User talk:Trovatore

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Previous discussions:

• Archive 1 (1 July 2005 to 13 October 2005):
• Archive 2 (17 October 2005 to 14 February 2006):
• Archive 3 (12 February 2006 to 4 December 2006):

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Contents 1 Nuclear crime and other related acts 2 Re: Boolean Algebra 3 Empty product woes 4 Combinatorial principle 5 About Vitamin A and bots.. 6 Vitamin A 7 Vitamin A , Retinol 8 Aleph_n 9 Vitamin A Part 0 10 Bollocks 11 Thank you 12 Hilarious 13 Uncountable set 14 I *still* think you're wrong 15 1/0 (literal translation) 16 Un soprannome italiano 17 minimal model disambiguation page 18 infinitas 19 Ph.D. 20 Prime Minister of Italy 21 Cantor 22 Cantor's diagonal argument 23 Godels Theorem and TOE 24 Axiom still sounds more accurate 25 Bot on Boolean algebra/logic 26 Peano axioms up for A-class rating 27 Xenu lecture 28 The Cberlet mediation

[edit] Nuclear crime and other related acts

Hi, I notice that you have found the nuclear crime page and you have an interest in the name. I have seen the use of the term nuclear crime on the web. Have you any ideas for a short alternative title for the page devoted to the subject ? Cadmium

I don't think it should have a short title, frankly. What you're trying to do here is create a category of crime, in a way that it isn't standardly categorized. That's borderline original research or original journalism. But it wouldn't be such a problem in a list article; we have lots of lists according to obscure, or even bizarre, selection criteria. So I stand by list of crimes involving radioactive substances (though there are many similar titles that could also work). --Trovatore 23:02, 9 December 2006 (UTC)

I saw your comment about "googlewacking", I would say that would be fair if one searched for the words nuclear and crime, but the yahoo search I used was one for "nuclear crime" which forces it to select only articles with the phrase "nuclear crime". I would also say that because the UN have used the term (IAEA used it) that it is a valid phrase which is not a neogism. Do you thin that if the UN use a phrase that it is then something which has entered the english langauge fully.Cadmium

Sorry, I don't buy it. I think the terms are being used in their usual meaning, according to usual English semantic rules, not as an attempt to identify a new category. Try doing the same search for "bicycle crime". You'll get plenty of hits, but it doesn't justify singling this out as a separate category of crime worthy of a WP article. But you could do a list of bicycle crimes and it'd probably be OK. --Trovatore 00:20, 10 December 2006 (UTC)

[edit] Re: Boolean Algebra

I see your point, and now I notice what I hadn't before which is that these are already in Boolean Logic. I've seen these topics treated different ways and the sort of person looking up boolean algebra in Wikipedia is most likely a student i'm guessing, so when they look at Boolean Algebra, it would be nice for them to be able to see all of these properties at once. if i had looked harder and seen them in the logic article i probably wouldn't have posted them. it seems that anyone who would ever need to use boolean algebra would need these identities, but i don't know everything and there are probably a myriad of used for boolean algebra that don't require manipulation like this. In the end though, i do still see your point and if my edit hasn't been reverted, i'll do it. Another thing i find interesting is that different sources use different axioms to begin with. I have one text that begins with 0*0=0, 1+1=1, 1*1=1, 0+0=0, 0*1=1*0=0, 1+0=0+1=1, if x=0 x compliment = 1 and if x=1 then x compliment =0. these are not the same as the article or other texts. Reading the article again i see how it fits together, i just wonder at how there is a distinction being made between "algebra" and "logic". When you look at the boolean logic article it seems to be about symbolic logic.

thanks for your comment and sorry for the long ramble.

--The Talking Sock talk contribs 05:09, 15 December 2006 (UTC)

Ah, see, this is the thing. The Boolean algebra article is not about "Boolean algebra". It's about a type of object called a Boolean algebra, which is a thing like a group, but more complicated. It's an easy misunderstanding, and has caused lots of trouble in the history of the article. We still haven't come to a good method of avoiding the confusion. --Trovatore 05:40, 15 December 2006 (UTC)

[edit] Empty product woes

Hello Trovatore. I came here because of your posting at WP:WPM. I see the problem over at Empty product. To save time, perhaps you could nominate the article for deletion, for lack of reliable sources? I see a lot of theorizing on the Talk page, but the claim made at the top of the article surely needs direct citations to back it up, and it has none. Of course, documenting things either way would be a chore, because many math books avoid this issue. But the people who believe the empty product clearly has the value 1 ought to have the burden of proof. EdJohnston 17:54, 15 December 2006 (UTC)

Hm, the basic notions seem standard enough to me. It certainly needs cleanup and sourcing but I'm not convinced it should be deleted (I'm sure sources can be found). Really I have no problem with taking the empty product to be the multiplicative identity (at least in contexts where this exists and is unique); what fails to convince me is that therefore 0^0 must be 1 in the context of real-number exponentiation. --Trovatore 18:58, 15 December 2006 (UTC)

[edit] Combinatorial principle

Hi, please have a look at my suggestion in Combinatorial principles (talk). Thanks! --Aleph4 21:57, 16 December 2006 (UTC)

[edit] About Vitamin A and bots..

Bots are trying to connect a normal page to another normal one, and a disambiguation page to other disambiguation one. The easiest way is to make Vitamin A into a normal page. -- ChongDae 09:30, 2 January 2007 (UTC)

[edit] Vitamin A

I've seen both english article, and the one called "Retinol" is closer in subject to the french one, which use almost indifferently both terms Vitamine A and Rétinol. But even if it wasn't the case : there's no sens to make the change if all the Bots make it in reverse the day after... When there will be two articles in french, the interwikilink may be changed, but meanwhile, I think the interwiki to the english "Retinol" is better. Yours, Blinking Spirit 10:46, 2 January 2007 (UTC)

Ditto for Spanish language Wikipedia. es:Vitamina A is about retinol, so it should better point to enwiki Retinol (and back) than to enwiki's desambigation Vitamin A.

Unfortunately a disamiguation does not tells me why "Vitamin A" and "Retinol" are two different concepts. The nature of a disambiguation page is to say: Well Vitamin A and Retinol are the same, but "Vitamin A" could also mean other retinoids or caretinoids. On the other hand is "Vitamin A" is somthing that is defined differently than just a molecule (e.g. Vitamine A is any of a set of molecules that had the following characteristics: ...), to which Retinol is just an example of that, that should not be a disambiguation page.

Vitamines are beyond my field of expertice, so I might not attempt to mimic enwiki into eswiki into the disambiguation think. I might be relexing false cognates, and just a disambiguation would not be appropiate. But just as fr:Vitemine A, es:Vitamina A should point to Retinol and back, for the moment. Probably it would be the same for most other wikis.

— Carlos Th (talk) 13:11, 2 January 2007 (UTC)

Actually, es:Vitamin A is not about retinol. It says that it's found in col verde. The only way you'd find retinol in col verde is if it's part of a meat dish. That's really the basic problem with having this stuff at articles called "retinol". --16:41, 2 January 2007 (UTC)

Response to your message on my talk page :
Well, then when all the bots agree with you, and there really is an english Vitamin A which is not only a disambiguation page, I'll see what I can do (probably make two french pages, too). In the meantime, would you be so kind as not to revert the bots on the french page ? An good interwiki, even if it is not THE best one, seems quite easy to tolerate for some time. Yours, Blinking Spirit 19:08, 2 January 2007 (UTC)

The main thing that's wrong with it is that the bots keep looking at it, and are reinforced in their wrong idea. Or at least maybe. I don't really know how the bots work. But the principle of "when all the bots agree with you" is wrong. The bots' opinion is not worthy of any respect; in fact, it's not an opinion at all, just some state saved somewhere on disk. If it weren't for that, I'd agree with you that the link from the French page would be tolerable. --Trovatore 19:14, 2 January 2007 (UTC)

I apologize : I was perfectly aware that the bots, having no opinion, can not "agree" to anything. My formulation was akward. What I meant was that as long as there isn't any better interwiki ANd the bots don't stop reverting you, there's no sense making the change, for it won't last no matter what. Thus my demand. Yours, Blinking Spirit 22:40, 4 January 2007 (UTC)

But what you're not taking into account is that the bots look at the link from fr:Vitamine A to Retinol and, because of it, link other foreign-language "vitamin A" articles to Retinol. While I haven't found out how to stop them, policing the links in the other wikis does seem to slow them down. --Trovatore 22:44, 4 January 2007 (UTC)

[edit] Vitamin A , Retinol

Sup man

How's it going.

About the changes, I hope you mean the ones that my BOT is making, not I.

I think the problem lies in the interwiki links on ar wikipedia, which point to retinol , and because of that, my bot reads the links, and appends the missing one. I have fixed the problem on ar wikipedia, and altered the interwiki on en wikipedia. So you shouldnt get a problem from any of the ar wikipedia bots.

However, I noticed that some other wikipedias point to the wrong interwiki link, so you might want to fix these or let the bot masters know about them, here they are:

[[ca:Retinol]] [[de:Retinol]] [[id:Retinol]] [[it:Retinolo]] [[ku:Rêtînol]] [[lb:Retinol]] [[nl:Retinol]] [[ro:Retinol]] [[sk:Retinol]] [[sv:Retinol]] [[uk:Вітамін A]]

Cheers. --Lord Anubis 14:48, 2 January 2007 (UTC)

[edit] Aleph_n

Thanks for giving the general (non-AC) definition of an aleph number. I actually flipped the definition, with the general case first and the implication of AC in a parenthetical. I also hyphenated well-order, as I think this is more usual. I hope you don't mind.

CRGreathouse (t | c) 05:20, 3 January 2007 (UTC)

Actually I prefer it the way it was. I won't fight over "wellorder", but AC is the standard assumption. --Trovatore 06:12, 3 January 2007 (UTC)

Certainly I also don't want to fight or get into a revert war (I try to follow the 1RR), but I don't think there's a good reason to state the special case definition when the general definition is so easily derived from it. Instead of having two definitions, which is unwieldy, there's one... and the parenthetical isn't even needed, as such.

What about this: You can edit it back to the way it was, AC first and wellorder (or leave it well-order, either way), and add an example for non-AC. That would satisfy your feeling that ZFC should be standard even in the theory of transfinites*, and would satisfy mine that "the next bigger well-ordered cardinal" might be confusing. What do you think?

* I think spelling out axioms is rather common is set theory in general, and in the study of infinite quantities in particular, so while I generally accept AC (in fact my user page even says that I'm a mathematical realist who believes that ZFC is 'true') I don't think its assumption here is justified.

CRGreathouse (t | c) 06:21, 3 January 2007 (UTC)

[edit] Vitamin A Part 0

I don't know what is confusing the interwiki bots!. I'm saying bots because I took a look at Retinol's article history and found out that it is not only my bot but probably all of them. also I noticed that you had/having great time reverting them ;).. I think(might be wrong) that there is a certain wikipedia(may be more) are responisble for that mistake(confusing the bots).. I don't really know the problem.. anyway, on the other hand; I think that Retinol is Vitamin A and that is just a different name..sorry for any inconvenience I've caused you! if you know how to fix that problem, please leave me a note...--Alnokta 21:03, 4 January 2007 (UTC)

See, I don't agree that retinol and vitamin A are the same thing. Retinol is one specific molecule; vitamin A is a nutrient. You can get vitamin A from plant foods, but you can't get retinol from plant foods. At least that's one widespread usage of the term "vitamin A" in English. It may not be so in all languages, which is why it's possible that there should be two separate articles in English wikipedia, but only one in some other languages. Unfortunately the bots don't seem to be sophisticated enough to deal with that situation. --Trovatore 21:28, 4 January 2007 (UTC)

Well, you may be right but let me write you an entry from a dictionary I have. but it may be an outdated information.

Vitamin A (axerophthol, retinol) A vitamin found in fish liver oils
such as cod and halibut oil, dairy products, egg-yolk and several
vegetables. A deficiency of vitamin A leads to abnormalities in the
mucous membranes lining the respiratory and alimentary canals,
softening or dying of the cornea and no regeneration of visual purple
in the rods of the retina. dictionary of science, P.Hartman Petersen,
J.N. Pigford 1985

Do you have any ideas on how fix the issue with the bots?--Alnokta 11:53, 5 January 2007 (UTC)

I don't know how to fix the bots.

As for the dictionary, note that it says vitamin A is found in "several vegetables". That means it can't be synonymous with retinol. Vegetables have no retinol. --Trovatore 17:58, 5 January 2007 (UTC)

Would it be correct to say that Vitamin A consists of retinol together with other substances which can be converted into retinol in the human body? JRSpriggs 06:15, 6 January 2007 (UTC)

I don't know. I'm not an expert on this by any means, which is why I haven't refactored the articles myself (plus the detail that the dab page first needs to be deleted so retinol can be moved). I just noted that there are lots of things called vitamin A (at the very least, on U.S. nutritional labels) that are not retinol, and that it was strange to put a lot of information on the retinol page that was, explicitly, not about retinol. --Trovatore 09:01, 6 January 2007 (UTC)

All right, I tried to fix the links manually on the wikipedia languages, the bots should behave correctly now(I hope so).. why there is a merger tag on the retinol page? please remove it.I'm alnokta. —The preceding unsigned comment was added by 81.10.39.199 (talk) 12:45, 6 January 2007 (UTC).

There are at least three closely-related compounds that some people will refer to as Vitamin A: retinol (the alcohol), retinal (the aldehyde) and retinoic acid. Beta-carotene may be labelled as Vitamin A as well, although it contains two retinal groups end-to-end and it behaves somewhat differently in the body. The Vitamin A that I take in my multivitamin is listed as 'retinyl acetate and beta carotene'. However if there's a retinol page, it should not be merged with anything else. It might be sensible to enhance the Vitamin A article to explain that it's a functional concept (it is whatever behaves like Vitamin A in the body) and not a specific molecule. EdJohnston 16:20, 13 January 2007 (UTC)

Left a further comment at Talk:Vitamin A. An article found on the web says that Vitamin A activity is assayed biologically. See [1] for details. EdJohnston

It's not exactly a merge I was arguing for, but rather moving some content from the retinol page to the vitamin A page. I think we should have both articles, but retinol should be much shorter and vitamin A much longer, with only information specific to retinol left at the retinol page. --Trovatore 19:35, 13 January 2007 (UTC)

Yes, that sounds right. There is a lot of stuff on the retinol page that is actually true of all the bioequivalent forms of Vitamin A, and there's no reason not to move it. EdJohnston 02:19, 14 January 2007 (UTC)

[edit] Bollocks

Didn't know that. I always assumed the reason that WP:BOLLOCKS and WP:BALLS resolved to the same page is that they meant the same thing. Fan-1967 23:23, 4 January 2007 (UTC)

[edit] Thank you

Thank you for your recent edit on exponentiation. I really should have thought of that. I'm trying to stick my nose into that article no more often than once a day, to preserve my sanity.

Thanks also for the examples of in-line css coding on your user page. I borrowed some of that the other day, and then got curious enough to poke into the "Template" namespace, and in the process I got a better idea of how the html tags and css coding fit together on Wikipedia. So you've taught me a couple of things, indirectly. Oh – if you want to, you can integrate your in-line code into the "userbox" template with <div></div> tags. Have a great day!  ;^> DavidCBryant 18:27, 17 January 2007 (UTC)

[edit] Hilarious

About once or twice a year I go through my old messages, and I always end up re-reading a bit of the AfD we participated in. I laugh aloud everytime I re-read your final line: "I admit the possibility that I may be serving the interests of a dark cabal that seeks to keep the common man ignorant by holding down a truly enlightened man, but at this point I'll take my chances with that."

Classic.

--Michael ^(talk) 05:27, 19 January 2007 (UTC)

Thanks, nice to be appreciated. --Trovatore 05:48, 19 January 2007 (UTC)

[edit] Uncountable set

If you have a moment, could you comment at Talk:uncountable set#Uncountable in non-AC settings. The question is how "uncountable" is defined in settings where AC is not assumed. CMummert · talk 21:09, 21 January 2007 (UTC)

[edit] I *still* think you're wrong

Moved to User talk:Trovatore/Capitalization.

[edit] 1/0 (literal translation)

You recently said this article seems to be nothing but a single pun. Will you please tell me what you mean by that? If you mean to say that the English language is nothing but a pun, please explain. Also, is your expertise in Math or English? —The preceding unsigned comment was added by Alphanon (talk • contribs) 05:12, 29 January 2007 (UTC)

I thought you intended the article to be a joke. If you did, it's not worth an article. And if you didn't, it's still not worth an article. If you insist, you can remove the {{prod}} tag, and I'll take it to AfD, where it will probably be speedy deleted, and people will get mad at me for wasting their time by not just requesting speedy deletion. --Trovatore 05:16, 29 January 2007 (UTC)

What is the reason for it to be deleted? Do you have a problem with the English language? I assure you, you should not take it out on my article. Afterall, it's not my fault that 1/0 literally means "one thing divided by nothing." That's you mathematicians who failed to realize it. —The preceding unsigned comment was added by Alphanon (talk • contribs) 05:18, 29 January 2007 (UTC).

[edit] Un soprannome italiano

Un anglosassone che parla un buon italiano e sceglie per soprannome un'opera lirica... affascinante! Come si spiega questa bizzarria? Ciao! Fabioman 83.103.74.23 07:52, 30 January 2007 (UTC)

[edit] minimal model disambiguation page

Shoule something from set theory be added to minimal model? Michael Hardy 23:53, 6 February 2007 (UTC)

No, I wouldn't say so. The term could come up in various contexts, but pretty much with the words' natural meaning rather than as a term of art. --Trovatore 23:56, 6 February 2007 (UTC)

I added an entry saying "The minimal model of set theory is part of the Constructible universe#L is absolute and minimal.". JRSpriggs 09:47, 7 February 2007 (UTC)

Hm, I don't really agree with that, strictly speaking. I'd be OK with it if you replaced "set theory" by ZFC, though even then I don't see the necessity of the entry. --Trovatore 17:15, 7 February 2007 (UTC)

Shouldn't there be a unique countably infinite minimal model? Michael Hardy 22:12, 7 February 2007 (UTC)

Well, "model of what?" is the question. L has built-in Skolem functions, so you can take the Skolem hull of the empty set inside L (equivalently, look at the collection of all sets in L that are first-order definable without parameters in L), and then take its Mostowski collapse -- that's a good "minimal model of ZFC" for a lot of purposes, and it's certainly countable.

But what if you want more or less than ZFC? L[U] (take a κ-complete ultrafilter U on a measurable κ, and then do the L thing, but allowing U as a predicate) also has built-in Skolem functions, so you can work the same trick on it, and get a minimal model, in the same sense, of "ZFC+there exists a measurable cardinal". (This model too is countable.) Why doesn't this model have an equal claim to being a "minimal model of set theory" as the previous one derived from L? I don't like seeing "set theory" conflated with ZFC; ZFC is just a fragment of set theory. --Trovatore 22:38, 7 February 2007 (UTC)

By "set theory" you mean the theory of V? CMummert · talk 04:56, 8 February 2007 (UTC)

Well, ordinarily when I use the term I don't mean any set of formal sentences at all; I mean "what set theorists do". So it doesn't really make sense to speak of a "model" of that. But if I did mean something you could speak of a "model" of, then yes, I suppose it would be the theory of V. --Trovatore 05:32, 8 February 2007 (UTC)

I think I was reading too much into the word fragment in your message above. "What set theorists do" does make sense there. CMummert · talk 12:55, 8 February 2007 (UTC)

[edit] infinitas

Hic est Cantor!--Ioshus^(talk) 03:15, 12 February 2007 (UTC)

Thanks much! Looks good. More could certainly be said but at least the topic is now represented there. --Trovatore 03:59, 12 February 2007 (UTC)

I will be happy to make adjustments, but I am more of a latinist than a mathematician. If you could provide concrete suggestions about what to add/translate, it would be alot easier. I don't mean to underrepresent the topic, but at the same time I don't want to devote the whole article to Cantor.--Ioshus^(talk) 04:01, 12 February 2007 (UTC)

No, certainly his work shouldn't take over the article. But I think maybe a little bit might be said, in the "mathematics" section, about the aleph numbers. --Trovatore 04:06, 12 February 2007 (UTC)

[edit] Ph.D.

See this edit. This robot said in its edit summary that it was correcting the spelling of "committe" to "committee", but it also (and quite incorrectly) inserted a space into almost every occurrence of "Ph.D.". --Trovatore 16:44, 21 February 2007 (UTC)

Fudge fudge fudge! You're completely right. The intent of this particular part of the script ws to fix the cases where users didn't put a space after the period at the end of a sentence. Clearly Ph.D. is an exceptional case I should make note of. I'll fix up my script to handle this. Thanks so much for pointing it out. Cheers, CmdrObot 21:47, 21 February 2007 (UTC)

[edit] Prime Minister of Italy

Hi, I happened to be browsing this and found it includes the statement that the Prime Minister occupies the fourth most important state office. As I can't read italian, I've no idea whether this came from the translation or somewhere else...do you have idea who occupies the second and third most important? A link would help the curious... Chrislintott 15:34, 22 February 2007 (UTC)

It came straight from the Italian; I just translated it. My guess is that the second and third (formally) most important are the presidencies of the Senate and the Chamber; not sure in which order. Of course this is a purely formal thing -- I couldn't even tell you who those presidents are. --Trovatore 17:52, 22 February 2007 (UTC)

Hello, I'm Italian and maybe I can help. The first state office, as the Constitution says, is the President of the Republic (Presidente della Repubblica), currently Giorgio Napolitano. The second office is that of President of the Senate (Presidente del Senato), which can become temporary president in absentia of the President of the Republic. Currently this office is held by Franco Marini. The third office is the President of the Chamber of Deputies (Presidente della Camera dei Deputati), held by Fausto Bertinotti. It's correct to say that the Prime Minister (but in Italian we seldom call it so; we prefer President of the Council of Ministers) is the fourth most important office. Now this office is held, ça va sans dire, by Romano Prodi. --Gspinoza 08:31, 10 March 2007 (UTC)

[edit] Cantor

I completely agree with you. It was getting really crazy and I just felt someone had to step up already. Of course, Cantor's article isn't the only one plagued by this sort of ethnic-lobbying. A while ago the first sentence in the Johann Philipp Reis article was something along the lines of "Reis was a German inventor of Portuguese Jewish background." After some brief research it was shown that that was pretty much completely made-up, precipitated on some Portuguese blog. --Tellerman

As expected, Gilisa returned it. --Tellerman

Thanks for the support. I'm trying to reason with Gilisa on this issue. --Tellerman

[edit] Cantor's diagonal argument

Hi!

Sorry to just keep editing and re-editing, but I'm new around here and just discovered that these "Talk" pages exist. I hope I'm using them correctly. Anyway, re: your last edit, you say

still no good; not clear what a computable enumeration means. To output a single infinite sequence takes infinite time, so the naive image suggested by the phrase doesn't make sense.

S is enumerable if there's a total function f from N onto S. S is computably (I would rather say effectively, but the entry at enumeration says 'computably') enumerable just in case there's a computable such f. A function is distinct from an algorithm; while it may "take time" for the latter to give some output, it doesn't take any time for the former to give an output. So for example take the two-membered set S, where:

s₁ = the sequence <a₀, a₁, ... a_n, ...> s.t. a₀ = 0 and for all n > 0 a_n = 1; and

s₂ = the sequence <b0, ... , bn, ...> s.t. b0 = 1 and for all n > 0, bn = 0.

Now the set S is enumerated by the computable function:

f (0) = s₁, and for all n > 0,

f (n) = s₂

...and so S is computably enumerable. I believe that all this jibes with the entries at enumeration and computable function.

I apologize if I've been overly pedantic. —The preceding unsigned comment was added by Futonchild (talk • contribs) 22:45, 28 February 2007 (UTC)

So what exactly do you mean by saying f is or is not computable? You've given one example; it's not clear you have a definition. (Note that even if you did have a definition, it wouldn't help, unless it's a standard definition that can be found in published references.) --Trovatore 22:53, 28 February 2007 (UTC)

You're right; I don't have a definition. It's the same "computable" that appears in the Church-Turing thesis. It's the same sense in which a goedel numbering function is computable. Just as a sequence generated by recursive function is computably (again, I prefer effectively) enumerable, so is a finite set of such sequences (or any finite set of anythings, for that matter). And so is the set of all finite sequences of ones and zeroes. What Cantor's proof shows (from the viewpoint of one who does not accept arbitrary functions) is that the set of infinite sequences is not effectively enumerable. When S is a set of natural numbers, then (assuming Church's thesis) we can switch out "effectively enumerable" for recursively enumerable; but where S contains a member that's not a natural number, we have to fall back on this intuitive notion of an effective or computable enumeration. Goedel numbering functions are computable functions, in this sense, though they're not mu-recursive functions (since the range is not a set of natural numbers). --Futonchild 00:27, 1 March 2007 (UTC)

I'm sorry, you haven't even expressed a clear informal notion here. The Church—Turing thesis asserts an equivalence between certain formal notions (all provably equivalent to one another), and an informal notion of computability, expressed by Turing in terms of a worker applying a completely mechanical decision procedure. How you would apply such a decision procedure to deciding whether an infinite sequence is in your set, is obscure, even at an informal level. --Trovatore 01:08, 1 March 2007 (UTC)

Well, "decision procedure" is of course something different; there need not be a decision procedure for membership in S when S is recursively enumerable. The worker presumably manipulates numerals and we can provide whatever semantics we want to the inputs and outputs (the numerals). Standardly '111' is interpreted to mean 2 when given as output, and so on. So imagine the worker follows these instructions:

If the input is '1', then output '1'.

If the input is longer than '1', then output '11'.

This is the algorithm for manipulating numerals. But what function does the algorithm (or the worker following the algorithm) compute? According to the standard interpretation, an output string of n '1's means n-1. And so the algorithm computes the function f(0) = 1; for all n > 0, f(n) = 1. According to an alternate (but equally unambiguous, computable, effective) interpretation, the output string '1' means my s₁ (above) and '11' means s₂, and so our algorithm computes the function f(0) = s₁, and for all n > 0, f(n) = s₂. And so the set S is effectively (or computably) enumerable, though, strictly speaking, we cannot say that S is recursively enumerable, since it's not a set of natural numbers. Goedel numbering functions essentially provide alternate (and again, effective and unambiguous) semantics for the output: a string of n '1's = the string whose goedel number is n. Turing's idea, I take it, was that a function (of natural numbers) is computable if there is a set of instructions that meets certain criteria and, under the standard interpretation (n '1's = n), computes the function.

There's nothing special about the standard semantics, though. What's important, for Turing-computability to be a model of computability for functions of natural numbers, is that the semantics be effective or intuitively computable at the meta-level. We can broaden Turing's definition so as to encompass other kinds of functions: a function (of whatever kind) is computable if there is a Turing machine that, under some effective interpretation, computes the function. (Circular, I know; but them's the breaks.)

So to sum up: from the perspective of computability theory, there is no substantial difference between recursive enumerability and "computable" enumerability; it's just that the latter is restricted to sets of natural numbers (because the range of a recursive function is a set of natural numbers). It's not as though the sense in which my computable function is computable pales in comparison to the sense in which a recursive function is computable. They're both computable in exactly the same sense. They just have different kinds of things as their values. So if there's a clear sense in which recursive functions are computable, there's a clear sense in which my function (the one enumerating the 2 sequences) is computable. (sorry this is so long) --Futonchild 02:46, 1 March 2007 (UTC)

I'm sorry, you still haven't explained what you mean by this in the context of giving a computable enumeration of infinite sequences. Even at the intuitive level. I'm not at all a formalist and I don't object to informal explanations if they're clear. But you have not even come close to meeting your burden here. --Trovatore 03:06, 1 March 2007 (UTC)

A set S is enumerable if there's a function...etc. Let us say that a set is effectively enumerable if there is a computable function f such that S is the range of f. "Computable" here means: there's an algorithm for computing f(s) for all s in S--if that's helpful. (Is it objectionable?)

Every two-membered set is effectively enumerable. Proof: let f(0) be the first member and for all n > 0, let f(n) be the second. That looks like an algorithm for computing f(n) for arbitrary n.

I take it you think that f can't count as computable if "the first member" and "the second member" are the infinite sequences s₁ and s₂. I think you're wrong. Ask yourself: what's f(3000)? The answer should come quickly: s₂. It's easy to compute the answer: just follow the alogorithm.

I think you're confusing the value of a function with the linguistic representation of that value relative to some convention (like confusing numerals for numbers). You don't need to write down the terms of the sequence s₂, in order, to indicate that it is the value of f(3000). You can name the sequence using the symbol, "s₂," as we've been doing in this discussion.

Think of it this way: you can name the number 2 by writing down the numeral '2'. Or you could attempt to write down the infinite sum '1 + 1/2 + 1/4 + ...'. Of course you can't write down an infinite number of terms. But you don't need to, to indicate that the value of some function is 2. Similarly, the value of the function f given input 3000 is the sequence s₂; but you needn't convey that fact by actually writing down the terms of the sequence, in order. You can do it by writing down ' s₂ '. We have set down the convention that the symbol ' s₂ ' names the sequence <b₀, ... b_n, ...> where b₀ = 1 and for all n > 0, b_n = 0.

It occurs to me that you think recursive functions are properly "computable" b/c their outputs (natural numbers) can be represented by finite strings. My point is: so can the outputs of f. Just as '23' names 23,

'<b₀, ... b_n, ...> where b₀ = 1 and for all n > 0, b_n = 0'

names

<b₀, ... b_n, ...> where b₀ = 1 and for all n > 0, b_n = 0.

Even if the sequence so-named is infinite, its linguistic representation doesn't have to be.

Given that, why do you think the function f can't be called "computable"? --Futonchild 05:33, 1 March 2007 (UTC)

All you've done so far is give examples of functions you argue are computable in your sense. But to make the notion clear, you have to explain what makes a function computable, or not. I haven't seen anything in your description that would allow you to argue that such a function f is not computable. Without that, the notion might be trivial; saying that f is computable might not tell you anything about f whatsoever. --Trovatore 08:19, 1 March 2007 (UTC)

(A) Do you maintain that these paradigms are not computable? The sense of 'computable' I'm employing is the one you learn in any introduction to mathematical logic: there exists an effective procedure for calculating the value of the function for arbitrary input, which procedure is guaranteed to terminate in a finite number of steps, etc. etc. And that's as far as the explanation goes. It's the 'computable' that appears in the antecedent of the Church-Turing thesis:

Every computable total function of natural numbers to natural numbers is primitive recursive

And in that sense of 'computable', the functions I've been calling "computable" are computable. (Though not recursive, since the range is not a set of natural numbers.) If you don't agree to that, I don't know what to tell you, other than your notion of "computability" differs from the standard one. If you do not accept that a finite set of infinite sequences can be enumerated by a computable function, then you do indeed have a non-standard notion of computability. Of this I am certain.

(B) Normally, to show that there isn't a computable function of a certain kind (e.g., one whose range is the set of infinite sequences of ones and zeroes), you assume that there is such a function and derive a contradiction. And to the constructivist whose viewpoint I thought should be noted, that's exactly what Cantor's diagonal proof does. And in that respect, from this viewpoint, the set of such sequences is no different than any other non-effectively enumerable set (e.g., the set of invalid sentences of first-order logic), whereas from the platonist perspective (which is, naturally, the only one represented in many of the wiki articles relating to set theory), there are more elements in the latter set than in the former. The platonist interpretation of the cantor proof requires the acceptance of non-computable or "arbitrary" functions (e.g., any function enumerating the set of invalid sentences of FOL). From the constructivist viewpoint I intended to represent, such functions simply do not exist (do not make sense), and so there is no justification or thinking that there are "more" sequences than natural numbers. --Futonchild 18:47, 1 March 2007 (UTC)

If you want to speak in terms of restricting to computable functions, then you have to give a criterion for what functions are computable, and which ones aren't. You have so far said nothing at all about the latter question. If you don't, then maybe all functions are "computable", in which case the word "computable" adds nothing, and we're just back in the classical case of saying that there is no function whatsoever from the naturals onto the set of infinite sequences of zeroes and ones. --Trovatore 21:11, 1 March 2007 (UTC)

A function isn't computable if it's not computable, i.e., if it is not the case that there exists an algorithm for computing the value of the function given an arbitrary input, which alogrithm is guaranteed to terminate after a finite number of steps, etc. etc. As I said above, you prove that a function is not computable by supposing it were (e.g., supposing you had an algorithm for computing a hypothetical function that enumerates the infinite sequences of ones and zeroes) and deriving a contradiction (e.g., that there would be a sequence of zeroes and ones that was not in the range of the function, contra the hypothesis).

From the constructivist view, to say that "there is no computable function" is just to say that "there is no function"--so I guess in that sense, there can be no criterion for which functions aren't computable, since they all are computable. So yes, Cantor's proof shows that "there is no function that enumerates the sequences." But in the constructivist context, we also say that "there is no function that enumerates the set of all invalid sentences of first-order logic", whereas the platonist says: there is such a function; it's just not computable. (Because if it were computable, we'd have a decision procedure for the valid sentences of first-order logic, and then we'd be able to build a Turing machine that would tell us whether an arbitrary Turing machine M halts on arbitrary input n, and then we'd be able to build a Turing machine H that halts on input h if and only if it does not halt on input h.)

So the significance of Cantor's result varies from constructivism and platonism. In the latter context, it shows that not even a non-computable function from the natural numbers can enumerate the set of sequences; whereas to the constructivist, it simply shows that (as the platonist would say) no computable function enumerates the sequences. But to the constructivist, this is no reason to go on to say that there are "more" sequences than there are, say, invalid sentences of FOL (as the platonist would). Neither set is effectively enumerable, and that's all there is to say (for the constructivist), and there is no sense of enumerability beyond effective enumerability. --Futonchild 22:20, 1 March 2007 (UTC)

Which branch of "constructivism" are you talking about here? There are several different programs of constructivism, and they disagree greatly about things like whether "every function is computable". Can you point to some well-known school of constructivism or author whose writings we can use to understand your claims about "constructivism" in context? CMummert · talk 22:31, 1 March 2007 (UTC)

Constructivism is a red herring. The point is that, in order for one to interpret Cantor's result as showing that there are "more" sequences of ones and zeroes than natural numbers, one must accept the existence of functions which are, in principle, not computable--so-called "arbitrary functions". (Few people, in my experience, notice this.) Now, why might one not accept the existence of such functions? Well, one might object to them on broadly constructivist grounds. Constructivism just provides a motivation for calling the existence of such functions into question; I tried to be very careful about not claiming that all constructivists reject the existnce of arbitrary functions; but some would. Anyone who is suspicious of the Axiom of Choice probably ought to be suspicious about the existence of arbitrary functions. --Futonchild 00:01, 2 March 2007 (UTC)

You have not said what it means for there to be an algorithm computing, in a finite number of steps, an infinite sequence of zeroes and ones. Until you explain that I don't see the point in discussing the rest of it with you. Please quit adding these long discussions until you answer the first question. By the way I do understand intuitionism/constructivism; I spent a year at University of Padua studying with Giovanni Sambin. --Trovatore 22:28, 1 March 2007 (UTC)

You can call a sequence computable if, for every n, a_n is computable. That is, there's an algorithm for computing a_n that terminates in a finite number of steps.

So an infinite sequence is no different from a function with domain N. Is that ok?

I have a hunch that you're confusing sequences and terms denoting them. The set {s₁, s₂}. above, is effectively enumerable. (Any finite set is effectively enumerable.) It's enumerated by the computable function f, where f(0) = s₁ and for all n>0, f(n) = s₂. You're thinking: "But f isn't computable! To compute f(3000), for example, you have to write down all the terms in s₂, and that you can't do in a finite number of steps!" But this is not right. To compute f(3000), you go to the rule I gave you for computing f(3000), and see what its value is. It's value is s₂. If I were to ask you to write down the value, you could write down ' s₂ '. If I wanted you to tell me the value, you could vocalize the sounds, "ess-sub-too". To calculate the value o f(3000)--let alone to convey what the value is--=it is not at all necessary that you write down the infinitely long sequence of terms in s₂.

Is that not clear? --Futonchild 22:59, 1 March 2007 (UTC)

It's completely wrong. By that standard, the solution to the halting problem is computable. Let s_n be 000... if Turing machine number n halts, and 111... if it doesn't. Now I have an easy computation solving the halting problem -- given n, my program will spit out the literal string s_n. By the standard you seem to be using, that's a solution, even though I can't actually figure out effectively which string of zeroes and ones is represented by the symbol s_n.

I am not confusing symbols and their denotations -- you are using the wrong one. I do hold a PhD in mathematical logic (and I feel safe in saying that you do not) so you might consider the possiblity that I know what I'm talking about here. --Trovatore 00:16, 2 March 2007 (UTC)

No, your function is simply not computable. The fact that the values of this function are infinite sequences is irrelevant. If you make the outputs 1 and 2 instead of 000... and 111... (I assume you're indicating infinite sequences of numbers here? And not infinite strings of numerals?), your function still is not computable. If it were, then we could build the machine H that halts on input h if and only if it doesn't. I don't understand why you think that if my function f is computable, then so is your s. The difference between your function and mine lies in the predicate in the antecedent of the instructions: where I have "If n=0, then..." and "If n>0, then..." whereas you have "If machine n halts on input n". I assume you meant to include the bit about the input being n). The predicates I employ are decidable (computable by a Turing machine, say) whereas yours are not. --Futonchild 01:34, 2 March 2007 (UTC)

You didn't say anything about being able to compute what s_n represents. You just said that it would output the string s₁ or s₂, not anything about what you would have to do to get from those strings to their denotations. I really don't think you've thought this through carefully.

But if you continue along the lines you're going, you might be able to come up with an honest criterion for what you mean by a computably enumerable set of reals. We still can't put it in the article, because it isn't a standard terminology, but you might have a genuine answer to the question I've been asking. Why don't you see if you can finish it off? You're not there yet; you still have to pull things together from various places to come up with a precise criterion, but I don't think you need any actual new ideas. --Trovatore 01:44, 2 March 2007 (UTC)

You didn't say anything about being able to compute what s_n represents.

You are confusing denotation with thing denoted. Your s_n is an infinite sequence of infinite sequences of numbers. (s₁ = <0,0,0...>, say; s₂ = <1,1,1,...>). It doesn't represent anything--at least, not any more than you represent something, or the moon does, etc. ' s_n ' is a symbol. It represents s_n--a sequence. But you're right if you mean that we cannot, in principle, come up with a computable function that would enumerate the terms of s_n (and this has absolutely nothing to do with the fact that those terms are themselves infinite sequences; it would be true even if those terms were natural numbers). But none of this has anything to do with my function f which is computable. So I don't understand your complaint.

Also, it should be clear that the set N of natural numbers is a computably enumerable set of reals, if you grant that every natural number is a real. And so is the set {pi} U N; let f(0) = pi and for all n>0, f(n) = n - 1. As I've said before, the fact that the infinite decimal expansion of pi cannot be written out is immaterial. --Futonchild 02:17, 2 March 2007 (UTC)

No, I am not confused; your presentation is confused. You claim to show that a finite set of sequences is always computably enumerable, and you support this claim by arguing that a program can spit out symbols for the sequences. Then when I do the same thing, you balk on the grounds that I can't effectively get from the symbols to the denotations. But you never said that was a requirement.

So what are your requirements? Finish it off; state them explicitly. In what sense must the transition from symbol to denotation be effective?

You're not really that far from a defensible definition. As I say, we can't put it in the article, because it's not a standard definition, but at least we'd have something precise to work with here on this talk page. --Trovatore 02:24, 2 March 2007 (UTC)

You are mistaking the crucial difference between my f and your s. Imagine this symbol-manipulation algorithm (i.e., the inputs and outputs are strings):

If the input is '1', then output ' s₁ '.

If the input is longer than '1' (e.g., '11', '111', etc.), then output ' s₂ '.

Relative to the following semantics, this algorithm computes my f:

a string of of n '1's = n; ' s₁ ' = the sequence <a₀, ... , a_n...>, where a₀ = 0 and for all n > 0, a_n = 1; ' s₂ ' = the sequence ... etc.

This semantics is effectively computable in the sense that it provides an unambiguous denotation for every input- and output-string.

For your s, the algorithm might look like this:

Given an input <string of n ones>, [here you describe the steps in the algorithm that turns the input into the nth algorithm that qualifies as a Turing machine], and run the input through that alogrithm.

If you complete the first step, output '0000000000000000000000000000000...'

If you do not complete the first step, output '11111111...................'

There are two things wrong with this algorithm. First, there is no, and there can be no, guarantee that you will ever get to the second step (the algorithm might throw you into an infinite loop). Second, the output is an infinite string, so even if you finish the first step, you will never finish the second. And obviously, in no case will you execute the third step. Following the algorithm, therefore, does not constitute an effective procedure.

We can fix the second problem by changing the output at step to to be '1'. And now we can provide an effective semantics:

'1' = the infinite sequence <b_n>, where b_n = 0 for all n.

This semantics provides an unambiguous denotation to the output '1'. So the second problem is averted. But the first--the fact that you may never get to the second step, remains--and that is why your function is not computable.

And as for a criterion for "effective semantics", what I've given here is the best possible explanation. No explanation in terms of, e.g., Turing-computability will work, for a Turing machine (an idealized symbol-manipulator, not a function) computes a function only relative to a semantics; and the function so-computed can be called "computable" only if the semantics is effective.

But all of this is neither here nor there with respect to my original point about Cantor's proof. I am going to clean up the article at Computable function (I've had a long discussion about it with CMummert) and then I will probably re-insert my (very brief) comments on Cantor's proof. If you (or someone else) thereafter reverts, I will consider it unjust but I will not expend any more energy on it. The point I wish to make is simple and even trivial, though not often recognized. --Futonchild 03:14, 2 March 2007 (UTC)

And BTW, there is a paper on the topic of "effective semantics" and its significance for interpreting Church's thesis in an upcoming issue of the Notre Dame Journal of Formal Logic. --Futonchild 03:18, 2 March 2007 (UTC)

I am not mistaking anything, and in particular I am not mistaking the fact that you have not answered the question, preferring just to give examples instead of making precise and general what you think is the difference between the examples. Your long ramble above appears to be a plea to let you not answer it, because there isn't any answer. But you're wrong about that; there is an answer, and it's quite simple and would probably satisfy you, and you could give it in much less space than you've devoted to the above. --Trovatore 03:34, 2 March 2007 (UTC)

Good-bye. —The preceding unsigned comment was added by User:Futonchild (talk • contribs) 2007-03-01T22:46:50.

You have mentioned this upcoming paper in two places now. Would you care to explain exactly what you are trying to achieve by mentioning it? I am beginning to suspect you are playing a joke of some sort. CMummert · talk 04:00, 2 March 2007 (UTC)

[edit] Godels Theorem and TOE

Hi Trovatore,

A few months back you were doing some edits on the Theory of everything page and deleted the section on Godel incompleteness theorems and their relevance to TOE. I have just found a transcript from a lecture by Hawkings about this very topic, somewhat in support of the position you were questioning - "Gödel and the end of physics"[2]. Sadly I'm no degree level physicist so thought I'd ask you to take a gander and look into adding the section back if need be. Links are on the Talk page of TOE. Thanks for the help on this Pluke 14:30, 4 March 2007 (UTC)

[edit] Axiom still sounds more accurate

In terms of set theory, i've never heard them refered to as propositions, but rather as axioms; propositions to me suggest that it has some kind of true-or-falseness to it, whereas axioms refer more to the formal system and not just the concept of a proposition. Still, did you revert it due to the accessibility, or ? Just questioning, is all. James.Spudeman 19:32, 6 March 2007 (UTC)

Well, whether they're true-or-false or just consequences of an axiomatic system is not really the point (I have my views on that score but I'm not going to argue with you about it). The point is that they are not usually taken as axioms, but rather proved as theorems. But "theorem" is too big a word for facts so small, and we have a bunch of words for mini-theorems ("proposition", "lemma", "fact", "observation", "claim", "remark"). There are no precise divisions between the various levels of mini-theorems. --Trovatore 19:37, 6 March 2007 (UTC)

I guess so, it's semantics of mathematics, as with other things which aren't neccesarily the important thing; still, i'll search around some of my algebra and set theory books to see if i can find anything to reference the terms as "propositions"; if not, i'll add a small note to explain the reasoning. Cheers, James.Spudeman 19:53, 6 March 2007 (UTC)

[edit] Bot on Boolean algebra/logic

Thanks for the notification. By the way, would it not be better to make boolean algebra a disambiguation page, given that I am probably not the only person who when hearing the term Boolean algebra thinks of simple:Boolean algebra? - Andre Engels 19:50, 18 March 2007 (UTC)

The idea has certainly come up. I wouldn't necessarily be against that, and I'm not sure anyone would be deeply opposed to it, but neither does there seem to be a clear feeling in favor of it. --Trovatore 00:19, 19 March 2007 (UTC)

[edit] Peano axioms up for A-class rating

Hi Mike. It seems you're busy in your new life. I'm not sure how much you're still following Wikipedia matters, but we at the mathematics WikiProject have set up a process to grant articles that deserve it an A-class rating at Wikipedia:WikiProject Mathematics/A-class rating. Recently, our article on the Peano axioms was nominated. Unfortunately, there are no comments from anybody who really knows logic (this was true when I wrote this message, but in the mean time Ryan Reich has commented). I was hoping that you could have a look at the article, see whether there is anything there that would embarrass us, and leave a comment on Wikipedia:WikiProject Mathematics/A-class rating/Peano axioms. By the way, you should blame Paul August for this spam (proof). -- Jitse Niesen (talk) 12:46, 30 March 2007 (UTC)

[edit] Xenu lecture

Thank you for your very stern and well adjusted correction of me on Xenu talk, much needed and indeed very gratifying, California mathematical cyclist boy. Useful! MarkThomas 08:10, 31 March 2007 (UTC)

[edit] The Cberlet mediation

See my talk page. - Jmabel | Talk 16:30, 4 April 2007 (UTC)