User talk:Trovatore/Archive02

DO NOT EDIT OR POST REPLIES TO THIS PAGE. THIS PAGE IS AN ARCHIVE.

This archive page covers approximately the dates between 17 October 2005 and 14 February 2006.

Post replies to the main talk page, copying or summarizing the section you are replying to if necessary.

I will add new archivals to User talk:Trovatore/Archive03. (See Wikipedia:How to archive a talk page.) Thank you. Trovatore 01:11, 18 February 2006 (UTC)

Cantorian set theory

I'm no Cantor expert: I've never read anything by him, and the only exegetical treatment of Cantor's work I've read is those passages in Penelope Maddy's Realism in Mathematics. I don't feel confident to judge the issue at stake on the Russell's paradox page. --- Charles Stewart 01:05, 17 October 2005 (UTC)

Rats. I was hoping you could shed some light on this. I find that the treatment of this topic on pretty much all our pages that mention "naive set theory" versus "axiomatic set theory" have a serious problem in this regard. See my remarks to Paul at User talk:Paul August#systematic error in set theory pages --Trovatore 02:00, 17 October 2005 (UTC)

You might find the following two references interesting:

• Maria J. Frapolli. Is Cantorian set theory an iterative conception of set? (abstract only)
• Allen Hazen. Iterative Set Theory: Historical References (post to FOM list)

To summarise, Hao Wang argued that Cantor's intuition was similar, or the same as, the cumulative one. The above show that there is no consensus that he was right, Frapolli arguing that Cantor's view was rather naive (in the sense that you are arguing against). --- Charles Stewart 14:22, 17 October 2005 (UTC)

Thanks, this is interesting. Maybe I can find some way to work it into some of the discussions. Of course the bigger problem isn't really the historical one, but rather the way the terms "axiomatic set theory" and "naive set theory" are used in WP, giving the "axiomatic" label to considerations that are really more about the iterative hierarchy, and leaving the impression that workers in the field consider sets to have no content beyond the axioms. That's harder to fix, because it's built into the division of the material into articles. --Trovatore 17:39, 17 October 2005 (UTC)

Hi Trov. As I was reading the above I wondered what you thought of the views presented in Stanford Encyclopedia of Philosophy: "Set Theory" and O'Connor and Robertson's "A history of set theory"? —preceding unsigned comment by Paul August (talk • contribs) 19:05, 17 October 2005

Scanning the Jech article (the Stanford one), there's a lot I like about it, but he seems to give insufficient attention to the cumulative hierarchy and particularly to the notion that it can be described as an interpretation of the axioms rather than a consequence of them.

The relevance of the O'Connor and Robertson piece to the discussion seems to be showing that Cantor wrestled with versions of the antinomies. I knew that, of course. Any complete discussion of Cantor's views would have to show an evolution, and the development of the "limitation of size" concept. --Trovatore 02:46, 18 October 2005 (UTC)

Taverna

Mike,

I have expanded this article explaining its significance in Greek cuisine and culture. I would be grateful if you could take a look. Capitalistroadster 11:11, 21 October 2005 (UTC)

Wikipedia:Requested articles/mathematics - Convex analysis/optimization vs Optimization

Not all optimization problems are convex (wouldn't that be nice). Perhaps convex analysis should be listed under optimization. Just a thought. ERcheck 03:47, 25 October 2005 (UTC)

Maybe so. I think you know a lot more about this than I; I just noticed that there were the two categories and that there was a chance something should be done about it, though not by me. --Trovatore 04:16, 25 October 2005 (UTC)

Spent many a year studying the problems. It's nice to see that someone is noticing activity on the math front. ;-) ERcheck 04:34, 25 October 2005 (UTC)

Trovatore, thanks for fixing Mathematics after I messed up. ERcheck, welcome here. It would be great if you could write some of the articles you listed. Optimization is part of my grand plan, but I expect that it takes at least five years before I reach it. I read the heading "Convex analysis/optimization" as: all articles in either of these field should go here. I did not read it as implying that optimization is a part of convex analysis. But feel free to change it. -- Jitse Niesen (talk) 20:45, 25 October 2005 (UTC)

L in terms of infinite computation

Thanks for your interest in describing L in terms of computation. I can't really describe the result in depth without essentially recapitulating sections from a paper I am currently working on. Once I have finished writing it I'll send you a pdf copy if you like. I don't really think its particularly pertinent to the article in question though. Barnaby dawson 22:12, 26 October 2005 (UTC)

Thanks, I'd appreciate that. --Trovatore 22:13, 26 October 2005 (UTC)

Standard Borel space

Short response on my talk page.--CSTAR 00:32, 31 October 2005 (UTC)

Sure, no problem--Trovatore 00:39, 31 October 2005 (UTC)

Thanks

Thank you for helping somewhat with my understanding. I didn't mean to cause any offence whatsoever in changing the article. I hope there's no hard feelings! Thanks once again. --Celestianpower ^háblame 18:26, 8 November 2005 (UTC)

Article Deletion ?

The article:Contact Consequences depict what MAY happen, should someone, like NASA find alien life, aliens find Earth. The article depicts both the positive and negative effects of such events, is a attempt to address a possibly unaddressed concearn. Imagine for instance that NASA has found something, then it is leaked to the press, or that alien ships have positioned themselves in a manner to carry out some plan for humanity. IF that happened, WHAT are your plans ?Martial Law 04:47, 9 November 2005 (UTC)

Do you have any idea how much research went into those plans? You expect me to cough them up for free? Seriously though, it's just not an encyclopedia article, completely independent of the merits of the arguments. --Trovatore 06:20, 9 November 2005 (UTC)

Re: Peano Arithmetic

First, please pardon me for my late reply.

While I can see that the article you linked is relevent to Peano axioms, I am actually not versed with Mathematics, and thus it is difficult for me to convert it into something understandable by normal readers.

I can always copy the article's content into a new section, however; would that suffice? Jeekc 17:44, 15 November 2005 (UTC)

If you're not a mathematician, I suppose we shouldn't put the burden on you. I'll take care of it, sooner or later, if no one else does. Thanks for your help with the article. --Trovatore 17:46, 15 November 2005 (UTC)

After studying the article further, I found that it is understandable and added the section - Does it look alright to you?

I've done a little touchup. I think it'll be OK now. Thanks for adding this. --Trovatore 19:30, 15 November 2005 (UTC)

PS: How does one call a number x which, when multiplied by another number y, always produces x? For instance, anything multiplied by 0 will give 0; what is the class of number 0 is in? Jeekc 19:19, 15 November 2005 (UTC)

Well, what do you mean by "number"? For the usual sorts of things called numbers, 0 is the only number with that property, so it doesn't really need a name. In a more general algebraic context (say, rings) it could be an interesting property, but I don't know a name for it. (In a ring, it would imply that the x with that property is a zero divisor, but the converse wouldn't hold.) --Trovatore 19:30, 15 November 2005 (UTC)

Having given it a little more thought, I can say that in a ring (at least, a ring with unity), the only x that can have that property is 0. Could still be interesting in some more general algebraic structure. --Trovatore 21:32, 15 November 2005 (UTC)

computability

Yes, however I would suggest that we create a new category called Category:Recursion theory and put some of the articles which are now in Category:Theory of computation in there. As far as I could tell the articles in Category:Computability (and Category:Computation) were a bit messed up already. I'm sorry if I made the situation even worse. —R. Koot 00:23, 26 November 2005 (UTC)

Could you sort this out or should I undo the move? —R. Koot 00:26, 26 November 2005 (UTC)

Do you have a list of the articles that used to be computability theory? The other point is that you might be wading into a terminological war; the fashion recently, started by Soare, has been to use the term "computability theory" for what used to be called "recursion theory". It makes more sense than I thought it did when I first heard about it; Denis Hirschfeldt more or less talked me into switching. However I don't object to the name "recursion theory". But someone might. --Trovatore 00:34, 26 November 2005 (UTC)

As computability theory would be ambiguous, commonsense would dictate not to use it for either recursion theory or the theory of computation. A list of pages moved is at Special:Contributions/NekoDaemon. —R. Koot 00:40, 26 November 2005 (UTC)

You also might want to undo the edit to recursion theory, somebody made this into a redirect to computability theory which is on the theory of computation. —R. Koot 00:31, 26 November 2005 (UTC)

Category for deletetion (mathematical model)

In relation to this please see Talk:Model (abstract)#Dispute. Thanks. --CSTAR 16:43, 27 November 2005 (UTC)

Intentionally blank page

Would you be able to write an equivalent mathematical statement that would cover the use of this phrase? -- BRIAN0918  21:33, 4 December 2005 (UTC)

Not per se. I think you may be overdoing it a little with the math-logic analogy. --Trovatore 22:44, 4 December 2005 (UTC)

Oberwolfach 2005

Hi,

your name was mentioned in a conversation at the current Oberwolfach set theory conference as a frequent contributor to set theoretic topics here. We were surprised at how much time you are spending with Wikipedia... But if you still have too much free time, you may be interested in the following list of Oberwolfach participants. I think you have created the (so far) three blue entries, but most are still red... (Note: some of them are students or very recent PhD's).

Uri Abraham, Alessandro Andretta, David Aspero, Joan Bagaria, Tomek Bartoszynski, Andreas Blass, Jörg Brendle, Riccardo Camerlo, Neus Castells, Benjamin Claverie, James Cummings, Patrick Dehornoy, Natasha Dobrinen, Mirna Dzamonja, Tapio Eerola, Ilijas Farah, Qi Feng, Matthew Foreman, Sy Friedman, Tomas Futas, Su Gao, Stefan Geschke, Martin Goldstern, Joel Hamkins David, Bernhard Irrgang, Steve Jackson, Ronald Jensen, Peter Koepke, Menachem Kojman, Bernhard Koenig, John Krueger, Jean Larson, Paul Larson, Andreas Liu, Lopez-Jordi Abad, Alain Louveau, Benedikt Löwe, Menachem Magidor, Heike Mildenberger, William Mitchell, Itay Neeman, Ernest Schimmerling, Ralf-Dieter Schindler, David Schrittesser, Assaf Sharon, Slawomir Solecki, Lajos Soukup, Otmar Spinas, John R. Steel, Katherine Thompson, Stevo Todorčević, Jouko Väänänen, Boban Veličković, Matteo Viale, Agatha Walczak-Typke, Philip Welch, Hugh Woodin, Jindrich Zapletal, Stuart Zoble.

--193.174.3.59 18:53, 7 December 2005 (UTC)

Thanks for the gentle kick in the pants. You're right; I should cut way back on WP, or cut it out entirely if necessary like any other drug. It's not a total waste though; I do learn a lot from it and it sometimes gets me thinking about concepts that I need to review. --Trovatore 22:51, 7 December 2005 (UTC)

As if a kick in the pants can be gentle.

Right about WP though. I am also attempting to cut down (went from 3100 pages on my watchlist to 1263 for the moment). Wikipedia is fun but a timesink. Oleg Alexandrov (talk) 02:06, 8 December 2005 (UTC)

Adminship nomination

I've nominated you to be admin: Wikipedia:Requests for adminship/Trovatore. If you want to accept, please say so on that page, then put {{Wikipedia:Requests for adminship/Trovatore}} at the top of the list of current nominations on Wikipedia:Requests for adminship. If you can't handle the great responsibility and frightening power that adminship entails, just say you decline on the subpage.

Cheers! --- Charles Stewart 21:44, 8 December 2005 (UTC)

Travatore please except, you would make a fine admin. Paul August ☎ 23:55, 8 December 2005 (UTC)

• Support Oleg Alexandrov (talk) 01:07, 9 December 2005 (UTC)

Just maybe?

I saw your very gracious declining of my nomination: just maybe you might reconsider? In terms of commitment, the only real difference being an admin will make is that you will be asked to read up on some more policy, and you will receive more requests to tackle this and that on your talk page. If you are disciplined about resisting the idea that WP needs you more than your next article, this is a light burden, and shouldn't really make any difference to the amount of time you spend on WP. I don't want to pester you, but I don't think you have much to lose by reconsidering the nomination. --- Charles Stewart 04:00, 9 December 2005 (UTC)

Trovatore, I'd also like to see you adminned. I'd already watchlisted the nomination page so I could support as soon as possible. My own experience is that I'm not spending much time with the admin priviledges (I sometimes close AfD discussion just before I go to bed or while watching TV). However, I must say I also feel the issue with the all-important publication list, and I'm sure Charlest and Oleg do too (as probably everybody in academia). -- Jitse Niesen (talk) 12:00, 9 December 2005 (UTC)

Trov, yes, as an admin you don't need to do a big deal. The most important thing is not abusing your new powers, which I am sure you won't. As far as the amount of time Wikipedia is taking you, I very much understand. I declined an admin nomination a long while ago partially for this reason. So, if you wish, we can stop bugging you for a while until you feel you are comfortable with how much time you spend here versus other things (and yes, the publishing hell). Just know, at any time, now or later, we will be more than happy to support you for admin. Oleg Alexandrov (talk) 17:02, 9 December 2005 (UTC)

Thanks to all, really. I think I need to find a better balance wrt the amount of energy going into WP, and promotion just doesn't seem like the way to go about that at this time. Thanks again. --Trovatore 17:07, 9 December 2005 (UTC)

Rename of category?

Replied on my talk page, but one quick question: perhaps Category:Topological spaces should be renamed Category:Examples in topology or Category:Topology examples. I'll do the article moves, if you agree in general. linas 20:04, 10 December 2005 (UTC)

New to Wikipedia

Hi Trovatore. I want to add more to the article on shattering. Wikipeople will not let me move an article; I am too new. So, where should I start typing. Hope I talked to you properly, on the right page, in the right way. Anyway, I need some guidance with the math markup language used here. Any hints? Thanks. MathStatWoman 18:26, 16 December 2005 (UTC)

For now, edit shatter, not shattering. The shattering article needs to be deleted so there'll be a place to move shatter to. --Trovatore 20:41, 16 December 2005 (UTC)

As to the math markup language—for the moment most math is done in HTML. Look at the source of some math articles to see how it goes. If you need something that can't be done in HTML, you can put things between <math> and </math> tags and use (a fairly large subset of) LaTeX, but unfortunately this gets rendered inelegantly with the current setup, so most prefer to avoid it except when it's on a line by itself. --Trovatore 20:46, 16 December 2005 (UTC)

second order language?

I do not understand the assertion that KM is done in second order logic; it certainly doesn't have to be, and I wouldn't. Perhaps you are thinking in terms of using different letter variables for sets and classes? That would make it second-order, after a fashion. Maybe I'm not understanding your terminology? I did remark that second order ZF looks like KM but has different semantics; if the spirit moves me I may add the remark you suggest, and of course you could yourself :-)

Randall Holmes 20:52, 16 December 2005 (UTC)

Yes, I meant the different letters for sets and classes. That is the second-order language of set theory, as I use the term. (I didn't say it was done in second order logic, just in a second-order language). --Trovatore 20:54, 16 December 2005 (UTC)

IMHO, the use of different letters is optional, but I'll try to add something about the option of using second-order language (but I'm not sure I would call it that myself). Randall Holmes 02:03, 17 December 2005 (UTC)

thanks

Sorry about writing at the top of your talk page. Thanks for all the info to me, a noobie. What would Donald Knuth say about how the Wikipedia looks. Yes, I have been using HTML and the math-font stuff, but I have not been able to make it pretty or to get all the characters that I want. Ok, so I will edit the "shatter" page, and hope for the best. The probability theory page needs work. What a geekess I am, caring about all this! Thanks for all the help, o fellow mathophile! MathStatWoman 21:40, 16 December 2005 (UTC)</math>

Eilenberg

Hello. At Cantor–Bernstein–Schroeder theorem, just after the words "Here is a proof", you will find a commented-out question. Can you enlighten us on this matter? Michael Hardy 03:14, 18 December 2005 (UTC)

Sorry, I don't know to whom the proof should be credited. I'm visiting family in California and my copy of Kunen is back in T.O.; otherwise that would be a natural place to look. --Trovatore 04:14, 18 December 2005 (UTC)

NBG

I wrote an axiomatization of NBG closely analogous to the Morse-Kelley set theory axiomatization. In this theory, I left things in a two-sorted (second-order language) state, which I think is more appropriate in NBG, since the classes are not "first-class citizens" in quite the way they are in Morse-Kelley. I also provide a finite subset of the comprehension scheme... Randall Holmes 23:06, 22 December 2005 (UTC)

"Constructing" objects in theories

First of all, thank you very much for your comments on Implementation of mathematics in set theory. You are asking the hard philosophical questions which need to be asked. I have added a section Implementation of mathematics in set theory#Preliminaries discussing the issue of what it means to "construct something in ZFC (or NFU)" which should cover my uses of such language. Randall Holmes 17:50, 23 December 2005 (UTC)

Does the discussion in the preliminaries section address your concern?Randall Holmes 19:37, 6 January 2006 (UTC)

Well, it looks reasonable if the reader actually reads it. My preference is really not to use the locution at all, though; it's hard enough in general to get people to make syntax/semantics distinctions.

Another issue, in principle, is under what circumstances it's accurate to regard a formula as defining the same object "in" two different theories, even if both theories prove that the formula defines a unique object.

I'm afraid I haven't spent the time to really digest the article, but my overall impression is that it reads like something between a textbook and a monograph or expository journal article. That's not really the "house style", as Charles Matthews puts it. It would be better if it could be made more "modular", so that someone could understand sections from the middle without reading the whole thing, and maybe a bit less "conversational" (remember that WP is fundamentally a reference work). Also you tend to wikilink the same article over and over again; I think the official style is to link each title only on the first occurrence. (But there's also a "use common sense" policy; in my view it's OK to re-link in a new section, particularly if the earlier link was much earlier.)

Finally, there may just be too much information here for a single article. You might look to see if there are parts that would naturally make articles on their own, and then link to them from the main article. (Remember, though, that each article is supposed to stand on its own; if the reader can understand it only if he comes from the main article, that's not desirable.) --Trovatore 03:46, 7 January 2006 (UTC)

no second order language in Kelley

It is interesting to note that Kelley uses strictly first-order language in the appendix to General Topology, which is one of the main sources for this theory: there is not even a conventional notational distinction between set and class variables. Randall Holmes 15:19, 30 December 2005 (UTC)

Interesting. I'm not actually sure where I picked up the different-sorted variables, but I had thought it was standard. The thing is, if you don't make the sort distinction, then there would be a universal set, and that would obviously be bad, right :-) ? --Trovatore 19:04, 30 December 2005 (UTC)

no, the class $U=\{x:x=x\}$ contains only the sets which satisfy $x=x$, as usual. But sethood is a predicate of classes, not an indication of a different sort of object. I think that the different letters may come from other primary sources: for example, I have never seen Morse. Or they may be imported from NBG, where they are IMHO more appropriate (if you look at my axiomatization of NBG you will see that I use two-sorted language there). Randall Holmes 19:14, 30 December 2005 (UTC)

Well, I mean if you use lower-case letters for classes, then one is tempted to call them "sets" (and maybe call the predicate "being a small set" or some such). It's a psychological point. --Trovatore 19:19, 30 December 2005 (UTC)

I noted this in myself: I failed to notice the first time I read Kelley's axiom of subsets that it entailed separation as well as power set, because it didn't occur to me that it said that there was a set which contained all subclasses of any set, (thus enforcing sethood of all subclasses of sets). But it is clear through Kelley's whole appendix that for him the classes are all first-class citizens of the world. Randall Holmes 19:26, 30 December 2005 (UTC)

Does it work now?

There was an error in my first edit does the footnote links work now? Paul August ☎ 22:53, 5 January 2006 (UTC)

Works now. Indentation is a little peculiar, though. --Trovatore 22:56, 5 January 2006 (UTC)

Well like I said feel free to change it back if you want. Paul August ☎ 23:00, 5 January 2006 (UTC)

"Infinity" cat

That's fine. I was wondering what supercategory to put it in. Category:Mathematics seems like the best choice. —Keenan Pepper 19:15, 8 January 2006 (UTC)

unicode

Does the symbol ⊊ not even show up with the {{unicode}} template? Perhaps we need to modify the stylesheet. What browser/OS are you using? -- Fropuff 05:11, 20 January 2006 (UTC)

Shows up as a square with "22" in the upper half and "8A" in the lower half. I'm using Mozilla 1.7.8 on Debian Sarge, but I assume the real issue is not the browser, but what fonts I have installed, and unfortunately I don't know enough about fonts to answer that question. I can tell you I'm using XFree86, though, not X.org (I know I should switch over at some point, but it never seems to be top of the stack). --Trovatore 05:16, 20 January 2006 (UTC)

Check out [1]. I wish these came standard with Mozilla. I always install them right away. Having Code 2000 on your system can't hurt either. -- Fropuff 05:23, 20 January 2006 (UTC)

Can these font installations be done reversibly? I'd like to be able to read articles with these characters for myself, but I don't want to write anything that most people can't read. So it'd be nice to be able to toggle the font access somehow (maybe with a symlink or something). --Trovatore 05:26, 20 January 2006 (UTC)

Not very easily, I don't think. Mozilla's character selection algorithm is somewhat complicated. You can always delete the fonts and reinstall them, but that's a pain. I just use an alternate browser with the default settings. The browser must be stupid enough to take all characters from a single font — I use IE ;) -- Fropuff 05:49, 20 January 2006 (UTC)

I am running Debian testing with X.org; perhaps this is sufficiently similar to Trovatore's system. I'm fairly sure that the unicode template only helps for IE and that Trovatore simply has no fonts containing the symbol. The Code2000 font contains the symbol; I don't think the fonts at the Mozilla site help in this case. Anyway, to install the font, I copy the file CODE2000.TTF to /usr/local/share/fonts (or a subdirectory thereof), run fc-cache, and restart Mozilla. To uninstall it, remove the file, run fc-cache, and restart Mozilla. An alternative is to install the Debian package ttf-paktype, which for some reason also includes the symbol. -- Jitse Niesen (talk) 23:04, 20 January 2006 (UTC)

Morse-Kelley to Morse–Kelley

I'm curious why you renamed Morse-Kelley set theory. I have no objection, but I also don't see the point. (I read your comment, but don't fully understand it.) Arthur Rubin | (talk) 01:14, 25 January 2006 (UTC)

It seems to be the way it's done these days. See the discussion at the talk page for Banach-Tarski paradox. --Trovatore 02:02, 25 January 2006 (UTC)

Trivially true

Hi. Trivially true has had an ugly accuracy dispute tag on it for almost a year. Can you say what to do with the article? My guess is that it should be deleted, per the aside at the bottom of Talk:Trivially true. But I guess logicians have plenty to say about "trivially true"; perhaps, there are even books about the difference between "almost trivially true", "ε-trivially true" and "supertrivially true" ;) -- Jitse Niesen (talk) 19:24, 26 January 2006 (UTC)

I'd say redirect (with slight merge) to vacuous truth; I guess it's not exactly the same thing, but it seems the natural place to discuss the concept. There are no substantive links to trivially true, so there should be no real ripple effect. Another place they could both be discussed is relevance logic. --Trovatore 19:41, 26 January 2006 (UTC)

Good call. I like the statement "There exists a set S such that every infinite subset of S has seven elements" in vacuous truth.

I redirected trivially true to vacuous truth and I said that "vacuous truth" is sometimes called "trivial truth", which is my experience. However, this page says that p → q is "vacuously true" if q is true and "trivially true" if p is false. Unfortunately, I cannot explain this in the article because to me, p → q is the same as ~q → ~p. -- Jitse Niesen (talk) 20:27, 26 January 2006 (UTC)

types

So a module is a mathematical object with a lot of structure. It is an abelian group and a ring, and an operation of the ring on the group. If you remove the abelian group, and are left with only the ring, what are you doing? I said you're removing a type, but since I'm not too sure what a type is, I wasn't too sure of my answer. I thought maybe you were a guy who knows more about stuff in that neighborhood of math, so maybe I could ask you. Do you know? -lethe ^talk 23:07, 26 January 2006 (UTC)

I can't off the top of my head think of any way to describe that as "removing a (model-theoretic) type"; that doesn't mean there isn't one, just that I don't see it. In category-theoretic terms it could be described as "applying a forgetful functor". --Trovatore 00:42, 27 January 2006 (UTC)

It could also be described as removing a sort, I suppose, in the sense of multi-sorted logic (where you have different variables for different kinds of thing. Is that more what you had in mind? --Trovatore 01:06, 27 January 2006 (UTC)

This question actually came up during my edits of forgetful functor. The problem is that forgetful functors can be distinguished into those which "forget axioms", which are fully faithful functors; those which "forget predicates", which are faithful functors; and those which "forget types" (as I had wanted to say it), like the example above. These functors need not be full or faithful. Some people over at forgetful functor wanted to define a forgetful functor as a faithful functor, and I don't think that can work. In fact, the category theoretic definition of a forgetful functor is problematic, which is why I wanted to get at it from the model theoretic angle.

OK, so if you remove the abelian group from a module, you don't think this is properly described as forgetting a type?

What I had in mind is this: a simple mathematical structure is a triple of a set, some predicates on the set, and some axioms about the predicates (is structure the right word to use here, in the model theoretic sense? is predicate?). Forgetful functors are easy here: forget some of the axioms or forget some of the predicates (and of course any axioms using those predicates). The first case gives you a fully faithful forgetful functor, the second case gives you just a faithful functor.

The third kind of functor arises when your structure is more complicated. Like the module. Here, I guess you could still make your list, but instead of starting with a set of points, you would start with an ordered pair of a ring and an abelian group. Maybe you could call those substructures? Or subtypes? I dunno. A forgetful functor which removes one of those guys is more general than the above type of functor, and need not be faithful. I know these functors exist, but I don't know the right language to describe them, or better yet, to define them precisely. But I have the feeling that the language of model theory is the right language to define forgetful functors. -lethe ^talk 01:50, 27 January 2006 (UTC)

OK, so I'm pretty sure what you have in mind here is sorts, not types. In multi-sorted logic, models may have more than one underlying universe, one for each sort (the sorts in your example would be the ring and the group). Each variable and constant symbol has a sort, and each function symbol and relation symbol has a signature telling you what sort to put in each slot and (in the function case) which sort it returns. Equality does not exist between terms of different sorts.

The thing is, it would usually be considered a bit of an inessential detail whether your module is to be thought of as a two-sorted model, or as a one-sorted model with a predicate to tell you whether an object comes from the ring or from the group. To make your formulation work, you'd apparently have to insist on the two-sorted version. But that's OK; people would be free to think of it as they liked except when applying your formulation. --Trovatore 14:49, 27 January 2006 (UTC)

What I know about sorts is that they're used to turn second order languages into first order languages. I can see that sorts might be the right thing to talk about here. So the module will be a two sorted model and the functor will remove one of the sorts and any relation whose signature includes the sort. Actually, I'm not clear on how this is different from a type. Types, sorts, structures, models, I'm confused about them all.

As for the question about treating a two-sorted set as a single sorted set with a predicate, insisting that we choose the two-sorted language makes me uncomfortable. They're equivalent, right? So there should be a way to say it there as well, even if not as cleanly. Like, the forgetful functor forgets the predicate, and replaces each set with the subset determined by the predicate. -lethe ^talk 09:24, 30 January 2006 (UTC)

Oh, one more thing in your question that I wanted to comment on: No, a mathematical structure is not properly defined as "triple of a set, some predicates on the set, and some axioms about the predicates". Drop the specific set, and you have a class of structures with a natural category-theoretic interpretation. But to make it a specific structure, you need more than the specific set. You need to know the exact behavior of the predicates; that is, on exactly which tuples from the underlying set they return true. A finite (or even r.e.) collection of axioms is rarely sufficient to specify that. (By the way, this paragraph is for relational structures; if you have constant or function symbols, the corresponding remarks apply.) --Trovatore 15:02, 27 January 2006 (UTC)

I think I see. It's the difference between a predicate symbol and a relation. A predicate symbol can only take variable symbols, whereas a relation can either contain or not contain specific values from the set in place of the variables. Syntax versus semantics. Is that your purpose there?

Thanks for the help, by the way. -lethe ^talk 09:24, 30 January 2006 (UTC)

Wikimedia Canada

Hi there! I'd like to invite you to explore Wikimedia Canada, and create a list of people interested in forming a local chapter for our nation. A local chapter will help promote and improve the organization, within our great nation. We'd also like to encourage everyone to suggest projects for our national chapter to participate in. Hope to see you there!--DarkEvil 15:12, 5 February 2006 (UTC)

For You

If you think I did something wrong then you don't need to critize me in public. You are too young and motioned. Improving will be OK. --HydrogenSu 22:43, 12 February 2006 (UTC)

For any third parties that wander by here and wonder what this is about, take a look at Wikipedia:Reference desk/Mathematics#Infinite Sums (Wikipedia:Reference desk archive/Mathematics/February 2006#Infinite Sums, after archival) and judge for yourselves. --Trovatore 23:50, 12 February 2006 (UTC)

Hmmm....Not good

Why do I have to judge myself? Maybe this action shall be let someone who critized me in public to do. Critizing a person in public is just a bad behavior.--HydrogenSu 21:50, 14 February 2006 (UTC)

Hmmm..

[2]

on it, You said: No, I don't really ‘’’think’’’ that's true; most solutions include some probability of finding the electron within the nucleus. Maybe the question is, can the electron then ever "stick" to a proton, making a neutron? In fact, that is possible -- see electron capture. --Trovatore 17:56, 5 February 2006 (UTC) Oh~~sounds wonderful,doesn't it? There may exist such possible. "Everything in the world is possible". For ex:UFOs might travel through stars at the speed greater than that of light. If UFO "existed"...no. If UFO exists...oh~yes! I think every possible in the world must its own existed condition. Maybe this is the law which God made.--HydrogenSu 18:27, 5 February 2006 (UTC)

Then…….Sorry. Did you really understand what the web you linked is going to say? physical conditions of Electron captures are not the as same as standing-wave conditions. And in fact your “think” does not matter to Topic Within atoms, why do the electrons not crash into the nucleus? You seem to do not good at a jugement method of condition in physics. --HydrogenSu 21:42, 14 February 2006 (UTC)

Sorry, HS, I'm not really sure what you're trying to say here. What I said was correct, and I stand by it. Solutions of the Schrödinger equation include some probability that the electron will be found within the nucleus. If the energetics are favorable, the electron can be captured by the nucleus, turning a proton into a neutron (there's also a neutrino involved). --Trovatore 21:51, 14 February 2006 (UTC)

I meant:For he asked,was about a steady electron,why does it not crash on nucleus. --HydrogenSu 21:56, 14 February 2006 (UTC)

No, I still don't see what you're trying to say. It may be a language problem. --Trovatore 22:04, 14 February 2006 (UTC)