User talk:Dfeuer/Archive/Prehistory

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I note that you're enthusiastically banging in the BPIT template all over the place in pages which in several cases already have AoC etc. templates in them. I have reservations about this for several reasons:

a) It has not completely been established exactly what BPIT actually is. Yes, there is a partial proof in place, but it might or might not work, but at the moment it is unfinished and, ultimately, incomprehensible because we don't even know what a prime ideal is yet.

b) Places where the BPIT already exists do not indicate exactly where it is used, and why it is relevant, and what the point is. If you look at existing pages using AoC (where the work has been done properly, of ocurse) you will see that there is a specific point in the proof where it is invoked, or if not, in the use of the template there is an indication of which result in that proof does use the AoC (e.g. "... by Zorn's Lemma" etc.). The BPIT work does not.

c) Even when we have sorted all that out, the philosophical significance of BPIT remains obscure. It's couched in technical language about a collection of objects that need considerable work to understand, and even when you do understand them all, it's like: so what? The AoC is a short, pithy fact that can be put into a single sentence's thought and therefore means something. The BPIT in consequence just looks like something clever-sounding that we read off the internet and include because it makes us look smart. Personally I don't think it *does* make us look smart.

Thoughts? --prime mover (talk) 06:24, 14 December 2012 (UTC)

Most set theory is pretty strange, but the set theorists keep making it anyway. The Boolean Prime Ideal Theorem is really no stranger than Zorn's Lemma. It apparently has several forms, some of them simpler than the one I started the page on, which all turn out to be equivalent (somehow), the simplest of which that I've seen is the Ultrafilter Lemma. It appears to have been studied extensively at least since the 1960s (and the Ultrafilter Lemma since the 50s), so there's plenty of source material out there—it's not some new experimental axiom that only a handful of specialists have heard of. Personally, I enjoy how it's helping me explore set theory some more to try to understand why different bits relate to each other in certain ways. I don't see how acknowledging on a page that a proof from weaker principles is possible and we'd like to have one makes us look stupid. As long as ProofWiki is a work in progress and not a finished product to be shown off in a library that seems fine to me. --Dfeuer (talk) 07:45, 14 December 2012 (UTC)
It would be a good idea to at least indicate that we are still looking for proofs from e.g. BPIT. This can be accomplished by amending the template. Upon establishing more relevant context for the BPIT it will be torn out of obscurity; and if not, there's always the statement that it is a consequence of AC so that any willingly ignorant visitor can content himself with accepting AC and not bothering. For the interested visitor, who may be pleased by accepting a varied assortment of axioms these sections contain valuable and interesting information. There should be more rather than less, IMHO. --Lord_Farin (talk) 12:14, 14 December 2012 (UTC)
What might be useful is a "overview" page like we have for, for example, Definition:Separation Axioms in which the various axioms and their relative strengths are presented and explained briefly. Extracting that brevity may be a challenge. In the meantime I recommend we complete the train of thought which goes towards the proof of BPIT (including the definition of the various objects - Prime Ideal comes to mind) and make an attempt to justify the claims that are made in its name. Finally, there are various pages which state "This theorem depends on BPIT" without any indication as to which particular point in the proof it is needed. IMO this is of paramount importance.
Demonstrations of relative strength and weakness of the axioms should ultimately be illustrated by examples of objects / constructs which are satisfied by one axiom but not another, proving that while A --> B there are objects that are B but not A. Again, this will be challenging, but should ultimately just be a literature search.
I understand the drive to raise the reader's awareness of these things - they fit into the strategy of where we want ProofWiki to go - but without making a solid attempt to justify the underlying foundations there's a danger of the information on this site becoming disconnected. In the extreme would fall into danger of becoming an alternative wikipedia - a conglomeration of cherry-picked interesting / important / noteworthy theorems with no solid underpinning. --prime mover (talk) 14:23, 14 December 2012 (UTC)

Page deletion

For the record if you want a page deleted, please invoke the "delete" template. --prime mover (talk) 09:16, 16 December 2012 (UTC)

Thank you

Thank you for recovering my extension of the Schur's Theorem (Ramsey Theory). That was very nice of you, and skillful!

BTW, why can't one edit a fragment only, why this necessity of editing the totality of the text?! It's ridiculous. Can it be true?!

Best regards, Wlod (talk) 02:27, 26 December 2012 (UTC)

I refer you to Help:FAQ#Why the extra blank line? --prime mover (talk) 09:41, 26 December 2012 (UTC)

Supremum of Suprema

I replaced your work with essentially a new page. This was easier for me at this hour of the day than the tedium of rewriting your sentences. No offence intended - it was simply quicker. --Lord_Farin (talk) 23:31, 2 January 2013 (UTC)

I don't generally mind, but could you explain what made mine so horribly awful that it needed such treatment? Also, I'm not a big fan of mathbb for general purposes (it tends to make me think some "standard" set like $\mathbb C$ is intended. Would you mind switching to mathcal or mathscr? --Dfeuer (talk) 23:36, 2 January 2013 (UTC)
Nothing wrong with mathbb. It looks a lot better than mathscr, which is nearly as bad as fraktur for impracticality. Reverted to mathbb. There is a precedent - several pages already use mathbb for subsets of the power set of a set. --prime mover (talk) 06:17, 3 January 2013 (UTC)
Would you mind if I switched it to mathcal? --Dfeuer (talk) 06:22, 3 January 2013 (UTC)
I tried mathcal but $\mathcal T$ is too similar to $T$. --prime mover (talk) 06:26, 3 January 2013 (UTC)
Changed to mathscr per suggestion. Changed it because I felt uncomfortable with the choice of letters and other symbols, and the amount of detail skipped. On a different time of day I might have simply tidied your version. Nothing special. --Lord_Farin (talk) 23:50, 2 January 2013 (UTC)
Thanks --Dfeuer (talk) 23:45, 2 January 2013 (UTC)
Unrelatedly, we need infimum of infima, and we need right distributive implies left distributive, and all results about ideals and prime ideals in ordered sets need their duals for filters and ultrafilters in ordered sets (which are related to, but not the same as, filters and ultrafilters on sets, and blah blah. There's got to be some way to deal with these duals without duplicating all the text and then struggling to keep them matched s they're edited. Even the cop-out approach of having a proof do little more than link to another strikes me as more practical.—Dfeuer (talk)
This stuff is pretty high on my wish list for new schemes. I have a feeling that we can do little more than put up both results because they may arise in very different contexts. If you have ideas on this I'd be very eager to hear about them - don't worry too much about practicality at this point. --Lord_Farin (talk) 23:49, 2 January 2013 (UTC)
Well, I already mentioned the impractical option of writing these theorems as templates parametrized by the relevant sensitive words, but I suspect that would lead to much confusion. --Dfeuer (talk) 23:52, 2 January 2013 (UTC)

Category and links

I understand about your physical infirmities, and I appreciate the fact that the page Strictly Positive Integer Power Function Unbounded Above‎ is barely started yet, but as a matter of courtesy to other editors it is good to get into the habit of adding the category for every page you write, as you start it rather than as an afterthought at the end, and also, as you proceed through the writing of it, to link to each concept as you go. --prime mover (talk) 06:26, 3 January 2013 (UTC)

I don't have terrible physical infirmities, but I sometimes have technological ones, and sometimes I don't know the right category to put something into. --Dfeuer (talk) 06:27, 3 January 2013 (UTC)
If you don't know the category to put something into, then I suggest you explore to see what categories are available. If nothing else it will increase your familiarity with this site, and if you plan on sticking around it's more than just a good idea that you get to grips with how the site is structured - and indeed what is already up on the site. As for "technological restrictions", I suggest you get access to a computer with a decent sized screen and a proper keyboard. Then you'll be able to use accessibility tools to expand the size of the font so you'll be able to see it better. --prime mover (talk) 06:31, 3 January 2013 (UTC)


It is suggested that work which is in process of being developed be done in your sandbox. This can be set up as the page User:Dfeuer/Sandbox. Only when your work is complete and you have a page which is presentable (by this meaning that the argument is complete) would it then be deployed to the wiki proper. (See User:GFauxPas/Sandbox for a model instance of how these things are used.)

This does not mean to say that a "stub" page should not go onto the wiki as a placeholder (this is perfectly acceptable practice), but so that a whole series of half-developed proofs do not clutter up the history page, and potentially confuse or irritate users of this site. Basically, it keeps it neater all round.

It would also be hoped that the links would also be complete by this time, and (dare I say it) the house style would be adhered to.

Please feel free to evaluate the above to determine whether it fits in with your own strategy for contribution towards this website. --prime mover (talk) 09:59, 3 January 2013 (UTC)

Notwithstanding the above (which contains valid nudges towards being a better contributor), it be mentioned that we highly value the time you devote to ProofWiki. --Lord_Farin (talk) 16:19, 3 January 2013 (UTC)
Thanks, Lord_Farin. I just created User:Dfeuer/Product of Positive Element by one Greater than One. Do you think you can check the math and suggest a better name? Or is this theorem already on ProofWiki somewhere I haven't found yet? --Dfeuer (talk) 16:23, 3 January 2013 (UTC)
Proof correct; a better (since more consistent with other pages) name may be "Product of Positive Element and Element Greater Than One". There is a lot of house style violation left on that page - you may want to try to sort that out yourself, though. --Lord_Farin (talk) 18:08, 3 January 2013 (UTC)
I did some tidying and moved it to the main space. Next up: User:Dfeuer/Strictly Positive Power of Element Greater than One Not Less than Element
Again, proof correct, style not so. Please try to tidy and add a tag before moving to main. --Lord_Farin (talk) 19:20, 3 January 2013 (UTC)


No, that's not what we do. We wait for one of them to do something illegal.I don't see why it would make you nervous. In the grand cosmic scheme of things it's a low-priority concern. Deleting someone's account just because they don't use it is fascism. --prime mover (talk) 06:06, 9 January 2013 (UTC)

Reflexive Reduction of Ordering

It would appear that we can do a better job than introducing $<$ as the reflexive reduction of $\le$ every time. I'm thinking along the lines of a definition for this concept specifically tailored to orderings. Maybe Definition:Strict Ordering of Ordering. --Lord_Farin (talk) 10:28, 9 January 2013 (UTC)

I'm open to suggestions. I've been trying hard to cut down on the boilerplate, but I may well have missed opportunities. The trouble is that whether you're basing it on a compatible relation or an ordering, you still need to invoke the theorem that the "strict ordering of ordering" or the "reflexive reduction of the relation" is compatible if the original relation/ordering is compatible. --Dfeuer (talk) 10:34, 9 January 2013 (UTC)
An alternative, of course, is to prove the ordered group theorems from preceding ordered group theorems, rather than from the compatible relation theorems. That's less boilerplate, but more actual repetition. --Dfeuer (talk) 10:35, 9 January 2013 (UTC)
It was accommodation to the reader's intuition ("... surely $<$ is the strict ordering corresponding to $\le$? Why don't they call it that?" which could easily lead to "This site sux, I'm outta here.") that made me propose this. Indeed, it does not resolve reference issues but it does allow to use one name for a concept. Such allows for reading and searching convenience. --Lord_Farin (talk) 10:41, 9 January 2013 (UTC)
Gotcha. I'll be glad to work on that, but I should have been asleep hours ago..... --Dfeuer (talk) 10:43, 9 January 2013 (UTC)


You've surpassed me in number of edits over the last thirty days. I think such hasn't occurred since I started to contribute significantly, approximately one year ago now. Keep it up :). On a side note, it'd be interesting to see how many characters a user contributed rather than how many edits. However, that metric is not available atm... Who knows what the future brings? --Lord_Farin (talk) 22:51, 9 January 2013 (UTC)

Thanks, Lord_Farin! I don't think it's the most meaningful metric, but I'll take what I can get! On the other hand, this use of the word "metric" doesn't match the mathematical one, so we should probably use another name. Measure, perhaps? --Dfeuer (talk) 04:44, 10 January 2013 (UTC)
The name "measur" doesn't match "the" mathematical one either. "Metric" is perfectly good, it's just a case where the word has tw separate meanings. I suggest that as this is an informal use of the term we do not atm add a disambiguation page. --prime mover (talk) 06:09, 10 January 2013 (UTC)
I do hope you realize I was joking. --Dfeuer (talk) 06:10, 10 January 2013 (UTC)
No I didn't. It wasn't obvious. --prime mover (talk) 11:20, 10 January 2013 (UTC)

New categories

You have enthusiastically created a bunch of new categories with associated definition categories. Two remarks on that:

  • I had intentionally put lattices and bounded lattices together because they have a huge overlap. Arguably that was a bad call on my part, so I've chosen to go with the way you organised things.
  • Putting e.g. Definition:Distributive Lattice only in the category Category:Definitions/Distributive Lattices is not the correct practice. You may note that many categories do not have an associated def category. This is because the def category is intended to be used for definitions specific to the notions the main category is about.

I may have more points occurring to me later, but that's it for now. --Lord_Farin (talk) 09:27, 10 January 2013 (UTC)

Please keep in mind prime.movers comment about creating virtually empty categories (especially definitions categories). Some of them may well be superfluous because there are no (well-known) concepts defined specifically for a certain structure. --Lord_Farin (talk) 11:20, 10 January 2013 (UTC)

I'll bear that in mind. It does seem likely sensible for definition categories. I guess I got a bit over-eager. --Dfeuer (talk) 11:24, 10 January 2013 (UTC)
Useful constructs around generation of categories are Template:DefinitionCategory, Template:LinkToCategory, Template:SubjectCategory, Template:DefsLink. I hope most of them have some explanation on the page. --Lord_Farin (talk) 11:26, 10 January 2013 (UTC)

On Definition:Distributive Lattice

I have finally found a quite satisfactory and good-looking approach. See Equivalence of Definitions of Distributive Lattice as well. If you agree, I'll delete the pages on join and meet distributing (I've already flagged them for deletion). --Lord_Farin (talk) 16:13, 11 January 2013 (UTC)

double redirects

I have gone through and cleaned up all the double redirects except those from your own (and Lord_Farin's) pages - this is to alert you that you might want to sort them out. --prime mover (talk) 07:50, 17 January 2013 (UTC)

Argh! I've tried to fix those as I move pages. Sorry I've missed some. I hope there weren't too horribly many. --Dfeuer (talk) 07:51, 17 January 2013 (UTC)
You're all right, just a handful. You can find them via the "Special pages --> Double redirects". --prime mover (talk) 08:10, 17 January 2013 (UTC)


Could you take a look at the new section of User:Lord_Farin/Sandbox? Literature seems to distinguish between Boolean lattice and Boolean algebra (with resp. without ordering) and is mostly vague on the connection between the two. I posted a connecting definition at Definition:Boolean Lattice and want to verify that it is correct before proceeding further with the refactoring (and, incidentally, resuming with Givant/Halmos). TIA. --Lord_Farin (talk) 22:27, 17 January 2013 (UTC)

I can look at the math, but I can't tell you if the definitions are standard. If the only difference is whether or not you impose the natural ordering, that hardly seems worth worrying about. --Dfeuer (talk) 22:30, 17 January 2013 (UTC)
It's the math; your sentiment about the ordering seems to be bon ton among authors. --Lord_Farin (talk) 22:32, 17 January 2013 (UTC)
Incidentally, it provides a really nice way of recovering the lattice ordering (since neither distributivity nor complements are used in the proof). Might consider adding that to Definition:Lattice (and it's a quite nice conclusion of my pondering "what can we say of an ordering compatible with $\vee$?"). --Lord_Farin (talk) 22:34, 17 January 2013 (UTC)
I had some trouble figuring out what was going where. As soon as you have a semilattice you have two orderings. If you have two semilattices on the same set that obey the absorption laws, then if you label one a "join" and the other a "meet" semilattice, they induce the same ordering and you have a lattice. I thought we'd already dealt with how you recover the ordering...--Dfeuer (talk) 00:02, 18 January 2013 (UTC)

Category theory

You expressed contemptuous ignorance of category theory in a post to another contributor soon after your joining this project. Now you're making fundamental architectural changes based on your noticing the similarity between the words used to define two completely different concepts. Give over, already. Can you undo all the work you have done in this area? I really have better things to do with my time. --prime mover (talk) 06:12, 18 January 2013 (UTC)

Fundamental architectural changes? Most of what I did was put definitions of different kinds of isomorphisms into a category. I would frankly be astonished if I miscategorized more than three pages in total. I need to go to bed now. --Dfeuer (talk) 06:20, 18 January 2013 (UTC)
There's a good pageful of changes that relate to what you've done in this area. --prime mover (talk) 06:21, 18 January 2013 (UTC)


Please curb your enthusiasm for creating categories for entities until they have at least been defined - and preferably until there are enough things to go into them. Is there really a need for a category called "topological division rings"? --prime mover (talk) 06:01, 30 January 2013 (UTC)

Obviously, that category is unlikely to have many things in it that won't go in topological fields. But if we have topological fields and not topological division rings, then any stray topological division ring theorems will get stranded in the topological rings category, which is too general for them. One might argue, then, that we should instead drop the topological fields category instead of the topological division rings one. But then that's crazy. So I don't really know. I don't feel very strongly about it. What I am much more concerned about is that the Continuity category is structured upside down. There should really be a "continuous functions" category as a subcategory of both "analysis" and "continuous mappings", but instead all the continuous functions are in Continuity, and Continuous Mappings is a subcategory of that. --Dfeuer (talk) 06:09, 30 January 2013 (UTC)
Even so, my request stands. Please refrain from creating categories until there is a need for them.
"One might argue, then, that we should instead drop the topological fields category instead of the topological division rings one." Certainly - that sounds like an excellent idea. Drop them both, till it has been demonstrated that there exists a need for either.
"I don't feel very strongly about it." In that case you will see the wisdom in following the suggestion (which has in fact been made to you before) to hold back on your category creation activities.
I concede that the Continuity category may be considered by some to be suboptimal, but I don't see it as "crazy". As it is, it's convenient. I just see it as something which has evolved in a particular direction. Categories are groupings of convenience only. Bad move to structure them into a form that makes them difficult to find things in. If there are too many categories it becomes difficult to find stuff in them.
And there should be no need for "continuous functions" to be a subcategory of "continuous mappings" - a function is a mapping. How confusing and pointless would that be? --prime mover (talk) 06:21, 30 January 2013 (UTC)
I personally see no reason whatsoever to distinguish mappings from functions, but since that is a point on which you stand very strongly I think we need to fit it into the structure properly. What we have now is that continuous functions go in Category:Continuity, and continuous mappings that are not about numbers go in Category:Continuous Mappings, which is a subcategory of Category:Continuity. My own view is that the "natural" structure is something more like this:
  • Topology > Continuous Mappings > Continuous Functions
  • Analysis > Continuous Functions
Why should I have to dig deeper to get to more general theorems about continuity? I think if we are to have a Continuity category at all, it should be to cover any notions of continuity that may or may not exist (I don't know) that don't correspond to topological continuity. --Dfeuer (talk) 06:28, 30 January 2013 (UTC)
"if we are to have a Continuity category at all" - feel free to delete it if it causes you pain. --prime mover (talk) 09:15, 30 January 2013 (UTC)

feelings of being ridiculous

"I feel ridiculous every time I use the name Definition:Continuity/Topology/Open Sets."

What you need to do, when you find a definition which has / in them, is to do a "what links here" to see what pages redirect to the page in question. If there is more than one, use the most popular one (it's probably superseded a less optimal one).

In this case, the link you want is Definition:Continuous Mapping (Topology). --prime mover (talk) 09:05, 30 January 2013 (UTC)

Why not MOVE Definition:Continuity/Topology/Open Sets to Definition:Continuous Mapping (Topology) and let the one with fully two slashes in it be the redirect to this incredibly important definition? --Dfeuer (talk) 09:07, 30 January 2013 (UTC)
So that the subpage structure is not violated. Simple. --Lord_Farin (talk) 09:10, 30 January 2013 (UTC)
Here's a big problem with the structure as it stands: this key concept can be defined in MANY ways (any topology book will have a collection). Open sets, closed sets, bases, subbases, various things about closures and interiors, etc. Our current primary name is "open sets", which doesn't actually describe all of these things, and does not gracefully leave room for multiple equivalent definitions. --Dfeuer (talk) 09:14, 30 January 2013 (UTC)
I agree that there is ample room for improvement here. I don't think that moving the page to somewhere else is improvement; how that will be approached is not up to me - basic topology isn't as close to my heart as some of its subfields. --Lord_Farin (talk) 09:18, 30 January 2013 (UTC)
In order to gather all the various threads together that define "continuity": in the various fields (topology, metric spaces, the various number fields and (arguably) most importantly on the real numbers, a certain amount of compromise had had to be accepted. It is important that all the definitions are available to be accessed immediately from a "common" core page from which all definitions are presented. It is also necessary that for individual fields of mathematics (e.g. real analysis) a definition of continuity is available directly (therefore the redirect page from, for example, Definition:Continuous Real Function to Definition:Continuous Function (Real Analysis), which itself has subpages definition subtleties of definition). This is how it is, and it has evolved into that format through four years of gradual refinement, via various techniques which have been tried and have failed, to what we have now which is perfectly adequate.
Continuity is usually defined in a topology text by its "open set" property. This is easy to state and completely unambiguous and rigorous. The fact that it can equally well be defined based on closed sets is equally valid, but rarely done as a primary definition because topologies are themselves "usually" defined based on open sets. And the fact that they too can also be defined in terms of closed sets is equally straightforward, and there is indeed a link to where they are so defined and that the definitions are mutually equivalent.
There is indeed room for multiple definitions of continuity in topology, and in due course all these can and should be added, and proofs be developed to demonstrate their equivalence.
But the original question was that linking to Definition:Continuity/Topology/Open Sets made you feel "ridiculous". Stong word that - in other words you think you're being laughed at. I am at a loss to understand what your problem is: if you don't like that particular definition, why not link to one of the definitions that you do like? What is wrong with using Definition:Continuous Mapping (Topology)? This is after all (as has been pointed out by L_F on another thread and in another context) mandatory in order to comply with house style.
If you particularly need to access a different definition of the concept then there should be no difficulty in constructing the actual definition that you need - but in the context of the basic definitions you are putting together that should not be a concern. --prime mover (talk) 09:52, 30 January 2013 (UTC)

categories again

You're doing it again. Please refrain from creating categories for things that do not yet exist. I mean that. It's not just an idle whim. Your personal predilections are in danger of reducing the usability and therefore quality of this site. --prime mover (talk) 09:55, 30 January 2013 (UTC)

On a more explanatory note, the approved method (when not advancing into a completely uncovered field of mathematics - *not* a subfield) is to first populate categories, then as they have grown sufficiently, the operation of creating subcategories may be started with. In that light, your efforts on separating results about continuous functions from the Category:Continuity are valued, while your creation of new and empty categories is not. I confess to have created some of these in the past as well, but I have come to the conclusion that it's not a good idea. --Lord_Farin (talk) 10:00, 30 January 2013 (UTC)

If you folks are talking about the uniform continuity category, there are loads of pages up already that need to go into it. --Dfeuer (talk) 10:06, 30 January 2013 (UTC)

My response is intended to apply in general, at least the first sentence. Please take it to heart. --Lord_Farin (talk) 10:07, 30 January 2013 (UTC)
So is the category "uniform continuity" or "uniformly continuous mappings"? And is it general, or is it specifically for metric spaces? Methinks this can do with being thought through a little bit more - and if you get obvious redlinks on your usage of the templates, suggest you need to go back and fix what you did wrong before continuing. As it is there is a danger of fragmenting the categories so much that material is going to get seriously lost. --prime mover (talk) 10:11, 30 January 2013 (UTC)
Sorry, I got that mixed up. For now I want uniformly continuous mappings and uniformly continuous functions. --Dfeuer (talk) 10:12, 30 January 2013 (UTC)
Or should I call that first one Category:Uniformly Continuous Mappings (Metric Spaces) to leave room for uniform spaces? --Dfeuer (talk) 10:15, 30 January 2013 (UTC)
Not at this point. If and when we get to it, a refactoring operation will be instigated. --Lord_Farin (talk) 10:16, 30 January 2013 (UTC)
I reiterate: the "correct" assignation of categories is not a primary concern. Too many fragmented categories causes it to be difficult to find stuff. Please stop creating categories. Whether they are or are not useful is at this stage beside the point - the fact is your judgment on this matter is up for question. --prime mover (talk) 10:22, 30 January 2013 (UTC)