User talk:JRSpriggs/Archive 1

This is the file "User talk:JRSpriggs/Archive 1" which archives User talk:JRSpriggs.

Zero Sharp

OK, mixing in here. There's an extra technical condition you have to add to indiscernibility, called "remarkability" if I remember right, that guarantees that the indiscernibles are "Silver indiscernibles". With that extra condition they're unique. If indiscernibles exist then so do Silver indiscernibles, and I believe you find them just by taking the smallest ordinal that's one of a proper class of indiscernibles, and then the smallest that's one of a proper class of indiscernibles containing the first one you fixed, and so on. Or something like that. See Jech for the formal statement of the condition—something to do with every ordinal being expressible as a term in terms of indiscernibles, such that the value of the term doesn't change if you change indiscernibles above the ordinal the term denotes. --Trovatore 06:55, 10 April 2006 (UTC)

Brent's method

Thanks very much for correcting my error in the algorithm in Brent's method, especially since it was the kind of mistake that could have sat there for years without anybody noticing it. -- Jitse Niesen (talk) 02:42, 24 April 2006 (UTC)

You are welcome. JRSpriggs 04:28, 24 April 2006 (UTC)

Finding multiple roots

Hello. I saw you started the article Finding multiple roots, thanks. It is always nice to see somebody getting infected with the Wikipedia bug :) I do have some questions about the article though:

1. Why did you decide to start a new article, instead of adding an extra section to Root-finding algorithm? I prefer to have a few long articles with a couple of people working on them and correcting each other, rather than spreading out our works on many small articles (but this is partially a matter of taste).
2. Ideally, every article should be supported by a couple of references. Please add one or two.
3. I'm especially interested in references, because it is not clear to me how to compute the gcd in the presence of rounding error. If the example you give in Finding multiple roots is computed with floating-point arithmetic, then the remainder might not be zero.
4. There is a host of style issues, as explained in Wikipedia:Manual of Style (mathematics), but those are of minor importance.

Since you (rightly) criticized my work on Brent's method, I hope you don't mind if I return the favour. ;)

Thanks again for your work, and I hope you are enjoying it.

All the best, Jitse Niesen (talk) 06:55, 29 April 2006 (UTC)

Thank you for your interest in my article. I appreciate your feed-back. I responded on the discussion page for that article. JRSpriggs 12:27, 30 April 2006 (UTC)

how to merge

All work on wikipedia is licensed under the GFDL license, which ensures that the work is always free to be edited and copied. This license is a cornerstone of the way that wikipedia works. One of the stipulations of the license is that all authors be credited. This is accomplished on wikipedia via the "history" function. Copy-paste edits are tricky, because they kill all authorship information. Nevertheless, copy-paste edits are sometimes necessary. The accepted method for dealing with this is this: when you do a copy-paste merge, you must mention in the edit summary where the text comes from. Doing so ensures that anyone who needs to can follow the trail of edits and see the author of the original text. The ability to track edits in this way is a legal requirement of the GFDL, so merges which are not done in accord with this principle must be reverted. I note that in this edit (where you carry out the merger), you merge and mention the source article. Thus your actions are within the terms of the license, and your merger is done correctly. I will suggest that, although not required, it is convenient if you wikify your edit summary (thus instead of an edit summary of merge in text of "Finding multiple roots", you could have used an edit summary with a text of merge in text of [[Finding multiple roots]]). This is a matter of convenience, not a legal requirement. It seems to me that your merger meets the legal requirements, and so is perfectly acceptable. For more information, visit Wikipedia:Merging and moving pages. -lethe ^{talk +} 07:24, 22 May 2006 (UTC)

Also, if you look at Talk:Root-finding_algorithm and go down to the section on "Merger with the 'Finding multiple roots' article", you will see that I explain how to get to the history of the old article. JRSpriggs 07:38, 22 May 2006 (UTC)

In the past, when I have performed mergers, I also merged the talk pages. It seems to me to be quite disorganized to have a single article with many different relevant talk pages. Thus I think it could be preferable to also merge Talk:Finding multiple roots into Talk:Root-finding algorithm. Nevertheless, there is no policy which dictates that this must be done. It is simply my own personal preference. Feel free to ignore it. -lethe ^{talk +} 07:54, 22 May 2006 (UTC)

Lethe, I took your advice and merged the contents of the talk; converting "Talk:Finding multiple roots" into a redirect to "Talk:Root-finding algorithm". I also added direct pointers to the histories of "Finding multiple roots" and "Talk:Finding multiple roots" at Talk:Root-finding algorithm. However, when I tried your suggestion of putting a pointer into the edit summary there, it did not work because the pointer was redirected back. JRSpriggs 09:11, 23 May 2006 (UTC)

I did not react on all your merging activities because you are doing very well. Yes, the link to Talk:Finding multiple roots in the edit summary redirects back, but if you want to go to that talk page, you can follow the "redirected from …" link on top of the page which you get when clicking on the link to Talk:Finding multiple roots. -- Jitse Niesen (talk) 09:58, 23 May 2006 (UTC)

Thanks, I did not know that I could get back to the redirect that way. That will be helpful. JRSpriggs 10:20, 23 May 2006 (UTC)

Math formatting

Hi JRSpriggs. Thank you for the additions at Newton's method. I did some formatting there, using math tags, <math></math>. They are available on the toolbar at the top of the edit box. One should use either that, or html tags. So for example, variables should always be italic, either ''x'' or <math>x</math>, but never an x by itself (compare $x$ vs x vs plain x). Just a tip. You can reply here if you have comments. Cheers, Oleg Alexandrov (talk) 15:12, 1 June 2006 (UTC)

Answer two months latter

from your comment in cardinality, its my initials... cjrs... Cjrs 79 21:06, 14 June 2006 (UTC)

Inaccessible cardinal

Re the countability/uncountability of inaccessible cardinals: I am not an expert on set theory, but at least the online sources I could find with google (e.g. Springer Encyclopedia of Mathematics, PlanetMath, several others) define inaccessibility without the requirement for uncountability. Could you provide a reference to convince me that the definition of inaccessibility with uncountability is "more standard"? — Tobias Bergemann 10:24, 27 June 2006 (UTC)

For me, the main point is that I would have to keep making exceptions every time I talk about inaccessibles, if I defined it to include aleph-null. JRSpriggs 10:29, 27 June 2006 (UTC)

True, but an encyclopedia should report about inaccessibles as they are most commonly defined in the relevant literature. However, your references below are good enough for me. Sorry for belaboring such a minor point. — Tobias Bergemann 11:13, 27 June 2006 (UTC)

"Mathematical Logic" by Joseph R. Schoenfield defines inaccessible (on page 304) in a way which requires them to be uncountable. On pages 208 & 209 of "Simplified Independence Proofs" by J. Barkley Rosser, he says "Some people hold a firm conviction that some inaccessible cardinals exist ..." which implies that inaccessibles are uncountable since the existence of aleph-null is not controversial. Suppes defines them to include aleph-null, but he also includes 2 as an inaccessible, which is ridiculous. JRSpriggs 10:55, 27 June 2006 (UTC)

Thanks. — Tobias Bergemann 11:13, 27 June 2006 (UTC)

cc

From Talk:order (group theory) Ahhhhh. I see. I have forgotten how to distinguish capital and small letters.

Stupid, stupid, stupid.  :) Thank you. --VKokielov 03:30, 8 July 2006 (UTC)

I have another question, if you have the opportunity. I understand that there are groups of infinite order in which every element generates a finite subgroup. Would you offer an example? But only if you know it at once; I don't want you to construct it for me. The algebra book says $Z_{p}[x]$ is one like that, but I don't know how to read that symbol. --VKokielov 03:30, 8 July 2006 (UTC)

That notation means the set of polynomials with coefficients in Z_p. So for example 2x³ + x², and remember that the coefficients are regarded as numbers mod 3. This polynomial has order 3 in the additive group, as does every other element in the group. -lethe ^{talk +} 04:11, 8 July 2006 (UTC)

I assume that you are taking p = 3. JRSpriggs 04:46, 8 July 2006 (UTC)

Thank you very much. --VKokielov 05:35, 8 July 2006 (UTC)

Translational motion doesn't contribute to heat energy

• JR: Please see Thermodynamic temperature: Discussion Thermodynamic temperature discussion page. Greg L 18:20, 15 July 2006 (UTC)

• I understand now what you intended. I've responded in the discussion page. Greg L 20:26, 16 July 2006 (UTC)

Merge of ultrafilter lemma

Thanks for fixing the double redirects there. It completely slipped my mind that there might be pages that redirected to it. CMummert 12:15, 17 July 2006 (UTC)

Beth numbers

You wrote:

Notice that for any cardinals κ and μ, there is an ordinal α such that

$\beth _{\alpha }(\kappa )=\beth _{\alpha }(\mu ).$

That's true in ZF, and in ZFU+AC, but not necessarily in ZFU, and it's not obvious. Perhaps rephrasing and/or citation is in order. — Arthur Rubin | (talk) 15:10, 25 July 2006 (UTC)

Done. JRSpriggs 04:15, 26 July 2006 (UTC)

Ineffable cardinal

Ineffable cardinal is considered to be a topic of number theory. --GoOdCoNtEnT 02:45, 26 July 2006 (UTC)

If you want to add it to the number theory category, go ahead. But I think that having more than one targeted stub is excessive. JRSpriggs 02:54, 26 July 2006 (UTC)

Photon in the box

Please do not pass judgement on things that you clearly do not understand. Try reading the modern treatment of "mass" in relativity. Until you understand, please abstain in making decisions. See here, fresh from Harvard:

http://www.courses.fas.harvard.edu/~phys16/Textbook/ch11.pdf

Have a look at chapter 11.8.....

After you read, you will understand that you reverted to an incorrect and outdated view of things. If you are a scientist, you will hopefully understand.....Ati3414 05:39, 8 August 2006 (UTC)

You, sir, are a fool who thinks he knows what he does not know. I am quite familiar with both special and general relativity and have solved the equations of general relativity in some cases. I know what I am talking about and you do not. JRSpriggs 05:42, 8 August 2006 (UTC)

The fact that you are getting abusive demonstrates that have run out of logical arguments. Try reading the recommended chapter, it is standard fare in universities nowadays, it may have been some time since you studied the subject and things have changed.Ati3414 05:49, 8 August 2006 (UTC)

JR, please keep your cool. Ati3414 will quickly exhaust community patience and be banned; there is no need to risk sanctions yourself. Ati3414, for your part, I'd just as soon you didn't keep your cool; you can work out why for yourself. --Trovatore 05:44, 8 August 2006 (UTC)

Dear Il Trovatore, I suggest that you take your frustrations with the new view, try reading the chapter from the current curriculum at Harvard: http://www.courses.fas.harvard.edu/~phys16/Textbook/ch11.pdf , have a look at 11.8

or Cornell: http://www.lassp.cornell.edu/~cew2/P209/part11.pdf or MIT, or Stanford.... Ati3414 05:49, 8 August 2006 (UTC)

Trovatore: Thanks. I will try not to do anything rash. I just felt that someone had to confront him with the facts of the situation.
Ati3414: You may not understand why you are wrong, but I suggest that you desist from promoting views that clearly conflict with the overwhelming consensus of editors here. JRSpriggs 06:17, 8 August 2006 (UTC)

Thanks

Thanks, but he just moved it from my userpage, some IP user posted it and he thought it was funny =D --mboverload@ 08:38, 15 August 2006 (UTC)

As I saw a moment later when I looked at your user page. Why have you not replied to my messages on your user page about you "correcting" words to the wrong spelling? JRSpriggs 08:41, 15 August 2006 (UTC)

<insert name here> operator

Are you aware that you did not move μ operator (originally mu operator) back to its former name, but rather to the nonsense μu operator? -- EJ 05:41, 19 August 2006 (UTC)

No. I thought I put it back correctly except for a problem with redirects. I do not think that a name with an actual mu in it is even possible. What proof do you have? JRSpriggs 05:47, 19 August 2006 (UTC)

Just click on mu operator, you'll see "Μu operator (Redirected from Mu operator)". All Unicode character are valid in article names (well, almost). -- EJ 05:51, 19 August 2006 (UTC)

That is the redirect problem. There is a redirect with the same name as the article which is taking precidence over the article and then redirecting you to the article. I cannot fix that. I asked User talk:Oleg Alexandrov to fix it. We will just have to wait for him or another administrator to do it. Blame CyberSkull for vandalizing the article in the first place. JRSpriggs 06:01, 19 August 2006 (UTC)

No, wait. I know what happened. When I move the article back I just added a "u" to the title after what looked like an "M", but it must have been a capital mu "Μ" instead. JRSpriggs 06:08, 19 August 2006 (UTC)

(Edit conflict) Yes, that's what I am saying. In fact, you couldn't move the article back, because it is impossible to overwrite an existing article by a move (it has to be deleted first, which requires an admin), and the redirect created by the first move counts as an existing article. -- EJ 06:26, 19 August 2006 (UTC)

I tried again but it did not work. Now I really do not know. Is there a way to see whether a character is a capital M or a capital mu Μ? JRSpriggs 06:22, 19 August 2006 (UTC)

Use a font where they are distinguishable, or look at the URL (where non-ASCII characters are encoded by % escapes). If "tried again" means you tried another move, it cannot work for the reasons above (the redirect has to be deleted first). -- EJ 06:29, 19 August 2006 (UTC)

Oleg just fixed it. But according to Wikipedia:Merging and moving pages, "If the destination does exist, but it only contains a redirect without any history, the move will still work — the designers of the MediaWiki software recognised this as a special case in which no information will be lost if a move is performed.". Therefor, I thought that the redirect at Mu operator would just be overwritten. JRSpriggs 06:44, 19 August 2006 (UTC)

I see, I was mistaken on that point. Presumably the history was non-empty after all the changes, as per "This is especially likely to happen if there is a history of moves from one name to another." Anyway, all is good now. -- EJ 16:57, 19 August 2006 (UTC)

Note on talk page about 4 forces

I have some concerned about some of the remarks you made about 4-forces in the article on "special relativity". Rather than interrupt an active edit session, I've put my concerns on the talk page.Pervect 07:52, 28 August 2006 (UTC)

Virtual particle

If virtual particles were unobservable by their very nature, they wouldn't be physics. They are very much part of physics, if they didn't exist then QED wouldn't work properly; and the observables there would be different.

So they have been detected, photons and electrostatics are both detectable, it's complete nonsense.WolfKeeper 07:13, 12 October 2006 (UTC)

You are missing the point. They cannot be observed INDIVIDUALLY (as single quanta). They have COLLECTIVE effects which are observed. JRSpriggs 07:17, 12 October 2006 (UTC)

AIV

Hello. Thanks for keeping a check on vandals. Please note that WP:AIV is only for ongoing cases of vandalism. Before listing a vandal here make sure that "The vandal vandalized within the last few hours and after the final warning". Some of the vandals you have listed recently have not edited since some time and some not since their last warning. Take note of this next time. And keep up the good work. - Aksi_great (talk) 10:20, 26 October 2006 (UTC)

re: Werdnabot Timing

Thanks for that! Randfan 15:17, 18 November 2006 (UTC)

Comment on the "M operator" redirect page

If there is such a problem, why do you not request full protection of the page, so that only administrators can modify the redirect. Whenever you leave a comment on a page that contains a redirect, it always ends up showing that there is a broken redirect. When that happens, user like me will go around and fix the problem. I suggest tha you do something else to let people know about the problem. --Willy No1lakersfan (Talk - Contribs) 15:26, 19 November 2006 (UTC)

I was not aware of the problem you mentioned; and I apologize. It appears that the "nowiki" you added fixes that. Thank you. JRSpriggs 04:27, 20 November 2006 (UTC)

You are very welcome. I am sorry if I did anything that you felt was wrong to the redirect. --Willy No1lakersfan (Talk - Contribs) 20:37, 21 November 2006 (UTC)

Metric Signature

Hi JR. I liked some of 207.200.116.139 edits. The one that really got under my skin, though, was the signature convention. I started a little discussion on Talk:Special relativity. I'd really like to see a consensus formed if there isn't one already. I thought you might like to comment. --MOBle 16:25, 20 November 2006 (UTC)

Reply to message

I have replied to your comment on my talk page. Rawling 12:22, 24 November 2006 (UTC)

Should I Make A Redirect?

I use Werdnabot, and checked my talkarchive template and noticd that the "current talk page" link is User:WarthogDemon/Archive . . . which doesn't exist. Can I redirect that page to User talk:WarthogDemon or would that mess Werdnabot's archiving job up? -WarthogDemon 21:46, 25 November 2006 (UTC)

To WarthogDemon: Apparently, the "talkarchive" template assumes that the current talk page is the immediate parent of the page on which the template is located. So if you want that pointer to go directly and correctly to your current talk page, then you should have made your archive be "User talk:WarthogDemon/Archive 2" instead of "User talk:WarthogDemon/Archive/Archive 2". Most users have their archive pages directly under their talk pages. Oddly, Werdna himself does it the way you do.

You could try making "User talk:WarthogDemon/Archive" into a redirect to "User talk:WarthogDemon". But I cannot tell you what would happen, since I have never worked with a redirect which has subpages. On the other hand, you do not really need to use the pointer created by the template, because there is already a pointer provided on each subpage which in your case appears in small print directly below "From Wikipedia, the free encyclopedia" and has the form "< User talk:WarthogDemon".

On another subject, the reason why your section "Bartells" did not archive is that Werdnabot needs at least one (and perhaps two in some cases) time stamp(s) in a section in order to know when to archive it. You can fix this by making it a practice to add a time stamp to any section which lacks one by using "~~~~~" (five tildas). JRSpriggs 10:03, 26 November 2006 (UTC)

Thanks. ^_^ (And sorry for the late thanks.) If you're interested, I decided to make the redirect page. What happens is that the User talk and the Archive links appear side by side each other in the top left. In case you wanted to add that on Werdnabot's page or something. Thanks again. :) -WarthogDemon 04:47, 3 December 2006 (UTC)

You are welcome. Thanks for the feed-back. JRSpriggs 08:00, 4 December 2006 (UTC)

Derivative

Yes, but, as Dr. Johnson said, people need reminding more than informing. I thought regular-sized fonts might help, but maybe next I'd be wanting regular-sized everything. In the meanwhile, I might just use what you have for my new Qualitative economics article & learn more on tex. Thx. BW, Thomasmeeks 13:08, 26 November 2006 (UTC)

Werdnabot

Actually I'm manually archiving because Werdnabot is not working on my talk page. There was stuff there over a month old. Guy (Help!) 09:11, 28 November 2006 (UTC)

Well, as you can see from your revision history, it is working now. The only reason it had not worked, since I changed the "Talk" to "talk", was that you were pre-empting it by archiving manually before the section's most recent entry was old enough to cause Werdnabot to archive it. JRSpriggs 06:51, 29 November 2006 (UTC)

P.S. Let me emphasize that Werdnabot decides whether to archive a section based on its MOST RECENT message, not on its oldest message. So you might very well have sections over a month old as judged by their oldest (i.e. earliest) message. JRSpriggs 07:03, 29 November 2006 (UTC)

φ(y, x₂, ..., x_n) versus φ(y, x₁, ..., x_n)

Actually I changed these to reflect the form that Kleene (1952) p. 219 uses (i.e. the first form φ(y, x₂, ..., x_n)). We see the usage repeated elsewhere (e.g. p. 237) so it is not a typo. Kleene is a very precise author -- the book was on its 10th printing in 1991 and errata had been fixed over the years. But I don't suppose too many readers will catch the sublety here (that y is taking the place of x₁, that these x and y are really place-holders for parameters and require natural numbers to fill them and they are not to be manipulated by algebra). Whether or not its worth changing them back to Kleene's form is another question. wvbaileyWvbailey 15:30, 2 December 2006 (UTC)

In Mu-recursive function, the preceeding line said and still says "* (5) Primitive recursion operator: Takes functions $g(x_{1},\ldots ,x_{k})$ and $h(y,z,x_{1},\ldots ,x_{k})$ and returns the unique function $f$ such that". I changed the "x_2" to "x_1" to be consistent with that notation. Notice that the "y" and "z" arguments of "h" are not replaced by "x"s. See the criticism by User talk:CBKAtTopsails at Talk:Computable function. JRSpriggs 07:50, 4 December 2006 (UTC)

Observable universe

You added a reference for the term billion as used in that article. Isn't billion commonly denominating "one thousand million" rather than "one million million" in English-speaking countries? Kncyu38 14:38, 3 December 2006 (UTC)

There are British people who insist on using "billion" to mean million million. And someone previously changed the "billion"s in the article to "thousand million"s for that reason. So I thought that to prevent confusion and a possible edit war, it would be best to specify what we mean in a foot-note. JRSpriggs 07:42, 4 December 2006 (UTC)

Ok, I see what you mean. It's really not a question of wrong or right, but of different approaches to the same problem. I happen to think it might be better to standardise the usage of such an important numeral throughout Wikipedia than to arbitrarily decide upon usage per article, so as not to confuse readers with different usage in different articles. But since I have no idea where or how to propose something like this, you may as well revert my latest change. Better to have clarity over the usage in this article, than no clarity at all. Kncyu38 13:06, 4 December 2006 (UTC)

I put both our methods together. JRSpriggs 04:21, 5 December 2006 (UTC)

Thanks, I had the same in mind. Kncyu38 08:46, 5 December 2006 (UTC)

Debatepedia.com - on minimum wage

Noticed your good contributions on the "minimum wage article" and am curious if you'd be interested in the minimum wage debate article on Debatepedia.com, and helping develop it. Minimum Wage Debate Loudsirens 21:25, 5 December 2006 (UTC)

My position is that the minimum wage is purely destructive and should be repealed. Anyone familiar with Economics knows this. Those who support the minimum wage are being deceptive. Their actual motives are malevolent, not benevolent as they pretend. JRSpriggs 10:18, 7 December 2006 (UTC)

Continued fractions

Hi, JRS!

I'm responding to a comment you made over here.

To DavidCBryant: You said "...well-formed continued fractions for functions like ln and sin and arctan converge a whole lot faster than the best of their series counterparts.". I would like to know more about this. I was working on the pages for Natural logarithms and Inverse trigonometric functions and I was disappointed at how slowly the series converge.

Well, I wish I could supply some specific examples, but I can't. Here's how I know about this, though.

I spent many years programming IBM mainframes in assembler language. Occasionally I had to interface an assembler routine with a FORTRAN library subroutine, to return say a natural logarithm in long floating-point form. So I acquired a copy of the FORTRAN math library logic manual from IBM. The manual didn't really belong to me; it belonged to my employer. So I don't have the book any longer. But I remember quite a bit of what it said.

The typical description of the mathematical logic underlying one of these subroutines ran something like this: We developed a continued fraction expansion for log₂ x that is valid in the range aaa through bbb and determined how many terms of that fraction must be used to reduce the approximation error below 2⁻⁴⁸ everywhere, then converted the truncated continued fraction into a rational function of x.

Come to think of it, the manuals didn't specify the coefficients of the two polynomials that entered the rational function. Those coefficients were buried in the machine language versions of the programs themselves. One other thing I remember is that they restricted the range of input values considerably. For instance, they might take the 48-bit mantissa of a floating-point number and interpret it as a fraction in the range 1 ≤ x < 2, then do a multi-precision division by √2 to further restrict the range to 1 ≤ y < √2. They'd feed y into the rational function algorithm, then adjust the result for all the powers of 2 that had been ignored up to this point. They'd switch the result from base-2 logarithms to some other base with a simple multiplication. (OK, OK. The IBM machinery was really base 16, not base 2. So they might have had as few as 45 bits of precision, and not really 48. But the concepts are right.)

I think that John Napier might have used a similar process, taking something like the 128th root of 2 (as the sqrt of the sqrt of the ...) and then calculating that logarithm ... multiplying the result by 128 gave log_e 2.

Anyway, I worked through something like this once and got pretty good results using just 5th-degree polynomials in the numerator and denominator to represent ln x in a fairly limited range. I'm sorry I don't have the details handy ... I'm not very good about hanging on to stuff like that. Just in case you want to play with them, though, here are some pretty neat continued fractions for the natural log and the arctangent.

$\log(1+z)={\cfrac {z}{1+{\cfrac {1^{2}z}{2+{\cfrac {1^{2}z}{3+{\cfrac {2^{2}z}{4+{\cfrac {2^{2}z}{5+{\cfrac {3^{2}z}{\ddots \,}}}}}}}}}}}}\,$

That expansion is valid in the cut complex plane, with the cut running from −1 to −∞. The partial denominators are just the natural numbers, in order, and the partial numerators (after the first one) are just n²z, where each perfect square appears exactly twice. That one probably doesn't converge too quickly for large z, because the partial numerators start getting bigger than the partial denominators too soon. But I think it works pretty well for z near zero. Another thing you might do with it is run it for +/- z, then use the (1 + z)/(1 - z) trick that's so often used with the logarithm series.

Here's another one.

$\arctan(z)={\cfrac {z}{1+{\cfrac {z^{2}}{3+{\cfrac {4z^{2}}{5+{\cfrac {9z^{2}}{7+{\cfrac {16z^{2}}{9+{\cfrac {25z^{2}}{\ddots \,}}}}}}}}}}}}\,$

This one is also valid in the cut plane, but this time there are two cuts, from −i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It probably works well for real numbers running from -1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just (nz)², with each perfect square appearing once.

Oh, yeah ... both of these were developed by KF Gauss, utilizing the hypergeometric functions. Have fun! DavidCBryant 00:31, 7 December 2006 (UTC) —The preceding unsigned comment was added by DavidCBryant (talk • contribs) 00:24, 7 December 2006 (UTC).

Thanks for the information. I will try out the arctan continued fraction and compare it to the series. JRSpriggs 10:10, 7 December 2006 (UTC)

The rational polynomials from the continued fraction come closer to the true value than those from Euler's series for arctan when corresponding degrees are compared. JRSpriggs 06:44, 8 December 2006 (UTC)

I put your continued fraction for arctangent into Inverse trigonometric function. JRSpriggs 10:40, 8 December 2006 (UTC)

A good book on this is Jack Crenshaw's "Math toolkit for real-time programming " —The preceding unsigned comment was added by User A1 (talk • contribs) 14:08, 10 January 2007 (UTC).

Your revert to InuYasha

Please try to write in a measured, nuanced, encyclopedic and factual way. Excessive and tendentious edits might disturb the work of other editors and be reverted. You might find reading WP:POV useful in this respect. Thank you. This message is in regards to your major revert to InuYasha in which you had reverted every single contribution made by editors after you. I would suggest reading WP:Civility. Power level (Dragon Ball) 15:59, 7 December 2006 (UTC)

Most of those edits were clearly vandalism. Most of the others were inadequate attempts to repair that vandalism. Your edits were the only ones I considered saving. However, you insisted on putting in a non-word, "titicular", or a very obscure word, "titular", either of which could be considered sexually suggestive when there was a perfectly good text saying the same thing already there. So I reverted you as well. JRSpriggs 06:38, 8 December 2006 (UTC)

P.S. My edit summary was "rvv to my last" ("rvv" meaning reverting vandalism). There is nothing uncivil about that. It is merely informative. JRSpriggs 06:41, 8 December 2006 (UTC)

Combinatorial principles

Hi, please have a look at my suggestion in Combinatorial principles (talk). Thanks! --Aleph4 17:28, 17 December 2006 (UTC)

Yes. It was your mention of Diamondsuit there which gave me the idea of adding a pointer to Statements true in L which includes pointers to several of your infinitary combinatorial principles. JRSpriggs 05:17, 18 December 2006 (UTC)

Formula editing problems

I have some problems when editing scalar form of "Coulomb's Law". I don't know how to demonstrate the magnitude of vector electric force. Can you give me some instructions about the code of math formula? Thank you._AbelinCAusesobad 13:23, 20 December 2006 (UTC)

I looked at the formulas there and my first impression is that they are correct as they stand. Could you be more specific about what is bothering you? You can look at Help:Displaying a formula and Wikipedia:How to edit a page for more information on how to make the formulas look like you want them to look. JRSpriggs 03:20, 21 December 2006 (UTC)

Re: Your bot screwed up our archives!

Sorry about that! My fault for celebrating Christmas rather than clearing up after my bot. I'll try to sort it out later today. See the noticeboard thread for my full reply – Gurch 12:42, 26 December 2006 (UTC)

The five affected pages are now fixed. Automated archiving is not allowed on article talk pages, and most automated archives are User Talk archives, which the bot didn't move (as far as I'm concerned people can do what they like with them). So the only pages that could potentially have been affected are the 30 or so automatically archived Wikipedia Talk pages; of these, most use a different style of archiving or don't have any archives – Gurch 18:47, 26 December 2006 (UTC)

WP:AIV

Hi, I wanted to make sure that you were aware that blocking is a preventative action, not a punitive one. As such, we generally refrain from blocking anonymous users that have stopped vandalizing, like 74.103.26.79 (talk · contribs · deleted contribs · filter log · WHOIS · RDNS · RBLs · http · block user · block log) has. As such, I am removing this listing from AIV. Warmest regards, —bbatsell ¿? 05:34, 27 December 2006 (UTC)

Mathematical induction revisited

Hi, if you still remember, I was the one mentioning about the sequences starting from 1 rather than 0 in the induction. I was rather noobish on using Wikipedia at that time, so I didn't care to respond.

I was browsing the Manual of style (dates and numbers) (for some arguments about if one-word numbers should be in letters or numerals concerning the consistency of all numbers in a context, which is kinda stupid), and saw in a section that mentions the definition of "natural numbers". The quotation follows:

Natural number has two meanings:

• positive integer, or
• non-negative integer (including zero).

When referring to such numbers, explicitly use one of the above phrases rather than "natural numbers", unless it does not matter which interpretation is chosen.

So back to what you have explained to me: maybe the use of the phrase "natural number" is after all confusing at some level. It would end up being about the religious beliefs and everything as the other person (Trovatore) commented in my talk page. I asked my math teacher about this issue, and he reported to me after a discussion with his colleagues that the sequence should always start at 1. Not that I'm saying that I or my teacher is correct; rather, I was wondering if this ambiguous definition for "natural number" should be cleared up since the article (Math. ind.) currently favors your explanation by assuming 0 to be included in the sequence. Regards, Vic226_(chat) 02:46, 28 December 2006 (UTC)

Did your teacher give any good reason why zero should be excluded? As you probably know, I believe that zero MUST be a natural number because the natural numbers are the cardinalities of finite sets (or order types of finite well-orderings), and the empty set is certainly a finite set. As far as using "non-negative integer" or "positive integer" is concerned, "integer" itself is a derivative concept compared to "natural number" so that would be a circular definition. JRSpriggs 05:27, 28 December 2006 (UTC)

While I completely understand your viewpoint in terms of set theory, do you know if there is any valid reason for the number theorists to stick to using sets starting from 1 instead of zero? Also, I'm not sure if topics like math. ind. (which uses the "natural numbers") needs some info about this since it looks like it is favoring one usage over another (in this article, set theory over number theory). Regards, Vic226_(chat) 15:23, 28 December 2006 (UTC)

I do not know enough about number theory to know why they oppose including zero. If I have to guess, I would guess that it is because zero does not fit well into the free commutative monoid of positive integers under multiplication.

When talking about mathematical induction, one must start somewhere. I prefer to start where one must start when defining the natural numbers, i.e. at zero. Once one has defined it, one can generalize to an arbitrary chosen initial value, such as one. JRSpriggs 06:03, 29 December 2006 (UTC)

*

Answered in email.Nina Odell 14:40, 30 December 2006 (UTC)

Likewise. JRSpriggs 03:25, 1 January 2007 (UTC)

Keeping nonsense in discussions

You have restored the my question raised due to my inattentiveness. It makes no sense, cloggs the internet traffic while reduces readability. What is the reason to keep it? --Javalenok 23:05, 9 January 2007 (UTC)

To my talk page adoptee

I don't know you, but you keep replying to people somewhat usefully on my talk page. Thanks, it's nice to have somebody answer the more common queries on my user talk page before I get around to it :-). — Werdna talk 12:30, 14 January 2007 (UTC)

You are welcome. I am glad that you appreciate my efforts, rather than being offended by them. I took over archiving Wikipedia talk:WikiProject Mathematics and Wikipedia talk:WikiProject Physics. So I thought that I could do a better job if I learned more about archiving in general and Werdnabot in particular. The best way to learn is by doing, so I have been trying to help others with it as much as I can. By the way, what is the best way to contact you (since you seem to be hard to reach)? I am hesitant to download more client software just to use epstone. JRSpriggs 12:41, 14 January 2007 (UTC)

You can email me, and I'll usually reply fairly quickly. I don't know what you're talking about as to "download more client software just to use epstone". — Werdna talk 02:42, 15 January 2007 (UTC)

Well, you say "If you want to join us on IRC, point your client to #werdnabot on EpstoNET (irc.epstone.net) or use our cgi-irc client.". That makes it sound like I have to have some new software, "our cgi-irc client", on my computer in order to use epstone. JRSpriggs 04:10, 15 January 2007 (UTC)

Veracious Rey

Thanks for the fix on my talk page, but my first discussion post, dated Jan. 1, still is not being archived. No big deal, but I'd like to find out the problem. Thanks for your help. Veracious Rey talk • contribs • review 04:48, 19 January 2007 (UTC)

To Veracious Rey: Werdnabot archiving is not instantaneous. Werdnabot usually runs once per day between 10:00 and perhaps 10:45. Sometime in that interval, it should archive your page provided that it does not encounter an edit conflict or some other problem. Check back sometime after 12:00 noon. You can see its contributions at Special:Contributions/Werdnabot. JRSpriggs 04:55, 19 January 2007 (UTC)

Got it. Thanks! Veracious Rey talk • contribs • review 05:16, 19 January 2007 (UTC)

Job search engine

Doh! Missed the extra cats. Thanks for spotting that. Kuru ^talk 04:21, 20 January 2007 (UTC)

You are welcome. JRSpriggs 04:23, 20 January 2007 (UTC)

Your userpage

I have redirected it to your talk page so that your signature does not turn up as a redlink. You might also want to make some changes in your preferences so that your signature directly links to your user talk page. Best regards, — Nearly Headless Nick 14:55, 20 January 2007 (UTC)

Thanks you JR

For the useful link you gave me. I'm sort of inactive at the moment, so I'll look at it as I have time. I might have some questions in the future related to this. Again, thank you for your kindness and consideration. NinaOdell | Talk 15:19, 20 January 2007 (UTC)

Please review

Hi, JR.

I just wrote this article to explain how the complex argument can be defined without reference to geometry, or trigonometry. (Well, I still have to add the references, which I'll get around to soon.) I value your feedback; when you have the time and the inclination, please take a look and let me know what you think. Thanks! DavidCBryant 19:24, 20 January 2007 (UTC)

I answered on your talk page. JRSpriggs 06:05, 25 January 2007 (UTC)

About the Kagome Higurashi Vandalism Warning

What I'm doing is NOT vandalism. If you've looked at my edits, you'll see that I've used no inappropriate words, etc in the Kagome Higurashi section. You'll see that I've protrayed Kagome in both a positive and a negative light )ie: I've shown her to be both beautiful and kind, but I've also portrayed her as having a bit of a temper. If you persist on irritating me with unwarranted claims, I will alert the REAL authorities. There I've signed in. Hope that satisfies you. LucifaelsBride 12:49, 22 January 2007 (UTC)LucifaelsBride

I am not harassing you. I have been more than patient. If you want to seek mediation or other intervention from an administrator, be my guest. However, you should have the courage to log in as yourself when you put messages like this on another person's talk page. JRSpriggs 04:33, 22 January 2007 (UTC)

Congratulations...

... on finally somebody not bearing it anymore with your redlinked user page and making it into a nice blue redirect! :) Oleg Alexandrov (talk) 05:41, 22 January 2007 (UTC)

Thanks, I guess. Apparently, Sir Nicholas de Mimsy-Porpington (talk · contribs) felt that I had been around too long to take advantage of the forbearance we accord to newcomers. JRSpriggs 05:56, 22 January 2007 (UTC)

Stop vandalizing my talk page with warnings about Kagome Higurashi

Again, that was not inappropriate stuff, it give both negative and positive (ie: neutral POV). Oddly, this is the way I've worded all my other edits in other section (refer to my contribution) and you did not seem to find a problem with any of them, only Kagome. This is shows that you're just a disgruntled fan of Kagome who hates seeing her portrayed in anything other than a flawless light.

LucifaelsBride 12:51, 22 January 2007 (UTC)LucifaelsBride

When you stop committing vandalism (or get banned which would have that effect), I will stop warning you about it. I suggest that you stop. Also, this kind of attack on those such as myself who are trying to enforce the rules will only get you in more trouble. JRSpriggs 12:23, 22 January 2007 (UTC)

This is not an attack. I don't see you doing anything to Ned Scott and he deleted the whole personality section from KAgome Higurashi. Why? Because you think he has a point about Kagome having no personality? That makes no sense. You're just trying to bully me, because I'm newer than you. You don't have the courage to confront someone else who knows Wikipedia as well as you do.

LucifaelsBride 12:51, 22 January 2007 (UTC)LucifaelsBride

As TheFarix (talk · contribs) pointed out on your talk page, the issue is your original research. In other words, it is not whether you are pro-Kagome or anti-Kagome or balanced. It is whether your contributions are: accurate, verifiable (i.e. you can point to documents which support them), and relevant to Kagome's part in the anime and manga InuYasha. Your very hostile attitude towards me and the other people who try to help you does not incline us to continue helping you or otherwise cut you any slack. JRSpriggs 06:27, 25 January 2007 (UTC)

Halley's method

Hi!

I've seen you cited me as the author of the French article. It was not necessary, as my contributions are public domain (see fr:Utilisateur:Lachaume).

Moreover, don't you you have a template on the English Wikipedia to automatically add links to the list of authors and current version in order to comply with the GFDL? fr:Modèle:Traduit de and fr:Modèle:Référence/Traduction add something like "parts of this article are a translation from the GFDL article hu:zene (version of January 17th, 2007), see the list of authors".

Cheers.

Régis Lachaume 17:59, 23 January 2007 (UTC)

Kinetic energy

Imagine a page on velocity which doesn't mention that it's a concept which only applies between two or more named objects, and has no meaning on its own, for ONE object? Don't you think somewhere in the beginning of the kinetic energy article (or indeed ANY place in the article) it should be noted that the concept of kinetic energy is more or less a relative (and therefore, except in special circumstances, somewhat arbitrary one), like velocity? SBHarris 14:45, 30 January 2007 (UTC)

The two paragraphs you added were both incorrect and misplaced. Incorrect, because you are confusing dependence on the frame of reference with being a collective property of a system of objects. Misplaced, because the lead is supposed to be elementary for the students who are just learning the subject for the first time and your paragraphs, even if they were true, would just confuse them. I put a short section acknowledging that kinetic energy is relative to the reference frame at the end of the article. JRSpriggs 06:17, 31 January 2007 (UTC)

• I assure you that I have no such confusion regarding dependence-on-frame with collective properties. Some collective properties have frame dependence, and some don't. The one we call invariant mass, which includes some (but not all) of the kinetic energy in systems, does not. That means some kinetic energy is frame dependent (i.e., not objectively real, because it merely depends on your point of view, like velocity), and some is not. Which is NOT like velocity. You might not consider this an important enough fact about this interesting form of energy to put in the LEAD, but it's surely not trivial enough to stick into the end. This is, after all, energy we're talking about. Either it's objectively real or not, or both. In this case, it's both. Some is objectively real (observer non-dependent) and some isn't real (is observer dependent or merely relative to point of view). Is is incorrect to simply say that kinetic energy is observer dependent. That is wrong. So how about we compromise and move it up front to the section following lead.? SBHarris 11:47, 2 February 2007 (UTC)

I disagree with your belief that things which are not invariant are consequently not objectively real. Kinetic energy is objectively real even though it is not an invariant. Your statements about it would just confuse a naive reader of the article. Moreover, to understand your comments about kinetic energy, one must first know what kinetic energy is. Therefor, any mention of your ideas should be at the end. JRSpriggs 04:38, 3 February 2007 (UTC)

Some kinetic energy is invariant. For example, all the kinetic energy in a can of ideal monatomic gas is invariant. All observers do and must agree on its value, for it is the summed energy of the gas in the COM frame (some observers may have to transform to that frame to get the number), minus the rest masses of the gas particles. We call that kinetic energy "heat." It contributes to invariant mass, and in fact is the invariant mass which isn't the sum of the particle rest masses. You can define kinetic energy any way you like, and you'll still come up (or should come up) with the immediate question of kinetic energy relative to WHAT?

One more very simple example. Two equal mass astronauts in space do total work = F x d = E to push off from each other in opposite directions (to be clear, each astonaut only does E/2 amount of F x d, because each pushes through only half the distance that ultimately separates then when they part). Yet each then sees the kinetic energy of the other as 2E. Some kinetic energy has appeared, from each person's view, without being paid for in potential (which totaled E not 2E). Clearly, kinetic energy is a system property, not a property of any given object, unless you clearly define your inertial frame (which is not done in the opening LEAD of this article). Then, it actually IS a system property, and a well behaved one, too. Both invariant and objective. SBHarris 14:30, 3 February 2007 (UTC)

Thanks for the Werdnabot help.

(from User talk:Werdna): To PurpleRain: You set the retention period to sixty (60) days. The only message on your talk page which is over sixty days old is the first one. And that message is not in a section because there is no section header above it. Thus Werdnabot is not supposed to archive anything from your page yet. In other words, Werdnbot is working the way it is supposed to work. If you think that some sections should have been archived, then you should either reduce the retention period or put a section header over that first message. JRSpriggs 04:57, 26 January 2007 (UTC)

Thanks for the help. I forgot to add the Werdna talk page to my watchlist, so I missed your helpful comment until now. I was aware that only the first message was old enough to be archived, but couldn't figure out why it wasn't being archived. Thanks for explaining. —P^urpleRA^IN 20:49, 31 January 2007 (UTC)

You are welcome. There is an option in one's preferences (which I use) to automatically add any page you edit to your watchlist. You might want to use it. (Or not, because it can cause your list to grow alarmingly fast so that you have to keep pruning it.) JRSpriggs 04:25, 1 February 2007 (UTC)

As far as I can tell, that option only changes whether a certain box is checked by default or not - by carefully unchecking the box before saving some edits, you can avoid adding unwanted articles to your watchlist. CMummert · talk 04:46, 1 February 2007 (UTC)

Yes, that is true, but I almost always forget to uncheck that box (on those occasions when I do not want to watch the article). But I figure that it is better to default to watching than not watching, if I forget to do something with that switch. JRSpriggs 04:52, 1 February 2007 (UTC)

I've tried that option, but it does cause the watchlist to grow alarmingly fast. I usually remember to "watch" all the pages I need to; this was just an exception. Thanks for the advice, though. —P^urpleRA^IN 17:00, 1 February 2007 (UTC)

There is a javascript function you can put in your monobook.js that adds an "unwatch" link to each entry on your watchlist. You can find it in the Scripts project or in my monobook.js if you are interested. CMummert · talk 16:26, 7 February 2007 (UTC)

wp:AIV comment

"If you are going to have a template or templates, it/they should be informative -- what can the reporter of vandalism do which he/she has not done? For example, if the vandalism is stale, then you should point him to who can deal with long-term vandalism. If the vandal has already been adequately warned for the offense, then suggest the reporter check whether a warning has already been given. etcetera"

Thanks for your comment on AIV. Just to check - how else should someone deal with long term vandalism? AndrewRT(Talk) 23:36, 5 February 2007 (UTC)

I just report vandals. I do not work on the other end of the process. So I do not know what the possibilities are, which is one reason I thought of asking for more information in the messages.

Personally, I would like to see us take a stronger position against vandals, even if that drives away a few potentially useful editors. (I am sure that we lose good editors because of vandalism.) I doubt that many vandals become constructive editors anyway. And speaking of IP-vandals, groups which hide the identity of their members should be held responsible for their actions. JRSpriggs 04:10, 6 February 2007 (UTC)

Gravitational Constant revert

Thank you very much for reverting gravitational constant. I was wondering all yesterday why my acceleration due to gravity kept turning out to 9.786 x 10^22. Greatly appreciated. --Aa35te 00:08, 6 February 2007 (UTC)

You are welcome. We had a very dangerous type of vandal -- one who changes the values of constants. This is potentially much harder to detect and fix than someone who just blanks the page and replaces it with "I am the greatest." or some such. Fortunately, I saw his work on another article where I knew what he put in was wrong. So I investigated his recent edits and found that he was changing a lot of mathematical and physical constants. I tried to revert the ones which had not already been reverted by others. Unfortunately, sometimes vandals assume another identity and pretend to revert themselves without really fixing the problem (or it might be another vandal taking advantage of the situation). So it is hard to know what to do sometimes.

I guess that this shows that you should not put too much trust in anything on Wikipedia, because anyone can change it for good or for ill. Just use it to try to get a general understanding and then check the crucial facts elsewhere. JRSpriggs 04:21, 6 February 2007 (UTC)

total lack of understanding from Biedermann 12:13, 22 January 2007 (UTC)

Well Sir, i guess it was you who diagnosed my total lack of understanding. This certainly couldn’t help me to some better understanding. So, let me try to explain in more detail what I think to be my understanding with respect to what Gödel’s term Z(y) could be in the symbols of Gödel’s System. Without such knowledge the Gödel number n of formula (x)~Dem(x,sub(y,19, Z(y))) cannot be determined. But this is essential for the progress of Gödel’s argumentation; just to name it ‘n’ , does not serve the purpose, n not being a symbol of the System. From Gödel’s definitions 16 an17, together with 8 and 9, we know Z(n) to be (to BE, as Gödel says expressis verbis, not to Designate, as some people seem to believe!) the Gödel-number of the number n in the symbols of the System. With the claim that sub(n, 19, Z(n)) be obtained from sub(y, 19, Z(y)) through substitution of 19 (i.e. fffffffffffffffffff0 ) by Z(n), of necessity Z(y) would have to BE the number 19, and the term sub(y, 19, Z(Y)) would in fact be sub(y, 19, 19)), from which the term sub(n, 19, Z(n)) certainly does not result through simple Substitution of n for y ! To my poor understanding, Gödel’s argumentation depends on the assumption that Z( ) could be a function sign expressed in the symbols of the System, in which you may freely exchange the arguments n for y etc., and which could simultaneously yield the mathematical function ‘Gödel number of n’ on the argument n, and the deliberately chosen non-mathematical relation 19 for the argument y ! To my poor understanding it seems evident that such function cannot exist. Would you please be so kind as to explain me what you think to be your better understanding! With kind regards Biedermann Biedermann 12:13, 22 January 2007 (UTC)

Gödel numbering makes the primitive recursive functions involved very messy and the numerical values of the codes very large. Like many "proofs" in mathematics, his proof is really just an outline of how a formal proof could be written, if one had the virtually unlimited resources necessary to carry out such a formal proof. That is one reason why his argument uses variables to refer to constant numbers which result from the encoding. Another reason is that the details of the encoding, beyond the fact that it is primitive recursive, are really unimportant. You could switch to any of an infinitude of coding schemes and achieve the same outcome.

For me to explain this to you in a way that you could understand, I would have to walk you through the details of the Gödel numbering and compute some of those codes. It is just not worth my time to do that, just to help one person, when I could be improving articles which help thousands of people. If you really want to pursue this topic, you should read the book recommend by CMummert. JRSpriggs 06:16, 25 January 2007 (UTC)

Well Sir, thanks for recommending me a book that is out of my reach. But, isn’t it funny: as I formulated my concerns in a few straightforward sentences with direct reference to Gödel’s Original, certainly any good answer to them should be feasible to be formulated in just as straightforward reference to that paper. But you say: that cannot be done! You wouldn’t believe it, but I take this your comment as an excellent corroboration for the justification of my concerns. Reading your book couldn’t serve but to divert from the problem at hand. Besides, to my great surprise, the author of that article does not even mention the decisive problem in Gödel’s invention: how to insert into a formula its own Gödel number. That almost smells of some lack of understanding! On top of all this, I am glad to find at the beginning of that article the sentence: ‘Roughly speaking, the Gödel statement, G, can be expressed: ‘G cannot be proven true’ (although I would prefer the simpler statement: ‘G cannot be proven, i.e. derived from the axioms’!), where by inserting the meaning of G for the letter G we obtain: ‘’G cannot be proven true’ cannot be proven true’’, and repeating this manoeuvre over and over again, we will obtain an infinite nested structure of ‘’’’’’’G cannot be proven true’ cannot…’…’…’’’’’, but we will never get rid of that letter G and never come to know what it should be that ‘cannot be proven true’! But if such a silly nonsense structure is at the core of Gödel’s invention, why do you take it for serious? I am sorry, I cannot! And I am glad that, according to my analysis, such selfreferent nonsense cannot even correctly be formulated in the symbols of Gödel’s Formal System! Biedermann 18:18, 2 February 2007 (UTC)

Perhaps you are a victim of an over-simplified "translation" to English of the Gödel formula. Rather than translating it as G = "G cannot be proven from the axioms of arithmetic.", it would be more correct to translate it as G = "The sentence which results from replacing all occurrences of the last letter of the alphabet by 'The sentence which results from replacing all occurrences of the last letter of the alphabet by 'z' in 'z' is not provable from the axioms of arithmetic.' in 'The sentence which results from replacing all occurrences of the last letter of the alphabet by 'z' in 'z' is not provable from the axioms of arithmetic.' is not provable from the axioms of arithmetic.". I hope you see that this is not an infinite regress. Rather it is a statement which refers to itself indirectly by describing a way of constructing itself. JRSpriggs 04:54, 3 February 2007 (UTC)

Well, Sir, thanks for your answer! However, having just two concerns with Gödel’s Original, I have some more with this answer. 1.) You suggest me being ‘a victim of an oversimplified translation….’. No, Sir, as I clearly said, I found that ‘translation’ in the Wiki-article, so, if anybody was victimized, it was the author of that article! I just extended on the nonsensical consequences of that translation, of which that author apparently was not aware. 2.) Now to your phrasing between those two double-apostrophes: did you take it from your recommended book, or is it your own invention? In any case, this doubly nested structure, although omitting the Gödelization, is so much longer and tedious to read than Gödel’s simple term ‘sub(y,13,Z(y))’ on which I concentrate my two concerns, that, to me, it is but a lengthy diversion from ‘the problem at hand’! 3.) With ‘the last letter of the alphabet’ you are introducing a concept that cannot be expressed in the symbols of the Formal System! That certainly is not an improvement over Gödel’s Original, where the interpretation of Z(y) to be the Gödel number of y cannot be expressed in the symbols of the System! 4.) Your phrasing ‘from replacing all occurrences of the last letter of the alphabet by 'z' in 'z' ‘ , does not make sense since it implies that the second ‘z’ contains occurrences of z (the last letter in the alphabet). But how can z contain z? Certainly, the second z must have another identity than the z that is assumed to be contained in it, in crude violation of Russell-Whitehead! Otherwise it would yield another infinitely nested structure! 5.) Conclusion: your lengthy tedious ‘translation’, notwithstanding the omission of the Gödelization, suffers from precisely the same flaws that I spotted in sub(y,13,Z(y)). Did you not notice that? 6.) By the way, what would be your comment to my further objection: if Gödel’s construct were assumed to be flawless, then Gödel’s term ‘sub(n,13,Z(n))’ defines a natural number, and his statement simply says that this number cannot be derived from the axioms. But certainly, only statements, and no natural numbers can be derived from the axioms! What good is all the fuss about this great invention of Gödel’s? With kind regards 195.23.182.195 11:36, 7 February 2007 (UTC)

(1) I assumed that your main concern was to either improve your own understanding of the theorems or, alternatively, to convince me that I am incorrect in thinking that they are theorems (rather than mistakes or hoaxes). As to the article Gödel's incompleteness theorems, it is not the work of any one author, but a mish-mash of contributions from many people. Consequently, I do not want to try to take responsibility for its correctness. And frankly, I must admit that it has been decades since I read an English translation of Gödel's paper which was originally in German (although the translation I was referring to in my previous message was from Mathematical symbolism to common English), so my own understanding may be less than perfect.

(2) My phrasing is my own and I do not claim that it is perfect. I just wanted to give you the general idea of how it works, so that you can see that no infinite regression is involved. I admit that G as I describe it is a convoluted and, one might say, pathological utterance. However, the question is not whether it is a desirable thing to say, but whether it is a possible thing to say in the formal language of arithmetic. But I think that you must first understand what I have said above before you can hope to understand what Gödel is saying.

Regarding ‘sub(y,13,Z(y))’: (without going back and checking Gödel's text) I think that y corresponds roughly to the Gödel number of "The sentence which results from replacing all occurrences of the last letter of the alphabet by 'z' in 'z' is not provable from the axioms of arithmetic.". Z(y) corresponds to the Gödel number of the numeral for that Gödel number. And 13 corresponds to my phrase "the last letter of the alphabet" which describes the variable for which y is to be substituted without using that variable itself. The result of the substitution is the Gödel number of the entire monsterous sentence which I called G above.

(3) The mapping, Z, of a number to the Gödel number of its numeral is certainly a primitive recursive function which can be encoded in the language of arithmetic with the help of the Chinese remainder theorem.

(4) It is a mistake to try to understand the meaning of "The sentence which results from replacing all occurrences of the last letter of the alphabet by 'z' in 'z' is not provable from the axioms of arithmetic." in isolation; it has none. The entire sentence G must be understood as a whole, reading the outer parts first, and performing the operations indicated on the included quotes.

(5) Rather, I would say, that when you are able to comprehend my translation, then you will hopefully be able to comprehend the original. Neither has the flaws which you imagine they have.

(6) The Gödelization process includes not only individual symbols of the language of arithmetic, but also the composite structures built from them: numerals, sentences, proofs, and the axiom system itself. So when he says that a number is not derivable from the axioms what he means is when that number is translated back to the sentence of which it is the Gödel number, that sentence is not a theorem of arithmetic. JRSpriggs 05:05, 8 February 2007 (UTC)

Well, Sir, thanks again for your response! You are so quick in answering. Last night, when falling asleep, I realized that with my question No.6 I must have had a serious blackout. I always knew that the Gödelization translates the axioms equally into numbers, so only specific numbers can be ‘derived’ from them. Sorry! I readily accept your confession that your ‘own understanding may be less than perfect’. Who can claim to be perfect?
To No.2 : I am surprised at your conviction that I would first have to understand your ‘pathological utterance’ before I could hope to understand Gödel’s reasoning, which after careful study of the German Original seems so much clearer and easier to understand. From that knowledge I feel sure, that your assumptions about sub(y,13,Z(y)) and what its components could correspond to, are definitely wrong! From Gödel’s Original “y” does not “correspond” to anything, it is nothing else but one of the free variables of the Formal System and “13” is the Gödel number of this symbol! The term “sub(y,13,Z(y))” simply reads “the number that results from the (unknown!) number y through substitution (a quite intricate arithmetical maneuver!) of the number 13 by the number Z(y)”, where Z(y) cannot be anything but the Gödel number of y, i.e. “13”, although Gödel does not say so expressis verbis. If you prefer: in English language it reads “the Gödel number of the formula that results from the (undetermined) formula with Gödel number “y” through substitution (normal substitution!) of the variable with Gödel number “13” by the variable y”! No “pathological utterances” whatsoever are required to express this! You cannot miss the fact that the variable y appears here in two different identities!
To No.3 : But haven’t you read my comment from January 22 ? The recursive function Z( ) as defined by Gödel in the cited definitions (16) and (17) certainly yields “The mapping of a number to the Gödel number of its numeral”. But that function does not map the symbol y to the Gödel number “13”!
To No.4 : Hey, I rejoice at your acceptance of my judgment that your phrasing of replacing “z” by “z” in “z” is NONSENSE! But what kind of mathematical logic is your declaring it a mistake to try to understand the meaning of part of your argumentation?
To No.5 : I feel sure that you cannot arrive at a proper understanding of Gödel as long as you refuse to come down to the simple facts in Gödel’s Original! With kind regards Biedermann 13:40, 8 February 2007 (UTC)

I looked through my library trying to find a copy of Gödel's paper. Then I remembered that I had borrowed it from and returned it to one of my professors back in the 1970s. All I have left is a hand written summary which I wrote back then. And the version I read was in English, but the abbreviations used in the mathematical formulas were based on the German words. My lack of complete documentation is one reason that I have tried to resist getting into the details of the formulas.

According to my notes, 17 was the Gödel number used for the formal variable m₁ and x was an informal variable corresponding to it. Similarly, for 19, m₂, and y. Your "sub(y,19,Z(y))" is an expression whose value is the Gödel number of the result of substituting the numeral for y for the variable m₂ in the formula whose Gödel number is y. Notice that by "numeral for y" I mean the numeral for the number which is represented by informal variable "y", NOT something which in anyway represents the formal symbol, variable, or letter "y".

q was an informal variable standing for the (enormous and very difficult to calculate) Gödel number of "~ x B sub(y,19,Z(y))" which means "x is not a proof of the result of substituting the numeral for y for the variable m₂ in the formula whose Gödel number is y". p was an informal variable standing for value of the expression "17 Gen q" which means "For all x, x is not a proof of the result of substituting the numeral for y for the variable m₂ in the formula whose Gödel number is y". r was an informal variable standing for the expression "sub(q,19,Z(p))" which means "x is not a proof of the result of substituting the numeral for p for the variable m₂ in the formula whose Gödel number is p", i.e. "x is not a proof of the result of substituting the numeral for the Gödel number of "For all x, x is not a proof of the result of substituting the numeral for y for the variable m₂ in the formula whose Gödel number is y" for the variable m₂ in the formula which stands for "For all x, x is not a proof of the result of substituting the numeral for y for the variable m₂ in the formula whose Gödel number is y".". Does this not look a lot like (albeit not identical to) my monsterous sentence G above? 17 Gen r (the crucial expression) just adds "For all x, " to the beginning of that.

You are wrong when you say "...substitution ... of the variable with Gödel number “19” by the variable y". The variable m₂ which is represented by 19 is not replaced by a variable y, it replaced by the NUMERAL for the number which y represents. Z does not map "the symbol y" to ANYTHING. It maps the number which y represents to the numeral for that number.

If I said "the sequence which results from replacing all occurrences of 'h' by 'abc' in 'ehfhg' ", would you do the replacement and get "eabcfabcg" or would you try to figure out what I meant by "abc" and "ehfhg"? The two inner sequences have no meaning in isolation, but the overall sentence has a meaning which is very important. JRSpriggs 05:01, 9 February 2007 (UTC)

part 2

Well, I guess my proposal to come down to Gödels Original was not the best idea; the mix of two types of variables will not serve to clarify the situation. Hopefully I can come back on that later. So let’s first return to your favourite, that pathological utterance which I can assure you to understand (in contrast to your assumption!) perfectly well! However , I would prefer a somewhat more perspicuous formulation: Since it makes to me much more sense to say that some sentence results from some other sentence through some substitution and not ‘the sentence results from some substitution….’ The inner part of your convoluted sentence would be “there is no proof to the sentence that results from sentence “z” through substitution of the (eventually therein contained ) last letter of the alphabet by the symbol “z” “ . Now you cannot ignore the appearance of “z” here in two quite different identities, once as symbol in the sentence and once as name for that sentence, (in violation of Russell-Whitehead! and as in your sentence G: G cannot be proven), and then this sentence obtains a perfectly good meaning. Remains the second defect, the term “the last letter of the alphabet”, which is not expressible in the Formal System, and its meaning to be nothing else but the symbol “z” resulting only from our outside knowledge! Just as with the “Z(y)” in the term sub(y,13,Z(y)) ! How will you manage to ignore these defects? Now to those variables: you are calling x, y, q, and p indistinguishably “informal variables” which certainly says: they are not symbols of the Formal System, and no Gödel numbers are attached to them. Yet there is a decisive difference: q and p stand as names for two supposedly previously determined exact numbers, whose numerals are consequently equally fixed and their Gödel numbers derivable. Nothing the like for “x” and “y” ! However, when you say: q was an informal variable standing for the… Gödel number of "~ x B sub(y,19,Z(y))", you should realize that you cannot determine a Gödel number for a symbol sequence that contains “informal variables”, you have to recognize “x” and “y” as the formal variables, and the number “19” as the Gödel number of “y” ! Remains my initial question, of what sequence of symbols of the System do you suppose the term Z(y) to be? Since there is NO number that is represented by the variable "y", with your own definition, Z(y) is the Gödel number of the numeral for NOTHING, so Z(y) does not exist, as I said from the beginning! By the way: I could impossibly claim to grasp the meaning of some compound as long as I do not grasp the meaning of any one of its components. Once more: With kind regards Biedermann 15:08, 10 February 2007 (UTC)

Change that to -- “There is no proof to the sentence that results from string “z” through substitution of the last letter of the alphabet by the string “z”.”. Notice, that if in this we replace “z” by “There is no proof to the sentence that results from string “z” through substitution of the last letter of the alphabet by the string “z”.” (i.e. the same string), we get “There is no proof to the sentence that results from string “There is no proof to the sentence that results from string “z” through substitution of the last letter of the alphabet by the string “z”.” through substitution of the last letter of the alphabet by the string “There is no proof to the sentence that results from string “z” through substitution of the last letter of the alphabet by the string “z”.”.”. This resulting sentence effectively says that there is no proof of itself. It does not say "myself", rather it describes itself as the result of the process by which it was constructed. To make this clearer, let me make it shorter. If I replace “z” by “AzBzC” in “AzBzC”, I get “AAzBzCBAzBzCC”. And we are interpreting the first occurrence of “AzBzC” in the result as the string to be substituted for “z” and the second occurrence as the string into which the substitution is to be done.

Using the same symbol to stand for two different things is usually something one should try to avoid because it makes it easy to commit a logical fallacy by changing the meaning of a symbol and thus the proposition in which it is contained after getting agreement on the original version. That is not what is happening here. Here, we are not attributing ANY meaning to “z”. We are simply replacing it with other strings as required by the surrounding text in order to construct a certain sentence.

To see why “the last letter of the alphabet” is not a problem, we need to get into the Gödelization process for which we have no shared reference document (which is why I thought this conversation futile from the outset).

Yes, I should have said "~ m₁ B sub(m₂,19,Z(m₂))" rather than "~ x B sub(y,19,Z(y))".

"Z(0)" refers to the Gödel number for "0". "Z(1)" refers to the Gödel number for "s0". "Z(2)" refers to the Gödel number for "ss0". And so forth. Do you doubt that there is a primitive recursive function which maps 0 to the Gödel number for "0", 1 to the Gödel number for "s0", etc.? JRSpriggs 05:03, 12 February 2007 (UTC)

Thanks again, you are certainly right that my “symbol z” can be replaced by “string z”, thus the two different meanings of “z” in these positions being eliminated. Then all my critique concentrates on your “last letter of the alphabet” which is but a circumscription for the symbol “z”, that relies on the knowledge of your readers who know the alphabet! And this “z” does not have the same meaning as in the “string z”. Next: what is the difference whether we use the symbols “x”, “y”, “z” as the free variables in the Formal System or the “m1”, “m2”, etc ? “Using the same symbol to stand for two different things” is to me a strict violation of Russell-Whitehead, not just “something one should try to avoid”; and using two different ways to describe one and the same thing in the intent to treat it differently in different positions is to me just as bad!

Why do you over and over again put in question whether I understood there a primitive recursive function Z( ) to exist that describes the Gödel numbers of the numerals “ffff…0”. I several times referred to Gödel’s definitions No 16 and 17! To be more precise: the interpretation of the numbers defined by Z( ) as the Gödel numbers of the “fff…0”s depends entirely on our outside knowledge of which Gödel numbers were attributed to the symbols “f” and “0” ! To get Gödel’s argument operative, the term Z( ) would have to be a Function Symbol, a Symbol of the Formal System with it’s own Gödel number attached to it, where we can freely substitute the arguments “n” for “y” or vice versa “y” for “n” . Then “Z(y)” and Z(fff0) were just these symbol sequences, Z(fff0) would not be the Gödel number of “fff0” , it only would be but a definition for that number ! That is what you are evidently assuming when you say “Z(0) refers to the Gödel number of 0” , but Gödel says expressis verbis “Z(n) is the Gödel number of the numeral for “n”” . Biedermann 12:55, 12 February 2007 (UTC) Biedermann

Although I am sure that you are frustrated at my failure to understand you as you want to be understood, I am also frustrated that you appear to be trying to mis-understand everything I say rather than to understand it.

The letter "z" is NOT being used in two different ways PERIOD. You are insisting on interpreting at least one of its occurrences in a way which is not what I intended.

YOU are the one who insists on the importance of Z (the numeral forming function). I am just trying to figure out what your problem is with it. Z is not a primitive symbol of Peano's arithmetic. It is an abbreviation for a primitive recursive function which would require several lines of symbols to give explicitly in the language of Peano's arithmetic.

If you want me to understand you, do not say things like "Gödel’s definitions No 16 and 17". Since we do not have a shared reference text, this is meaningless to me. You must explain what you mean explicitly and clearly.

There is nothing wrong with us using our "outside knowledge" to construct and interpret the formulas. The only thing which we must be careful to avoid doing with it is "proving" results in the formal system with it rather than with Peano's axioms. JRSpriggs 05:17, 13 February 2007 (UTC)

OK. I just discovered that I have a copy of Gödel’s paper in my library after all. Not the one I read before, but another translation from German to English in "The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions" (1965) by Martin Davis. So now, I can better see what you are dealing with.

The system (of "Principia Mathematica") which he was working with is a hybrid between set theory and number theory which makes some of the formulations a little odd by modern standards. Since you are concerned with Z primarily at this point, I will give a possible partial translation of it to the language of PM. Suppose Z(n) occurs in a formula, say P(Z(n)), we can convert this to (Ex)(Ey)(P(y) & x({ {n},{n,y} }) & (r)(s)(x({ {r},{r,s} }) → ((r=0 & s=2) v (Et)(Eu)(r=ft & s=8*u & x({ {t},{t,u} }))))) where x is a variable of type 4 representing a set of ordered pairs of natural numbers and the others are variables of type 1, i.e. natural numbers, and "*" is the concatenation of sequences operation which he defines in definition 8. I hope this helps clarify things for you. JRSpriggs 06:04, 14 February 2007 (UTC)

Referring to your comment of 13 February: Well, it is obvious that we are mutually misunderstanding one another, but I definitely reject your suggestion that I were trying to mis-understand everything you say rather than to understand it.

Let me try to explain my problem with “Z( )” at the simple example of the definition of the function “faculty” as I know it: “ 1! = 1 and (n + 1)! = n!∙(n+1)”. For this definition I require the function symbol “!” and only then can the term “y!” make sense to me. And this symbol must have its own Gödel number, otherwise I could not determine the Gödel number of that definition. Evidently that function symbol must appear within that “primitive recursive function which would require several lines of symbols to give explicitly in the language of Peano's arithmetic”. How do you think you could avoid it?

And please try to answer my initial question: what symbol sequence do you think “Z(y)” to be, keeping in mind that “Z(n)” is, with Gödel own words, nothing but the Gödel number of the numeral for n; so he unmistakably precludes the assumption it to be the lines of symbol sequences he had introduced for the definition of that number!

And please comment my question: “Next: what is the difference whether we use the symbols “x”, “y”, “z” as the free variables in the Formal System or the “m1”, “m2”, etc ?”

And please comment my “….and using two different ways to describe one and the same thing in the intent to treat it differently in different positions is to me just as bad!”

Sorry, I will never again “say things like "Gödel’s definitions No 16 and 17". I just thought this hint would suffice to indicate you that I am familiar with that primitive recursive function.

I unmistakingly agreed that the letters z in your two “strings z” are NOT being used in two different ways! There is NO need to insist on that with your POINT. But what about that further z that you address as “the last letter in the alphabet”, a phrase that cannot be expressed in the symbols of the Formal System? I still try to understand your “pathological utterance” as interpretation of a formula expressible in Gödel’s Formal System after its amplification to accept “strings” in general as arguments, not just those representing numbers. There Gödel’s “sub(y,19,Z(y))” turns to the simple Substitution function “Sub(y,y,y)”, with NO room whatsoever for any trickery with “the last letter…”, with which you are leaving the Formal System, so you can no more claim to reflect therewith Gödel’s invention!

Honestly, I cannot see any HONEST way of obtaining a selfreferent formula in a Formal System!

With kind regards Biedermann 11:16, 14 February 2007 (UTC)

First, let me reverse my previous request that you avoid referring by number to definitions or propositions in Gödel's text since I now have a copy of it in hand. I am sorry that I did not find it earlier.

As he says in his footnote #6, "By a 'formula of PM', we always understand here and in the sequel a formula written without abbreviations (i.e. without the use of definitions). Definitions serve only to make the writing briefer and are therefore theoretically superfluous.". Nonetheless, he uses definitions VERY extensively in his argument. He intends that we are to understand his argument with the help of the definitions, BUT if we want to Gödelize part of his argument we MUST convert all the definitions back to the given symbols of the language of PM which are: "0", "f", "~", "v", "Π", "(", ")", and a countable set of variable symbols of each type.

To begin at the beginning, "p → q" is converted to "(~(p))v(q)". "p & q" is converted to "~((~(p))v(~(q)))". "x = y" is converted to "z Π ((~(z(x))v(z(y)))". And so forth. It would be excessively tedious to convert such a complex expression as the factorial function all the way back to the language of PM. So from now on, I will convert expressions just far enough to eliminate the problematical term and leave it to you (if you wish) to convert it the rest of the way. However, I should note that functions which have numerical values present a special problem. They cannot be converted "in place" as it were. Consider "P(x + y)", I cannot simply replace "x + y" with something in the language of PM. I must instead do something like this -- "(Eu)(Ew)(P(u) & <y,u>εw & (a)(b)(<a,b>εw → ((a=0 & b=x) v (Er)(Es)(a=fr & b=fs & <r,s>εw))))". JRSpriggs 06:08, 15 February 2007 (UTC)

17.2.07 Thanks for this comment. Yet it leaves me as confused as before, with Gödel’s remark: “Z(n) is the NUMERAL denoting the number n”, precluding it to be the definition No.16 -17, or something much worse like the definition that you offer for “P(x + y)”. I see perfectly well how in these definitions the argument “n” could be substituted by “y” (to obtain Z(y)) and vice versa, but nothing the like can be done in that long sequence “ffff……0” that “the NUMERAL denoting the number n” IS.

In any case, there remains my concern about the violation of Russell- Whitehead, which I see in Gödel’s formula just as well as in your verbal version: Just as we read “sub(n,19,Z(n))” as “the Gödel number of the formula that results from the formula with GN n via substitution of the free variable y by the number n” so is “sub(y,19,Z(y))” the “GN of the formula that results from the formula with GN y via substitution of the (supposedly therein contained) free variable y by the symbol y”, (assuming Z(y) to be the GN of y) . I guess nobody can ignore the two different identities in which the “y” is here used: as name for the GN of some (unknown) statement form, as well as for the free variable therein!

Would you please comment on what I said on 14/2:

“….and using two different ways to describe one and the same thing in the intent to treat it differently in different positions is to me just as bad!”

I unmistakingly agreed that the letters z in your two “strings z” are NOT being used in two different ways! There is NO need to insist on that with your POINT. But what about that further z that you address as “the last letter in the alphabet”, a phrase that cannot be expressed in the symbols of the Formal System? I still try to understand your “pathological utterance” as interpretation of a formula expressible in Gödel’s Formal System after its amplification to accept “strings” in general as arguments, not just those representing numbers. There Gödel’s “sub(y,19,Z(y))” turns to the simple Substitution function “Sub(y,y,y)”, with NO room whatsoever for any trickery with “the last letter…”, with which you are leaving the Formal System, so you can no more claim to reflect therewith Gödel’s invention! With kind regards Biedermann 15:38, 17 February 2007 (UTC)

I do not see anything to be gained by continuing this mis-communication. So please do not send me any more messages on this subject. JRSpriggs 10:41, 18 February 2007 (UTC)

22.2.07 Well, Sir! That leaves me but to thank you for this your declaration of retreat! You certainly would not leave the field of action at this point if you could still see a chance to give me a better answer to my simple question: what is Z(y) ! Certainly, with your working hypothesis: start with a non-sense phrase, and do not ask for a meaning in your argument, I should not expect any better. Nevertheless, what I learned therefrom is, that your bold diagnosis of my total lack of understanding originates in your own deliberate refusal of any understanding of the term “sub(y,19,Z(y))”! After all your futile diversions from “the problem at hand”, all I can recommend is to copy this stuff to the discussion page of the Wiki article, so other readers may have their fun at it. Biedermann 10:21, 22 February 2007 (UTC)