User talk:Jochen Burghardt

Subsumption lattice

Hello Jochen, You replied in the correct way. As for the supposedly incorrect review, the page that you mentioned in your message to me does contain similar language, however that does not mean that it is the correct way to write an article. The reviewer of the article may have been mistaken. Good luck for any future articles you propose! Thomas85753 (talk) 12:15, 22 August 2012 (UTC)

Hello again,

Try Mass-energy equivalenceThomas85753 11:06, 23 August 2012 (UTC)

Hi Jochen, I moved your article into the main namespace, after some minor copyediting. Cheers, —Ruud 13:26, 4 May 2013 (UTC)
Many thanks!!! I'm a bloody novice among Wikipedia authors and felt unable to meet Thomas85753's critics. Jochen Burghardt (talk) 19:02, 9 May 2013 (UTC)

A belated welcome!

Sorry for the belated welcome, but the cookies are still warm!

Here's wishing you a belated welcome to Wikipedia, Jochen Burghardt. I see that you've already been around a while and wanted to thank you for your contributions. Though you seem to have been successful in finding your way around, you may benefit from following some of the links below, which help editors get the most out of Wikipedia:

Also, when you post on talk pages you should sign your name using four tildes (~~~~); that should automatically produce your username and the date after your post.

I hope you enjoy editing here and being a Wikipedian! If you have any questions, feel free to leave me a message on my talk page, consult Wikipedia:Questions, or place {{helpme}} on your talk page and ask your question there.

Again, welcome! (talk) 08:48, 7 June 2013 (UTC)

"Size" for cardinality

Hi Jochen, I agree with the point you made. "Number of points" needs to be avoided when talking about infinite cardinalities (even with the quotes). I chose "size" to replace it because that seemed to be the alternative used elsewhere in this article. It is certainly not a perfect choice and not one that I would normally use in my own writing, but for consistency it seems to be the best choice. While it is possible that some readers could confuse size and length, it is unlikely that anyone who does would be sophisticated enough to understand the meaning of a one-to-one correspondence. Thank you for the addition. I didn't check to see if you created the image, but if you did I would suggest that you increase the line width (to 2 or 4 pixels) otherwise the colors are too pale and the point you are making with the colors is weakened because they are hard to discern. Bill Cherowitzo (talk) 04:06, 8 June 2013 (UTC)

punctuation

Please see my edits to Anti-unification (computer science).

right: pp. 74–83
wrong: pp. 74-83

Ranges of pages, years, or other numbers, or of letters of the alphabet, use an en-dash, not a hyphen. This is codified in WP:MOS. Michael Hardy (talk) 21:02, 30 June 2013 (UTC)

Table floating layout

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In the articles

I used the "float: left;" style parameter to arrange several tables in a nice way. However, I'd like the text following the tables to be ordinarily left-aligned (as would be usual in the absence of floating tables) rather than floating around the tables. In particular, the 2nd article ("Word_problem ...") looks very ugly now - its "See also" section should begin below the tables rather than right to them.

How can I achieve that? I didn't find any appropriate hint in the Help:Table article. Many thanks in advance. Jochen Burghardt (talk) 17:11, 11 August 2013 (UTC)

What you want is the {{clear}} template. I have added it to those articles. However, Confluence_(term_rewriting)#Motivating_examples still doesn't really look good to me, and I wonder whether it wouldn't be better to put the group axioms and all the small proofs into a large table to neatly arrange them. Huon (talk) 18:34, 11 August 2013 (UTC)

Thumbnailing animated GIFs

There's not trick to it, and it's not even intentional that the sieve illustration on Prime number is not animated in the thumbnail. It's a feature of the MediaWiki software: It does usually create animated thumbnails, but if the overall image size exceeds a certain configurable limit, it only generates a still-image thumbnail. The overall image size is calculated from the geometric extents of the image but also the number of frames in the animation; so if you have a relatively large image and a relatively long-running animation (as in this case) there is no way to make Wikipedia create an animated thumbnail. Presumably this is so in order to reduce server load, because creating animated thumbnails can potentially require much computing time on the server—or so I would guess. --SKopp (talk) 10:41, 13 October 2013 (UTC)

Cantor's diagonal argument

Hi Jochen. I'm happy to see that you are editing Cantor's diagonal argument. In the past, I felt that the article could be improved but I only made changes to one section.

However, I think you may have made a common error in your exposition of Cantor's 1891 proof. Here's how Cantor's proof is presented on page 823 of Gray, Robert (1994), "Georg Cantor and Transcendental Numbers" (PDF), American Mathematical Monthly, 101.

3. CANTOR'S DIAGONAL PROOF. We now turn to Cantor's 1891 article [9], which contains his well-known diagonal proof. Cantor begins by discussing his 1874 article. He points out that it contains a proof of the theorem: There are infinite sets that cannot be put into one-to-one correspondence with the set of positive integers. Then he asserts that this theorem has a much simpler proof than the one given in 1874. His new proof uses the set M of elements of the form E = (x1, x2, … , xν, …), where each xν is either m or w. Cantor states that M is uncountable, and notes that this result is implied by the following theorem:
If E1, E2, … , Eν, … is any simply infinite sequence of elements of the set M, then there is always an element E0 of M which corresponds to no Eν.
Cantor proves his theorem by using the diagonal method to construct E0. Note that, once again, Cantor states a theorem that separates the constructive content of his work from the proof-by-contradiction needed to establish uncountability.

Of course, you may want to check Cantor's original article to make sure the above exposition is correct.

The above quotation was not written for Wikipedia readers and assumes the reader can fill in the proof-by-contradiction: Assume that the set M is countable. Then its elements can be written as a sequence Eν. Applying Cantor's theorem to this sequence produces an E0 that does not belong to the sequence. This contradicts our original assumption, so M must be uncountable.

I believe there are several reasons why Cantor's constructive theorem should be mentioned:

• It is historically accurate.
• It is important for the reader to understand that the diagonal method is constructive. After all, Gödel used it to construct an unprovable sentence in number theory. Also, Turing's answer to the Entscheidungsproblem uses the fact that the diagonal argument can construct computable numbers. (Turing constructs a computable real number that does not belong to a sequence of computable real number.)
• It shows readers how Cantor separated the constructive content of his work from his proofs-by-contradiction. Another example of this can be found in Cantor's first uncountability proof.

Once again, thank you for working on Cantor's diagonal argument, and I look forward to your edits on this and other articles. --RJGray (talk) 00:50, 8 December 2013 (UTC)

Hi Robert(?) J. Thank you for your elaborate and convincing explanation. When editing the article for the first time, I just felt its presentation should be improved, but didn't pay attention to the separation of constructive and indirect proof. Following your hint, I (only then) checked Cantor's original article, and found your(?!) 1994 AMM paper in perfect agreement with it. I tried to adapt the wikipedia article accordingly, also using your above text as basis for the indirect proof.
Reasons for some deviations were: "Binary digits" are more common than Cantor's original "m"/"w" symbols (he might have in mind the german "männlich"/"weiblich" = "masculine"/"feminine" - ?), maybe a "0"/"1" version of the picture should be used for consistency. I kept the set name "T" and sequence name "s" found in the article, where Cantor used "M" and "E", maybe I should adapt this. I used "i" as general index to avoid Cantor's Greek "ν" (less easy to understand). I used "enumeration" (of sequences) for a list of members of T and "sequence" (of bits) for a single member of T to ease reference to, say, vertical and horizontal coordinates in the diagonal argument's matrix. I tried to rephrase your indirect argument text in conjunctive form to emphasize the speculative nature.
Maybe each of the two proof parts should have its own subsection, but I couldn't yet find appropriate headings for them.
Best regards - Jochen Burghardt (talk) 11:37, 8 December 2013 (UTC)

Hi Jochen. Your rewrite is excellent−I especially like your clear and concise way of expressing the constructive and indirect aspects of Cantor's work. On the issue of changing T to M and s to E, I see no reason for changing them or for using Cantor's m and w. My MAA article (yes, I did write that article) is an historical article that contains quotations from Cantor's article, so it was best for me to use Cantor's notation. In the article Cantor's first uncountability proof, I used xn rather than Cantor's 1874 ωn notation. However, I suggest that you replace the index i with n since sn and "nth digit" is easier to read than si and "ith digit". Also, the section "Real Numbers" uses n as an index.

As for the picture with m and w, I think they can be changed to 0s and 1s. Also, the E can be changed to s (and renumbered to start at 1). Then you could make the first 7 elements in your proof identical to the first 7 elements of the picture so the picture would be an illustration of your proof.

I think that the two proof parts are fine together in one section. Since the uncountability result is an application of Cantor's constructive theorem, I regard it as belonging to the same section since it shows how Cantor's theorem can be used. As for writing the proof-by-contradiction in subjunctive form, I think this is a matter of taste. In English, the subjunctive is not used very much, so I tend to not use it. The section Square root of 2#Proofs of irrationality does not use the subjunctive, but the article Proof by contradiction does.

I like your addition of graphs to the section "Real Numbers". Since this section is about bisections, I don't think it's a question of whether they distract from the main theme of diagonalization, but a question of whether they enhance the "Real Numbers" section (which they do). However, I think some readers may find the left picture confusing since it does not illustrate a bijection from (0, 1) to (−π/2, π/2). Instead, it illustrates a bijection from (e1, e2) to (s1, s2) where e1, e2, s1, and s2 are positive.

You are doing excellent editing and it's a pleasure to communicate with you.--RJGray (talk) 02:19, 9 December 2013 (UTC)

Hi Robert. Thank you for your compliments. As you suggested, I changed i to n, and adapted the picture (however, I don't know yet how to produce Svg from LaTeX, so the Jpg thumbnail looks somewhat noisy). Concerning the "Real Numbers" section and its pictures, I meanwhile thought about using a rational function (as shown in File:Bijective map from interval (0,1) to R.gif; the function looks like being strictly increasing, but I didn't prove it)) to avoid tan and function composition, hence also to save one image. On the other hand, the current approach, using "components off the shelf" is more typical for mathematicians, so maybe we should keep it (I'd adapt the linear map image to the proper intervals in that case). I'm quite indecisive about that; what do you think? Best Regards - Jochen Burghardt (talk) 12:52, 10 December 2013 (UTC)

Hi Jochen. One small experiment for your illustration of Cantor's argument: Try removing the commas (and perhaps spacing the 0s and 1s closer together?). Then it may be simpler visually, and it's an illustration so you don't need the precision of the commas. I don't know how to produce Svg from LaTeX myself, but I noticed that the original illustration was created on Inkscape, which can be downloaded free from inkscape.org. Also, I noticed that you labeled the resulting sequence as sn rather than s.

As for the pictures in "Real Numbers", I suggest either doing one picture using the composite function tan(πx - π/2) or replacing the current linear one with πx - π/2. I feel that the most important thing about illustrations is that they should agree with the text. By the way, the text uses tan(x) because it's a common way to get a bijection from an open interval to R. Also, most readers should be familiar enough with tan(x) to realize that, restricted to (-π/2, π/2), it is a bijection (your tan picture will help readers here). Keep up the fine work! --RJGray (talk) 02:54, 11 December 2013 (UTC)

Cantors erster Überabzählbarkeitsbeweis

Hi Jochen. I'm happy to see that you are editing Cantor's first uncountability proof. Have you read the German version de:Cantors erster Überabzählbarkeitsbeweis? It does not follow Cantor's original approach of a constructive theorem followed by a proof-by-contradiction. It also contains a non-constructive proof of the existence of transcendental numbers instead of Cantor's original proof. In fact, the German version seems to have come from the original English version of "Cantor's first uncountability proof".

I am currently working on a French translation of "Cantor's first uncountability proof". Do you have any interest in doing a German translation? I think that you would do a great job, you are a native German speaker with excellent editing abilities. Of course, I understand that the article is a bit long with all its footnotes, but I think it would be great for Cantor's original approach to appear in the German Wikipedia since he was a German mathematician. --RJGray (talk) 02:30, 15 December 2013 (UTC)

I'm not sure I want to get involved with German wikipedia, where a lot of rules, templates, etc. are quite different. I'll think about it. - Jochen Burghardt (talk) 12:45, 16 December 2013 (UTC)

Sfrac template renders slash and horizontal line in mobile view

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In the article section Hoare logic#Conditional rule, I used the "sfrac" template to create a natural deduction-style rule (consisting of a horizontal line, some formulas above it, and some below it). The rendering looks fine in the desktop view. However, in the mobile view, an additional "/" is shown immediately above the horizontal line. I guess the "/" is shown due to a programming error in the template code, but I don't have a clue how the template code works. Many thanks in advance. - Jochen Burghardt (talk) 20:00, 13 January 2014 (UTC)

This is how sfrac looks like in desktop view (it is ok):

```{B ∧ P} S {Q}    ,     {¬B ∧ P } T {Q}
--------------------------------------
{P} if B then S else T endif {Q}
```

This is how sfrac looks like in mobile view (the "/" should not be there):

```{B ∧ P} S {Q}    ,     {¬B ∧ P } T {Q}
/
--------------------------------------
{P} if B then S else T endif {Q}
```
Thanks for your attention to this detail. For the relevant people to see it, please, ask at the mobile front-end feedback page or file a bug like linked at the top of the page. Thanks. ☺ Gryllida (talk) 20:24, 13 January 2014 (UTC)

Galois connection reference

I notice that in this edit you added a reference to one of your own papers. Personally I don't have a problem with that, but some editors don't like it. It's probably safest to suggest it at the article talk page first. Deltahedron (talk) 20:51, 1 February 2014 (UTC)

Thanks for your advice; I followed it. - Jochen Burghardt (talk) 21:13, 1 February 2014 (UTC)
I'm sure it will be OK. Deltahedron (talk) 21:17, 1 February 2014 (UTC)

Converting citation to cite doi

Please do not convert {{citation}} to {{cite doi}}.

• They have different formats: citation uses commas as separators, while cite doi uses periods.
• Citation allows author names to be spelled out, while cite doi requires them to be given using initials (this requirement is enforced by bot editing)
• Citation works with {{harv}} referencing, while cite doi in its default behavior does not.
• Changes to citatations made using citation are visible in watchlists, while changes to the separate pages made by cite doi are not, causing cite doi to be much less resistant to vandalism.
• Citation is much more flexible, while cite doi requires incompatible citation templates to be used for references that do not have dois.
• Per WP:CITEVAR, "Editors should not attempt to change an article's established citation style merely on the grounds of personal preference, to make it match other articles, or without first seeking consensus for the change."

David Eppstein (talk) 19:06, 16 February 2014 (UTC)

Hello David Eppstein. I naivley assumed that having a central site for e.g. an journal article would be valuable in any case. I therefore created a couple of "cite doi" pages, and replaced articles' references by ((cite doi)) templates. After I saw your reverts, I immediately stopped that activity. Please apologize the inconvenience. I guess, ((cite journal)) shouldn't be changed to ((cite doi)) either? Is there any acceptable way to centralize reference data, without causing problems like those you described above? - Jochen Burghardt (talk) 19:25, 16 February 2014 (UTC)
Here is a complete (except for those already reverted by you) list of articles I changed an the above way:
• citation-->cite doi (meanwhile all reverted):
• cite journal-->cite doi (I'll revert them too, if you wish):
• no template-->cite doi (I'll revert them too, if you wish):

- Jochen Burghardt (talk) 20:14, 16 February 2014 (UTC)

Thanks! I'll let you decide for yourself which of these you want to undo, but I think of cite journal to cite doi as being less of a problematic change — a couple of my objections to cite doi above are still valid (the ones about watchlists and author initials) but their formatting is pretty much interchangeable with each other. I would love to have a centralized database system like BibTeX where one could just refer to a reference by name and have it formatted automatically in whatever variation is appropriate for the article but I don't think we're there yet. —David Eppstein (talk) 20:15, 16 February 2014 (UTC)

Boolean Axiomatics error?

Hi Jochen, I'm not a WP regular, but I think I spotted an error (perhaps a typo or copy/transcription error) in an edit you made to http://en.wikipedia.org/w/index.php?title=Boolean_algebra_%28structure%29&oldid=571924694

In the edit, you very nicely added some proofs of basic laws from axioms. The error I think I spot is in the proof for Huntington's A1 theorem. The code "XIb" does not seem to refer to any previous theorem or axiom, and a quick google search didn't turn up anything to explain what "XIb" might mean. By looking at the step being justified, however, I believe the correct justification reference/code should be "Abs2". I'm going to make that edit now. If I'm wrong, feel free to revert. ~ RH — Preceding unsigned comment added by 24.57.4.82 (talk) 20:00, 1 April 2014 (UTC)

Hi RH, you are perfectly right, "XIb" should be "Abs2" - many thanks for recognizing and correcting this! The "XIb" originated from [Huntington, E. V. (1933), "New sets of independent postulates for the algebra of logic" (PDF), Transactions of the American Mathematical Society, American Mathematical Society, 35 (1): 274–304, doi:10.2307/1989325, JSTOR 1989325], where Huntington assigns on p.277 the numbers "Xa" and "Xb" to the absorption laws. Additionally to not replacing "Xb" by "Abs2", I had confused "Xb" with "XIb". - Jochen Burghardt (talk) 20:53, 1 April 2014 (UTC)

I have not "added copyrighted material" to the Aho–Corasick string matching algorithm article. Apart from minor layout edits, I only added a link to a pdf file. As far as I understood, WP:COPYLINK explicitly allows this (else, I certainly wouldn't have added the link):

Since most recently-created works are copyrighted, almost any Wikipedia article which cites its sources will link to copyrighted material. It is not necessary to obtain the permission of a copyright holder before linking to copyrighted material,...

If you still think my addition was illegal, please give a proper reason. If not, please undo your deletion. Thanks in advance. - Jochen Burghardt (talk) 20:17, 21 April 2014 (UTC)
I agree that this message was badly phrased. However, the issue is whether the PDF file that you linked to was itself a copyright violation: if you know or reasonably suspect that an external Web site is carrying a work in violation of the creator's copyright, do not link to that copy of the work. In this case it is certainly not clear that the web site owner was the owner of the copyright in that page. Deltahedron (talk) 21:17, 21 April 2014 (UTC)
You have linked to an apparent copyright violation. Just because you can find the article somewhere on the internet does not mean that copy is legal. There are no problems with citing copyrighted material or linking to authorized copies of the material. But you need to be sure the copy is authorized.
The added link was to a copy of a printed journal article. The article's first page clearly states "Copyright © 1975, Association for Computing Machinery". If you follow the doi link in the reference, you will see that the publisher, ACM, is still selling the article. The link you provided appears to be in the home directory for somebody named Watson and does not appear to be either Aho or Corasick (or even Bell Labs). There is no evidence that the link is an authorized copy. Glrx (talk) 23:12, 21 April 2014 (UTC)

Robert Recorde

Hi, just wanted to stop by to apologise for the edit I made on Robert Recorde. I should have given an edit summary explaining that I used another citation to reference the statement you had (quite rightly) highlighted. However, instead of pressing preview, I hit save. So, sorry about that. I shall endeaver to be more careful in future. All the best, Daicaregos (talk) 12:23, 6 May 2014 (UTC)

A barnstar for you!

 The Original Barnstar Inductive logic programming is amazing to me. I found this page and the related pages a complete revelation. I had just been thinking, "I wonder if this is a way of creating theory's/hypotheses that explain conditions, in a way similar to resolution". Thanks for your explanations. Thepigdog (talk) 08:50, 15 May 2014 (UTC)

Deductive lambda calculus

Thanks for your requests for clarification in Deductive lambda calculus. I have responded to your requests.

All criticisms and comments always appreciated.

Thepigdog (talk) 05:38, 19 May 2014 (UTC)

Bach audio edits

Hi Jochen, thanks for your efforts to add audio to lists of Bach's works. However, on the English Wikipedia we're generally quite conservative about when audio is added to a classical music article ... the sound file must be faithful to the original composition, be performed on acoustic instruments (in almost all cases), and must have appropriate licensing. That's why I've undone your additions of the audio files. We have a template specifically designed for music files (among others), {{listen}}. Graham87 15:00, 15 August 2014 (UTC)

Ok, I see. I quite naively thought it would be a good idea to have an overview page where each (available) file can be played in one click. I didn't think much about the subtleties you mention above. I was too bold this time - sorry. - Jochen Burghardt (talk) 15:09, 15 August 2014 (UTC)
And thanks for cleaning up my mess. According to my Contributions page, all my Bach edits are undone now. (I didn't touch other composers' pages.) - Jochen Burghardt (talk) 15:14, 15 August 2014 (UTC)

Speedy deletion nomination of Wayne Snyder

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You may want to consider using the Article Wizard to help you create articles.

A tag has been placed on Wayne Snyder requesting that it be speedily deleted from Wikipedia. This has been done under section A7 of the criteria for speedy deletion, because the article appears to be about a person or group of people, but it does not indicate how or why the subject is important or significant: that is, why an article about that subject should be included in an encyclopedia. Under the criteria for speedy deletion, such articles may be deleted at any time. Please read more about what is generally accepted as notable.

A barnstar for you!

 The Original Barnstar For work on Conjunctive normal form. Thepigdog (talk) 01:19, 9 November 2014 (UTC)

a small modification to Schröder–Bernstein theorem - update

You deleted my addition with the Statement that the proof does not use equivalence classes. I was not clear enough in my writing.

here is another go:

two elements in the set (A union B) relate to each other if and only if they belong to the same sequence. The proof called the equivalence classes that are created by this equivalence relation as a "sequence".

I just wanted to note that to complete the proof you don't have to find the bijection explicitly ,only show that for each sequence, the cardinality of elements of A is equal to the cardinality of elements of B. But this is trivial: if the sequence is infinite then the cardinality is countable infinity (for elements of A and of B). if the sequence has a finite number of elements then it must have an even number of elements : and therefore the number of elements of A is equal to that of B.

no need to build the bijection. no need to consider so many special types of sequences.

thank you for your time — Preceding unsigned comment added by 31.210.187.4 (talk) 13:53, 28 January 2015 (UTC)

My reason for deleting was that your text suggested ("Written concisely") to be a summary of the proof immediately before it, which it was actually not. My criticism wouldn't apply if you're going to present an alternative proof.
However, I still don't understand you above text:
1. If a sequence is infinite, why should "the" (i.e. its?) cardinality be countable infinity? After all, no restriction is made about A and B.
2. The notion of cardinality is usually defined after the Schröder–Bernstein theorem has been proven. The validity of the theorem is a prerequiite for the notion being sensible. In particular, the theorem is not about A and B having same cardinality in the first place, but about a bijection existing between A and B.
-Jochen Burghardt (talk) 20:25, 28 January 2015 (UTC)

Thanks for the quick response!. I get it now. let me rephrase with consideration of your comments: when you look how each sequence is constructed you can see that I have an element from A then B then A again and so on... . by the construction of the sequence itself it is clear that I can index the elements of the sequence - in otherwords there is a bijection from the sequence to the natural numbers. for "doubly infinite sequences" the simplest bijection is to the integers.

and it is clear that the even(or odd) indexes belong to elements of B and the odd(even) indexes belong to elements from A.

A bijection from even to odd natural numbers(integers) is trivial.

Notice that I did not claim anything about A or B only that the sequences are countably infinite/finite by construction.

If the sequence is finite my original argument (the number of elements in the sequence is even) stays.

About "prerequite for the notion (of cardinality) being sensible" : I just defined sets with equal cardinality with bijections. this "relation" between sets is an equivalence relation. I consider this introduction to infinity as good enough to use the term cardinality for a proof of this theorem. — Preceding unsigned comment added by 132.70.66.14 (talk) 23:18, 28 January 2015 (UTC)

For now, just a counterexample where an uncountably infinite sequence occurs: let A be the first uncountable ordinal in von-Neumann representation, let B=A\{0}, let f:A-->B be the successor function, and g:B-->A be the identity. Then there is exactly one sequence, which is an A-stopper (since 0 has no predecessor), and contains all members of A and of B; it hence is uncountably infinite. - Jochen Burghardt (talk) 23:57, 28 January 2015 (UTC)

Thank you very much. — Preceding unsigned comment added by 132.70.66.14 (talk) 07:58, 29 January 2015 (UTC)

Concerning your introduction of cardinality: The approach I've learned is to define two relations, say ≡ and ≤, on the class of all sets, with the intended intuitive meaning has the same cardinality as, and has cardinality less or equal than. The relation ≡ is defined via the existence of a bijection, and ≤ via that of an injection. It is straight-forward to show that ≡ is an equivalence relation, and that ≤ is reflexive and transitive. The anti-symmetry of ≤ (modulo ≡) is the difficult part, this is just the Schröder–Bernstein theorem. After that, i.e. when ≡ and ≤ has been shown the be an equivalence and a partial ordering, respectively, the notion of the cardinality of a set makes sense.
If I understood you right, your way would be to introduce ≡, prove that it is an equivalence, then introduce a (seemingly) weaker notion of cardinality (e.g. ℕ and ℝ would still have different cardinalities by Cantor's diagonal argument, but you couldn't tell yet which set is "larger"), then use that notion in the Schröder–Bernstein proof, then define ≤, state its ordering property, and establish the usual notion of cardinality. That may be a feasibly way, too. It seems, however, you'd have to establish that in the finite case your notion of cardinality coincides with the "obvious properties of counting" (reasoning about "even number of elements" etc.), and to solve the problem countable vs. uncountable in the infinite case. I don't yet see that his way would turn out more elegant than constructing the bijections directly. - Jochen Burghardt (talk) 11:58, 31 January 2015 (UTC)

I dont think you counter-example works : for the successor function to reach all elements you need a limit step in addition to n+1 step or successor step: the first uncountable ordinal contains the first countable ordinal+1 (omega plus one or second countable ordinal). and the successor can't reach the last elements in omega+1.

Ooops, you are right - sorry. No limit ordinal has a predecessor, so every limit ordinal is the start of an own sequence in my example. I'll have to think about this. - Or can you prove that each sequence has at most countably many elements? - Jochen Burghardt (talk) 17:11, 1 February 2015 (UTC)

I don't think I can present the proof in a better way then what is now. I'll need induction to prove that a sequence is countable infinite. maybe its worthwhile to add a comment to the proof about the "sequences" and call them as they are, if you think it adds a bit more understanding to the proof of this theorem. — Preceding unsigned comment added by 132.70.66.14 (talk) 18:19, 2 February 2015 (UTC)

equals sign

I thought that using the math "equals" sign in the Equation article made it a bit bigger and clearer than the version you used. But what the heck. That's not worth fighting over. However, in the context, the sign should certainly be between quotation marks.DOwenWilliams (talk) 21:10, 1 April 2015 (UTC)

I don't prefer a particular size of the sign; also, the quotation marks are fine. I'd just like to keep the link. Would you accept "${\displaystyle \scriptstyle =}$" ? (Observe that clicking on it links to another article.) - Jochen Burghardt (talk) 05:45, 2 April 2015 (UTC)

Philosophy of mind

You're very welcome! I'm always curious as to how different values of column width render in various accessibility situations. One guide that helps with this is found in Reflist's template documentation and probably could use some tweaks. There are still a lot of "hard" columnization in articles, so feel free whenever you see a "2" or "3" in a References or Notes section to change it to "20em", which so far seems to be the best. And I've found that from "15em" to "20em" is also good for See also columns, maybe smaller like "10em" if there is a Wiktionary or other template used. Thank you! and Best of everything to you and yours! – User:Paine Ellsworth 16:58, 17 April 2015 (UTC)

Hi, I don't habe any wikipolicy to quote, but I think that 20em columns are a bit narrow for rendering on desktops. I think the width I've most commonly seen is 30. It still renders well on tablets and it makes the references easier to read. T.Shafee(Evo﹠Evo)talk 08:58, 21 June 2015 (UTC)

I made some screenshots of Point mutation#References on different devices with different settings ("PC": firefox in a 1280x800 window on my PC, "TABLET": on my 8 inch screen diagonal tablet, "desktop": Wikipedia display option "desktop view", "mobile": Wikipedia display option mobile view). I've put a ruler showing centimetres at to bottom to indicate the appearance in the real world. Note that with "35em", the references are unreadable in the tablet in Wikipedia's "desktop" setting (font below my eye's resolution, lower mid image) as well as in "mobile" setting (right column off screen, lower right image). This is the reason why I changed the "35em" to "20em", as recommended by Paine Ellsworth above. Maybe Wikipedia's column rendering algorithms should be improved - on the other hand, I don't really see what problems people have with "20em" on a PC, the upper right screenshot looks fine to me. - Jochen Burghardt (talk) 13:18, 21 June 2015 (UTC)

PC (desktop) TABLET (desktop) TABLET (mobile)
20em 150px 150px 150px
35em 150px 150px 150px

I think 20em is awfully narrow for usual full references (full = includes title and date and author etc etc). 30em is the most common width and it looks great to me. It's not really something that I want to edit war over but I remember that I have reverted a few of your edits where you changed the column width to 20em with the explanation that it looks better so on tablets. I'm sorry but I think normal computers, i.e. desktops and laptops, should be prioritized. If you want to keep changing these to 20em then you should gain some sort of a consensus for that first. (Changing 35ems to 30ems though shouldn't be a problem, I think.) — Jeraphine Gryphon (talk) 13:43, 21 June 2015 (UTC)
At first, many moons ago, I began changing "3" (the usual number of columns I found in the Refs sections) to "30em". I don't remember who the editor was, perhaps it was Jochen(?), but I was told that the tablet they used did not render 30em very well at all, and that 20em worked much better for them. So I've been using 20em ever since. As this is an important accessibility issue, I would advise all to be as sensitive as possible to the needs of others. If this means that 30em is better for some but not all, and those some can live with 20em even though it's not quite as good for them as 30em is, then to be accessible to all, the choice should really be 20em, or perhaps slice it down the middle to 25em. Since I abhor edit wars, I would ask that everyone continue to stay on talking terms and not resort to such things. Thank you very much and Best of Everything to You and Yours! – User:Paine Ellsworth  13:56, 21 June 2015 (UTC)
We can't decide anything final here, this should really be discussed at a more proper venue, like the Village Pump maybe. — Jeraphine Gryphon (talk) 13:58, 21 June 2015 (UTC)
30 em (not 35) has been the standard for columns for quite some time (as far back as I can remember, in fact). If someone wants to change that, the discussion should indeed be centralized and not on one user's talk page. --Randykitty (talk) 14:01, 21 June 2015 (UTC)
I agree with Randykitty - 30em is the standard that I've always seen. 20em is too narrow. 21:14, 22 June 2015 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── As another piece of relevant information: I've just done some tests with the point mutation page and the see also section with 35em columns renders fine on my tablet (MS surface 3 in both chrome and firefox) and mobile (Samsung galaxy note 3). The columns just re-flow to a single column that fits itself to the width of the screen. What system are the images you posted from? T.Shafee(Evo﹠Evo)talk 04:01, 23 June 2015 (UTC)

My tablet is a TrekStor SurfTab ventos 8.0, running Android 4.1.1; the browser app is just called "Browser version 4.1.1-eng-root.20130502.193714". I couldn't find out more about it. Do you think it is a problem of the browser rather than of the Wikipedia page rendering algorithm? - Jochen Burghardt (talk) 14:03, 28 June 2015 (UTC)
Hmm, I've tried on a couple of other phones (iPhone 5, HTC One) and found that multiple columns are still flowed into a single column of the width of the screen with no obvious errors. You might be right that the TrekStor's browser is rendering the page weirdly. If you happen to come by a different device, could you test it out? T.Shafee(Evo﹠Evo)talk 23:28, 29 June 2015 (UTC)

Need help with a few diagrams

Hi Jochen,

I'm rewriting Cantor's first uncountability proof, which was nominated for a Good Article but failed because the editors who looked at it found problems with it. The editors did give excellent feedback, which I'm using for my rewrite.

I would greatly appreciate some help from you. Because it will take me at least a couple of months to do the rewrite, I'm in no rush. My problem is that I don't know how to make diagrams, and I'm too busy with the rewrite to learn. I remember the excellent diagrams you did for Cantor's diagonal argument in a proof I had written. Here are my diagrams in ASCII (please ignore the periods—I used them because Wikipedia shrinks all spacing to one space):

——(————|—————|——)———

aN .......... c ............... xn .... bN
Case 1: Last interval (aN, bN)

——|————(———|—————)——

xn .......... an ....... a ............. bn
Case 2: a = b

——(———[——|–——]————)———|—

an ...... a ... c ...... b .......... bn ....... xn
Case 3: a < b

A draft of the rewritten section is at User:RJGray/The proofs. Just look for the 3 cases the proof has. You can experiment with the page; I've set it aside for you. Also, any comments you have on the section will help me. Thanks, RJGray (talk) 20:26, 1 August 2015 (UTC)

Hi Robert, the pictures are no problem; I can provide initial versions of them in the next days. I guess each variable should be exactly below the corresponding v-bar or paranthesis, right? If I have modification suggestions for the page, may I make the changes there? - Jochen Burghardt (talk) 20:56, 1 August 2015 (UTC)

Hi Jochen, Glad to hear that you can do the pictures. Yes, each variable should be exactly below the corresponding parenthesis, or v-bar. I've played around with the ASCII pictures a bit more trying to make them more similar to each other. It seems that if the (an, bn) intervals were in the same position on each picture, it would be easier for a reader to see the differences between them, especially if we stacked them on the right of the page. I came up with this idea when I was converting my ASCII drawings into tables as an experiment to see how they would look to the user. I have added them to User:RJGray/The proofs for you to see. Also, any suggested modifications you have, just make them to the page. You and I will be the only ones working on this page. Thanks for your help. RJGray (talk) 13:51, 5 August 2015 (UTC)

Hi Jochen. Since you may be making suggestions to the rewrite I'm working on, I added the new lead and "The article" sections that I have also rewritten to User:RJGray/The proofs to give you the context. Feel free to make changes anywhere you want. Thanks - RJGray (talk) 17:43, 5 August 2015 (UTC)

Engaging in argument on talk pages

Hi Jochen,

I'm sure this edit was well-meant, but in my opinion it's not a good idea. I'm not perfect myself and occasionally yield to temptation, but I try to keep it on extremely technical pages where there's a reasonable chance that someone could actually benefit from some little-known exposition.

On problematic pages like the one on the diagonal argument, the halting problem, the incompleteness theorems, etc, we really need to hold the line and direct the querents to the refdesk. If you give them an opening, it can open up a thread that's very difficult to shut off, and the talk page becomes less useful for its intended purpose. --Trovatore (talk) 19:32, 13 September 2015 (UTC)

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Columns

In response to this edit summary, the fact that there is a bug in page rendering in some circumstances (which is what the screenshots clearly show) should not stop editors from applying the preferred formatting. Rather, the fact that such layouts are used should provide impetus to the developers (either of the website or the browser) to fix the problem. One could certainly file a bug report if the issue is not already known, but we should not be pre-emptively removing column formatting because of a side-effect in a subset of browsers. --Stemonitis (talk) 13:27, 3 February 2016 (UTC)

Thank you for your help on "Georg Cantor's first set theory article"

Hi Jochen, The article Georg Cantor's first set theory article is now up! Thank you again for all your help. To read my public thank you, go to Talk:Georg Cantor's first set theory article#The article rewrite and thanks to all those who helped me. Thanks, RJGray (talk) 01:16, 15 February 2016 (UTC)

Minor edits in equivalence relations

Hi Jochen,

The minor edits I made were converting the mathematics part in Equivalence Relations to proper math notation so that it will render more correctly (and beautifully). I have made no alterations in content.

I don't understand your point of reverting those edits. Would you like to explain?

Rushikeshjogdand1 (talk) 19:11, 15 March 2016 (UTC)

I left an answer on Talk:Equivalence relation#Math style. - Jochen Burghardt (talk) 22:04, 15 March 2016 (UTC)

Congratulations on being updated to extended confirmed user

Hi Jochen, Congratulations and what a coincidence! I was updated to extended confirmed user ‎just 28 minutes before you. The article Georg Cantor's first set theory article is doing fine. Editors have just started to make changes (only 2 so far and the first was reverted by the editor who did it). One of my lead sentences was deleted--it was flawed so I'm rewriting it. I'm happy that people are reading it, and when they edit, they are supplying informative comments. Thanks again for your help on the article, RJGray (talk) 18:23, 6 April 2016 (UTC)

AfC notification: Draft:Lottie Louise Riekehof has a new comment

I've left a comment on your Articles for Creation submission, which can be viewed at Draft:Lottie Louise Riekehof. Thanks! Robert McClenon (talk) 03:30, 2 July 2016 (UTC)
Books move done. - Jochen Burghardt (talk) 16:08, 2 July 2016 (UTC)

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Tree diagram placement help

Hi! I made this (subsequently reverted) edit on Tree (data structure) mainly because all the thumbnail figures are bunched together on the right, making the text on the left very narrow and difficult to read. Perhaps these could be arranged in a different way, like in a gallery? Thanks! J. Finkelstein (talk) 21:56, 7 July 2016 (UTC)

Hi! My reasons for reverting were: you didn't keep the node label sequences from the captions (like "undirected cycle 1-2-4-3") which help an unexperienced reader to identify what is meant; and you didn't make too clear which sentence refers to which image (i.e. above or below the text), this could be fixed by ending sentences with ":". Concerning the image arrangement, I'm impassionate; your inlining may be fine, or a gallery may be fine as well. - Jochen Burghardt (talk) 07:41, 8 July 2016 (UTC)

Might be worth a look

Made these I noticed you did some excellent polishing work on another article of mine:- Heinrich Scholz, Gisbert Hasenjaeger, Hans Rohrbach, Wilhelm Fenner, Wilhelm Tranow. Any editing would be apreciated. Scope creep (talk) 16:19, 15 August 2016 (UTC)

Heinrich Scholz

@Jochen Burghardt, thanks for the edits to Heinrich Scholz. Excellent work finding that reference. I Can you please take a look at my other German articles and determine if you can squeeze some extra quality into them. Thanks Scope creep (talk) 10:22, 26 August 2016 (UTC)

Nondeterministic PDA

Hey, seems like the PDA article is large enough that the content related to nondeterministic PDAs could be split into its own article. Just sayin'. 75.139.254.117 (talk) 17:37, 29 December 2016 (UTC)

There is no nontrivial common superclass of nondeterministic and deterministic pushdown automata (PDA). The formal definition of "deterministic PDA" is a special case of that of "nondeterministic PDA". Hence, every pushdown automaton is a nondeterministic one; some of them are even deterministic (and at the same time nondeterministic) ones. This naming is somewhat confusing, but has widespread use. It can best be understood as "nondet. PDA" meaning "PDA that is allowed (but not forced) to be nondet.", and "det. PDA" as "PDA that is restricted to be det.".
The article doesn't make these relations sufficiently clear. For these reasons, an own article "PDA" wouldn't have much to say beyond the "nondeterministic PDA" article. - Jochen Burghardt (talk) 19:19, 29 December 2016 (UTC)
Today, I tried to improve the article along the above line. - Jochen Burghardt (talk) 18:42, 30 December 2016 (UTC)

Dear JB, The image you added to the article on Damaris Cudworth Masham is undoubtedly that of a very beautiful woman and makes the page look lovely, but it doesn't look much like a painting of c.1700 to me. The difficulty is that the source (Find a grave) doesn't give any provenance or reason to think it is a picture of her, beyond the fact that someone has uploaded it there. Whatever the copyright questions attached to it might be (and the source doesn't help with that problem) it really does need some kind of authentication as being really her, such as artist, date, whether contemporary or a sort of retrospective imagination of her, etc etc, otherwise it is just an unsupported image which may have nothing to do with her, and ought to be removed. Can you supply any further information about it? It would be so nice if you could! Regards, Eebahgum (talk) 01:11, 21 February 2017 (UTC)

Dear Eebahgum, I found the image just by chance while trying to re-categorize the images in commons:Category:Philosophers by country. I suggested it in Damaris Cudworth Masham hoping some experts would check if it is appropriate there.
Unfortunately, I have no information beyond the data of its commons description page. I agree with your doubts about the Find a grave page. Some confidence may be gained from the image's appearance in pt:Damaris Cudworth Masham, sl:Damaris Cudworth Masham, and wikidata:Q2332520. However, they all may be wrong, too. - Jochen Burghardt (talk) 06:08, 21 February 2017 (UTC)
Apparently, the image was uploaded on 13 Dec 2016 by Commons and Pt editor ILuna10 (talk · contribs), and used half an hour later in the Portuguese article by an anonymous editor. Magnus Manske (talk · contribs) added it to Wikidata on 24 Jan 2017, from where it got automatically to the Slovene article. Maybe some of these people have additional information? - Jochen Burghardt (talk) 06:29, 21 February 2017 (UTC)
In the Hebrew article, the image was added on 15 ((can't translate month)) 2017 by Matanya (talk · contribs). However, I'm even unable to copy-and-paste the article title to here. There are more links shown on the commons description page, but most if them seem to be automatically generated lists of philosophers without local article. - Jochen Burghardt (talk) 06:41, 21 February 2017 (UTC)

Unambiguous grammar?

Hi, Can you give an example of an unambiguous grammar in which there are more than one derivation for a given string. 112.196.179.176 (talk) 08:44, 22 February 2017 (UTC)

I gave one yesterday at unambiguous grammar#Counter-example, in response of a (your?) recent edit of that page. - Jochen Burghardt (talk) 14:27, 22 February 2017 (UTC)

Usage of \mid

Hello there! On the mathematical induction page, you replaced the \mid LaTeX symbol with '.' Can I ask the reason why? The vertical bar is commonly used as an equivalent to 'such that', and '|' is reserved for absolute value usage, so \mid was instead. Aredaera (talk) 15:07, 31 March 2017 (UTC)

Hi! I often saw \mid in a set, but never in a formula after quantifier. For example, quantifier (logic) lists a lot of notations, but none using \mid. Set builder notation does use \mid. - Jochen Burghardt (talk) 16:54, 31 March 2017 (UTC)
Ah, I see. Then perhaps instead of the '.', a 's.t.' would be better instead? Also, is there by chance any LaTeX stylistic guidelines? (i.e., using \Rightarrow instead of \implies for implication)Aredaera (talk) 22:23, 31 March 2017 (UTC)
My personal opinion would be that one of the notations from quantifier (logic) should be used. I'd use "s.t." in an English sentence, but not within a formula. Concerning \Rightarrow: I guess you are right, and \implies is better, since it indicates the logic relation rather than the typographic symbol. As far as I know, any guidelines concerning mathematics are contained in (or linked from) MOS:MATH.

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Deutsche Mathematik

Hello, I saw that you re-added propaganda to the discipline section of Deutsche Mathematik. I removed it as it is not really an "academic discipline" or "field of study" (although maybe some people retroactively study it), but the references are good. I just wanted to explain. Thanks. — Preceding unsigned comment added by Hrodvarsson (talkcontribs) 15:02, 28 May 2017 (UTC)

Cantor's diagonal argument

In regards to your edit of 'Cantors diagonal argument' technically you people claim the one set is (base 2, which is the same as any finite 2 or greater) exponentially 'larger' than the other set, so much larger. You should have corrected rather than deleted. Further all presented versions of diagonal arguments are incomplete. Completed they produce ALL sets of bits not in the considered proposed count. For example all 'diagonals' as all one-to-one mappings of rows to columns not just one. Exponentially many verses one, a very important difference thus the previous version. If you had known this you could have made an addition and also changed 'many' to 'exponentially'. However diagonal arguments are wrong and easily disproved, the table width and height are forced the same whereas the width and height of any counting would be height exponentially larger than width AS THE PROBLEM IS STATED 'all possible sequences of bits' presumably the same length as the width of the table 'infinitly long'. The proof is uncountability due to size rather than uncountability. — Preceding unsigned comment added by Victor Kosko (talkcontribs) 04:53, 14 June 2017 (UTC)

Hi Victor Kosko! (I moved your note down to here, and added a section heading, to keep the chronological order of my talk page. If you answer, please append your text after my signature below.)
I wonder where you found the word "exponential(ly)" in the 'Cantor's diagonal argument' article? It doesn't appear there, and this notion doesn't make sense when dealing with infinite sets. Neither does the notion "many" make sense then, as I stated in recent my edit summary.
The Wikipedia article properly reflects the state of the art about the issue, as presented in mathematical textbooks. If you wish to challenge Cantor's proof, you should submit an article to a mathematical journal, presenting your counter-arguments. Wikipedia is not the place to do this, according to the policy Wikipedia:No original research.
Apart from that, I think that you are wrong: 'width' and 'height' is fixed by the problem statement, so you cannot change any of them in the proof. Best regards - Jochen Burghardt (talk) 12:56, 14 June 2017 (UTC)

The reference for ∞ly larger is the Wikipedia article:

```Cardinality of the continuum
```

In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers ${\displaystyle \mathbb {R} }$, sometimes called the continuum. It is an infinite cardinal number and is denoted by ${\displaystyle |\mathbb {R} |}$ or ${\displaystyle {\mathfrak {c}}}$ (a lowercase fraktur script "c").

The real numbers ${\displaystyle \mathbb {R} }$ are more numerous than the natural numbers ${\displaystyle \mathbb {N} }$. Moreover, ${\displaystyle \mathbb {R} }$ has the same number of elements as the power set of ${\displaystyle \mathbb {N} }$. Symbolically, if the cardinality of ${\displaystyle \mathbb {N} }$ is denoted as ${\displaystyle \aleph _{0}}$, the cardinality of the continuum is

${\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}>\aleph _{0}\,.}$

This was proven by Georg Cantor in his 1874 uncountability proof, part of his groundbreaking study of different infinities, and later more simply in his diagonal argument.

And also many other places in Wikipedia.

The phrase “as presented” in my previous .message and below refers to:

```   Uncountable set
```

In his 1891 article, Cantor considered the set T of ALL infinite sequences of binary digits

From the article, for example.

A clearer version of the disproof:

The diagonal argument, as presented, (for reals, sets of bits, or sets of naturals) cannot work even if its conclusion is true because

For proof A to prove B to be FALSE it must allow B room to be true.

Consider someone asking you to count all 1000 three digit numbers, on 3 lines so only 3 numbers fit! Or they ask you to count all 1 digit numbers and after you count 1 number they say count all 2 digit numbers and after you count a second number they say count all 3 digit numbers …!

The height of the list HAS TO BE the exponential of the width to make room for all the sets to be counted per the statement of the proof 'count all the...' for it to be a PROOF. Saying that doesn't count with ∞ because with ∞ one can do magic has to be PROVEN for the rest to be a proof!

Example of proper counting: In set theory (Cardinal) ∞ is ∞te increase without end. Set the rate of increase of the height of the sets, the real numbers, to normal, have the digits of each real number produced by separate algorithms for each line, small to large algorithms, they can produce true random bits. Force the first digits to be a count sequence so each line is different. Set rate of increase of number of digits of non-random reals to normal, random reals to the (same base) logarithm of normal, this will force the diagonal to the same rate. The sort is by algorithm, like a program but programing language very complex so will always produce infinite list of digits, and in reasonable time as function of the number of digits yet produced by the algorithm. The width of the random numbers relative to the number of numbers counted is the same as natural numbers. The height keeps pace with the increase of possible numbers per the size of the algorithms and number of bits of randomness used to make any random real. Thus π is not ∞ly far thru the list. The height counts as Aleph0 not Aleph1 per analogy to the height : width relation of the set of natural numbers. π , for example, would be as precise in digits as the number of reals counted. Victor Kosko (talk) 21:40, 14 June 2017 (UTC)

A page you started (Maycock (surname)) has been reviewed!

Thanks for creating Maycock (surname), Jochen Burghardt!

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Nice DAB page - I'm surprised no-one hadn't created it before now.

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Ordered pair: Cantor-Frege definition

Dear Jochen,

Cantor-Frege definition would indeed work in NBG "size-wise", but NBG defines relations in terms of ordered pairs!

(Sorry, if this page is a wrong place to discuss this, I have never done this before.)--nikita (talk) 19:29, 14 September 2017 (UTC)

Confluence (abstract rewriting)

Hi Jochen,

You reverted an edit I had made to this article as redundant, but my intention was to change the meaning of the sentence in question - since, as it stood, it was incorrect! I have edited the article again, this time changing the sentence structure more in the hope of making the intended meaning clearer. Please let me know if you disagree with the change I have made.

Thanks, Robin S (talk) 01:49, 3 October 2017 (UTC)

Hi Robin, your new version appears better understandable to me. In the previous version, b,c could just have been chosen luckily to deem a confluent; this is ruled out now. Thanks for noticing and fixing that! - Jochen Burghardt (talk) 05:14, 3 October 2017 (UTC)

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Sources

Hello, can you please add sources to Akadémiai Kiadó? I'm afraid German Wikipedia wouldn't count as WP:Wikipedia is not a reliable source. Thanks, Boleyn (talk) 13:55, 28 December 2017 (UTC)

Sorry, the German article doesn't cite sources, and I'm unable to read the Hungarian article. Maybe, we could wait if some Hungary expert provides reliable sources? Best regards - Jochen Burghardt (talk) 14:19, 28 December 2017 (UTC)

Trovatore's comments on "Cantor's diagonal argument"

Hi Jochen, I was just reading some of the Talk on Cantor's diagonal argument and came across Trovatore's comments in the section: "Cantor" as agent in the argument. It's very important to mention that the proofs are Cantor's, but I think that Trovatore makes a valid point when he says that Cantor is mentioned too much. (I think it's just a little too much.) So I've done a small amount of rewriting (see User:RJGray/Sandboxcantor1#Uncountable set) that just mentions Cantor at the start but not in the proofs themselves. I'd like to know your opinion. I also realized that in the second proof, the initial assumption is not directly contradicted. Instead, the statement "s being an element of T and therefore belonging to the enumeration" is being directly contradicted. I suspect that more often a proof by contradiction doesn't directly contradict the initial assumption. Anyway, tell me what you think of my changes and send me any improvements you come up with. Thank you, --RJGray (talk) 18:45, 10 January 2018 (UTC)

Hi Robert, sorry for the delay. I compared both versions and found you new one slightly better. Cantor's name needn't appear that often, and your new indirect proof is a little bit more clear. By the way: if Bois-Reymond used the diagonal argument before Cantor, this should be stated; so I don't quite understand Estill-math's deletion of 9 Jan 2018. However, when I recently glanced at the Math Ann paper, I didn't find the argument. - Jochen Burghardt (talk) 18:10, 13 January 2018 (UTC)

Hi Jochen, Thanks for looking over my proposed change. I've updated the article. Concerning du Bois-Reymond and his diagonal argument: It's mentioned on the Talk page: du_Bois-Raymond and Cantor's diagonal argument. Of particular interest is note 1 page 187 in Simmons, Keith (1993). Universality and the Liar: An Essay on Truth and the Diagonal Argument. This note claims that Bois-Reymond's diagonal argument is a different type than Cantor's.

I agree with the Talk page that something should be mentioned, but some research is needed beforehand. Ultimately, it would be nice to have a section covering both sides of this issue, which would include a comparison between the two proofs and what various mathematicians and historians say about the two proofs.

Here's some interesting material from Stackexchange (of course, we can't use this in a Wikipedia article, but it does say that the diagonal argument is in a footnote on page 365):

He [du Bois-Reymond] refers to [an] 1873 paper where only a particular case was considered. This time he gives a more general version, which Borel highly praised and termed du Bois-Reymond's theorem. A consequence of it, and the motivation, is the non-existence of the "ideal boundary" that can be specified by a sequence of converging/diverging series, as Bertrand earlier proposed (he was thinking of reciprocals to products of powers and powers of iterated logarithms as terms). The du Bois-Reymond's theorem would be the "diagonal argument":
"...if an unlimited family of more and more slowly increasing functions λ1(x), λ2(x), λ3(x) ... is given which for each r satisfies the condition lim λr(X)/λr+1(X) = ∞, one can always specify a function ψ(x) which becomes infinite with x, but more slowly than any function of that family".
The construction of ψ(x) is in a footnote on p. 365, and does show some "diagonality" if one looks hard. There is however no cardinality involved in du Bois-Reymond's setting (Hausdorff will relate gaps to cardinalities only later), so making it into the diagonal argument takes some reading in. In his 1882 book Die allgemeine Functionentheorie (The General Theory of Functions) du Bois-Reymond touches base with Cantorian set theory, and mentions that Cantor showed "continuum of the idealists" to be uncountable. He does not however point out any affinity between the diagonal argument by which it was shown and his earlier construction, let alone lay any claim to it. So whatever the relation between the two it was not apparent to du Bois-Reymond.

NOTE: There is an error in the next to last sentence: du Bois-Reymond book is dated 1882 and Cantor didn't give his diagonal argument until 1891, so du Bois-Reymond couldn't compare his argument with Cantor's diagonal argument. I came across this book years ago and du Bois-Reymond just gives Cantor's 1874 argument. Of course, it can still be argued that if du Bois-Reymond did have a diagonal argument similar to Cantor's later argument, then du Bois-Reymond could have used it to give a new proof of the uncountability of the reals rather than repeating Cantor's 1874 argument.

I'm interested in hearing your opinion of du Bois-Reymond's argument on p. 365 of his 1873 paper and how it compares to Cantor's diagonal proof. --RJGray (talk) 21:48, 14 January 2018 (UTC)

Thanks for the informations! I downloaded the article from GDZ; the diagonal argument is indeed somewhat hard to detect. In User:Jochen Burghardt/sandbox8, I gave a download URL and a viewer URL, and started a translation of the argument. I intend to complete it in the next days. - Jochen Burghardt (talk) 20:09, 15 January 2018 (UTC)
Finished a (sloppy) translation. However, I didn't understand some phrases about convexity which are essential for the proof. Also I'm not sure whether the λ functions are required to be monotonically increasing; the examples he mentioned on the previous page happen to have that property. - Jochen Burghardt (talk) 19:38, 17 January 2018 (UTC)
Found the paper cited on p.364; see User:Jochen Burghardt/sandbox8. Apparently, the appendix (starting on p.29-->88) of this paper establishes the context in which to understand the 1875 paper. - Jochen Burghardt (talk) 20:03, 17 January 2018 (UTC)

Math induction clarification

Hi, Jochen Burghardt! I′ve seen your recent edit at Mathematical induction. Please detail on the associated talk page opened sections what do you consider to be confused phrasing in order to improve clarity of the content to be inserted in article! Thanks--5.2.200.163 (talk) 14:51, 17 January 2018 (UTC)

Galois connections

Hallo Herr Burghardt,

danke für die "citation needed" - Tags im Artikel über Galois connections. Ich bin ein alter, aber kein allzu erfahrener WP-Autor und kenne die Etikette nicht gut genug. In diesem Falle müsste ich eigene Publikationen nennen, und ich zögere, mich in Wikipedia selbst zu zitieren. Deshalb habe ich nur zur WP-Seite über Formal concept analysis verlinkt, dort gibt es Zitationen (die allerdings noch aufgeräumt werden müssen, das haben wir demnächst vor).

Zur Formalen Begriffsanalyse, die ja als angewandte Theorie der Galoisverbindungen verstanden werden kann, gibt es mehrere tausend Publikationen. Da ist es angebracht, nicht einzelne Papers zu zitieren, sondern systematische Darstellungen. Dass alle Galoisverbindungen "bis auf Isomorphie" aus Relationen stammen, kann man als Teilaussage des Hauptsatzes der Begriffsanalyse ansehen. Dieser ist im Buch B. Ganter, R. Wille : "Formal Concept Analysis -- Mathematical Foundations", Springer gut nachzulesen. Außerdem gibt es ein Paper von mir ("Relational Galois connections", Proceedings ICFCA 2007, Springer), in dem der Begriff der Galoisverbindung so weit wie möglich verallgemeinert wurde, bis hin zu Galoisverbindungen zwischen beliebigen Relationen, und in dem dann gezeigt wurde, dass man all das auf Galoisverbindungen zwischen Potenzmengen zurückführen kann.

Es gibt in der Formalen Begriffsanalyse viele Papers zu Algorithmen, zu deren Komplexität etc., und es gibt auch einiges an frei verfügbarer Software. Ich habe vor einiger Zeit aus dem WP-Artikel über Formal concept analysis eine nach meinem Urteil viel zu langatmige Passage, in der allerlei Algorithmen (für ein einziges Problem) verglichen wurden, rüde gekürzt. Ich würde das Buch von Sergei Obiedkov und mir mit dem Titel "Conceptual Exploration" (Springer 2016) zitieren, das enthält immerhin 33 verschiedene Algorithmen mit sorgfältigem Pseudocode. Das sind Algorithmen zu einem Wissensakquisitionsverfahren (der "Merkmalexploration"), das aber wiederum mit Galoisverbindungen arbeitet.

Grüße, --Bernhard Ganter (talk) 20:09, 20 January 2018 (UTC) Ich hab' mal drei Zitate eingefügt. --Bernhard Ganter (talk) 14:56, 21 January 2018 (UTC)

Hallo Herr Ganter, danke für die Erklärungen und für die eingefügten Zitate. Selbstzitate sind in WP:SELFCITE "geregelt"; ich finde, Ihre Zitate sind danach ok. (Um solche policy-Seiten zu suchen, ist m.E. Wikipedia:Editor's index to Wikipedia sehr hilfreich.) Ich persönlich würde es sogar begrüßen, wenn Sie noch einen Link auf einen öffentlich zugänglichen Text (PDF) einfügen würden (vielleicht Ihr ICFCA'07-Papier?). —
Mit der neuen Einschränkung "antitone Galois connections between power sets" fällt es mir intuitiv leichter, die Darstellbarkeit über zweistellige Relationen zu glauben. —
In "Birkhoff, Ch. IV, §5" habe ich auf die Schnelle nichts über Galoisverbindungen finden können, sie werden in meinem Exemplar (3. Aufl., 1967) erst in Ch.V, §8 (vor Thm.20) eingeführt. Dort in Exercise 2 kommt immerhin eine zweistellige Relation vor und ein Rückbezug auf §5 (allerdings wohl von Ch.V), aber jedenfalls auf Anhieb konnte ich keine Verbindung zum Wikipedia-Abschnitt Galois connection#Connections on power sets arising from binary relations sehen. Könnten Sie die Stelle noch etwas genauer angeben?
Beste Grüße zurück - Jochen Burghardt (talk) 13:08, 23 January 2018 (UTC)
Danke für Ihre sehr konstruktiven Bemerkungen.
Tatsächlich ist die Einschränkung "between power sets" unnötig. Ich habe sie nur eingefügt, weil ich auf die Schnelle kein passendes Zitat zur Hand hatte. Das kann ja noch kommen. Die Argumentation geht so: 1) Die vollständigen Verbände, die aus Galoisverbindungen zwischen Potenzmengen entstehen, nennen wir Begriffsverbände. 2) Es ist einfach nachzuweisen, dass Galoisverbindungen zwischen Begriffsverbänden durch Galoisverbindungen zwischen Potenzmengen dargestellt werden können. 3) Jeder vollständige Verband ist isomorph zu einem Begriffsverband ("basic theorem on concept lattices"). Das zusammen ergibt die Behauptung. Ich werde mal stöbern, ob's irgendwo gut lesbar steht. Eigentlich steht es auch in dem genannten "ICFCA07-Paper".
Beim "Birkhoff" habe ich den Fehler gemacht, mich auf die zweite Auflage zu beziehen. In der dritten Auflage (die heute aktuell ist), ist das Ganze in der Tat in das Kapitel V ("Complete Lattices") gerutscht. Dort beschreibt der Abschnitt 7 ("Polarities") die Galoisverbindungen zwischen Potenzmengen, die aus Relationen entstehen. Der darauffolgende Abschnitt 8 ist dann den "Galois Connections" gewidmet. In einer Fußnote verweist Birkhoff darauf, dass dies auch schon in der ersten Auflage stand und dort offenbar als Erstveröffentlichung verstanden werden kann. Ein kleines Zuordnungsproblem besteht nun darin, dass der WP-Artikel auf die erste Auflage verweist ("Birkhoff 1940"), die aber heute kaum noch zu finden ist. Ich versuche mal, das zu bereinigen.
--Bernhard Ganter (talk) 15:09, 23 January 2018 (UTC)

Turing machine

Regarding a recent revert:

1. if a Turing machine is NOT a thought experiment, how about updating Computational complexity theory#Turing machine which introduces TMs as such? Perhaps this indicates a problem between distinguishing what something is, vs. how it is used or thought about.
2. I don't see TM under Category:Mathematical objects, although they are under Category:Mathematical modeling (of a hypothetical never realized) computing abstraction
3. Perhaps all mathematical objects (or even abstractions) should be categorized under thought experiments (but I wouldn't go so far).

Dpleibovitz (talk) 20:30, 13 February 2018 (UTC)

My line of thought was this: Formally, a Turing machine (TM) is defined (in Turing machine#Formal definition) as a particular kind of 7-tuple, with the latter clearly being a mathematical object, see Tuple#Definitions. In this sense, a TM is (can be encoded as) a set of sets, like e.g. a natural number (cf. Ordinal number#Von Neumann definition of ordinals).

I guess, since a TM is rather complicated and unfamiliar, and to distinguish it from a real computer, it has been called a thought experiment in the article. However, I think a thought experiment, in a narrow sense, describes some hypothetical situation, often including human acting (the article mentions "causes" and "effects", which is hardly applicable to mathematical models).

In a wide sense, I'd agree with your point 3 (for example, imaginary numbers can be thought of as originating from a thought experiment "what if we had something with its square being -1 ?"), but I wouldn't go that far either.

Best regards - Jochen Burghardt (talk) 20:55, 13 February 2018 (UTC)

Semiorder

My thoughts on a recent edit of yours: I don't think I've ever had an edit survive for 6 years and still be revertable with the "undo" button before! Anyhow, I don't know if the article asymmetric relation was ambiguous back in 2012, but (contra what that article says) I do think the adjective is ambiguous. That is, I do not think it is universally agreed that "asymmetric" means "irreflexive and antisymmetric," whereas I think the definitions of the latter two terms are unambiguous. I was thinking of adding "(that is, it is irreflexive and antisymmetric)" to semiorder, but since the sentence is already a "that is" sentence, that would be ridiculous. So, I guess this is all just a "here is my opinion but I'm not going to change anything" type comment. Best, --JBL (talk) 15:52, 15 April 2018 (UTC)

I was surprised that "undo" still worked, too. I agree that "asymmetric relation" can be misunderstood e.g. as "a relation that is not symmetric" by non-mathematicians. On the other hand, it is a common notions with a fixed meaning for mathematicians. Since the article appears to be of interest not only to mathematicians but also to economists, restoring your edit may have a point. So, if you like to revert again, I won't complain. But maybe we'd wait for another 6 years? ;-) - Best regards Jochen Burghardt (talk) 17:41, 15 April 2018 (UTC)
Ha! I think it is fine for now, but I will put a note in my calendar to revisit the question in 2024. Best, JBL (talk) 21:46, 15 April 2018 (UTC)

Non-trivial article

Hi, I looked at the history of learning automata, and your contributions, and guessed that perhaps you might be a good choice as co-pilot on an article. I stumbled across Tsetlin engine in a Norwegian newspaper a few weeks back, and now wrote a few sentences in a tiny stub article. This is about a new type of learning automata that could be very interesting, at least for rule based knowledge systems. Granmos article is the first I know of, but it could be other sources later this year. This is pretty heavy stuff! I guess there should be a proper article about Tsetlin automata before too much stuff goes into the Tsetlin engine article.

Any ideas? The field of reinforcement learning in this area is pretty new to me. Jeblad (talk) 20:26, 3 May 2018 (UTC)

Just a quick grammar note here: Automata is plural. The singular is automaton. I made a couple of corrections. --Trovatore (talk) 20:34, 3 May 2018 (UTC)
Thanks! Jeblad (talk) 21:20, 3 May 2018 (UTC)

@Jeblad: Sorry, but I'm not an expert in reinforcement learning either; in particular, I never heared about Tsetlin automata or engines before. When you look at the source code of learning automata, you'll see that I added comments about what I guessed from my background knowledge (which is mainly in formal language theory). — This said, I can have a look at the Arxiv paper and possibly add some stuff from it to your stub. — Concerning your distinction of "Tsetlin engine" vs. "Tsetlin automaton", I wonder whether it is supported by the Arxiv paper, since I couldn't find the string "engine" in it. If that word occured only in a newspaper I'd think some non-expert journalist just might have used it as a synonym for "automaton". Best regards - Jochen Burghardt (talk) 10:44, 4 May 2018 (UTC)

It might very well be that the correct name should be "Tsetlin automata" and not "Tsetlin engine". The few notions I found with the phrase "Tsetlin engine" has referred to Granmos learning algorithm. It seems to me that the algorithm is more or less an artificial neural network with alternate primitives, ie. just an implementation of a slighly more general form of an ANN. Jeblad (talk) 12:20, 4 May 2018 (UTC)
Seems like I got the "Tsetlin engine" from another source. Granmo uses "Tsetlin Machine". A quick search in the paper gives me 150 hits. Jeblad (talk) 12:28, 4 May 2018 (UTC)
There are a few references on the net to people that can't get quite the same error bounds as Granmo. I'm tempted to say there might be an error in the paper, but that would be OR! :D Some fun, and some rather harsh remarks at Reddit.[1] Jeblad (talk) 13:03, 4 May 2018 (UTC)

Anarchism

Hi Jochen Burghardt,

I saw your work on articles related to anarchism and wanted to say hello, as I work in the topic area too. If you haven't already, you might want to watch our noticeboard for Wikipedia's coverage of anarchism, which is a great place to ask questions, collaborate, discuss style/structure precedent, and stay informed about content related to anarchism. Take a look for yourself!

And if you're looking for other juicy places to edit, consider adopting a cleanup category or participating in one of our current formal discussions.

Feel free to say hi on my talk page and let me know if these links were helpful (or at least interesting). Hope to see you around. czar 11:39, 13 May 2018 (UTC)

Pure Function

I followed the policy WP:BRD. I don't intend to enforce a particular naming, but the article should be consistent. I'll continue the discussion at Talk:Pure function#alternate usage of the term: as soon as I find the time. - Jochen Burghardt (talk) 06:25, 6 July 2018 (UTC)
Please see the computer-science resources I cited. Please do not force and article titled "Pure Functions" to deviate from the standard usage in computer science. Cerberus (talk) 13:00, 10 July 2018 (UTC)
Let's continue the discussion at Talk:Pure function, not here. - Jochen Burghardt (talk) 15:26, 10 July 2018 (UTC)
I am hoping you will reply to my comment of 23 July. Thanks. Cerberus (talk) 02:39, 28 July 2018 (UTC)

Thank you

Thank you for contacting Michael Hardy after Georg Cantor's first set theory article was nominated for Good Article. I was on a vacation that started just a few days before Bilorv gave his assessment of the article. It all worked out fine. Michael informed Bilorv that I was away and started making some improvements to the article. As soon as I got back, I started making improvements. Yesterday, I finished the necessary improvements and Bilorv certified that it's a Good Article. So your excellent case diagrams now reside in a Good Article. In fact, they are an excellent contribution to the key proof in the article. —RJGray (talk) 18:36, 18 August 2018 (UTC)

Merger discussion for Terminal and nonterminal functions

An article that you have been involved in editing—Terminal and nonterminal functions—has been proposed for merging with Terminal and nonterminal symbols. If you are interested, please participate in the merger discussion. I noticed you participated in the last merge discussion we had on this topic. Thank you. Enterprisey (talk!) 05:33, 12 September 2018 (UTC)

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