu^*$ and $\Sigma^*$ be the outer measure and $\sigma$-algebra appearing in the proof of Carathéodory's Theorem. Describe $\mu^*$ and show that no $B \in \Sigma^*$ is such that both $B$ and $\R \setminus B$ are uncountable.

I have refrained from covering it. It exhibits non-measurable sets on this $\sigma$-algebra. It clearly defies concise description as $\Sigma^*$ is an involved construct relying on $\mu^*$ as well. --Lord_Farin (talk) 10:53, 2 February 2013 (UTC)


 * Okay ... thought needed. "Nature of Outer Measure of Co-Countable Measure on Sigma-Algebra of Countable Sets on Reals" and "Sigma-Algebra by Carathéodory's Theorem has no Uncountable Subset whose Complement in Reals is Uncountable"? --prime mover (talk) 11:15, 2 February 2013 (UTC)


 * Actually, $\Sigma^*$ arises in the proof of C's theorem in Schilling. There are undoubtedly other proofs avoiding it (since this $\sigma$-algebra is not canonical - it is larger than necessary). Still, even now my remark "defies concise description" stands. Also, this was merely an example; I've had a lot of these (also ones that weren't exercises but genuine results). --Lord_Farin (talk) 12:02, 2 February 2013 (UTC)


 * Maybe conciseness is the criterion which needs to be sacrificed? --prime mover (talk) 12:29, 2 February 2013 (UTC)