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 29 Division by zero 30 Proper use of talk pages 31 Karel de Leeuw 32 "Nuclear crime" 33 Wayne Chiang 34 Cantor Again 35 User Antidote 36 Category for deletion -- WHY? 37 Mathematics CotW 38 Thanks 39 Set theory 40 Reply 41 your last comment on my talk page 42 deleting off-topic section of Talk 43 a key three-letter word 44 greetings 45 wikimania 46 "clarify" template at well-order 47 capacitance 48 Logic template et. al. 49 Cat 1 50 Duality (mathematics) 51 Ordering sets with ideals 52 Your pov on Logic 52.1 Proof and derivation 53 Categories 54 Invite 55 user page 56 The game of Questions 57 Boolean 58 Cantor redux 59 MoS 60 A little help here 61 ...for you and geometry guy... 62 one more time, with feeling 63 question 64 WP:LGC Tags 65 Georg Cantor FA 66 Ref desk question removal 67 small set 68 Colloquiality 69 Asprin and salicylic acid 70 Continuum Hypothesis 71 HI 72 Phenomenon 73 Godel's incompleteness theorems article 74 could you straighten up the sets discussion near the end of the mathematics reference desk now? 75 Hi again 76 reducing arithmetic to logic ? 77 Zero Sharp's edits to Zero Sharp 78 Division by zero 79 Vandal user, or just confused? 80 Omlauts 81 secondo problema di Hilbert 82 two changes to Georg Cantor that merit attention; see G-guy's talk 83 Unpleasent Profession of Jonathan Hoag 84 Gödel's incompleteness theorem 85 Georg Cantor 86 About Wikipedia:Arguments to avoid in deletion discussions 87 "grounded relation" 88 Descriptive set theory 89 Thank you 90 Boolean algebra task force 91 Game theory FAR 92 Discussion at MoS 93 MoS 94 Strategic voting in Canada 95 Thanks for doing all the work 96 Measure-theoretic question 97 Powerset 98 Image copyright problem with Image:Los Altos.ogg 99 Lebesgue measure help -- thanks 100 reference search - invention v. discovery of mathematical objects 101 Inaccurate Template 102 Bitter Almonds 103 Automaton group

[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)

[edit] Division by zero

There is a question at Talk:Division by zero#Scientific trivia. Your comments would be most welcome. Jesper Carlstrom 08:16, 10 April 2007 (UTC)

It looks to me as though Jitse already did the right thing. It's what I was planning to do once I had time to read it more carefully and be sure. --Trovatore 19:57, 10 April 2007 (UTC)

[edit] Proper use of talk pages

Please do not make comments like that again on my talk page. WP:AGF, I won't ask how you came to find Gilisa's comment on that page, but to assume that he was referring to Georg Cantor rather than Albert Einstein made no sense. Also, please do not say that there are no reliable sources that call Cantor Jewish or document his mother's Jewish ancestry.--Brownlee 17:38, 14 April 2007 (UTC)

Well, it did make sense, because he had been part of the edit wars at Georg Cantor and Cantor fit the description he gave. I found his comment because I have your talk page on my watchlist, along with many others (basically everyone I've exchanged user-talk comments with). I didn't say there aren't reliable sources that call Cantor Jewish (don't know much about his mother; frankly that sounds a little shaky, basing things on oblique comments in a letter), but anyway as I say, I mainly want the sources to be cited, some accurate compromise wording reached, and then for every one to just let it go. It's absurd that this question, which seems to have had so little impact on Cantor or his work, has dominated the discussion at his page (especially when there are so many more interesting things to argue fruitlessly about in connection with Cantor). --Trovatore 19:33, 14 April 2007 (UTC)

[edit] Karel de Leeuw

The article on Karel de Leeuw says nothing about this person's notability. As best as I can tell from the article, he was an ordinary professor who was a murder victim. Could you please add something to indicate why this persn was notable? Otherwise, I will nominate the article for deletion (although I would like to believe that something more can be said about this person). Thank you, Dr. Submillimeter 20:52, 19 April 2007 (UTC)

I think he had plenty of publications to merit keeping. He wasn't in my field, but an "ordinary professor" in mathematics at Stanford has almost certainly done something notable. --Trovatore 20:54, 19 April 2007 (UTC)

Could you please add more material to the article on his academic work? Dr. Submillimeter 21:01, 19 April 2007 (UTC)

OK, I'll give it a shot. As I say he wasn't in my field, so I'll have to do some digging; can't promise it in the next couple of days. --Trovatore 21:03, 19 April 2007 (UTC)

[edit] "Nuclear crime"

Hi regarding List of crimes involving radioactive substances, I think that you are being unduely quick to dismiss events as not being crimes. I will explain my reasoning for some of the cases being criminal acts.

The loss of the source, under the criminal codes of the UK, Czech Republic, Slovak Republic, Brazil and US is covered by criminal law. If you doubt me please look at the following three souces (not radioactive) which show that the loss of a source (and or failure to report the loss) is an criminal act.

http://news.bbc.co.uk/1/hi/education/332525.stm http://www.judiciary.state.nj.us/charges/jury/crimmis2.htm page 7 of http://www.nea.fr/html/law/legislation/slovak.pdf

It is important to note that a court can take the jurisprudence produced by any court, for instance the logic of an english judgement could be applied by a US court to a similar case. So I would reason that if an act is a crime in one country then it is possible that after the first conviction that any criminal court in the world could choose to regard the act to be unlawful.

The transport accidents section deals with crime, in one british case the failure to transport the medical source correctly resulted in a conviction. So it is a crime to transport a source wrongly. The bus irradation is clearly a case of "reckless endangerment" which could (had the dose to the passengers been much higher) which could have resulted in a manslaughter charge.

Regarding Goiana at least one of the doctors who had owned the site where the cource was left (in an unsecure manner) did face a criminal court case. So I think that it is a case where some crime may have occured.

In the case of Radithor if you read the scientific american article you will see that this was a lanmark case which caused radioactive medicines to be regulated under criminal law. Also the FDA did obtain an order against the maker of the Radithor. Thus I would argue that it is case that a crime, or situation (which if repeated was a crime).Cadmium

Cadmium, please keep in mind that there is another sort of relevant law, which is libel law. Someone who wants to accuse people of crimes in a public forum had better be extremely sure of his facts. That's above and beyond the usual prohibition on original research, which you'd be flirting with in the above arguments even if there were no legal risk. --Trovatore 20:34, 20 April 2007 (UTC)

The absolute defense against libel is to prove that the damaging statement is not untrue, many of the cases which I have cited are cases where a criminal case has occured. For instance two UK cases leading to fines being imposed have proven the point that the loss of a source and the incorrect transport are criminal acts. Regarding OR I think that the OR arguement which you have raised is not right, reporting and recording the actions of others is not OR.Cadmium

By putting together laws and legal interpretations you find in one place, with facts you find in another, you are in fact engaging in original research by WP standards. The fact that the Wikimedia Foundation, or you personally, could at least potentially be sued over it in this particular instance, makes that especially a problem here.

Really this article was OR from the start and my first impulse was to list it on AfD. That still might be the best plan. --Trovatore 02:46, 21 April 2007 (UTC)

The article is a collection of facts which are gathered together from other articles on wikipedia (with a small amount of commentry), if you either edit the page to make it smaller or send it for removal then I think that little content will be totally lost. So if you do wish to recommend the article for removal I will not take offense. However I think that regarding libel that the page is OK, the cases mentioned are ones where the absolute defence applies or the defence of fair comment applies.Cadmium

[edit] Wayne Chiang

Please see the compromise suggested in the talk page. We have agreed to merge and redirect this page to Virginia Tech Massacre as there is information there about him. Kntrabssi 01:32, 21 April 2007 (UTC)

You can't make an agreement on the talk page that short-circuits an open discussion at AfD; you have to close the process there. Otherwise that process will continue, but people looking for the Wayne Chiang article won't know the process is going on. --Trovatore 01:58, 21 April 2007 (UTC)

[edit] Cantor Again

Hey Trovatore. In order to put the Cantor Jewishness debate back into equilibrium, I'll revert to the last version before Gilisa peacocked it: [3]. Here, we basically write what all important biographers and researchers of Cantor think. That, with the removal of the categories, should really be enough. --Tellerman

It looks like it died down on Cantor but unfortunately we have a whole new paragraph concerning his ethnic background which was what we were trying to avoid in the first place. Hopefully this won't transfer to other mathematician articles. --Tellerman

It's not a perfect solution, but I think it's acceptable if it ends the wars. At the very least the writing is better than Gilisa's (not entirely his fault; he's obviously not a native speaker, but you'd have thought some of the folks defending his text would have tried to clean it up a bit). --Trovatore 06:48, 8 May 2007 (UTC)

[edit] User Antidote

I do not consider that I am in a content dispute with him.--Runcorn 19:33, 1 May 2007 (UTC)

[edit] Category for deletion -- WHY?

Why'd you put Category:Isms up for deletion? It was just made, and has had no time to grow. It might need to be renamed to Category:-isms, but there is no reason to delete it so soon. --Wassermann 08:06, 9 May 2007 (UTC)

I put it up for deletion because I think it should not exist at all, no matter how many articles it might eventually have. The fact that the name for something ends in "-ism" is unimportant and not worthy of a category. --Trovatore 08:08, 9 May 2007 (UTC)

Why no proposal then to delete the poorly maintained List of isms or the ponderous List of philosophical isms? Also, having all (or most) of the -isms in one central location could be potentially useful, yes? See the deletion discussion for more reasons (glossary, etc). As for "not being worthy of a category," have you not seen the thousands of other ultra-obscure categories all over Wikipedia? --Wassermann 14:59, 9 May 2007 (UTC)

Deleting those lists might not be a bad idea, but I never see them so they don't bother me. When an improper form of categorization starts showing up on articles I'm watching, that's another matter.

Categories should describe the thing, not the name. So it's fine to put some of these articles in category:political philosophies or category:philosophical systems or category:dialects (say, for Bushism) or category:diseases (say, for Parkinsonism). Because their referents are political philosophies or philosophical systems or dialects or diseases. But they aren't in any important sense "isms"; that's a classification based on the name, not on any underlying commonality. --Trovatore 16:11, 9 May 2007 (UTC)

[edit] Mathematics CotW

Hey Trovatore, I am writing you to let you know that the Mathematics Collaboration of the week(soon to "of the month") is getting an overhaul of sorts and I would encourage you to participate in whatever way you can, i.e. nominate an article, contribute to an article, or sign up to be part of the project. Any help would be greatly appreciated, thanks--Cronholm144 00:20, 14 May 2007 (UTC)

[edit] Thanks

Just stopping by to thank you for improving/fixing some of my maths ratings (and adding a few of your own). Geometry guy 10:49, 14 May 2007 (UTC)

[edit] Set theory

I completely understand your point of view, but as long as Category:Set theory is such a broad category, making it a subcategory of mathematical logic isn't helpful. In particular, Category:Basic concepts in set theory is not mathematical logic. The same problem occurs for several other categories: for instance Category:Applied mathematics originally included Category:Probability and statistics as a subcategory. whereas it is more appropriate only to include Category:Applied probability and Category:Statistics (at most). There are similar problems with Category:Mathematics of computing, and I fear there will be more reverts than this as I gently try to point out that just because discrete mathematics and numerical analysis are subcategories according to the ACM, it does not mean they are for Wikipedia! Geometry guy 00:56, 30 May 2007 (UTC)

I'm sorry, I don't agree. It's true that there's a general problem with the usage of the term "set theory" to mean something different from "what set theorists do", but I don't see any objection to the basic set theoretic concepts (say, function (mathematics)) from winding up by inheritance as part of math logic. They are part of math logic, even if a very elementary part. --Trovatore 01:04, 30 May 2007 (UTC)

Maybe you should take off your professional hat here, and also click on some of the links to see e.g. the claims made by Category:Mathematics of computing? Do you think I should make Category:Differential calculus a subcategory of Category:Differential geometry, just because, hey, the derivative of a function is a very elementary example of the tangent space to a manifold? Geometry guy 01:11, 30 May 2007 (UTC)

Well, I have to say that I don't see that as entirely analogous; mathematical logic has certain claims to universality that differential geometry doesn't. You really can't do research in mathematics at all, in the modern era, without knowing something about it. Still, I wouldn't be too upset if you wanted to remove the "basic concepts" cat from category:set theory and put it instead in category:elementary mathematics; that would answer the objection you've specifically stated. Then we could let category:set theory be about what research set theorists consider to be set theory, which seems to me the natural way to populate the cat.

Part of the problem, I think, is the text at the top of category:set theory, which I don't like at all; it strikes me as coming from a very formalist point of view. By the definitions given there, much current research in set theory is in fact "naive set theory", because set theorists reason in terms of collections of objects rather than by directly applying axioms. However these distinctions are unfortunately embedded in a large number of articles (and, indeed, in their titles) so it's a huge project to fix it, one I haven't yet gotten up the nerve to tackle. --Trovatore 01:39, 30 May 2007 (UTC)

I agree with Trovatore. Set theory is part of mathematical logic and the categories should reflect that fact. JRSpriggs 08:30, 30 May 2007 (UTC)

Okay, there seems to be agreement that from the point of view of contemporary research set theorists, set theory is a subfield of mathematical logic. However, this is not what Wikipedia is about! The category is about what set theory covers both historically and currently, and from all points of view such as that of a school child, a layman, a mathematician in another area, a specialist in set theory, and, dare I say it, a formalist.

There is a natural tendency in all fields to maximise their scope. For example, I sometimes go so far as to say that geometry is not a branch of mathematics, but a way of doing mathematics. From this reasoning everything should be a subcat. I linked to other examples to try and illustrate the problem with populating categories in this way. Most of mathematics can be formulated and studied in terms of mathematical logic. This does not mean that everything should be a subcategory. If set theory is, how about category theory, general topology, group theory,...?

We could end up with a situation where everything is a subcat of everything else, and the category system becomes useless. Geometry guy 12:32, 30 May 2007 (UTC)

I think category theory is a subfield of math logic, though it's not one of the traditionally listed ones. the other two, no, because they have other more natural homes, geometry/topology and algebra, respectively.

The formulation I'm used to is that there are four top-level subfields of mathematics, namely algebra, analysis, geometry/topology, and logic/foundations. Obviously this isn't quite complete as there's nowhere to put number theory (among the four it fits best with analysis, but it's an uncomfortable fit). --Trovatore 17:20, 30 May 2007 (UTC)

Okay, but the top of the mathematics category hierarchy is finer than this: the top-level subcategories of Category:Mathematics are currently algebra, arithmetic, analysis, geometry, number theory, topology, category theory, mathematical logic, discrete mathematics, applied mathematics, probability and statistics, and several subcategories that cut across fields. I hope you can see why in such a setting, I find it natural to lift set theory to the top level.

An alternative would be to refine the logic/foundations field into Mathematical logic and Foundations categories, with Set theory as a subcategory of both. How does that sound? Geometry guy 17:46, 30 May 2007 (UTC)

I don't see any clear way to distinguish math logic from foundations; the terms are used almost synonymously. The reason we say "logic/foundations" instead of just logic or just foundations is to accomodate different philosophical approaches and aesthetic preferences, not different subject matter.

I hope it's clear that math logic is not really logic, it's mathematical logic, which is historically and etymologically, rather than *ahem* logically, related to logic. I think our article says that mathematical logic is more the mathematics of logic than the logic of mathematics, and that's a step in the right direction, but candidly a great deal of mathematical logic is hard to characterize even as the former. But the term "mathematical logic" (or when clear from context, just "logic" for short) is so well-established that it no longer really matters that it's not particularly related to logic in the older sense of "the science of making valid inferences". --Trovatore 17:55, 30 May 2007 (UTC)

To Geometry guy: You said "For example, I sometimes go so far as to say that geometry is not a branch of mathematics, but a way of doing mathematics. From this reasoning everything should be a subcat.". One should not make new rules based on hypothetical problems which might occur (but are not actually occurring) if people went to unreasonable extremes in applying the current rules. If it ain't broken, do not fix it. JRSpriggs 07:08, 31 May 2007 (UTC)

I agree, I am a practical guy: I raised this question because problems like this actually are occuring. They were occuring for the Applied maths category (and still are to some extent) which seems partly to take the point of view that any topic which can be applied is a subcat. I managed to improve that slightly without complaints so far. I don't think I will be so lucky with Category:Mathematics of computing. Take a look at it, if you haven't already, and tell me how to reconcile it with Category:Theoretical computer science, Category:Discrete mathematics, and Category:Numerical analysis.

Rules have not been mentioned up to now. No one is proposing new rules. As for "...in applying the current rules", I see little sign that the current rules (see WP:CAT) are being applied in making category decisions. In particular, the very first guideline states:

1. Categories are mainly used to browse through similar articles. Make decisions about the structure of categories and subcategories that make it easy for users to browse through similar articles.

It does not state:

1. Categories are mainly used to organize the hierarchy of knowledge. Make decisions about the structure of categories and subcategories in accordance with the general practice of experts in the field.

Geometry guy 09:42, 31 May 2007 (UTC)

[edit] Reply

I have no problem with it on AfD. If it gets deleted, so be it. —Kurykh 04:02, 5 June 2007 (UTC)

[edit] your last comment on my talk page

Please remove your warning from my talk page a.s.a.p, as Tellerman writing on the talk page (let me remind it again:"...the only people that called Cantor Jewish in the past were anti-Semites or fervent Zionists...".) is ,no doubt, an act of vandelism in violation of WP:TALK.--Gilisa 20:37, 5 June 2007 (UTC)

I will not remove it and am not interested in discussing it with you. Thus far you have not repeated your vandalism; as long as you do not, I consider the matter closed. --Trovatore 20:42, 5 June 2007 (UTC)

Ok, very well then, I will not repeat what you called, unjustly, vandelism, for now.I have alot of patience, and these matter, which is of extreme importance, is yet not closed . My lesson from you is that I cant treat it alone-otherwise I exposed myself to unjust charges. About your warning-it can stay for now, even if horrified me very much. And please, dont be so personal (it's actually a violation of wikipedia rules, not that I warn you-I dont an adherent of on-line warnings)-I'm not.Have an American nice day--Gilisa 05:15, 6 June 2007 (UTC)

[edit] deleting off-topic section of Talk

I'm pretty sure you can't do that.. per GDFL ... besides it will just irritate people... Ling.Nut 21:36, 5 June 2007 (UTC)

Archiving is a better solution--Cronholm¹⁴⁴ 22:01, 5 June 2007 (UTC)

There is no GFDL issue. Removing attribution would arguably give rise to a GFDL issue. Deleting page histories, when the content is kept, is a GFDL issue. Removing content is not.

There is precedent for removing off-topic material from talk pages. In this case, in addition to being off-topic, the material I removed constituted a personal attack and was potentially libellous (I say "potentially" because I don't know that it is false; I know only that Gilisa has not adduced adequate evidence). I tend to agree that it's better to archive than delete when it comes to stuff like crackpot theories, but not when it comes to defamation of character. --Trovatore 22:58, 5 June 2007 (UTC)

As I see it, Tellerman vilified any one who support the idea of Cantor Jewishness and any Zionist as well (oh, and me also-while I wasn't present at the discussion). I dont know, nor should I care, about Trovatore definitions for "adequate"-but I do know that he done nothing against Tellerman back then-so, the last word is still to be told.--Gilisa 07:15, 6 June 2007 (UTC)

Tellerman made it obvious that he was a bit irritated with you; I don't deny that. I don't see where he "vilified" you, though. Nor to my recollection did he say anything unambiguously against Zionists. You apparently interpreted the reference to "only anti-Semites and fervent Zionists" having said Cantor was Jewish, as claiming a moral equivalence between anti-Semites and Zionists. The sentence can be read that way. But I would suggest, in context, that a more likely interpretation is that references written by anti-Semites have a possible motive to ascribe or deny Jewishness to historical figures in a way that may not match the judgment of a disinterested observer, and that references written by fervent Zionists also have such possible motives. Therefore neither can be taken entirely at face value on the question.

If Newport's assertion that the Jewish Encyclopedia was not Zionist at the time it listed Cantor as Jewish is accurate, then it appears Tellerman was misinformed. But that doesn't make him anti-Semitic or even anti-Zionist.

I am aware, by the way, that the word "Zionist" is sometimes used as a codeword by anti-Semites, to mean something like "member of the International Jewish Conspiracy" or whatever nonsense it is they believe in. Again, I don't know that that isn't what Tellerman meant. But I see no evidence that it is, either. --Trovatore 07:34, 6 June 2007 (UTC)

So, before continue, as an act of good will, please remove your warning from my page-belive me that I ask it only for fixing the trust between us-I think that you been too personal about me, correct me if I wrong.--Gilisa 10:31, 6 June 2007 (UTC)

Do you agree not to use the Georg Cantor article or its talk page as a forum for your grievances against other users? --Trovatore 19:35, 6 June 2007 (UTC)

No problem with that, it's not realy about Cantor as I said. More, I never used Cantor article, or any article, for grievances against any user.Best--Gilisa 05:23, 7 June 2007 (UTC)

Long essay by Gilisa moved to User talk:Trovatore/Tellerman issue -- I will respond at my convenience.

[edit] a key three-letter word

Dauben 2004 (link to pdf in Cantor article) says: "In so doing he laid the groundwork for abstract set theory and made significant contributions to the foundations of the calculus and to the analysis of the continuum of real numbers." My problem is the word the before calculus. Does it mean "the foundations of the calculus of the continuum of real numbers," or is it an error and means instead "the foundations of calculus." Ling.Nut 00:37, 6 June 2007 (UTC)

I think it's not an error, but is synonymous with "the foundations of calculus". Often in formal writing "the calculus" means "the infinitesimal calculus", or what is less formally called "calculus" full stop. --Trovatore 02:44, 6 June 2007 (UTC)

Thanks. I hope you saw my most recent comment at Talk:Georg Cantor. Maybe tomorrow I can do some intensive ce... Ling.Nut

[edit] greetings

I've just got into editing Wikipedia recently and I noticed your trail in the logic websites that I have been looking at. so greetings! would like to chat with you about your views on graduate schools in logic but I think email is more appropriate. do reply at my user talk page!

another thing out of interest. how do you traverse the wikipedia web space looking for articles? do you do it by have a (supposedly long) watchlist? i'm asking you this since the article sentence is definitely not one of the more hot stuff to edit. Hope to hear from you!--DesolateReality 01:29, 11 June 2007 (UTC)

You could start with looking at category:mathematical logic and its subcategories. Not everything will be there. --Trovatore 01:53, 11 June 2007 (UTC)

[edit] wikimania

have you any idea how wikimania is like?--DesolateReality 01:36, 11 June 2007 (UTC)

No, don't know much about it. --Trovatore 01:51, 11 June 2007 (UTC)

[edit] "clarify" template at well-order

Hi Nadav,

could you specify what it is that you feel is unclear? --Trovatore 20:19, 19 June 2007 (UTC)

I added it at first because I wasn't sure I was interpreting the text correctly, but now I've removed it since it looks like I was reading it right after all (in line with the ref desk explanation that is [4]) nadav (talk) 23:18, 19 June 2007 (UTC)

[edit] capacitance

I'm not going to post this on the RD, but just now I put on a pair of tennis shoes, grabbed a screwdriver, walked into my bathroom, stuck the screwdriver into the hot side of the bathroom's electrical outlet, grabbed the metal shank of the screwdriver with my hand, and felt... nothing. I then held one lead of a neon electrical tester in my other hand, touched its other lead to the bathroom faucet, and observed its glow, indicating that I was, indeed, at high voltage. —Steve Summit (talk) 00:50, 28 June 2007 (UTC)

OK, good for you :-). I'm not going to try it myself. Let's just say I'm taking your word for it and leave it at that. --Trovatore 00:52, 28 June 2007 (UTC)

......It takes all kinds I suppose.....--Cronholm¹⁴⁴ 00:57, 28 June 2007 (UTC)

The experiment is not quite so reckless as it sounds. Tennis shoes act as electrical insulation, at least we hope so. If there is no water or air spark closing the circuit, then current cannot flow. After all, current does not "leak out" of electrical sockets, though we know the voltage is there. One old safety practice was to work standing on an insulating mat, keeping one hand in a pocket. But high voltages can be nasty, and very high frequencies travel peculiar paths. Standard domestic AC power passing through the heart can be lethal, so this definitely falls into the "Kids, don't try this at home" category. --KSmrq^T 04:43, 28 June 2007 (UTC)

A capacitor looks like a closed circuit to alternating current, and the human body is a capacitor. I haven't done the actual calculations to figure out how much capacitance it has or what impedance it would present at 60 Hz. --Trovatore 04:45, 28 June 2007 (UTC)

A closed circuit from what to what? Remember the tennis shoes. The body has such a high fluid content, essentially salt water, that gripping the metal shank guarantees the voltage will reach the hand, and can then travel through the body. (Really high frequencies, however, tend to stay on the skin.) If there is a capacitor to consider with respect to shock hazard, the tennis shoes would be the dielectric.

True story: A device I constructed in my youth could deliver a surprisingly large tingle. It was a diminutive box covered with two pieces of foil, one on each side, so picking it up would close a circuit through the hand. Inside was a small battery (probably 9 volt), some wire coils, and a screw that would drop as the box was lifted. This produced an intermittent contact that allowed the small voltage to be amplified and delivered to the unsuspecting lifter. (I got the design from a magazine, as I recall.) The effect was quite satisfying for me, and quite startling for my experimental "volunteers". Then a teacher insisted on picking it up despite my warning. :-)

Of course, the tingle box was designed to create a closed circuit in the hand. If experimenter Steve made contact with both sides of the outlet, even with one hand, the results might not be so entertaining. --KSmrq^T 10:56, 28 June 2007 (UTC)

A closed circuit to nowhere. I'm talking about self-capacitance. --Trovatore 19:22, 28 June 2007 (UTC)

I don't understand it either, but you (Trovatore) are right about this.

It would be easy to understand for DC: your body (plus your tennis shoes, if you like) are a capacitor, one side is grounded, one side is touching a live wire, and there's some nonnegligible inrush current as the capacitor charges.

I don't understand the equivalent argument for AC, but I know it applies. I've talked to commercial electricians who have described huge sparks when connecting new branch circuits to live wires. The new branch circuit isn't drawing any load yet -- it's just a big, long length of big, thick wire -- but it takes a bunch of initial current to "charge" it up to the supplying circuit's potential.

Similarly, linemen who work bare-handed on high-voltage wires take care to charge the insulating basket they're standing in (and thereby themselves) up to the line's potential, via a jumper cable, before they first touch the wire, so that the jumper cable can carry the inrush current, and not their hands and arms.

Also, User:Edison has described [5] the huge sparks that can be seen when performing live-line work on extremely-high-voltage cross-country transmission lines with a helicopter. —Steve Summit (talk) 01:49, 1 July 2007 (UTC)

We need to be careful, and not lump all these situations together. The human body can have a charge deposited on it, but with a DC source that would ordinarily be a fairly small current transient decaying exponentially. What the linemen are worrying about is not so much the capacity of the body to take a static charge, but the possibility that it may act as a conduit for the high voltage. And I wonder if that long length of wire is acting more as an inductor, storing energy in a field. --KSmrq^T 10:22, 2 July 2007 (UTC)

[edit] Logic template et. al.

Thank you for your correction on the cardinality of the continuum. I have posted a response to your earlier comments at Template_talk:Logic. Be well,

[edit] Cat 1

I was reading on le tour's discussion page about your journey up a 1 cat climb, impressive but what time can you do it in? I am still pleased with myself for climbing a 25% climb albeit a short UK climb still, 30 miles off road after that in the Yorkshire Dales is not too bad for 12 (at the time) year old--GOD 21:34, 5 July 2007 (UTC)

Oh, very slow. Takes me about an hour, not counting two rest breaks -- that's for gaining about 2000 feet of elevation over 5.3 miles. The hard part, the first 1000' of elevation over two miles, takes about 25 minutes without stopping, if I push it. But I figure most people probably can't do it at all, and anyway I'm not racing anyone. --Trovatore 21:45, 5 July 2007 (UTC)

To do that pushing yourself is a great achievement, I have to get in shape just being amateur and waiting for the summer but it aint coming over here, 6th July and waking up to rain!--GOD 07:38, 6 July 2007 (UTC)

[edit] Duality (mathematics)

Hello, after seeing your recent comments at WT:WPM, I became curious as to what would you think about the article of the title. Arcfrk 06:38, 7 July 2007 (UTC)

Well, I think it's problematic, though I'm not planning to make an issue of it. It might be OK in list format -- something like list of duality properties in mathematics.

But I don't think it's as bad as the proposed "small set" article, because the "duality" article is at least linking to existing, hopefully well-referenced, articles about concepts for which "duality" is the standard terminology. The "small set" thing seems to collect some concepts where "smallness" is a nonce term or an intuitive motivation rather than the usual terminology for something. --Trovatore 10:03, 7 July 2007 (UTC)

… in other words, "small set" suffers from the same problems as Natural topology. Yes, it's a very good point, and something to keep in mind for the future. Thank you, Arcfrk 19:57, 7 July 2007 (UTC)

[edit] Ordering sets with ideals

After our discussion on my page-in-progress, I realized a potential problem with making all the notions of smallness ideals. The Wikipedia definition of ideals makes the order over elements a partial order, and other definitions I've seen are at least as restrictive. The orders on my page are only preorders -- N - {2} isn't the same as the composite numbers but both have density 1. The obvious way out is to use equivalence classes over the subsets of the universe (generally natural numbers), but this would make for very trivial filters. Consider the combinatorial definition, for example:

Let be the reciprocal sum of the elements of X and define the preorder ≤ such that a ≤ b when f(b) = +∞ or f(a) < +∞. The equivalence classes needed to make this a preorder partition the power set of the naturals into only two sets... so while there is a filter, it is just another name for the filter {0} on the set {0, 1}. (Yes, the preorder could make a ≤ b when f(a) ≤ f(b), but that's less natural since the order depends largely on the small elements instead of not at all.)

Similarly the others have little structure as partial orders. What do you think?

CRGreathouse (t | c) 21:41, 8 July 2007 (UTC)

Um, I'm not quite following you, I'm afraid. The relevant ordering that translates ideals in the general order-theoretic sense, to ideals on sets, is always set inclusion.

That is, an ideal may be defined thus: I is an ideal on X if I is a subset of the powerset of X, such that:

1. if A ⊆ B ⊆ X, and B∈I, then also A ∈ I, and
2. if A, B ∈ I, then A∪B ∈ I

Condition 1 implies that we can interpret membership in I as "smallness" (if you take elements out of a small set, it should remain small). I'm afraid I'm not sure what you're getting at with these preorders. --Trovatore 21:52, 8 July 2007 (UTC)

[edit] Your pov on Logic

I appreciate the level of attention to these topic, but your pov is an issue. Your belief in the "discredited" logicists is a misunderstanding of the issue. I'm sorry, and I realize I am out numbered here, but this edit to Logic really is not appropriate.

Logic really is the foundation of mathematics. To the degree that it is taught or learned in math classes is really just incorporating philosophy concepts into the area appropriate to math. There is no one out there making an attempt at a "mathicist" account of logic because there is none. I think there is a certain amount of pride or loyalty to the home team here on both sides. However, this is a sincere plea to loosen the grip on this and several other articles of "overlap." There's no shame in identifying the foundations or meta- level as philosophy -as distinct from the 'object'-subject of mathematics.

Anytime you are talikng about the principles behind things you are talking about philosophy. It's a formal study so the distinction is meaningful. Soil scientists and archeologists both have perspectives on dirt, but I think I will listen to the soil scientist on the subject of "what soil is." Gregbard 01:42, 10 July 2007 (UTC)

Gregbard, I recognize that as a logician you may have different perspective than mathematical logicians. Surely the articles can be written in a neutral way. They should respect the following opinion, which is widespread in mathematical logic today: the "logicist" program to use logic alone as the foundation of mathematics failed, and the only way to make it successful is to incorporate "mathematics" into logic itself. I'm not certain that a neutral article can accurately claim that anything is "the foundation of mathematics". — Carl (CBM · talk) 02:08, 10 July 2007 (UTC)

Exactly, what Carl said. Claiming that something is the "foundation" of mathematics is foundationalism, which is itself a POV. I lean more to coherentism myself. But I don't write articles that claim coherentism is right and foundationalsim is wrong. --Trovatore 03:48, 10 July 2007 (UTC)

I would like to see the middle ground develop. However; stating that "Traditionally..." logic is studied under philosophy, and then stating "Since the 19th..." ..studied under foundations of math...; portrays this math bias I am talking about. I guess we are supposed to believe we've come to our senses or something? Most philosophy departments teach logic, and some teach little else. P.S. I myself am a coherentist.

I'm a little concerned we are going to have a meta-notability issue screw up the intellectual integrity of the wikipedia here. In raw numbers of people learning, and teaching, and using logic we will find plenty o' math people. There are precious few philosophers and they are quite jobless indeed by and large. So to all the world it will appear to people learning this stuff on the wikipedia that logic is a math world and that what really matters about it. However, in principle those factors are irrelevant to organizing the content.

I stand by the soil scientist v archaeologist analogy. I think we should defer to the philosophers on this one on principle. I'm sorry but that is a bias in the other direction that has been let run loose here for way to long. I'm certainly open-minded to a middle view. Let's see it. Be well,

Gregbard 04:50, 10 July 2007 (UTC)

OK, you're "traditionally...19th century" contraposition is a possible valid criticism, I'll allow. But you don't fix it by making the (to me outrageous) claim that logic is the foundation of math. Logic is part of the story, but not the whole story.

And by the way, very few philosophers of mathematics would say otherwise, so even deferring to the philosophers doesn't help you. In philosophy of math these days, your choices are generally between some version of formalism/nominalism/fictionalism, some version of realism/Platonism/empiricism, and (bringing up the rear because of its austerity) some version of intuitionism/constructivism/finitism. Logicism is, for the most part, not even on the radar screen.

That said, I'm willing to work with you on the language that you see as disparaging towards logic that's not used in the study of mathematical foundations. To me, the second challenged sentence simply reads as an additional use of logic, not as supplanting the traditional study, but if we can make that point clearer I'm in favor of it. --Trovatore 05:12, 10 July 2007 (UTC)

Incidentally, even if logic is not "the foundation" of math, it still is more general than math. Wikipedia articles should be organized from general to specific and therefore the "logical" perspective should appear earlier and preferably more prominently than it currently is in many mathematical logic articles.Gregbard 05:17, 10 July 2007 (UTC)

How is it "more general than math" if it's not the foundation? It's a tool used by mathematics; neither subsumes the other, so neither is "more general". --Trovatore 05:19, 10 July 2007 (UTC)

I thought I proved my point with the Theorem article, but it seems to have gone over everyone's head (or it's denial, or people just don't care). The definition of theorem to a logician is a more general definition than the one mathematicians use (and is portrayed at theorem). The article is back to talking about "truth" and "proof" in the opening paragraph. This is an aricle with a way too long "lede." btw.

Do you see what I'm saying? The definition of theorem INCLUDES the math definition not the other way around. It's really quite frustrating to watch. But it's okay. I'm an eventualist. Stay cool Trovatore. Gregbard 05:26, 10 July 2007 (UTC)

Greg, it isn't more general. There are two meanings of the word "theorem" in play here, one being a formal string that can be derived from formal syntactic rules, and the other being a natural-language statement that can be established by a stylized sort of argumentation. These are pretty much disjoint sets; though we elide the distinction between them when it's convenient to do so, neither can be called "more general" than the other.

We believe that the natural-language theorems, and their proofs, can be translated into the syntactic-string form, but no one ever does it (except in trivial cases, or occasionally as part of a computer-based project), and the translation is not at all canonical -- there are many arbitrary choices that would be made in any such translation. This is one thing the intuitionists got right in their arguments with the formalists. --Trovatore 05:34, 10 July 2007 (UTC)

No it' s not two definitions. It is the same concept. It is employed in both areas. When I say:

'In logic, a theorem is an expression ... in a formal language which is derivable from applying the rules of the language.'

There is NOT something different than this going on in those math classes.

Every 'natural-language statement' corresponds to some form of wff. 'Being established' just means 'being derived' and 'stylized argumentation' means 'derivation' Do you not see that this is a more general definition of the same thing?! They are not disjoint sets.

Every theorem as described in the first sentence of theorem can be expressed in terms of the more general definition I provided. However the theorem as described in the first sentence does not account for the example I gave in the formal language FS.

That's a more general definition and it belongs front and center so as to build upon it in later sentences and paragraphs. Oh Yeah Man.

Gregbard 06:01, 10 July 2007 (UTC)

Greg, this is simply just not true. "Established" does not mean "formally derived according to a set of syntactic rules". Do you think perhaps that there weren't any theorems before the syntactic rules were formalized? --Trovatore 06:08, 10 July 2007 (UTC)

"established" is an interpretation of the more general term "derived." Hey, being derived doesn't really tell us that it's neccesarily "established." It's just an intererpretation. "rules of the language" is just a more general way of saying "principles of mathematics." Math is a language, and logic is in large part a study of the languages we use to describe reality.

I guess I probably have three non-silly answers for that last question: no theorems? Hmm. Gregbard 06:16, 10 July 2007 (UTC)

You've got things backwards. Formal derivation was designed to model a pre-existing concept. Natural-language provability is prior to formal derivability, and the latter would be meaningless if not based on the former.

Perhaps you're arguing that formal derivations are "more general" in the sense that they don't have to be meaningful. Of course, neither do natural-language proofs, but I can see some sort of point in this argument -- there are formal derivations that can't easily be rendered in natural language because they have no ideas, just meaningless strings of characters. That's not the sort of "generality" that I would commend as a reason to emphasize them, though. That would be a little like saying that monkey scribblings are "more general" than literature.

Or perhaps you don't think any mathematics is meaningful; maybe you're a strict formalist. You're entitled to that viewpoint, but you can't expect others to base article organization on it. --Trovatore 06:30, 10 July 2007 (UTC)

No I really don't have things backwards. BTW, do you believe it was an intelligent designer that "designed" formal derivation or what?

Natural-language provability is NOT prior to formal derivability. They are both attempts to express true descriptions of reality consistent with reason. The whole point of a formal language is to be a model that represents the reality. The model is supposed to have all the important components of the reality being modeled. The model is not the same thing as the reality.

When you are saying that you have "proved" a theorem it is presumptuous. All you are really saying is that you have derived this expression using the rules. The rules allow you to write the new expression on a new line. There is no more than that going on with "derive." Saying you have "proved" a theorem is just an interpretation of "provided a derivation." It really is saying something additional than what is really there.

Formal derivability is not based on Natural-language provability. The derivation follows the principles that exist in our logical environment (the environment in which it is true that ~(p&~p), and modus ponens and all the rest --you know: reality). Our natural language also attempts to follow those principles. I don't really think of symbolic logic as "an interpretation" of my natural language. Since the formal language attempts to leave behind what is not important to preserve logical principles, and preserve what is important to preserving logical principles that makes the formal language closer to representing the actual principle.

When you say:

"Perhaps you're arguing that formal derivations are "more general" in the sense that they don't have to be meaningful."

...you have shown that you've missed the point. It's not that they don't have to be meaningful... syntax is the movement of the language WITHOUT REGARD for the meanings. So your jump to monkey scribbles may serve a rhetorical satisfaction, but it is exactly this more general sense that we are looking for in logic.

More general definitions are more useful in understanding. It's not that I think that mathematics is not meaningful, I think it is 'meaning laden' as compared to logic. My efforts on this subject are to get some acknowledgement of that fact in wikipedia articles. Gregbard 07:29, 10 July 2007 (UTC)

Greg, you have a strawman conception of mathematical proof that is simply not accurate. A natural-language proof does not correspond directly to a formal derivation; it requires a human intelligence as intermediary, one that can understand the meaning of the proof. In principle, yes, you could write out formal derivations in natural language, and call that a natural-language proof. But no one ever does, and almost no one ever writes out a formal derivation of a nontrivial result in any language, natural or formal.

Mathematical proof is not a well-defined formal construct of any sort. Rather, it is a way of communicating with mathematicians, and convincing them of propositions. Formal derivation is a different thing -- its purpose is not to convince anyone of anything, but rather to serve as an object of study in itself, a simplified and better-behaved emulation of proof, that we believe -- to some extent on faith -- to be capable of emulating any real (and correct) proof. Without such an emulation, we would have no hope of demonstrating independence results -- it's hard to say what can't be proved if, as in the natural-language case, there's no exact definition of "proved"; this is perhaps the most important reason to study formal derivations at all. --Trovatore 08:14, 10 July 2007 (UTC)

No, it really is not a strawman at all. Yes, natural language proofs must correspond directly to at least one formal derivation OR THEY ARE NOT PROOFS. My goodness! All the languages are equal, formal, natural or whatever. They are languages and so is math. Gregbard 09:53, 10 July 2007 (UTC)

[edit] Proof and derivation

G: I don't see any reference to truth in the first paragraph of theorem. Are you saying that the words "proof" and "derivation" don't mean the same thing to you? In mathematics and mathematical logic, except in the specialized jargon of proof theory, they are essentially synonymous. In the jargon of proof theory still neither one refers directly to "truth".

No. My whole point is that they are not the same. A derivation is just the thing that the rules allow you to write on the next line. That's it. The whole perspective is to see these things as things that move according to rules. Like boxes. A derivation is derived from moving around in the logical environment. Where you are at is a theorem, places you can't go on the map are not theorems. The idea that something is "proved" by this is an interpretation created by humans.

What is a proof if not the "proof" that you have "proved" of the truth of something? Saying a thing is a proof says something that we haven't been given to say. Don't you think readers would benefit to get this more open understanding, rather than prejudice them right away? -GB

The question of "what is a mathematical proof" is a question for philosophers (or sociologists), not mathematical logicians, but the answer is not the same as a formal derivation. The question of when a natural language argument is considered a "proof" by mathematicians involves a great deal of personal discretion and social convention, unlike the question of when a formal derivation is a "proof" in the sense of formal logic.

Sociologists? No, that sounds ridiculous to me. I'm wondering now if anyone knows what it is that philosophers do! The task of philosophers these days is clarification. Especially analytic philosophers, logicians among them. -GB

An even more significant problem for the opinion that all of mathematics can be reduced to logic is the question of whether ordinary mathematics is better represented by first-order logic or by higher-order logic, the latter being criticized as a disguised form of set theory.

It doesn't really matter. I think you guys are hung up on the fact that these logicist projects like PM have had successful philosophical challengers. This type of thing is quite normal in philosophy. Each philosopher tearing down some other philosopher's main point. But don't get bowled over by this! That's the simple analysis. People are sure still using arithmetic aren't they? No one is out there with the opposite project of "mathism." So in general it is just more reasonable to believe that the convienient interpretation of this world is that the math is going to be organized under the logic. And yes, my point about the theorem article should be an excellent one. You guys just can't bring yourselves to give in on it because it will carry over to discussions about all the other mathematical logic articles with this heavy lean to math and short shrift to the more general logical aspect. An encyclopedia article should be organized from general to specific, so all that math gets bumped over for some pointy headed esoteric blah blah blah. Well it shouldn't be so esoteric now should it? -GB

My viewpoint about WP is that our goal is to describe the viewpoints that exist, giving them due weight. The opinion that all of mathematics is just formal logic rephrased into natural language is certainly common, but it isn't the only significant viewpoint. The viewpoint that all of mathematics is based on Principia Mathematica is a very insignificant viewpoint among mathematical logicians today. — Carl (CBM · talk) 14:12, 10 July 2007 (UTC)

There are three sentences in the first paragraph of theorem. The first one is the one mathematicians use. The second one identifies the one that logicians, and philosophers use. The third says that the second one is important in a particular field of mathematical logic. -GB

The problem I have is that it seems to portray that the math that's not mathematical logic somehow gets a pass, and it doesn't. All those theorems are derivations too. Gregbard 21:57, 10 July 2007 (UTC)

I tell you what! The mathematicians should defer to the philosophers on the first paragraph of a lot of these articles. But those other ones... third, fourth, etc. The mathematicians are the unmitigated experts on most of those hands down. You guys should just hand certain parts over I would think. The resistence looks cultural to me. Silly silly. Gregbard 22:20, 10 July 2007 (UTC)

I have no doubt that there are cultural differences, and that each of us will be a strong defender of our own culture. To keep NPOV we can't choose either culture over the other: neutral articles will discuss both usages and, where there are reliable sources to draw from, compare and contrast them. My goal in the first para of Theorem was to describe both usages.

Here is an example of a common viewpoint in mathematical logic so you can see where Trovatore and I are coming from. The following is from Samual Buss, "Introduction to Proof Theory", which is available online [6]. Any spelling mistakes are mine, of course.

There are two distinct viewpoints about what a mathematical proof is. The first view is that proofs are social constructions by which mathematicians convince one another of the truth of theorems. ... Of course, it is impossible to precisely define what constitutes a valid proof in this social sense, and the standards for valid proofs may vary with the audience and over time. ... The second view of proofs is more narrow in scope: in this view, a proof consists of a string of symbols which satisfy some precisely stated set of rules and which prove a theorem, which must also be expressed as a string of symbols. ... Proofs of the latter kind are called "formal" proofs to distinguish them from "social" proofs. (p. 2)

Buss goes on to say that "a formal proof can serve as a social proof" and "in order for a proof to be socially accepted, it should be possible (in principle!) to generate a formal proof corresponding to the social proof". Note that, contrary to the viewpoint you have proposed, Buss claims that formal proofs (derivations, in other words) are "more narrow in scope" than social proofs, not more general.

Although I didn't look this up before editing theorem, there is an echo of Buss in the structure of the first paragraph. — Carl (CBM · talk) 23:25, 10 July 2007 (UTC)

I'm really astonished by this exchange here. I am reading the same paragraph that you are reading. It is very clear to me that brother Buss is incorrect. Furthermore any person reading this paragraph should be able to see that it is incorrect just by the meanings of the words used.

The notion of theorem as derivation is NOT more narrow. It is more broad. Just like when I say "over there in that room" instead of "make a right to the den." Where O where O where is the confusion?!?!?!? The one phrase ALWAYS includes the other. While they mean the same thing ONLY when there actually is a den on the right. That's what more general means. I am really having this discussion?

Theorems are ALWAYS derivations, they are only proofs when they are used to prove something. My example in formal language FS would really seem to end that debate. So are you standing by this Buss statement despite what has been shown and can be seen by all?

Buss says: a theorem is "... a proof consists of a string of symbols which satisfy some precisely stated set of rules." Does anyone deny that this includes math AND IS THEREFORE GENERALLY TRUE OF MATH? Haven't I demonstrated using FS that the idea of "proof" IS NOT GENERALLY TRUE IN LOGIC.

Aye yae aye. I am living in the twilight zone here folks. Somewhere in the literature there is a correction of professor Buss (or perhaps reverend Buss?). Gregbard 02:22, 11 July 2007 (UTC)

Greg, quite frankly there are three possibilities: (1) you haven't done many mathematical proofs and don't know what you're talking about, (2) you've done a few but haven't very closely examined what you're doing, or (3) you're blinded by ideology. What you seem to be claiming is completely unsustainable; it doesn't withstand even a cursory examination.

Formal, syntactic, derivation has existed (for human beings) since, I think, Hilbert. For all I know someone else might have had a similar idea earlier (certainly Leibniz's calculemus is a similar notion, but as far as I know he never put it into practice). So do you think there weren't any proofs before Hilbert? Or do you think, as you seem to be claiming, that the natural-language proofs before Hilbert were simply uninterpreted strings manipulated according to syntactic rules?

Because the latter possibility is just utterly false; if you think that, you're just plain wrong. There is no feasible set of syntactic rules that can determine whether a natural-language argument is a valid proof.

I think you should dial back your dogmatism a little bit. At your age, straight out of a bachelor's degree, some of your statements would have sounded reasonable to me, before a deeper examination changed my mind. You might also take into account the possibility that the two math-logic PhD's that you're arguing with, might have encountered these issues before now. Doesn't mean we're right; does mean you should think a little before flying off the handle. --Trovatore 06:16, 11 July 2007 (UTC)

I have no doubt that I am dealing with some very knowledgeable and intelligent people here. However, your three possibilties are still what is known in logic as 'unsupported claims.' Whereas, I have taken pains to demonstrate my claims.

The history is irrelevant to the understanding of the concept. It does not matter who discovered the idea logician or mathematician. It does not matter who uses the concept mathematician or logician.

Part of theoremhood is that what is derived now always was, and always will be derivable the same way. All natural language proofs are themselves interpretations of derivations.

Certainly there were plenty of proofs before Hilbert. Any case to be made as an convincing argument was an attempt to mirror the rules of language as revealed to us by reason. They were all also derivations. However, we only started thinking of them as derivations rather than as "proof" more recently. (Because we are getting smarter and making clearer distinctions).

You ask if I think ... "that the natural-language proofs before Hilbert were simply uninterpreted strings manipulated according to syntactic rules." No. They are all interpretations that each correspond to some derivation (gotten from manipulating strings according to syntactic rules.)

You also state that ..."there is no feasible set of syntactic rules that can determine whether a natural-language argument is a valid proof." The goal here isn't to get some algorithm to determine whether a natural language statement is a theorem (or valid, true, etc). The syntactic rules do determine how the valid form of an argument takes shape, the natural language attempts to mirror that shape.

I'm no PhD, however I think I know an "appeal to authority" in that last paragraph when I see one. No handle flying here. You have failed to convince me of your view which is supposedly the prevailing view. I'm open minded, but not only have I not seen it, but I have made great efforts to demonstrate my view which quite frankly have all the appearance of success in their goal. So I don't know what to tell you. It would be nice to have a bunch of philosophy PhDs to help me here so I don't get so beat up. (okay in my mind that last sentence was sarcasm -sorry)

Sincerely, I may be wrong, but I still have all the reasons I gave to believe that I am crystal clear on this issue, and I have seen no reasons that are left unanalyzed by myself from you guys. Lots of good questions though! Thanks for that. I'm still listening. I admit you almost had me with a moment of doubt in myself, but sorry. That passed, and I chalked it up to a propaganda effect of constantly being questioned. (Bravo btw.)

I suspect that philosophy people have these issues all the time. I don't know why the conflict though. Philosophy is the field that seeks to identify the foundations of things. Why is it such an affront to say that logic is more general than math? Philosophers set out to make sure that it is.

I have demonstrated my point repeatedly. So I don't know what's next. Be well, Gregbard 07:40, 11 July 2007 (UTC)

"Appeal to authority" -- not exactly. I allowed myself to get irritated by your remarks about your comments having "gone over some heads". That was probably an error on my part; I need better self-control, but I hope you can see that the remarks were provocative.

As regards the derivations that supposedly "determine how the valid form of an argument takes shape" -- how did they do that, exactly, given that the rules had not yet been formulated? Let's say that it's true that every valid argument can be reduced to such a derivation, and that therefore these derivations existed Platonistically, as abstract objects, before Hilbert, in the same way that the play Hamlet existed before Shakespeare. How exactly did that aid pre-Hilbertian mathematicians in formulating their natural-language arguments, to "mirror", as you put it, derivations that they knew nothing about?

What you might conceivably claim is that there's a deeper Platonic "essence of proof", of which both natural-language proofs and syntactic derivations are an imperfect expression. I might even agree with that. But that doesn't imply that natural-language proofs are derivations, which you formerly claimed (although your latest missive seems to have backed off on that claim a bit).

On the organizational question -- yes, "general to specific" is a good rule of thumb, and yes, purely in the sense that it is not as linked to meaning, logic is more general than mathematics. But what you don't seem to be taking into account is that it's the meaning, and not the logic, that's the point of mathematics. Starting with the point is a higher priority than starting with the greatest generality.

It's absolutely not true that "logic is more general than math" if what you mean by that is "math is a branch of logic". Mathematics is the study of mathematical objects; logic is its principal, but not its only, tool in conducting that study. --Trovatore 08:11, 11 July 2007 (UTC)

G: you say "Philosophy is the field that seeks to identify the foundations of things." The most commonly accepted foundational system for mathematics is set theory, and set theory is not logic, it's mathematics. Other people don't accept set theory as a valid sole foundation for mathematics, or reject entirely the notion that there is or needs to be a foundation (e.g. the book Foundations without foundationalism by Shapiro).

I disagree that most natural language proofs are "derivations", but here's a more extreme example. The image at the top of Theorem is a proof of the Pythagorean theorem. In what way is this image a derivation?

When you say "The syntactic rules do determine how the valid form of an argument takes shape, the natural language attempts to mirror that shape." I think you are reversing the priority of logic and language. Logic was invented to model natural language reasoning, and as Trovatore points out natural language reasoning existed long before anyone knew about syntactic rules. We have no difficulty reading (translated) natural language arguments from thousands of years ago. — Carl (CBM · talk) 15:19, 11 July 2007 (UTC)

[edit] Categories

Gregbard added Category:Logic to a lot of articles, not just Boolean Algebra, but I didn't have the energy to clean out Category:Logic right away. You're right, of course, that the articles shouldn't be in that category while they're in its descendent categories. — Carl (CBM · talk) 00:48, 12 July 2007 (UTC)

[edit] Invite

I notice you signed up as an early proponent for the creation of WikiProject Logic. Since then, the proposed project has been moved to it's own space and you have been listed as a Charter Member. Thank you for paving the way for the creation of this project. This is an invitation for your continued and perhaps renewed participation and correspondence. Be well.

P.S. Greetings again. I intend to respond to the engaging discussion above soon. I just need to take a break and think about it for a while. Meanwhile, I am reaching out to all the supporters of the WikiProject proposal to see if we can breath new life into it. Be well, Gregbard 03:11, 14 July 2007 (UTC)

[edit] user page

There is a strange edit to your user page by an IP editor. I don't know whether you approve or not. — Carl (CBM · talk) 00:49, 17 July 2007 (UTC)

When in doubt, cut it out. :) I reverted the thing for now. The quote is funny enough though. Oleg Alexandrov (talk) 02:57, 17 July 2007 (UTC)

[edit] The game of Questions

"Foul! No synonyms! One....all." -- MarcoTolo 02:19, 19 July 2007 (UTC)

What synonym are you talking about? --Trovatore 02:24, 19 July 2007 (UTC)

[edit] Boolean

You know that they have a tool for what you are doing. WP:AWB It's probably too late now but I thought I would let you know. Cheers--Cronholm¹⁴⁴ 20:04, 20 July 2007 (UTC)

I moved the talk archive pages. Also spotted the "Hilarious" thread: nice one - ROFL material. Geometry guy 21:05, 20 July 2007 (UTC)

Thanks, both of you. I've never gotten around to installing AWB or learning how it works. I suppose now might be a good time.... --Trovatore 21:18, 20 July 2007 (UTC)

Then again, maybe not. It seems it's not available for Linux. --Trovatore 06:15, 21 July 2007 (UTC)

...bummer, I could always pirate you a cracked copy of vista...(this comment is a joke and is no way meant to reflect upon me or Microsoft). G-guy, don't you use Ubuntu with AWB?--Cronholm¹⁴⁴ 16:39, 21 July 2007 (UTC)

[edit] Cantor redux

Hi, please see Cantor/phil/relig. Thanks! Ling.Nut 01:53, 21 July 2007 (UTC)

[edit] MoS

Sorry, I don't know what's happening. I'm in the process of creating chronological archives and had several windows open. I must have hit save on the wrong window. But now I can't read the diff to see what you wrote (the diff is all messed up looking), so I tried to revert to before I made that edit, but it's not saving. Sorry for the mess. :-( SlimVirgin ^{(talk)(contribs)} 08:23, 27 July 2007 (UTC)

[edit] A little help here

Logic has its own space for requested articles. I have a link in the math section for it. The same is true on the philosophy side. We really have to learn to work together here. That means that the articles have to be at least equally accessible to the math and philosophy folk. I'm not trying to be an asshole here, but I must assert for the sake of the WikiProject that I will revert again if this is changed again. The math people have had it under their category for so long I"M SURE you believe its math and that's that. Get over it. Work together in a common space. Be thankful for it. Please don't fight with me over it. Gregbard 11:13, 27 July 2007 (UTC)

[edit] ...for you and geometry guy...

• link: User talk:Geometry guy#..for you and Trovatore.2C as per Willow. thanks Ling.Nut 11:46, 27 July 2007 (UTC)

[edit] one more time, with feeling

• I added a longish paragraph about the philosophy of mathematics to Georg Cantor, after which my head promptly exploded. You might wanna quietly check it for accuracy... see the paragraph starting "Debate among mathematicians grew out of..." ... oh I know the article on finitism says that Wittgenstein denied holding that philosophy, but the ref I cited explicitly called his attacks on Cantor finitist... Ling.Nut 00:05, 30 July 2007 (UTC)

• PS I wondered if throwing Kronecker's name in there was an anachronism given the manner in which I characterized the constructivists' concerns... Ling.Nut 00:14, 30 July 2007 (UTC)

• PPS it's supposed to be an extremely simplified overview; the real details should be given in the relevant articles, such as Philosophical objections to Cantor's theory Ling.Nut 00:16, 30 July 2007 (UTC)

Well, the particular distinction made between intuitionism and constructivism is new to me, but that doesn't mean it's wrong. I haven't read much of the work of the early figures in those schools; my knowledge of them is from a more contemporary standpoint. I think some consider Kronecker to be "finitist" rather than "constructivist". --Trovatore 16:34, 30 July 2007 (UTC)

Whew, I'm relieved... those don't sound like strong objections/corrections. :-) I believe I sourced everything.. so I'll just leave it the way it is for now.. but of course if you come to the conclusion that something definitely needs to be changed, please do so! Thank you for your time and trouble! (I know I can talk a lot). Ling.Nut 17:21, 30 July 2007 (UTC)

[edit] question

Can you tell me please in which university/ academic institute you are taking position now? if you dont want to answer that, it wouldnt be surprised or offended.thanks--Gilisa 10:00, 31 July 2007 (UTC)

I am not currently associated with a university. However I maintain a strong interest in academic pursuits and have a paper currently under submission. --Trovatore 18:27, 31 July 2007 (UTC)

I thought that you should be now at the Kurt Goedel Research Center of Vienna university- which have a very good scientific relations with some Israeli universities, but I couldn't find you there, that's why I asked you. paciencia, buena memoria, determinacion y justicia son las mejores armas --Gilisa 09:38, 2 August 2007 (UTC)

[edit] WP:LGC Tags

Thank you for chiming in on the project tag issue. Your position is the intellectual way to look at it. Since the bot owner has stopped until he feels comfortable he can move forward without becoming collateral damage, could you become the point person for that? We'll do it any way you want. We can put it up for discussion for another week (or 2, or a month or whatever) if you want. I'd prefer it if you just asked SatyrBot to continue though. I am intrigued about the bot that the 1.0 assessment team has. It identifies most linked pages within a project. I was thinking an over inclusive project (to start at least) would provide us with some interesting information.

I also notice that you are revising the Mathematical logic lead. I am sad to see that. I think it's a bad decision. it appears to be the diametrically opposed position from what it was in response to me. The whole thing will be obvious to everyone. That's too bad. Being reactionary is not necessary. The Wikipedia is going to be around for a long time. Be well, Gregbard 23:15, 31 July 2007 (UTC)

I have publicized this discussion on the noticeboard as:

• Discussion about the fact that the bot has stopped tagging articles due to complaints.

Gregbard 01:05, 1 August 2007 (UTC)

Excuse me, is this your idea of being the point person? Weak no? Exactly what is going to happen to move this forward? You are holding it up, so please answer. Gregbard 21:31, 1 August 2007 (UTC)

My first comment on the subject was based on an analysis from the outside, asking whether there was any real harm in the tags, and I didn't think there was. When directly polled, on the other hand, on the question of whether it was a good idea, from the point of view of a participant in the project, to tag so many articles, I came to the conclusion that it probably was not, on the grounds that it was not very useful to the project and did impose some minor costs. --Trovatore 21:42, 1 August 2007 (UTC)

Tagging the articles is not useful to the project? That makes absolutely no sense. Gregbard 23:09, 1 August 2007 (UTC)

[edit] Georg Cantor FA

...looks like we made it! Ling.Nut 03:31, 1 August 2007 (UTC)

Congratulations, well done! --Trovatore 05:28, 1 August 2007 (UTC)

Ling is busy, so it is left to us to get Georg in the mainpage cue.—Cronholm¹⁴⁴ 15:21, 15 August 2007 (UTC)

OK, I added it. Hope you approve. For future reference, is there a template for this? I couldn't find one; I just did a bunch of cut-and-paste. --Trovatore 18:06, 15 August 2007 (UTC)

Looks good to me, I'm not sure about the template, but I will look for the next FA math article :) —Cronholm¹⁴⁴ 22:55, 17 August 2007 (UTC)

[edit] Ref desk question removal

You removed a ref desk question on the grounds of soapboxing [7]. Please discuss this at the discussion page of the Reference Desk where I have added an item about it . Often questions which have unsubstantiated or incorrect assumptions can be answered in a way which enlightens and is in accord with the purposes of Wikipedia, and this one sounded like it might be one such. The limits of battery capability vis a vis electric cars is an important issue. Thanks. Edison 02:31, 2 August 2007 (UTC)

[edit] small set

Look, if you want to continue to be fussy about this, fine. I'm not going to discuss this with you any more, because you'd rather have a bad version than anything that even resembles a version I've advocated. Not everyone on Wikipedia is going to let you make sweeping unilateral changes whenever you feel like it based on irrelevant personal preferences. I'm sorry if that offends you or makes you stoop to being uncompromising and uncivil. At this point, I am no longer willing to discuss this matter with you, due to your willfully editing in a fashion that negatively impacts the article (or disambiguation, as it were). You would rather see a bad version up than any good version that isn't yours? Is this how you normally do things, or are you making an exception because you don't like me? This has gone on long enough, and you are way out of line. Feel free to willfully degrade the quality of Wikipedia, in some sort of adult-version of a temper tantrum, but don't expect to solicit any more responses out of me. --Cheeser1 05:44, 29 August 2007 (UTC)

[edit] Colloquiality

Pls see my (too lengthy) comments on talk:set. I'd appreciate learning your view.—PaulTanenbaum 18:31, 1 September 2007 (UTC)

[edit] Asprin and salicylic acid

Hi I have posted an answer to your very old question on the Talk: Aspirin Page about the mechanisms of aspirin and salicylic acid. Better late than never. Dirac66 23:49, 20 September 2007 (UTC)

[edit] Continuum Hypothesis

Hello, I am the fellow who did the edits on continuum hypothesis which you reverted. I was wondering if the reason is because you were unhappy with the content, or because you think I was confused on the issues you brought up on the talk page. I believe I was fair to Cohen's points of view, although perhaps not as fair to more platonist views, but obviously you think I wasn't. Perhaps if you could tell me where you think that I went wrong, I could explain.

I am sure you misunderstood the correct statement that the continuum has different position in the heirarchy depending on the continuum hypothesis with the incorrect statement that the continuum hypothesis allows you to prove new theorems. The power of the continuum to prove new theorems is clearly an important issue for cantor and Hilbert, as I read it, although others might disagree, I don't know. Likebox 01:37, 28 September 2007 (UTC)

Well, I have both content and style objections. On the style issue, as I said in my edit summary, I think much of your added content is too chatty in tone, appropriate perhaps for the classroom but not for an encyclopedia. That also made the additions much too long.

On content, I believe that you are conflating models and the objects in them, with theories and propositions. The reason I haven't responded yet on the talk page there is that first I'd like to figure out if your claimed theorem, that ZFC-P+"the powerset of the naturals exists" proves the consistency of ZFC-P+"aleph_1 exists", is correct. Somewhat embarrassingly I haven't worked that out yet; it's an interesting question and there might just possibly be a way to interpret your argument so that it's relevant. As written, however, your argument just makes flat out category errors, as for example when you say

Now let me add countable powerset CP, "There exists a powerset for every countable set". I can ask the question, what can I prove in this system? I can prove "Consis ZFC-P", but can I prove consis "ZFC-P+A1"? This exactly depends on whether the continuum hypothesis is true or not. If the continuum hypothesis is true in this universe, then certainly not.

This phrasing confuses ZFC-P, which is a theory, with a model (or "universe" as you put it). Whether "ZFC-P + P(omega) exists" proves the consistency of "ZFC-P + aleph_1 exists" does not depend in any way on the model (models obtained by forcing over a wellfounded ground model are themselves wellfounded, and therefore are correct about all arithmetic statements, and therefore cannot differ on the truth value of arithmetic statements).

If your claimed theorems are correct, while they would be interesting, I still see no real connection with CH. They seem more like objections to the powerset axiom (or perhaps to the axiom of choice).

In any case, though, while this discussion is very interesting, your exposition goes far past the quote of Cohen's that I have seen on this point. Remember that we cannot include original research. If you could point me to actual published remarks of Cohen on this point, we could discuss how much of it is appropriate for inclusion, and with how much exposition. --Trovatore 05:47, 28 September 2007 (UTC)

Finally, I understand your points. Thank you for patiently explaining. As far as confusing models and axiom systems, I was being sloppy because the completeness theorem allows you to construct a countable model in a unique way. It's the model with the fewest symbols, and that's the model that I have in my head when I am thinking about axiom systems. So when I said "if the continuum hypothesis is true in this universe" I meant "If the continuum hypothesis is true in the model constructed by the completness theorem applied to these axioms". That model is the minimal model, and in set theory it obeys V=L. Its described in Cohen's book "Set theory and the continuum hypothesis" right before forcing. I can see that now that this can lead to misunderstanding, I'm sorry, I will try to use more precise language in the future.

I also now understand the source of the confusion about the CH stuff. Whenever I was asking "is CH true?" I am asking "Is CH true in the model constructed by the completeness theorem?". That question is not undecidable, since the minimal model has extra properties derived from V=L. When you add forcing axioms the way cohen does it, you can add maps that force that CH is not true. This is not quite the completeness theorem, because it also includes all the statements forced true. So now it is good to stick with the language of models.

I am not doing original research here. Everything that said comes from the only source which I have read and understood, which is "set theory and the continuum hypothesis". I have not read anything else about the subject, except the first chapter of "proper and improper forcing" by Shelah. I don't like any exposition other than Cohen's, because other expositions are not as radically finitist (although perhaps "discretist" is a better term).

Cohen's quote is from the last pages of "set theory and CH". This is it, as best as I remember:

"It is my belief that future generations will come to see the continuum hypothesis as obviously false... The continuum will be seen to be infinitely richer than any of the sets constructed by processes of successive collection and replacement starting from the empty set ... The continuum is given to us at one fell stroke by a bold new axiom, the powerset axiom ... so that the continuum should be thought of as bigger than or ... Perhaps future generations will see things more clearly and explain themselves more eloquently." (don't shoot me if its slightly wrong, I got the intent right. I think the italics are in the original)

(I found the exact quote) "A point of view which the author feels may eventually come to be accepted is that CH is obviously false. The main reason one accepts the Axiom of Infinity is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now is the set of countable ordinals and this is merely a special and the simplest way of generating a higher cardinal. The set C is, in contrast, generated by a totally new and more powerful principle, namely the Power Set Axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the Replacement Axiom can ever reach C. Thus C is greater than where , etc. This point of view regards C as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. Perhaps later generations will see the problem more clearly and express themselves more eloquently." (Set theory and the Continuum Hypothesis p. 151.)Likebox 18:37, 28 September 2007 (UTC)

This is a very radical point of view in the current climate, but in my opinion a sensible one. I think it needs representation. Since I have the "advantage" that Cohen is all I know, I am very close to his perspective, and I felt I could explain things his way.

Since this point of view is radical, some statements which are phrased by a person who thinks this way sound like nonsense at first. Forcing, for example. The point is that Cohen is saying that the powerset axiom pushes you infinitely high, higher than anything you can construct by ordinal processes.

Cohen also constructed models where the axiom of choice failed. This is just as easy as making CH fail, because it only requires "indiscernables" which you can't choose between. Cohen adds all these forced reals which could all be indiscernables. In this formulation, the axiom of choice (for subset of the continuum) is also obviously false. This is also in his book.

If you want my personal opinion, I personally don't like powerset. I like the idea that the real numbers are a proper class, bigger than all sets. Then you can take set theory to have V=L, and all the nice theorems, but the "true" powerset would include all of set theory and obey the axiom, "every subclass of the reals is measurable". But I would never write that on wikipedia, because there is no source.

Solovay did construct a model in which the real numbers are measurable.

You are completely right about the reference/exposition dichotomy. What do you think of starting pages which are all exposition? Like "exposition of continuum hypothesis" or "exposition of calculus" which would be more like wikipedia articles in other subject, with history and different points of view, not just reference material. I was thinking of taking all the new stuff into another page like that. —Preceding unsigned comment added by Likebox (talk • contribs) 16:32, 28 September 2007 (UTC)

Hello again, I hope I am not cluttering up your talk page too much. If you delete stuff later, I figure I did no harm. But to see that ZFC-P+countable powerset proves consis ZFC-P is easy if you accept that ZFC-P is the theory of all countable sets. Using countable powerset, you construct the continuum, then use the diagonal argument to show that the continuum is uncountable. Then use choice to well order the continuum, and this proves that there is an uncountable ordinal. The smallest uncountable ordinal is . The set of all sets of order less than which cannot be bijected with is a model for ZFC-P.

Similarly, ZFC-P+countable powerset+continuum powerset proves consis ZFC-P+A1, and so on up the hierarchy. It is easy to see why if you add CH (which is still equiconsistent since these are just truncations of ZFC), because then each step of adding powersets is the same as adding a new larger cardinal, and the set of all sets smaller than a given cardinal is a model for the previous theory. The question of where the continuum lies then is important, because it reveals how far up the cardinal heirarchy you go before you reach the continuum.Likebox 19:11, 28 September 2007 (UTC)

Wow, this discussion could go on a long time; there's a lot to talk about here. Let me touch briefly on a couple of points before getting into the technical discussion:

1. Yeah, I thought that's the Cohen quote you were probably thinking of. It's a point of view worth mentioning, but I think we shouldn't go beyond what Cohen actually said (and indeed, should condense it quite a bit). Right now the article mentions that Cohen "also tended towards rejecting CH"; I think we could reasonably add that he thought the continuum was "very large" as measured by alephs, and say that this view was motivated by the "boldness" of the powerset axiom.
2. The idea that P(omega) is a "proper class" is a recognized foundational position; various people (Scott, Steel) have mentioned it and talked around the edges of it, though I don't know anyone who directly advocates it. That could be discussed, if you can find cites, at some article on foundational philosophy of math, but it really doesn't have much direct bearing on CH (except, I suppose, that it would make CH meaningless, insofar as it contains terms that fail to denote).
3. You seem to have learned what is now called "ramified forcing", Cohen's original version. For the most part ramified forcing is no longer used; unramified forcing can, as far as I know, do everything ramified forcing can, and is generally considered simpler and cleaner. I don't think there is any philosophical distinction here; unramified forcing is just more streamlined. But I'm a bit vague on this because I never learned ramified forcing.

Now to the interesting technical part. I think I've got this worked out now. Your proof that ZFC-P+"P(omega) exists" proves the consistency of ZFC-P+"aleph_1 exists" looks good. But you appear to be claiming that, using forcing, the same idea can be extended to show that ZFC-P+"P(omega) exists" proves the consistency of "ZFC-P+aleph_2 exists". That's wrong.

(Note by the way that I've taken the liberty of replacing your "every countable set has a powerset" with the simpler statement "P(omega) exists"; they're equivalent in the presence of the axiom of replacement.)

Here's the refutation: Working in ZFC-P, if we assume aleph_2 exists, we can show that exists,where by I mean first compute in the sense of L (it could be smaller than the real ), and then compute the collection of all sets hereditarily of cardinality less than that, in the sense of L. But is a model of ZFC-P+"P(omega) exists". Therefore ZFC-P+"aleph_2 exists" proves the consistency of ZFC-P+"P(omega) exists", so by second incompletness, ZFC-P+"P(omega) exists" cannot prove the consistency of ZFC-P+"aleph_2 exists".

Here's what's wrong with the argument I think you were trying to make (though it's certainly possible that I've misunderstood your argument). What I think you wanted to do is something like this: Suppose we have a model M for ZFC-P+"P(omega) exists". Then force to make its P(omega) have size at least , and recover from P(omega).

But you can't do that, because M may not have an object it thinks is in the first place (that's what you're trying to prove, after all!) and if it doesn't, then you can't form the partial order inside M to add Cohen reals to the P(omega) of M. --Trovatore 05:54, 29 September 2007 (UTC)

Yes, you are absolutely right. That is a wrong arguement. That's why I had to go to all those ridiculous lengths on the CH talk page to use two iterations of powerset:

start with ZFC - P + P(omega)+P(P(omega)) This proves aleph2 exists, but not that aleph3 exists, But it now has powerset of powerset of the integers.

Now force aleph2 into P(omega). You already have aleph2 from P(P(omega)) independent of any new considerations.

The forcing model obeys the additional non-CH axiom aleph2<=P(omega), so taking this as an addition axiom, P(P(omega)) will now give you aleph3 and prove the new theorem Consis ZFC-P+aleph1+aleph2.

So with a forcing-consistent axiom, you can get aleph3 with only two iterations of powerset, where before you could only get aleph2. So the forcing allows fewer powerset iterations to prove the same theorems. I think this was intuitively understood all the way back to Cantor, but I am not a mindreader.

To be honest, my first thought was to do it with only one powerset, but then it is impossible because, like you said, you can't get to aleph2 even. You can only get around that by adding axioms which construct the alephs by hand. I thought the point was better clarified by the argument with P(P(omega)).

On a completely different note-- I do remember you from sci.math and such. I'm glad the discussions are still going on here.Likebox 20:36, 29 September 2007 (UTC)

Ok-- now I'm confused about this too, because I could go to aleph5 and then I get something

impossible--- take P(P(P(P(P(omega))))) 5 times powerset. Assume consis of this and prove consis

aleph5<P(omega). Now you should be getting aleph9 or something, and that's again impossible for

the same reason you brought up. In the P(P(omega)) case I thought it was ok because to get to

aleph3 I only need consis aleph2, and I have that anyway from the countable model that gives

the forcing map. But there should be no way to get to aleph9 from aleph5 without adding four

more Godel steps. Thank you for bringing this up, I guess there's a subtlety. I am totally

confused now, does this mean you can't force until you have all the cardinals? I will think

about it. I won't do any edits until I sort it out. Likebox 23:05, 30 September 2007 (UTC)

So I've been thinking about this most of the weekend, and I think I've got it worked out. I think your argument is wrong but it wasn't trivial to figure out where.

The first thing that should make us suspicious is that my earlier refutation still appears to go through. Working in ZFC-P, if exists, then exists (not completely trivial; uses that GCH holds in L), and is a model for ZFC-P+"P(P(omega)) exists". So it can't be the case that ZFC-P+"P(P(omega)) exists" proves the consistency of ZFC-P+" exists".

Yes, you are right, my bad. You can't prove consis ZFC-P+aleph1+aleph2 in a forcing extension. It should have been clear right from the start from incompleteness, as you said. Thank you for straightening me out on that. I could tell that I was confused because I couldn't figure out what was going wrong until you told me. As you explain--- the powerset doesn't exist in the extended model for sets of size aleph2, even though the old axioms lead you to expect that it would.

So I now think the right way to argue should be this: if you have P(omega) and P(P(omega)), you can make a forcing extension which makes P(omega)=aleph2, but which is no longer a model for the axiom system you started with. If you make an axiom system for which this model is a minimal model, and which also has a (now more powerful) axiom P(P(omega)), the system now gives you aleph3. Then you can repeat the process and get aleph4 and aleph5, but at each step, the same axiom P(P(omega)) is not provably equiconsistent using the old axioms.Likebox 18:25, 1 October 2007 (UTC)

Or look at it another way: Start with , and add as a singleton. Now start closing under Skolem functions to make ZFC-P hold, and additionally to witness, for every ordinal β, that there's an injection from β into , and round off to the full level of L every time you close under a Skolem function. This process will close off at some ordinal α below , so that L_α satisfies ZFC-P+" exists"+" does not exist". The last clause is because we made sure to witness that every ordinal had cardinality at most in the sense of the model.

Note that these witnesses are preserved in every forcing extension of this L_α, and also that forcing does not add any new ordinals (the height of the model remains the same), so every forcing extension of L_α also satisfies " does not exist". Also note that L_α satisfies "P(P(omega)) exists".

I totally agree that no forcing extension will ever produce new cardinals. As far as all this skolem function business is concerned, I don't know what a Skolem function is. I think you are just producing all sets in L with a birthday earlier than aleph2 steps of predicative definition. I don't know what a witness is either.Likebox 18:25, 1 October 2007 (UTC)

So when we run your forcing argument on this model, we've got a problem: The ground model satisfies "P(P(omega)) exists", and we add Cohen reals to the P(omega) of the model (which you're right, we can do), but the forcing extension does not satisfy " exists".

This I understand and completely agree with.Likebox 18:25, 1 October 2007 (UTC)

Thus it must be that the forcing extension does not satisfy "P(P(omega)) exists", even though the ground model does satisfy it. That's a bit surprising; I wouldn't have thought that likely. But if you try to figure out a name in the ground model that names P(P(omega)) in the extension, you'll see the problem. You need something roughly like the powerset of a name for P(omega). But what's the cardinality of a name for P(omega)? It has to take into account all the elements of the forcing poset, and that poset has cardinality (as calculated in L). So a name for P(P(omega)) should have cardinality at least , as calculated in L, and there just is no such name in L_α. I think. I haven't worked this out in fine detail.

I'm a little rusty on this stuff, but it's fun! I know that forcing works over models of ZFC-P, and if the ground model satisfies powerset then so does the extension, but it appears that it doesn't quite go level-by-level. I wouldn't have guessed that if you hadn't made me think about this. --Trovatore 04:19, 1 October 2007 (UTC)

I think we agree completely, although I need to independently understand exactly what is happening in the models, for my own edification. I have to clarify all sorts of stupid things for myself.

Just to take a stab at summarizing:

I think we both agree with the whole world that forcing does not produce new theorems, that powersets produce new theorems, and adding alephs produces new theorems.

I think we also agree that since you don't prove aleph-k exists in ZFC without k powersets, you will never be able to reduce the number of powerset iterations used in the ZFC proof of a theorem (I think this is your main point).

But I think we also agree that if we produce the aleph sequence by a different type of axiom other than powerset, like large cardinals for ZFC-P, then the continuum can be anywhere on the aleph chain. Then you need fewer iterations of powerset to prove the same theorems. In particular, you could put the continuum above any cardinal constructed by a piecemeal approach (This is how I interpret Cohen's quote).

Does this sound fair to you?Likebox 18:25, 1 October 2007 (UTC)

Um, to be honest, I'm not quite following what you mean by the above. A proposition's consistency strength is what it is; you can't magically increase it by forcing. On the other hand forcing may certainly be used in proofs that one proposition is consistency-wise stronger than another. However none of the proofs along the lines you've thus far suggested, really seem to work, as far as I can tell. It seems to me that you are still playing rather loose with the distinction between axioms or propositions, and their objects of discourse.

I really am trying to be precise, I hope that I am not playing fast and loose with anything. As I said, I agree that forcing does not increase the strength of axiom systems. I couldn't figure out why that wasn't happening in the example, but you cleared that up, although I still have to think about it more.

But assuming that you are right, and I think you are, what's wrong with the P(P(omega)) example fixed up?

Same initial axiom system: System 1: ZFC-P+P(omega)+P(P(omega))

This axiom system allows you to build by forcing a model with the properties:

1. ZFC-P holds. 2. P(omega) is in the model. 3. aleph2 is in the model and is less than or equal to P(omega)

taking these properties of the model as new axioms gives an axiom system:

system 2: ZFC-P + P(omega) + aleph2 exists + aleph2<=P(omega).

Since system 1 (plus consis system 1) built a model for system 2, system 2 is at best equiconsistent with system 1. It's clear it's exactly equiconsistent from L, but who cares. Now add the axiom P(P(omega)) to system 2, and get system 3

system 3: ZFC-P + P(omega) + aleph2 exists + aleph2<P(omega) + P(P(omega))

Now system 3 is stronger than system 1 and system 2, but still only has two iterations of powerset.

You can't do magic. You can't get aleph3 without a stronger axiom. But it seems to me that this stronger axiom never needs to be stronger than P(P(omega)). In terms of alephs, it is stronger. In terms of iterations of powerset, it is not. If you can get aleph3 without ever talking about P(P(P(omega))), it is clear that forcing does buy you something.Likebox 02:39, 2 October 2007 (UTC)

I agree that the forcing extension moves you to a model where the P(P(omega)) is no longer true, as you said. But if you then make it true, you are making the system stronger. It's not the forcing extension that did the magic, it's the powerset axiom.

I think that if you don't know what Skolem functions are, it would be to your advantage to go find out. I'd recommend learning just a little basic model theory and theory of L -- say, Skolem hulls, the Mostowski collapsing lemma, the condensation lemma. Take a look in Chang and Keisler, Model Theory, and -- well, I'd say Kunen, Set Theory, but I don't really like his treatment of L. But then again you probably already know the part of it where I don't like his exposition, so maybe it's OK after all for the Mostowski collapse and the condensation lemma (you should check your understanding of those by trying to follow the proof that GCH holds in L). --Trovatore 22:41, 1 October 2007 (UTC)

I appreciate that I have a lot to learn. However, life is short and if I wait until I know everything before I say anything than I end up never saying anything. I appreciate your background and expertise, and I am glad you know all that stuff. One day I will learn it. But as for GCH holds in L, Cohen's book has a self contained proof which is short.Likebox 02:39, 2 October 2007 (UTC)

The suggestion about GCH was not for its own sake, but rather for the technique. How do you know, for example, that does not satisfy " exists"? On the face of it it would seem that there might be an ordinal less than that believes to be . In fact, that can't happen, but to see why, you need the techniques I've been talking about. Or at least I don't know another way. If you knew these techniques you could more easily check some aspects of these arguments.

I'm not suggesting that you should shut up until you know everything. But I am saying I don't really follow what you're getting at a lot of the time, and I think it's partly because you're using nonstandard language to describe them, and partly because you're mixing together objects, on the one hand, with their properties or definitions on the other. I think learning just a little bit about these basic techniques would help with both issues. --Trovatore 02:47, 2 October 2007 (UTC)

Oh, I just saw that you added a new argument. I'll take a look at it. --Trovatore 02:47, 2 October 2007 (UTC)

OK, I'll try to sort through your latest post here. Let me know if I misinterpret something.

• First, you specify a theory (axiomatic system), namely ZFC-P+"P(P(omega)) exists". Let's call that T0 for easy reference.
• Now you argue that T0 proves the existence of a model (let's call it M1 -- the 1 because it's not a model of T0, but of a new theory), such that:
  • M1 satisfies ZFC-P
  • P(omega) is in M1. Now, do you mean the real P(omega), or do you mean that M1 satisfies "P(omega) exists" -- that is, that there's an element of M1 that M1 thinks is P(omega)? I'm guessing the latter -- that M1 satisfies "P(omega) exists".
  • You say "aleph2 is in the model and is less than or equal to [the cardinality of] P(omega)". Again I'm guessing you mean that this is in the model's opinion; that is, M1 satisfies the statement " exists and ".
• OK, so going with my guesses thus far, I agree that the consistency of T0 implies the existence of such an M1. M1 might be, for example, M₀[G], where M₀ equals (which is a model for T0), and G is M₀-generic for the partial order to add Cohen reals.
• Now, what's "system 2"? I think it means the properties you've specified for M1; that is, ZFC-P+"P(omega) exists"+, right? Let's call that T1.
• So I agree Con(T0) implies Con(T1).
• Now you mention a "system 3" (let's call it T2), which is T1+"P(P(omega))" exists.
• Now you say that T2 is stronger than both T0 and T1. Sure, that's true. T2 proves that exists, and therefore exists, which is a model for T0.
• So following everything up to this point, we have that T2 proves Con(T0), and Con(T0) implies Con(T1), so certainly T2 proves Con(T1), and we have a nice linear chain of consistency strengths.
• But now, what's the punch line? That's what I don't really get. You say "if you can get without talking about P(P(P(omega))) then...forcing buys you something". But we didn't use forcing to see that T2 proves the existence of ; that was directly provable with pre-forcing techniques. The only place forcing comes in in the above argument is to put T1 at the bottom of the chain. --Trovatore 07:00, 2 October 2007 (UTC)

What's the punchline

The punchline is that T2 still only has two powerset iterations. You have aleph 3, but you didn't ever talk about P(P(P(omega))).

The set P(P(P(omega))) is so enormous, it is beyond anybody's mathematical intuition. It is important to establish that this type of construction is completely irrelevant for producing alephs. I understand that this is not very profound in terms of actual theorems. But it goes some way towards explaining philosophically why Cohen says that powerset operations are stronger than any aleph.

Models and Axioms--- What Belongs Where

The reason you say I keep "confusing" models and axiom systems is that these are two notions which are not all that distinct. Each axiom systems has (many) associated models, and conversely each model has (many) associated axiom systems (listing it's properties). Anything you can do with axiom systems, you can do with models and vice versa, but it requires shifting perspective, because the association is not one-to-one. I agree that once you formalize the discussion, it is very important to keep them separate. But you can formalize the discussion in many ways, in some case you put some ideas into the models and in other cases you put the same ideas into the axiom systems.

Modern mathemticians have ossified forcing so that it happens in models and not in axiom systems. But historically, Cohen initially formulated forcing in terms of axiom systems and not in terms of models. He added axioms for new real numbers, and the forcing notion was a list of new axioms, not a list of model properties. Some of that thinking survives in most modern expositions. Stronger and weaker positions in the forcing poset correspond to stronger and weaker axiom systems in Cohen's approach.

The reason everybody shifted to talking about models is because when you think of forcing in terms of axiom systems you have to use a different logic. Instead of talking about true and false in the usual way, you have to include statements which are forced true (by the axioms, remember he isn't doing models) as true deductions. Nowadays people think in terms of models, not in terms of proof theory. But I think an exposition of Cohen's original approach makes everything philosophically clear.

So I'll describe Cohen's original approach, which is important and instructive. You have an axiom system, say ZFC and a smallest inaccessible cardinal C. You know you have a countable model. Consider the following axiom schema:

Axiom schema: for each element of the inaccessible cardinal C, there is a real number r_p between 0 and 1.

This allows you to build up a model step by step by the completeness theorem, associating a new symbol r_p with every unique p that is proved to be an element of C. The reason I say it's a schema and not an axiom, is because the r needs to have the name of the element attached to it, so that it keeps track of which element it is associated with. Now you want to show that assuming the r_p's are different does not lead to contradiction. To do this, the axioms needs to be specified more precisely, to define more properties of the r_p. So you add more axioms:

new axioms: the first digits of r_p are blah blah blah.

where blah blah blah is some list of digits. Now you end up proving some things are true about r_p, like r_p is between .921 and .922, but you never end up proving anything meaningful like r_p = r_q or r_p <> r_q. So Cohen invented a new notion of true and false.

A statement is forced true when it is a consequence of the axioms. A statement is forced false when no extension ever proves it true. He emphasizes this second idea very strongly, because it is not a construction you can decide in a program whether no extension will prove a statement true. But in certain limited cases you can prove it. This notion of forced false produces all statement which are false for randomly chosen numbers in probabilistic theories.

Now note that it is forcing true that all the r_p's are different. And now you have a map, by construction, in the model of the axiom system (ZFC+large cardinal + forcing axioms) which has deductions that come from both ordinary logic and from forcing logic.

The point is, since there are only countably many elements of C in any model, how could it possibly lead to contradiction to assume they all have a corresponding randomly chosen element of [0,1]? It obviously can't, since any map of a model of C into R is always measure zero.

I think now I get exactly what you objected to--- the fact that I was talking about forcing in terms of axioms and not in terms of models. That is not a confusion. It was historically the way in which forcing was developed. Cohen explains this in his book.

Continuum Hypothesis

The continuum hypothesis is not a mathematical question in ZFC since its provably equiconsistent. Its entirely a philosophical question. So to discuss it productively, you need to admit something more than "what theorems about the integers can I prove from ZFC by assuming this" as a philosophical criterion.

The continuum hypothesis is asking "What is the size of R in alephs". Cohen's reasoning is designed to show that this question is absurd. Because P(R) might as well be bigger than any aleph.

Since we agree entirely on the simple example, but you don't see the point, let me try a slightly more complicated example, which might make the point more clearly.

Start with ZFC-P and add the axiom "every set has a set of larger cardinality". By the way, for the purpose of every discussion, the Choice axiom takes the form "every set can be well ordered", so that there is no subtlety in requiring powersets to well order.

Now from the axiom of infinity, and the axiom of larger cardinalities, you get the aleph sequence of ZFC. In fact, I think you get the exact same aleph sequence as you do in L, and I think this system is equiconsistent with ZFC, but who cares. It is clearly consistent from consis ZFC. You can still construct, just using pairing,union, etc, and ordinal induction, certain subsets of omega, the L subsets.

Now add P(omega) as a new axiom. To be clear this is the axiom system:

ZFC-P + + "every A has A' which does not inject into A" + P(omega)

Where is P(omega)? Remember P(omega) is the first set whose elements are arbitrary subsets of Z, not constructed by inductive L-like procedures. The question arises, where does P(omega) belong in this heirarchy? Is it consistent to put it anywhere?

Cohen's proof makes it obvious that all the A's could inject into P(omega). You can add the axiom P(omega) injects all the A's. The moment you admit the real numbers, they could be above everything. This is not an accidental propery, and it's not going to go away by adding new axioms. Likebox 00:17, 3 October 2007 (UTC)

Well, no, they can't be above "everything". They can't, for example, be above the real numbers. It's true that every consistent theory has a hereditarily countable model, and a hereditarily countable set is basically a real number, so if you're only interested in first-order properties, then you can code all the structure you want inside the reals.

It depends on whether you construct the real numbers starting from P(omega), or if you construct the real numbers from L-operations. If you construct the real numbers from P(omega), your statement becomes a tautology-- the reals can't be bigger than the reals. But if you define the constructible reals, than yes, the P(omega) can be above P(omega)_L in the heirarchy, and both can coexist. That's true in any forcing model. The essnetial difference is that P(omega) can have numbers which are not generated by construction rules.

I'm tempted to give my objections to that approach, but it would take us off track here, because we're talking about CH, and the proposal to consider only the hereditarily countable sets essentially pushes the whole question out of the universe--if there is no set of all reals, then it doesn't have a size to evaluate. I really don't think that's what Cohen was proposing in that famous passage. I think the passage should be taken at face value -- it doesn't mean the continuum doesn't have a value as an aleph, just that that value is larger than you can write down by methods analogous to the ones he describes. Whatever that means to a formalist.

If it pushed the question out of the universe, I wouldn't even be discussing it. Even if you are a formalist, you deeply care about where you can place the real numbers, because that translates into an effective question--- can you make models of geometric points which can serve as models of set theory. That's a question about the relative strength of different axiom systems.

I do understand syntactic forcing; maybe not per se with the conditions being sentences, but I don't think that's going to change the discussion much. It really doesn't make any difference. You still aren't using forcing to establish that ZFC-P+"P(P(omega)) exists"+ proves that exists; that follows easily from elementary techniques. And it isn't true that higher iterations of powerset are "completely irrelevant for producing alephs" if by "producing" you mean "proving the existence of"; if (working in ZFC-P) you assume plus " exists", the best you're going to be able to prove is that exists. So you either need more iterations of powerset, or more alephs assumed. And no number of alephs that you can express with such techniques is going to recover for you the strength of the full powerset axiom, even for proving theorems about the natural numbers. So even by your own standards, which I don't really buy, you are not going to be able to show that "geometry is as big as set theory". --Trovatore 07:04, 3 October 2007 (UTC)

Alephs are Large Cardinals For ZFC-P

You don't need the powersets to ensure that the alephs exist. You do in ZFC, but not in mathematics as is usually done. The alephs exist for the same reason that large cardinals exist.

I assume that you have no problem with the axiom "there exists an inaccessible cardinal". Why not? I don't have a problem with it either. For me it's because of this.

If you consider an axiom system, like Peano Arithmetic (finite set theory), it doesn't prove consis PA. But you want to be able to prove that, because consis PA is intuitively true and, for a finitist, is the only kind of thing that is intuitively true. So you can imagine adding the axiom consis PA. But in model theoretic language, consis PA asserts that there exists a model for PA. This is the justification for the axiom of infinity. All that the axiom of infinity states is "there exists a model for PA". It is a minimal set theoretic extension of finite set theory that proves the consistency of PA.

But now you have PA+consis PA, and maybe a little more (I get hazy on this point, because ZFC-P might prove more, like Consis(PA+consis PA) I don't know). Whatever extra theorems you get from ZFC-P that you didn't get from PA are omega-consistent only because the axiom of infinity is equivalent to some ordinal consis-chain from PA.

If you didn't know that, then what reason would you give that the axiom of infinity is not omega inconsistent?

On a personal note, the reason I stopped studying mathematics as an undergraduate was because of the well ordering theorem. That was such an absurdity to me, the existence of a well order on R, that from that point on I could not trust that any of the theorems that you prove in set theory were going to be true. I wasn't thinking that ZFC was inconsistent. I was sure ZFC was consistent. But I wasn't sure if it was omega consistent.

For example, suppose someone gives you an argument "blah blah Powerset blah blah Choice blah blah powerset and therefore there exists an integer N such that P". How do you know that the integer N is a real integer? That's not an academic question if the property P is computable. If P is computable, I can check P(1) P(2) etc. But what if I never find P(N) is true? It is necessary to establish omega consistency.

So you can only accept axiom of infinity if it is omega consistent. Since you've never seen an infinite set, how would you know that you won't end up proving some theorem about the integers using infinite sets which says (there exists n) P(n), and then also find that ~P(1) and ~P(2) and ~P(3) etc.

The reason I learned to accept the axiom of infinity is that the axiom of infinity is the axiom which says "there exists a model for PA", which is a simple way of producing a set theoretic axiom system in which consis PA is provable. In such a system, nothing else is true except consequences of that (and maybe consis(consis PA)) up to some ordinal. I get confused how far up each step takes you). So I'm OK with that now, even as a radical finitist.

So that's why I also accept inaccessible cardinals, because you want consis ZFC to be provable in a reasonable system. What "superduperpower" set do you use to get to inaccessible cardinals? There isn't one. But large cardinals are just a minimal way of ensuring "there exists a set which models ZFC". That was explicitly Godel's motivation for introducing them.

Why accept the alephs exist? Because they are produced by the same steps: take ZFC-P and find a set theory in which you can make a model for it. To do this, you need a set which can serve as a model, and that set must not be bijectible with any proper segment of ZFC-P. So you make an axiom "There exists a set which cannot be bijected to the integers", and you find that you got your model. But the axiom "every aleph has a bigger aleph", is just of that nature. It produces new system by Godel steps. Each system proves the consistency of the previous one, and that's all it does, so the whole thing stays omega-consistent no matter how far up you go.

You could go on to inaccessibles, and then again, and again, and this is an old game. So the justification for the alephs has nothing to do with powersets. The justification is Godel's theorem. That's very important, otherwise Cohen's quote is incomprehensible. If the alephs were only justified by powersets, then how can you have an opinion on whether powersets are bigger than alephs?

These questions are important to anybody, formalist, finitist, platonist, because the alephs prove theorems about integers. These are real theorems about actual objects. If you are a radical finitist, you might phrase the axiom of infinity as consis(PA) and consis(PA+consis(PA)) etc. And you might phrase the axiom of higher alephs (powerset in ZFC) as consis(ZFC-P) consis(that) etc. So long as these are equiconsistent with an infinite set theory, there will be peace between all camps. This was Hilbert's response to Godel's theorem, in the Grundlagen der Mathematische (although I can't read German so that is only a secondhand account). He said that the proper thing to do is add reflection principles, essentially consis(PA) consis(consis(PA)) etc to prove the consistency of higher systems. These axiom systems only talk about integers, but, if iterated to a high enough ordinal, should be as strong as any set theory.

Unfortunately, aside from Gentzen, nobody took Hilbert seriously because they didn't understand Godel's theorem as deeply as he did (my interpretation, perhaps uncharitable). So after Gentzen completed Hilbert's program for PA and proved consis PA from induction to epsilon naught, nobody went and did the next logical thing which is to prove consis ZFC from some ordinal chain of consistency conditions starting from PA. I am not a set theorist or logician, and I get confused on elementary points still, as you probably have seen. Maybe one day I will understand enough to prove omega-consistency of ZFC from enough reflections.

Now comes powerset. In this Godelian game where you get higher and higher infinity, where does powerset belong? The answer is anywhere, which in practice means, above everything, because as high as you go by godel-steps of consis, the next step could be P(omega). The only reason you know this is true is because of forcing.

Rambling Tirade

If you want to know my personal opinion on how mathematics should be done--- it is in an axiom system something like this:

Set axioms: ZFC-P + axioms of alephs + axioms of higher infinity

Powerclass axiom: Every (well orderable) infinite set S has a powerclass P(S)

Measureability axiom: Every subclass of a powerclass has lebesgue measure.

Forcing axiom: Every map from a set to a powerclass has measure zero.

The set axioms would give you all the real theorems, the powerclass axiom allows you to do geometry with arbitrary points, and the forcing axiom makes sure that all the sets stay small so that you don't get nonmeasurable sets or a well ordering of R. Of course, you always have LP(S) for any set, meaning the L powerset of S, and that's a set. Since all the usual arguments about sets don't care if you're in L or not, you could have both an L powerset and a "real" powerclass at the same time.

These questions are not just academic. The fact that mathematics is done in ZFC makes my life miserable. It requires Borel sets to talk about Lebesgue measure, which is annoying as all hell. It is difficult to perform Borel constructions in function spaces. In particular, I want to construct a measure on a the space of all real valued distributions on R4. I know how to do it on a computer, because I have an algorithm that will pick a function at random, and I can show that it will converge to a distribution as the grid becomes finer. On the other hand, to prove convergence in measure, I need to construct a measure on distribution spaces. Isn't it awful that a random picking algorithm doesn't define a measure? And only because there "exist" nonmeasurable sets in some nineteenth century warped intuition?

That's the end of my tirade. I just wanted to get it off my chest. I assume you will erase this entire discussion eventually, so I don't have to worry about my chances for elected office, but it feels good to finally say it. I will read what you have to say, but I am doubtful that we will ever philosophically agree. Still, it was nice talking to you. I can tell you are intellectually honest and sincere, and have a valid point of view too.132.236.173.14 22:14, 3 October 2007 (UTC)

Hi Trovatore! Haven't heard from you in a while. Just so you know, I copied all our (to me interesting) correspondence onto my talk page, so feel free to delete everything if it is too long. I still haven't internalized all the proofs of theorems about L to my satisfaction. I understand why takes aleph3 steps to produce aleph3, and I have intuition that aleph3 steps are necessary, but it does require a lowenheim skolem type argument to actually prove that, and I have trouble reproducing it without looking in a book. But I will keep thinking about it. Hope all is well.Likebox 00:19, 5 October 2007 (UTC)

Oh-- I just realized you wrote something wrong in the last post--- you said "And no number of alephs that you can express with such techniques is going to recover for you the strength of the full powerset axiom, even for proving theorems about the natural numbers. " That is just not true. ZFC-P + "every set has a set which does not inject into it" is equiconsistent with ZFC, and has no powerset. ZFC-P + "every set has a bigger set" + "there exists an inaccessible cardinal" is stronger than ZFC, no powerset either. Since I went off on a tangent, I never responded to that. I am not confused on this issue.Likebox 00:50, 11 October 2007 (UTC)

Neither of those is a "number of alephs". --Trovatore 00:58, 11 October 2007 (UTC)

I see! You don't mean ordinal number. I meant "ordinal number". Thank you for clarifying. By the way, also thank you again for pointing out that embarassing gaffe in the proof of the incompleteness theorem. I was mortified. Likebox 23:58, 12 October 2007 (UTC)

No, ordinal numbers are fine. But if you want to speak about an ordinal number of alephs, then you have to identify that ordinal number as a pre-existing object; it can't depend on the model. So the hypothesis "" exists is an assumption of a certain number of alephs (because ε₅ + ω² + 1 is the same ordinal number in all wellfounded models, up to a unique isomorphism). But "for every aleph there's a bigger aleph" is not. --Trovatore 00:05, 13 October 2007 (UTC)

Ok. So you are good with countable ordinals effectively constructed. That does raise an interesting question. If you take a countable model for ZFC, you can prove the existence of a largest countable ordinal in that model, call it "Z". I think you would consider this ordinal as absolute-up-to-ismorphism, since it's just some countable ordinal effectively constructed. It's the limit ordinal of all countable ordinals in the model. It is not clear to me that assuming only powerset iterations up to this ordinal "Z" isn't as strong as assuming arbitrary powerset iterations. This is one of those questions I can't sort out, because I never can get straight how many ordinal steps of induction an aleph-operation buys you, but even if there was some small correction, I think it is true that there is always a countable ordinal measuring the strength of an arbitrary axiom system, although these ordinals are so big that it's hard to keep them in your head.Likebox 00:24, 13 October 2007 (UTC)

On reflection my last comment didn't make too much sense; we're talking about strength in terms of implying arithmetical statements, but all wellfounded models satisfy exactly the same arithmetical statements. My point is mainly that I think you have been confusing definitions of ordinals with the ordinals themselves (and more generally, definitions of objects with the objects themselves). I suspect that's what's going on in your last comment as well, but I'll have to think about it when I have more time. --Trovatore 00:30, 13 October 2007 (UTC)

Wow, zippy response! I think I see what you are saying. If you take a platonist view and consider the integers as an existing god-given set, then all well-founded models satisfy the same arithmetic statements. That's the problem with talking to a Platonist! I always have to translate. I agree with that. But that wasn't the question, and I appreciate that you are busy, don't feel the need to respond, this is all for my own education. I just like thinking about this stuff, because it is interesting to me. But the question was about provability. The claim you seemed to be making is that no number of aleph-axioms up to a "fixed and definite" countable ordinal (meaning, I assume, an ordinal which orders the integers effectively using a computer program) can reproduce all the theorems of ZFC. That may be true. But what I was suggesting is that aleph-axioms up to "Z", the upper bound of countable ordinals in a countable model of ZFC, might be enough. Or maybe up to "Z^Z^Z..." or something. Nobody was claiming anything about properties in the models, just more theorems from the aleph-exists axioms.Likebox 00:48, 13 October 2007 (UTC)

(Unindent) Not sure I'm following. How would you formulate the "aleph-axioms up to Z"? What exactly would you add to ZFC-P? Maybe some axiom like

There exists an M such that M is a countable wellfounded model of ZFC-P, and for every ordinal β below the height of M, exists (though not necessarily in the sense of M)

Is that what you had in mind? --Trovatore 02:05, 13 October 2007 (UTC)

Yeah, exactly. You said it much more precisely than I could say it. I can't wrap my head around this, because I am shaky on L stuff still. Maybe you can see something there.Likebox 05:25, 13 October 2007 (UTC)

OK, so that's clearly weaker than ZFC with full powerset. It should be the case (I haven't checked this in detail, but I would be really surprised if anything went wrong) that ZFC proves that is a model for ZFC-P satisfying the above axiom. --Trovatore 07:57, 13 October 2007 (UTC)

Yes, I think it proves that too. I think ZFC just doesn't prove that ω₁ exists as well-founded ordinal. Remember that ω₁ is constructed by a countable model of ZFC. To get it, you need the assumption "there exists a model for ZFC". Likebox 17:10, 13 October 2007 (UTC)

I misread your statement last night--- you said M is a well-founded countable model of ZFC-P. I meant to take M to be a well founded countable model of ZFC, Not ZFC-P. Or equivalently, take M to be a well-founded countable model of ZFC-P+"every set has a bigger set". Of course I agree that you can't prove ZFC theorems from such a small ordinal as the height of a countable model of ZFC-P.Likebox 17:17, 13 October 2007 (UTC)

You keep claiming to have a hard time following what I'm saying, but you usually have no trouble immediately translating it correctly into obscure jargon. If your goal is get me to start talking in jargon, I'ts never gonna happen. My religious faith does not allow me to use obscure jargon. But, because you want a precise statement, I'll provide a translation into jargon: Consider the following axioms:

1. ZFC- P

2. There exists a countable ordinal ω₁ which is the upper bound of countable ordinals in a countable transitive model of ZFC. Note I don't need powerset to talk about or construct countable transitive models.

3. Aleph-\omega_1 exists.

Is this equivalent to ZFC? That's the question.Likebox 17:37, 13 October 2007 (UTC)

First, I should say that I got it a little wrong -- is the wrong model; it doesn't satisfy the axiom of replacement. I should have used .

As to your next question -- no, this is not equivalent to ZFC. But in terms of consistency strength, it's stronger than ZFC -- has to be, because it assumes that there's a model of ZFC. All the stuff about alephs and the ω₁ of the model is irrelevant to that question; you've made the theory stronger by brute force. If that's your goal, you could just add Con(ZFC) to PA, and get a stronger theory (consistency-wise) without talking about sets at all.

On the jargon and translation issue: It really is generally not clear what you're talking about when you start talking about iterating axioms into the transfinite. Taking this literally you're going to get either infinitely many axioms, or infinitely long axioms. I'm not certain that you're keeping clear in your head the distinction between, say, adding the axiom " exists", and requiring that the model should think that exists, where α is the real ω₁. You seem to want to draw conclusions from the latter when doing only the former. Or something like that.

(deindent) You can add Consis(ZFC) to PA, and be done with the discussion! For a platonist, that's enough. That's why Godel could do so much against the whole Hilbert school. He could easily prove theorems, because he didn't bother trying to prove that ZFC is consistent. So the platonist outdid everybody else. It's not a coincidence--- Platonism lets you take stuff for granted.

But I'm a finitist, and that leaves open the question "is ZFC consistent?". I do think ZFC is consistent, but I would like to be able to prove it by finding a finitary equivalent. The statement "This particular medium sized countable ordinal is well founded" is a finitary statement. It doesn't talk about uncountable infinite sets, and it can be translated to a statement about the halting of a complex computer program, similar in complexity to the program that searches for a contradiction in ZFC. With a big enough countable ordinal, it should be strong enough to prove consis(ZFC). The pressing issue for me is whether there is a purely textual way, involving only talking about computer programs which "obviously" halt, to complete the Hilbert program--- to prove that ZFC is consistent using only well-orderings of the integers which are explicitly given by a computer program, and where the process of counting down halts.

Of course it won't be possible to formally prove that the down-counting halts without going through many Godel-style (consistent ergo no halt) steps starting from PA. Just at no point should you use any uncountably infinite sets to justify the non-halting. This is what I think needs to be done, so that mathematics can finally jettison the theology of enormous sets.

That's why I am harping on the largest countable ordinal constructed by ZFC-P + consis(ZFC). While this already assumes what I wants to prove, it is useful for figuring out which ordinal to try to construct.

Sorry about the jargon dig... it was just anti-establishment instincts! I appreciate your very precise language, and I was slightly ambiguous in the theory specification because I wasn't using it. Also, thank you for listening and talking. I don't usually get to communicate with people who are interested in this stuff since they're few and far between. I really appreciate all the time you have taken so far.Likebox 20:13, 13 October 2007 (UTC)

Part of the reason you have a hard time following me is because I think your statement about the "real" is an imprecise string of words with no meaning. When you say that, I hear "The in Trovatore's favorite countable model of ZFC". I don't know what the other properties of Trovatore's favorite model are. When I say I usually mean precisely the in the countable V=L universe constructed by the completeness theorem applied to ZFC. If I am using a different system, I mean the object constructed by the completeness theorem applied to those axioms.Likebox 20:24, 13 October 2007 (UTC)

• I think you have it exactly backwards. It is absolutely not enough, for the Platonist, to "add Con(ZFC) to PA, and be done with the discussion". The Platonist wants to find out what is true about the real universe of sets, the one out there, independent of our reasoning about it. Con(ZFC) is already a "finitary" statement; it's not about sets at all. It appears, empirically, to be true. But treated as a statement about the naturals, it's just some silly nonsense with no intuition behind it. There's no reason for it to be true, unless sets actually exist and the ZFC axioms are really true when interpreted as talking about sets. Thus the failure to find a contradiction counts as empirical evidence that sets really exist.

You're repeating an old party line. I don't agree with this party line, and that's what I have been trying to say this whole discussion. I have intuition that is completely finitist, no uncountable sets at all, and that tells me that there is no contradiction in ZFC. That in fact there is a reason for ZFC being consistent which has absolutely nothing to do with infinite sets "existing" in any way but textual. The reason ZFC is consistent is because some large perfectly computable countable ordinal is well founded.Likebox 05:24, 14 October 2007 (UTC)

• You should be careful when talking about "the" model constructed by the completeness theorem. The completeness theorem does not give you a canonical model for a consistent theory, just some model -- in fact to prove the completeness theorem in general requires the axiom of choice. When working with theories in a countable language you don't need AC, but the properties of the model will be sensitive to irrelevant details like the particular order in which you consider atomic formulas, so there's certainly nothing special about that model.

Look, you don't have to sell me the infinitary party line. I have heard it many times and I'm not ever going to buy it. It's convenient if you accept the whole infinite sets deal. I don't. That's the whole point. I don't accept that there is an unambiguous definition of an uncountable set, and so I don't accept your uncountable axiom system or uncountable formal language as unambiguous. If you tell me you have an uncountable language, I will first ask you whether you are working in ZFC. If you say "yes", then the language that in your mind is uncountable becomes, in my mind, some warped language which is mapped to an "uncountable" ordinal in ZFC, which I secretly know is a countable set in the canonical (yes canonical) countable model for ZFC.

If I can write down all the axioms as algorithms to generate statements from objects already constructed, as I can for ZFC, the completeness theorem generates a model which I consider canonical. If you don't consider it canonical, OK, but I do. It's a philosophical difference, there's no point in arguing about it.Likebox 05:24, 14 October 2007 (UTC)

• But the last two bullet points are gettting off track. You still haven't said what you mean to start with an infinite non-computable ordinal, like the height of a wellfounded countable model of ZFC, and then come up with an axiom that there are that many alephs. I think you thought there was some clear meaning to that statement, but it does not appear that you have yet specified one. There are some candidates for what it could mean -- I thought of one that seems pretty natural, and it seems to me that it has surprising properties (that probably aren't going to be what you want). Let me know if you want to hear about it. --Trovatore 02:10, 14 October 2007 (UTC)

Of course I want to hear about it! I have been struggling to understand this stuff for many years now.

But I personally don't think it's a non-computable ordinal. I don't know exactly what the height of a wellfounded countable transitive model of ZFC is, in an unambiguous finitary way, because that requires knowing how big the (secretly-countable, secretly-computable) ordinal that your model thinks is ω₁ is. If you switch models of ZFC, like collapse aleph2 to aleph1, then the value of this (still secretly countable, still secretly computable) ordinal ω₁ changes.If it turns out that your mind or your axioms tell you that the height of a countable model is a non-computable ordinal, it would be the wrong ordinal to look at. I am not sure if the height of a countable model of ZFC is computable or not. Just because it is proven to be non-computable in ZFC does not mean it is in any computational sense non-computable. ZFC also proves that the ordinal ω₁ isn't countable in any model, even the smallest countable model when I know very well that it is countable and not even very big.

I think that the right ordinal to look at for a completion of Hilbert's program is a computable ordinal. I think it's the limit ordinal of all countable ordinals proven to be computable in ZFC, which is a computable ordinal, the first one that you can't prove is computable in ZFC. I'm guessing, because I don't have any proof, that it's the same as the height of the model in ZFC, when the value of all the secretly computable, secretly countable ordinals are revealed. Let Z be the computable countable ordinal which is the limit of all computable countable ordinals proven to be computable in ZFC. If you assume Z (or something like Z^Z^Z.. ) is well founded, (this can be given an intuitive justification without using infinite sets) then I think that ordinal induction up to Z can give you a finitary proof that ZFC is consistent. And I think you don't need any "intuition" about infinite sets, just this big computable ordinal.Likebox 05:24, 14 October 2007 (UTC)

[edit] HI

Dear Professor Trovatore, Some one has sabotage my talk page, he/she did it from a public domain and so I cant live a message and ask him\her not to do it again without my permeation. As you are a very kind and fair person (and a Ph.d in mathematics+ a programer), I assume that you can help me about this (when youll be back in California)- and if you don’t, thanks any way.

Best Wishes

--Gilisa 14:47, 7 October 2007 (UTC)

Are you talking about these edits? You can certainly leave a message at the talk page for the IP address, User talk:158.123.229.2, but there's no guarantee that the next time the person logs in, he or she will have the same IP address. If you want to find out where these people are posting from, you can click on the link to the user talk page and scroll down to the bottom, and click on the word WHOIS. In this case it appears that the internet provider is in the US state of Rhode Island, for whatever that's worth. --Trovatore 16:54, 8 October 2007 (UTC)

[edit] Phenomenon

I'm sorry- I didn't know you were making those edits and thought I just left that section on. I actually moved the Kant section to Phenomenon (philosophy). Beast of traal T C _Beast of traal

Ah -- well, I don't think that should be split out. There's a difference between the most common usage of a word, and the usage most appropriate to an encyclopedia. In my opinion it's precisely the philosophy stuff that justifies having an article about phenomena in the first place (otherwise, there's just not that much to say). So the philosophical material should appear prominently in the main phenomenon article. --Trovatore 01:52, 10 October 2007 (UTC)

Okay, but it will need to be shortened. Beast of traal T C _Beast of traal

It will need to be copyedited, I think. But shortened? Why? The article is quite brief, and treats an important concept from philosophy. (I think by the way that phenomenon (philosophy) should be redirected back to phenomenon). --Trovatore 03:00, 10 October 2007 (UTC)

Phenomenon is a very brief subject. "Phenomenon (philosophy)" is a different subject. A short section on the Phenomenon page is acceptable, however half of the original section was a list of philosophers and is not needed in a page on Phenomenon. Beast of traal T C _ 03:30, 10 October 2007 (UTC)Beast of traal

No. The philosophical content is the main subject. The rest of it is dictionary definition, and WP is not a dictionary. You wanted the merge, fine, it's merged, might be better that way. But the article under the word phenomenon must prominently and extensively treat the philosophical concept; that's the only part of it that merits encyclopedic treatment. --Trovatore 03:35, 10 October 2007 (UTC)

They are undisputedly different topics. The word has two completely different meanings. There is a separate page for the word's second meaning- that is where the information about the word's philosophical meaning belongs. Please stop reverting these edits or I'll have the page protected. Thank you, Beast of traal T C _ 11:34, 10 October 2007 (UTC)Beast of traal

If they are different topics, then the article you want to write should not exist at all, as it is not an encyclopedic topic but merely a dicdef. -15:51, 10 October 2007 (UTC)

Alright. If you can write a comprehensible section on Phenomenon (philosophy) you can put it on. It should not include a list of figures in phenomenology or a definition on a noumenon. Until then I will remove it from the page as it is curently incomprehensible. Beast of traal T C _ 16:16, 10 October 2007 (UTC)Beast of traal

No. I'm not a philosopher and don't have the training to do that. You never got consensus to remove the philosophical material and I am going to restore it. If you want to improve it, solicit help at Wikipedia:WikiProject Philosophy. --Trovatore 16:40, 10 October 2007 (UTC)

You are correct, I did not get a consensus. But I didn't remove it. I just moved it. Beast of traal T C _ 18:28, 10 October 2007 (UTC)Beast of traal

You moved it out of its proper place. There is no justification for a "phenomenon" article without that material. It's not entirely clear that there's a justification for the article even with the material (see e.g. Larry Sanger's comments on the talk page) but that's a separate question. --Trovatore 18:31, 10 October 2007 (UTC)

How about this:

• put in a section on philosophy (temporarily one that is simpler until a philosopher has a chance to expand it because the current section is incomprehensible- I'll leave a message on the philosophy talk page).
• update the section on physics use (the current one has too many redundancies).

I can do this very easily Beast of traal T C _ 18:42, 10 October 2007 (UTC)Beast of traal

I have left a message on the WP:PHILO talk page. The "use in philosophy" section is now in their hands- so we don't have to worry about it. Beast of traal T C _ 20:19, 10 October 2007 (UTC)Beast of traal

[edit] Godel's incompleteness theorems article

I scanned the latest archive and the current talk page. I sense that you feel the article needs work, but like most everyone is too busy to work on it, and are feeling a bit cautious about emendations. After my poor experience approaching the Function (mathematics) page I can see why you might be cautious. In fact, that is why I am writing here, rather than at the article's talk page. (It's okay by me if you want to move it off your talk page to there ... but I'm wondering what your thoughts are.) Here are mine:

The first 2/3 of the incompleteness article didn't help me much. It seems muddy, as if written by someone who didn't quite have a grasp of their material (I hope I didn't insult you just then, it's just my reaction...). Yet I read the article's talk archives/page and see that there are contributors, yourself included, who clearly do have a grasp. But no one knows how to approach the article. I have some thoughts:

New stuff in Gödel's own words: Now that Dawson has mostly published Godel's nachlas, new stuff has come out. For example, here is Gödel's own explanation in his own words, a wonderful quote from the first half of the same Balas letter I quoted on the talk page:

"The occasion for comparing truth and demonstrability was an attempt to give a relative model-theoretic consistency proof of analysis in arithmetic. (For, an arithmetical model of analysis is nothing else but an arithmetical ∈-relation satisfying the comprehension axiom:) (∃n)(x)[x∈n ≡ φ(x)]. Now, if in the latter "φ(x)" (is replaced) by "φ(x) is provable", such an (∈-)relation can easily be defined. Hence, if truth were equivalent to provability, we would have reached our goal. However (and this is the decisive point) it follows from the correct solution of the semantic paradoxes that "truth" of the propositions of a language cannot be expressed in the same language, while provability (being an arithmetical relation) can. Hence true ≢ provable.) (The ≠ is actually a triple "identical to" with a bar through it. The typography is a challenge because this letter had inserts and crossouts and etc, and notes at the bottom by Dawson, p. 10. For example I used ∈ which I believe is correct, I have no exact symbol (it looks like a Greek epsilon).)

What sources and presentation-style worked for me might work for others: Nagel-Newman (years ago, my first exposure) was a tough slog, just like a first exposure to Turing in the original. But Nagel-Newman helped especially with the notion of "Godelization" and by use of their examples. More recently, I found Rosser's 1939 An Informal Exposition of Proofs of Godel's Theorem and Church's Theorem (in Undecidable, p. 223ff), in particular re the issue of where exactly the "diagonalization" is buried. Also a careful reading of Kleene 1952's Chapter IV A FORMAL SYSTEM (i.e. a system for number theory, looks almost identical to Gödel's 1931 development ... written in typical Kleene language (pardon the pun), sharp, clear, precise.)

German abbreviations: The symbol-dictionary (i.e. German to English) on page 33 of Meltzer and Braithwaite is a great help.

Knowledge of Godel's typography: The Meltzer-Braithwaite translation is no good, and I've warned readers away from it in the annotated references. For some godawful reason they changed the typography and used c instead of κ, for instance. Godel used small capitals to distinguish metamath from formal math (or whatever... is confusing ... I'm confused as I write this).

Elucidate the notion of type elevation: The notion is not a newbie notion, probably a major source of confusion to anyone who isn't a logic professor. Gödel uses subscripts to identify type. Once you get this, it's easier.

Describe, somewhat thoroughly but briefly the notions in "the formal system" as defined by Gödel and then elucidated by Kleene 1952: List the symbols and notions behind them: logical ⊃, & , V, ￢, ∀, ∃; predicate =; functions + , * and '; 0 (zero); variables, parenthesis (, ). Define juxtaposition (aka concatenation). Describe notion of formation rules: terms, formulas. Describe the notion of abbreviations as "subroutines" defined by the formal symbols and formation rules: e.g. ≠, <, >. Notion of free and bound variables with reference to "substitution". Describe the notion of "transformation rules" in particular "the rule of inference" A & (A -> B) => B. 8 axioms of the propositional calculus. 4 axioms of predicate calculus. 9 postulates of number theory (the Peano axioms, addition and multiplication defined). Finally, the notion of (primitive-) "recursion".

Interesting how, in what takes Kleene 69-85, Godel spills out in about 2 pages. From this comes the definitions of immediate consequence and provable and formally provable formula' or 'formal theorem. And there is another key unmentionable: classes of formulas κ and Flg(κ), the consequence set of κ. And: where oh where has the diagonal gone? Oh where oh where can it be?

The problem of Theorem VI: Speaking for myself alone here, this is the nut that must be cracked. I get it all up to the details of Theorem VI, and then all hell breaks loose. About a page and a half of mass confusion. First off, ω-consistency muddies the water. So forget that. Then the arcane symnbols and having to learn German words Bew and Gl and all that. I know diagonalization is buried there somewhere, but where and how exactly? Rosser noted in a footnote that Godel had intended to write "the second part of the paper ... Due to ill health, Godel has never written this second half).(p. 223)

Thoughts? (My sense is: To do this "right" -- meaning "elegantly" and "effectively" -- will be a challenging undertaking. Might take years, for example.)

In the talk pages I read that you claim to be a substance dualist? Shame on you. Nobody admits to such failing in public setting. Tsk. (Just kidding....) Bill Wvbailey 20:08, 10 October 2007 (UTC)

It will indeed be a challenging undertaking. I think, though, that there is no need to make it both harder to do and harder to understand by trying to be faithful to Goedel's original notation. The exposition has been much streamlined since then. I think Enderton and Smullyan would both be likely candidates for sources (though I don't have either book so I can't be sure).

Note also that we want to limit ourselves to saying just a little about the proofs, just enough to give the reader an idea of their broad structure. We definitely don't want to get into arithmetization of syntax, beyond simply mentioning that it can be done and is a necessary component of the proofs, although not in the end a very interesting part of them. --Trovatore 21:31, 10 October 2007 (UTC)

I hear you. I agree that Gödel's original notation should not be pursued (except in a footnote as to where a reader might find assistance in deciphering it). Even Kleene 1952 eschews it. I have Enderton 2001 A Mathematical Introduction to Logic, but not Smullyan. I will explore Enderton's treatment. I have a few other books as well. My sense is that computer-sci folks and software folks and hardware electrical engineers (all those who know about assembly and machine language, anyway) already know the notion of "arithmetization of syntax" from that angle, so it isn't much of a surprise, unlike in 1931. Bill Wvbailey 22:24, 10 October 2007 (UTC)

$124 "and fifty cents" for Smullyan. I hope the library has it (someone ripped off the last two volumes of Dawson's Godel's Completed Works, probably for the same reason -- $$). Bill Wvbailey 15:07, 11 October 2007 (UTC)

[edit] could you straighten up the sets discussion near the end of the mathematics reference desk now?

Thank you!! —Preceding unsigned comment added by 84.0.126.201 (talk) 21:30, 12 October 2007 (UTC)

[edit] Hi again

I wanted to respond, because I am worried about something--- perhaps you are right about this. I get what you're saying. If you have something too expository, you are taking a role which is inappropriate. but I am afraid that the next generation will not be able to read the proof of Godel's theorem anywhere, and it frightens me. You know, it's this practice (unconscious) in academia of making stuff obscure, and the only antidote is clarification, reinterpretation, and constant vigilance. I worry because I think ideas like Godel's theorem need to be understood by everybody. I am worried that a layman can't understand it. It needs to be understood. I am sorry I am not eloquent.Likebox 05:52, 16 October 2007 (UTC)

I had another comment of a certain personal nature--- I had such a hard time understanding the Friedberg Muchnik arguments about smaller degrees and the Spencer arguments about minimal degrees. If the books would just write a computer program in a decent modern language, the proofs would all be so trivial. The only trivial ones now are the Turing arguments about the oracles, and those have an incomprehensible counterpart in the arithmetic heirarchy which is exactly the same thing in logic gibberish. I can translate heirarchy statements now to oracle statements, Post proved they're equivalent, but I still don't have a clear grasp of Friedberg/Muchnik or Spencer or any of the incomparable degree arguments.

Another thing which I am totally confused about is the Shore/Nerode argument (or maybe this one's just Shore) that the automorphisms of the Turing degrees are countable.Likebox 20:53, 16 October 2007 (UTC)

Honestly I do not know that much about degree theory. Was Shore the one who claimed to have proved that the degrees have no notrivial automorphisms, but ten years later no one can say whether the proof is correct?

I don't know all his ouvres. I only know this one because someone mentioned it offhand once, and it confused the dickens out of me, becuase the turing degrees are just real numbers. I couldn't figure out why I couldn't force a bajillion real numbers into R which are all indiscernable and then automorph the new degrees into each other. I went to his office and tried to ask him about it, and he kicked me out for being a crazy person. I thought about it since, and it is not clear that if you make a forcing extension where a and b are automorphable as turing degrees that they aren't just equal as Turing degrees. But I'm still confused, and there's no road to enlightenement except thinking about it again and again and again.Likebox 19:43, 17 October 2007 (UTC)

So if you add two Cohen reals, they are certainly of incomparable Turing degree. But the point is that the new reals are there only in the forcing extension. What we want to do is prove outright that there are incomparable Turing degrees, not that you can force the existence of incomparable degrees.

But I believe you can modify the forcing argument to prove it outright, and this is supposed to be a much simpler and cleaner way to see that there are incomparable degrees than the original priority argument. What you do is figure out which dense open sets you need a filter to meet, in order to recover the incomparable reals from them. If there are only countably many of these, then you can get a filter in the ground model that meets all of these, and Bob's your uncle.

This is simpler, of course, only if you already know forcing. The priority argument is probably the fastest way to prove the result to someone who doesn't.

Oh, I just noticed that you were talking about automorphisms of the degrees, not incomparable degrees. I'll have to think about that. --Trovatore 20:04, 17 October 2007 (UTC)

I certainly agree that we should be as clear as possible within the constraints we face. But we can't turn WP articles into textbooks. It's a difficult line to walk. --Trovatore 21:06, 16 October 2007 (UTC)

I hear you. I'm trying my best.Likebox 19:26, 17 October 2007 (UTC)

[edit] reducing arithmetic to logic ?

Hmm. I don't quite know what that means ... do you mean "natural number arithmetic" or arithmetic over the real numbers, or what exactly? And by "logic" do you mean, for example, a simple counter machine built out of nothing but NAND operators (implemented in any form whatsover -- mechanical, electromechanical or pure mechanical) or implication operators ~a V b, or NOR gates ... we need a bunch of two-input devices with inversion (NOT) in there somewhere. When we're done our logic will "do (natural-number) arithmetic" on unary "marks" (because that is how you and I have defined the marks and the behavior of the mesh of logical operators). Where this gets dicey is (i) the need for a sequential "event" called "the clock" -- (my guess is "the clock" is (i.e. has for an analogue) the y variable in the primitive-recursion schema f(y',x) = g(y,f(y,x),x). But we can dispense with "the clock" and get along by punching two or three clock-buttons in a repeating sequence "0 to start" then 1-2-3-1-2-3-1-2-3... and watch our machine compute, "cycle by cycle" , (ii) our stick-built machine is of necessity finite, whereas what the mathematicians want machines with unbounded registers. To get around this we engineers would add more "memory" (paper, tape, whatever...) until at last the speed of light (per Gandy's Principle of mechanisms) begins to intrude, at which point we have to "slow down" our clock. Bill Wvbailey 19:17, 16 October 2007 (UTC)

No, I don't mean "logic" in the sense of "computer logic"; I mean "logic" in the classical sense of the word, "the science of drawing valid inferences" or some such. To "reduce something to logic" should mean that you don't have to either assume or discover any facts about the subject matter; all conclusions of the theory should be analytic truths, "true by virtue of their meaning", as "every red car is red" is true by virtue of its meaning.

Somewhat strikingly, at least to me, it's not really at all clear exactly what any of this means -- this, for an idea that is supposed to make everything else crystal-clear. --Trovatore 21:10, 16 October 2007 (UTC)

[edit] Zero Sharp's edits to Zero Sharp

Hi Trovatore: The link to 'mathematical' was already there, I just failed to remove it. I just felt 'mathematical set theory' was an awkward phrase and expanded it to 'the mathematical discipline of set theory'; as I said in the edit summary, what other kind of 'set theory' is there but 'mathematical' (at least that's relevant to the article.) But I agree, linking 'mathematical' is probably not necessary. Thanks! Zero sharp 03:03, 20 October 2007 (UTC)

[edit] Division by zero

I've requested semi-protection for Division by zero and Theory of everything. — Loadmaster 23:32, 25 October 2007 (UTC)

[edit] Vandal user, or just confused?

• DanaLynn07‎ (talk · contribs · deleted contribs · logs · block user · block log) -- I am not sure what to do about this individual. Clearly they know their way around Wikipedia, but are doing something non-constructive anyway. Should I report this to WP:AIV, somewhere else? What do you think? Curt Wilhelm VonSavage 00:56, 2 November 2007 (UTC).

[edit] Omlauts

How do you make that o with two dots? I keep trying, but I have to cut and paste from some other place.Likebox 01:07, 6 November 2007 (UTC)

When you edit a page, there's a box at the bottom with various symbols. Or if you're using Windows you can press Alt, key 0246 on the numpad, and release Alt. --Trovatore 01:36, 6 November 2007 (UTC)

With the right setup, [Ctrl]+":" then "o" is even easier. Similarly, the accented "e" in "Poincaré" uses the easily-remembered and typed [Ctrl]+"'" then "e". Mozilla FireFox users will find Zombie Keys a convenient solution. --KSmrq^T 20:15, 8 November 2007 (UTC)

[edit] secondo problema di Hilbert

ciao, riporto qui quello che ho appena inserito in it:Discussione:Problemi di Hilbert, ti prego di contribuire.

nella tabella si dice: soluzione parzialmente accettata, ma questo "parzialmente" non viene spiegato in alcun modo nella sezione relativa. chi è che non accetta la soluzione di Gödel (o, più probabilmente: chi ritiene che il teorema di Gödel non risolva il problema)? Giorgian 01:22, 6 November 2007 (UTC)

Beh, veramente non saprei fare dei nomi. Pero' spero che tu veda il problema: Hilbert chiese se fosse possibile o no dimostrare la coerenza dell'aritmetica, ma non specifico' in cosa consistasse una dimostrazione della stessa. Oggigiorno abbiamo due risultati dai quali si potrebbe arguire per conclusioni opposte, a seconda del tipo di dimostrazione che si desidera. --Trovatore 01:31, 6 November 2007 (UTC)

[edit] two changes to Georg Cantor that merit attention; see G-guy's talk

Hi Trovatore,

There have been two changes to Georg Cantor that merit attention. I've mentioned both of them on G-guy's talk; this one in particular is something I think should really be verified. Thanks! --Ling.Nut 06:24, 8 November 2007 (UTC)

Hmm, I have seen that before on the Cantor talk page, but no, I don't have the reference to check. But even if it's correct I'm not sure I think it really merits mention in the article. It's some sort of evidence, I suppose, though I'm not sure exactly what it's evidence for, and in any case it doesn't settle anything. --Trovatore 07:22, 8 November 2007 (UTC)

The entire "he was jewish" argument rests on one or perhaps two stray comments in one or two letters. It musdt be included, since the debate over his jewishness is significant (as wiki-debate has evidenced). --Ling.Nut 14:06, 8 November 2007 (UTC)

The issue does need to be mentioned, but we don't have to let the discussion grow without bound. Given that (for reasons obscure to me) there seems to be a fair contingent of editors whose main interest in the Cantor article is this point, we need to be pretty vigilant about keeping it in check. I'm for text that minimally states what's definitely known, and then drops it. I wouldn't necessarily be against an ethnic background of Georg Cantor article where all the bits and scraps of evidence one way or another could be collected. --Trovatore 18:09, 8 November 2007 (UTC)

It's clear that speculation about Cantor's Jewishness has occurred in Wikipedia's talk pages, but has it also occurred in publications by reliable sources? That could help decide whether such speculation is important enough to cover. Whatever is written about this issue (if anything) in major biographies of Cantor is presumably noteworthyEdJohnston 18:47, 8 November 2007 (UTC)

I don't really agree with that. The biographies are hundreds of pages long. Not everything covered there belongs here; we're entitled to be, and should be, somewhat selective, as long as the selectivity is not used to favor one point of view. Granted, if reputable biographies spent whole chapters on a point we could hardly ignore it, but I haven't seen that as regards the current issue. --Trovatore 18:56, 8 November 2007 (UTC)

Sorry to butt in here. I disagree with the quavering stance. Why bring in "religion" and "ethnicity" into an article unless there's a damn good reason to (e.g. he was persecuted, had his career ruined, was a persecutor). I've stomped on attempts to claim Al-Kwarizmi's supposed religious heritage a number of times (at algorithm) and will continue to do so unashamedly and without mercy. What does "Jewishness" mean? Genetic makeup? For all any of us know 100% of the world's population is "Jewish" by that definition. As religion of a child is a matter of both parents' heritage and choice, and the bios say his parents were practicing Christians and he was raised a Christian, that leaves only his personal choice as a matter of debate. Who cares if it's not an issue in his life? I'd recommend a block any attempts to insert innuendo, even if it means an edit war, at which point put a quasi-permanent block on the article. It's an FA ferchrisake so what's the problem? I can barely express how neo-nazi extremist innuendo of this sort pisses me off. This same sort of shit happened at Hilbert a couple years ago; the perpetrator was finally blocked for a year as I remember. Bill Wvbailey 19:04, 8 November 2007 (UTC)

Well, there is a good reason to mention Cantor's personal religious beliefs (which were unambiguously Christian), because he himself explicitly connected them with his mathematics. Now I can't think of anything in this connection that really depended on the differences between Christianity and Judaism -- the Jewish conception of God would probably have worked just as well for his remarks about the Absolute Infinite and about transfinite ordinals existing because God could conceive of them. However his attempt to save the Catholic Church from its Thomist errors as regards the theology of the infinite are, I think, worthy of note (Cantor himself was Lutheran, but philo-Catholic).

I would be really careful about suggesting that there is any neo-Nazi or anti-Semitic sentiment at work among those who want to include material suggesting Cantor was Jewish. I haven't seen that. I've seen people accused of anti-Semitism for not wanting to include such material (this of course is also nonsense). --Trovatore 19:44, 8 November 2007 (UTC)

I'm curious: what would be someone's motivation (point, intent, mission) to "suggest" (insuinuate, produce "evidence") that Cantor was "Jewish" (whatever that means) given, if you are indeed correct in this, he was unabashedly self-avowed Christian and his religious beliefs were a factor in his professional life? If all the historical evidence also corroborated his behavior and his writings that he was Christian? Because his name is "Cantor", as someone suggested? My dictionary shows the word derived from L. singer cantus and the 1st definition is choir singer: PRECENTOR. The 2nd definition is "a synagogue official..." The answer: "I don't read people's minds, is a cop-out: As much as action, intent is admissible in a court of law: Sometimes something that smells like shit really is shit. (I guess I've had my say on this one, huh?) Bill Wvbailey 21:05, 8 November 2007 (UTC)

There are those who would label someone Jewish to denigrate them; there are others who would claim someone as Jewish to promote Jewish heritage. The same can be said for other labels, like "Mormon" or "Hungarian" or "mathematician". As for relevance, consider Madeleine Albright; she was raised Roman Catholic, yet around the time she became the first female U.S. Secretary of State she was startled to learn that her family was originally Jewish. Her discovery became a major news item; yet clearly Judaism had no conscious effect on her life before that. It baffles me why some people are obsessed with such things, but I'm sure my enthusiasm for mathematics could also be considered baffling. Or look at the passion on both sides about Erdős number categories. --KSmrq^T 21:49, 8 November 2007 (UTC)

(edit conflict) First, let me specify the context -- I'm talking about editors here on Wikipedia; I'm not denying that there may have been anti-Semites historically who painted Cantor as Jewish with the intent to disparage him. But the editors that I've seen on WP that want to portray Cantor as Jewish mostly seem to be Jewish themselves (for example User:Gilisa), and the motivation appears to be "hooray for our team". That said, it is pretty hard to interpret Cantor's letter about his "israelitische Grosseltern" in any way other than as a reference to Jewish heritage, so it's not just the surname. --Trovatore 22:37, 8 November 2007 (UTC)

• HELLO guys, no offense, but you're missing the boat. I personally don't give a flying cr*p if Cantor was Jewish or not. Moreover, it is almost certainly wholly irrelevant to Cantor's success. However there is and always has been sizeable academic debate on this point.. [as the article and talk points out; noted historians, some of whom were also noted cranks, have weighed in repeatedly] and the debate itself is inherently noteworthy. So all of our comments and opinions on this topic are pointless (plus a HUGE waste of time and a HUGE argument magnet); this topic must stay in the article. Later! --Ling.Nut 00:28, 9 November 2007 (UTC)

I agree it needs to stay in the article; I just think the article doesn't need to say everything about it that could possibly be said (even with references). We can justifiably hold it to a bare minimum, and we should. --Trovatore 00:32, 9 November 2007 (UTC)

(undent) Slow to reply: There are two issues I adressed on g-guy's talk.. first the Jweishness quote... If my memory serves the additional text was only a score or so words over the old version; perhaps less. It did seem a bit more perspicuous than the older version... however, first of all my orig. question was merely to see if you could verify the text.. Secondly, I defer to your judgment, Trovatore. :-) If you wanna restore the older version, then restore away... The other issue I mentioned on G-guy's talk may be closer to your heart.. some text was removed the WP:LEDE that may be quite germane to the article (G-guy thought it was).. see farther up his Talk, but hurry, before he archives it ;-) Later --Ling.Nut 06:23, 10 November 2007 (UTC)

Ooops he has already archived it; see this. later --Ling.Nut 06:43, 10 November 2007 (UTC)

This is a response to Ling.Nut's question about a source for the "paradigm shift", at least I think that is what is in question. I found these quotes in Anglin 1994:213 from his brief chapter about Cantor (when I read the article it did not seem to emphasize Cantor's importance as much as I would have expected. So here are some strong quotables):

"Empiricist philosphers, such as Hobbes, Locke, and Hume, had convinced some mathematicians, such as Gauss, that there is no infinite in mathematics. Thanks to Georg Cantor (1845-1918), however, almost every mathematician now accepts the infinite. Georg Cantor single-handedly produced a clear and ocomplete theory of the infinite that answers all the objections previously raised by anti-infinity philosphers, and which has become the basis of contemporary mathematics."(p. 213)

"History, however, has judged Cantor to be one of the most original and important mathematicans of all time. The opening sentence in Michael Hallett's Cantorian Set Theory and Limitation of Size is not an exaggeration:

" 'Cantor was the founder of the mathematical theory of the infinite, and so one might with justice call him the founder of modern mathematics. '" (ibid)

I'd recommend a strengthening of the "lede" with something to the effect of these quotes. They came from W.S. Anglin 1994, Mathematics: A Concise History and Philosophy, Springer-Verlag, NY, ISBN 0-387-94280-7. Bill Wvbailey (talk) 18:38, 29 November 2007 (UTC)

[edit] Unpleasent Profession of Jonathan Hoag

I understand your comment on the difference between Hoag and Matrix/13th floor, but I do not agree. Both are studies of epistomology and a universe that is not real. As Hoag article states, -The art in question is their entire world, created by an "artist" as a student project.- And suggesting that Hoag is about religion and Matrix is not is also not supported. However I shall find a ref before I revert.Obina (talk) 21:44, 28 November 2007 (UTC)

Well, we clearly have different interpretations of Hoag. I don't see any clear indication that Ted Randall's world is not "real". As far as I can tell it is real -- remember that Hoag says that he uses the term "art" without fear of abusing it, because it can pretty much mean anything. I take the Artist to be just another name for God, though not exactly God the way the Abrahamic religions see him, given that there are apparently other gods that God wants to impress. Heinlein returns to this theme in his later work (especially in Job, a Comedy of Justice).

But in any case I think the idea of using "see alsos" to point to works with arguably similar themes is shaky. American Gods reminded me very strongly in some (perhaps superficial) ways of The Long Dark Tea-Time of the Soul, but I wouldn't put see-also links between them. --Trovatore (talk) 22:22, 28 November 2007 (UTC)

Um.... I wonder if those who write "epistomology", with an "o" in the third syllable, write "epistomic" instead of "epistemic". Michael Hardy (talk) 23:49, 28 November 2007 (UTC)

Heh Heh I'm a legendary poor speler, thanks for noticing.Obina (talk) 19:15, 29 November 2007 (UTC)

[edit] Gödel's incompleteness theorem

> sorry, what do you mean by "prove the statement is consistent"? The point is that if the system proves the system is consistent, then the system is inconsistent.

By ‘the statement’, I mean the various possible constructions of what is in spirit, the Gödel sentence. I just found the two occurrences of ‘it’ jarring to read: to me, if I expand out the two ‘it’s, the sentence reads as: “Furthermore if the system can prove that the system is consistent, then the system is inconsistent.” which sounds counter-intuitive on first reading. (I mean, I do understand what it's trying to say, but it's not quite making the point clearly.) Whereas I would expect it to say: “Furthermore if the system can prove from within itself (e.g. via the Gödel sentence) that it is itself consistent, then we can show that in fact, the system itself is inconsistent.” But that's much longer and arguably more confusing to read. Change it to my version at your discretion. I'm only a compsci, not a logician. :3 —Liyang (talk) 02:04, 5 December 2007 (UTC)

Actually I don't really see the difference between your two versions. It is counter-intuitive -- but nevertheless true -- that if the system can prove the system is consistent, then the system is inconsistent, and I think that's precisely what was intended.

Now that's not to say that that paragraph doesn't need improvement (or, perhaps, deletion). The fundamental problem with the whole Gödel's incompleteness theorems article is that it's a patchwork mess -- people keep adding stuff to the middle without concern for the overall flow. What it really needs is a near-complete rewrite by one person that everyone will trust to get the basics right, and who will then listen to criticism from everyone else and adjust it accordingly. --Trovatore (talk) 02:20, 5 December 2007 (UTC)

[edit] Georg Cantor

...are you attached to the word "denumerable" as opposed to "countable"? Someone just did a wholesale substitution; dunno if it matters. Ling.Nut (talk) 01:09, 11 December 2007 (UTC)

Personally I never use "denumerable", only "countable". However in a historical context I'm not sure. If Cantor historians all say "denumerable" then we might want to be a little careful. --Trovatore (talk) 01:34, 11 December 2007 (UTC)

The other issue is that there's this running usage argument about whether finite sets are "countable". The most standard usage is that finite sets are countable, but the other usage is also reasonably well attested. But I think it's rarer to consider finite sets as "denumerable". --Trovatore (talk) 01:37, 11 December 2007 (UTC)

I leave it in your capable hands. :-) If countable is OK, then no worries. BTW, there was a thread a while back about various changes to poor Georg's article. It started with me asking questions about other peoples' edits... Someone copy/pasted it to the article's talk but I'm not sure anything was done.. i also leave that in your hands... :-) Ling.Nut (talk) 01:44, 11 December 2007 (UTC)

[edit] About Wikipedia:Arguments to avoid in deletion discussions

"That's only a guideline or essay" is a specific argument to avoid in deletion discussions. Zenwhat (talk) 17:10, 7 January 2008 (UTC)

Wow, I hadn't seen that. That's completely ludicrous. This "arguments to avoid" thing has gotten way out of hand; much of it is wrong, and people present it as though it were holy writ. --Trovatore (talk) 17:31, 7 January 2008 (UTC)

I disagree. The "arguments to avoid" is based on logic. "It's just an essay" is an ad-hominem that says absolutely nothing about whether the essay is true. Zenwhat (talk) —Preceding comment was added at 21:20, 8 January 2008 (UTC)

It doesn't say anything about it being true, no. But that's not the point. The point is that when people quote it in AfD discussions, they typically give the impression not just that they believe it's true, but that they are quoting an authoritative document. Which they're not; they're quoting an essay that makes some good points and some bad ones and some in-between ones, none of which have been through the process that establishes consensus for WP policy-generation purposes.

As I say, it's fine to refer to the document for the purposes of giving more detail on arguments you don't want to repeat every single time. It's not fine to leave the impression that the document is settled policy. Whether that is the intent or not, that is the message I often see coming across when people link to the essay at AfD. And I think it's entirely appropriate, in those circumstances, to remind people that it is only an essay, not established consensus. --Trovatore (talk) 21:46, 8 January 2008 (UTC)

If you don't like WP:ATA then you'll enjoy reading its opponent, WP:BASH. EdJohnston (talk) 16:07, 9 January 2008 (UTC)

[edit] "grounded relation"

Hello. Please take a look at Wikipedia:Articles_for_deletion/Grounded_relation#Grounded_relation. Michael Hardy (talk) 06:12, 19 January 2008 (UTC)

[edit] Descriptive set theory

I expanded the article some this evening, but it is still very rough. I'm sure you are much more familiar with the area than I am; any input or improvements there would be welcome. I do realize I should edit it to use the pointclass terminology. I also noticed while I was typing it that a bunch of stuff about the Wadge hierarchy and determinacy seems has written in the last six months; did you notice that? — Carl (CBM · talk) 02:11, 23 January 2008 (UTC)

Actually I thought you did a pretty nice job. I had thought about expanding that article various times but could never seem to figure out what to include -- I'm not very good at writing articles on huge, diverse topics. Determinacy is the widest topic on which I ever started an article and took it beyond a stub, and I never did finish that one. --Trovatore (talk) 03:49, 23 January 2008 (UTC)

[edit] Thank you

Thank you for correcting my error. You are also welcome to comment on the content. As you can see, my English is not really good. May I ask you sometimes to check my edits? Thanks again for help.Biophys (talk) 20:39, 27 January 2008 (UTC)

[edit] Boolean algebra task force

I'd like to invite you to participate in the Boolean algebra task force that I am forming. Despite the name, a task force is just an ad hoc subcommittee of a wikiproject to work on a particular issue. In this case, I think that our articles on various aspects of Boolean algebra, propositional logic, and applications would benefit from some big-picture planning of the organization of material into various articles. The task force would not require a great time commitment. The main goal is to work out a proposal for how the material should be arranged. A second goal is for the focus to remain interdisciplinary, including computer science, logic, and mathematics. — Carl (CBM · talk) 16:13, 28 January 2008 (UTC)

[edit] Game theory FAR

Game theory has been nominated for a featured article review. Articles are typically reviewed for two weeks. Please leave your comments and help us to return the article to featured quality. If concerns are not addressed during the review period, articles are moved onto the Featured Article Removal Candidates list for a further period, where editors may declare "Keep" or "Remove" the article from featured status. The instructions for the review process are here. Reviewers' concerns are here. —Preceding unsigned comment added by Peter Andersen (talk • contribs) 21:59, 6 February 2008

[edit] Discussion at MoS

I strongly support your take on things. Keep it up. If I had discovered WP two years ago, I would be giving you (and a few others) all the support I possibly could. I'm so involved with robotics now that I don't have the time to get into every conversation I want to, but please call on me for support at any time. - Dan Dank55 (talk) 17:04, 7 February 2008 (UTC)

[edit] MoS

The tone of the conversation at the MoS page is going downhill quickly. Part of it may be because of polemical accusations of "power-grabbing" ([8]). It might be better to move the personal conversation to user talk pages, so that the MoS page can remain productive. — Carl (CBM · talk) 14:22, 8 February 2008 (UTC)

It wasn't polemical. I called it like I saw it. --Trovatore (talk) 16:53, 8 February 2008 (UTC)

[edit] Strategic voting in Canada

In the absence of any good answer to the question at hand, I'll just ramble.

I think I can best address the parenthetical point. I believe the talking heads predict a Tory majority if a general election were held now, so the opposition doesn't want to trigger an election. With the opposition unwilling to trigger an election by having the Govt lose a vote of confidence (IIRC, this happens automatically if the Govt loses a vote where budgeted money is involved). The view is that the party perceived as being responsible for a new election will be penalized by voters, who apparently don't want to endure another Federal election again so soon after the last one. So Harper casts all possible votes as being votes of confidence (e.g., don't ask me explain how that works, I'm not sure that it does...) and the opposition parties fall over each other to comply...

As for why there's a preference for minorities over coalitions, I *think* the Governor General is obliged to make the offer of minority rule to the party with the most seats after an election, and that coalition govts may only form after such an offer has been declined (clearly, if I wasn't such an unpatriotic nomad I'd either know the answer, or care enough to look it up). An alternative answer, perhaps more game theoretical, is that if voters have a bias to voting out incumbent parties (especially ones that claim to be in power but are perceived as being impotent), then if a coalition looks like it has little chance of accomplishing party goals, that more long-term benefit exists by waiting for the next election and campaigning as the party of change, rather than participating in a coalition.

Cheers, Pete.Hurd (talk) 18:46, 8 February 2008 (UTC)

Thanks, this does make it a bit clearer. I was wondering the extent to which the Harper government does favors for the NDP in exchange for their continued abstention, and whether it was reasonable to describe the government as being in effect a Tory-NDP coalition. Which to be honest doesn't sound that farfetched to me, as those strike me as the parties most afraid, as Mencken might have said, that someone, somewhere, is enjoying himself. --Trovatore (talk) 22:23, 8 February 2008 (UTC)

Naah, I wouldn't describe it as Tory-NDP coalition, more like three kids at the edge of the pool each daring the others to jump in, but with only the Tories wanting to be in (so long as they're not first). I think it's more like the NDP and Liberals united in a desire to postpone another election, acting outraged by the Tories, but suddenly retreating to not-so-outraged-afterall, lest they seem compelled to bring the Tories down. I don't see any desire by the Tories to do favours in return for staying in minority... My (totally biased, I'd stop it if I could) impression is that the prospect of someone else forcing an election makes them swagger with a Bushian Innercircle Neo-con kind of an entitlement to power. It's brinksmanship, how far can the Tories go before an election call looks like their fault? While the NDP and Liberals are in a race of the slow retreats: how meek can they be without appearing totally craven compared to the other?

I'm still hoping to come across a RS source for a clear definition of Strategic Voting. I think it needs a civics textbook, I don't have those on my bookshelves. Cheers, Pete.Hurd (talk) 00:23, 9 February 2008 (UTC)

Yeah, a source would be good. The article isn't bad, except that I still don't think the "compromise" style vote is "strategic voting", unless the strategy is to vote for the person or party you intend to elect. Usually a good strategy, but not really a very subtle or deep one. By that definition it's "strategic voting" any time you don't write in the name of the single person in the world you most want to win, whether he's running or not. I do admit there are different levels here though -- for example in my scenario above with the NDP sympathizer who votes Conservative, he actually does intend to elect the Conservative MP, just not the Conservative Party. And I'm not sure how I'd describe it if you voted against your party because you thought they'd become complacent and a few years in the wilderness would make them stronger in the long run. But just simply voting for your second choice because you think your first choice can't win--that just seems too ordinary and prosaic to call "strategic voting".

By the way, "neo-conservative" is another term that seems to mean something different in Canada, and I'm not sure exactly what. But I don't think it has much to do with The Weekly Standard. I talked to someone at York about it once and he suggested it meant someone who had conservative views without coming from the sort of family that one would expect to have conservative views. (That's at least more specific than what it seemed to mean at UCLA, where students seemed to think that "neo" was just part of the word "conservative".) --Trovatore (talk) 04:20, 9 February 2008 (UTC)

Yeah, I think variation in the definition of neo-con is more a function of it changing over time, rather than geography... I also thought of that infinite regress argument: "I've never heard of a candidate I agree with on all issues" must mean that all voting I've ever done must be "strategic"... but that being said, it is as you say here in multiparty-landia "Strategic voting" is used to mean the ordinary and prosaic thing you've identified. Cheers, Pete.Hurd (talk) 05:06, 9 February 2008 (UTC)

[edit] Thanks for doing all the work

...on the User:Btyner {{lowercase}} spree. Robert K S (talk) 05:52, 19 February 2008 (UTC)

[edit] Measure-theoretic question

Hey, Trovatore, how 'ya doin? I just posted a Measure Theory 101 question about the definition of measurable functions. Since that discussion page doesn't seem to be the world's most active, I thought I'd explicitly ask if you'd mind having a look and fielding my freshman befuddlement. Regards, PaulTanenbaum (talk) 01:51, 28 February 2008 (UTC)

Looks like Carl already got there. --Trovatore (talk) 02:33, 28 February 2008 (UTC)

[edit] Powerset

Do not blank this article again without discussion! Powerset (no space) is a notable company, and this is the proper spelling. A disamb link to Power set at the top is proper and appropriate. The correctly-spelled company is more notable than the incorect spelling of the mathematical principle. - Realkyhick (Talk to me) 21:52, 5 March 2008 (UTC)

At least give me some time to move to "Powerset (company)" or something similar before you go off and delete the whole article, for pete's sake! - Realkyhick (Talk to me) 21:54, 5 March 2008 (UTC)

Please let me handle this!!! You've screwed up umpteen links as it is. I'll put your precious math article back in place. I don't want "(search engine developer)" as the disambiguation - it's too verbose. I want "Powerset (company)" instead. - Realkyhick (Talk to me) 21:59, 5 March 2008 (UTC)

Dude, if you had taken a moment to see what I had actually done in the first place, we wouldn't be in this mess. I didn't delete anything -- I moved the article, then retargeted the redirect. --Trovatore (talk) 22:01, 5 March 2008 (UTC)

[edit] Image copyright problem with Image:Los Altos.ogg

Thank you for uploading Image:Los Altos.ogg. However, it currently is missing information on its copyright status. Wikipedia takes copyright very seriously. It may be deleted soon, unless we can determine the license and the source of the image. If you know this information, then you can add a copyright tag to the image description page.

If you have any questions, please feel free to ask them at the media copyright questions page. Thanks again for your cooperation. NOTE: once you correct this, please remove the tag from the image's page. STBotI (talk) 07:39, 17 March 2008 (UTC)

[edit] Lebesgue measure help -- thanks

Thanks for sorting my math problem. Very interesting. I see that you're a Caltech alumnus? I'm a mathematics undergraduate at Caltech (Avery), although I'm currently taking a year off to do research at (coincidentally) University of York (the one in the UK, not Toronto). Eric. 81.157.254.39 (talk) 15:32, 24 March 2008 (UTC)

[edit] reference search - invention v. discovery of mathematical objects

On the complex numbers page, I changed a claim that the complex numbers were "invented" to say they were "discovered", and someone asked for a reference for that... I can find lots of references for either view, but I don't know of any good reference that explains the issue of invention v. discovery with suitable mathematical perspective, but is accessible for an untrained reader. I have a sense you are more well-read on these things than I am - do you happen to know any good explanation of the various philosophical threads that underly the question of invention v. discovery? — Carl (CBM · talk) 02:52, 10 April 2008 (UTC)

No, sorry, I can't think of any references on that. --Trovatore (talk) 02:53, 10 April 2008 (UTC)

In this context it might be worth mentioning that even Paul J. Cohen, a self-avowed formalist, said that he "discovered" rather than "invented" forcing. He wrote a paper with the title "The discovery of forcing". (But this might have been due to modesty, not to his actual beliefs.) --Aleph4 (talk) 12:28, 10 April 2008 (UTC)

See my note on the talk page. In particular:

John Cunningham, 1965, Complex Variable Methods in Science and Technology, Van Nostrand, New York, no ISBN, card catalog number 65-20159.

This is a very dense book (on page 11 he discusses Jacobians). But he starts off his Chapter 2 Complex Numbers like this:

"The concept of an 'imaginary number' such as the square root of minus one was first introduced, with great scepticism, late in the sixteenth centruy. Mathematicians and physicists soon discovered that the device led to many simplifications in difficult problems, and having proved its worth in practical situations the imaginary number became a reputable and powerful mathematical tool...

" 2.1 The Square Root of Minus One The art of mathematics is largely concerned with symmetries and patterns. Let us pick up any school textbook on the solution of quadratic equations. we are likely to find the following type of statement 'The equation x^2 - 9 = 0 has two roots x = +/-3, but the eqution x^2+9 has no roots'. The mathematician finds this sort of assymetry rather unpalatable ... this can in fact be achieved by adding to the real number system the imaginary number which is the square root of minus one [etc -- he goes on to discuss the solutions to ax^2 + bx + c = 0].... In this way the oriignal notion of an imaginary number gives way to the concept of a complex number.(p. 27)

So: Like any scientific theory, the notion of square root of minus one was proposed, tested in real applications -- solution of the quadratic, etc -- and found good. This led to further developments (extensions) of the method and its general acceptance. Bill Wvbailey (talk) 16:13, 10 April 2008 (UTC)

[edit] Inaccurate Template

Hi I'm an italian user. Forgive me but I wanted to put in the page President of the Council of Ministers of Italy that you have review a Template that shows only a little revision of page's contents regard recent events, but I have made an error. Unfortunally I have inserted that template in some pages of english Wikipedia, but I'm not able to find all them. I salute thanking you for your patience. By --151.48.167.52

Not a problem. I hope my edit summary didn't sound too agressive. --Trovatore (talk) 22:51, 16 April 2008 (UTC)

[edit] Bitter Almonds

There seems to be a large discrepancy here - one article claims that bitter almonds are from a variant of almonds, the other that they are in fact apricot kernels. Various other sources seem to call apricot kernels bitter almonds too. Perhaps they are both referred to as bitter almonds, and both contain amygdalin. This needs further investigation. Halogenated (talk) 16:52, 17 April 2008 (UTC)

This page [[9]] states that bitter almonds are in fact a variant of almonds (Prunus dulcis), where as this one [[10]] displays three types of bitter almonds, one of which is referred to as apricot kernels. This page [[11]] mentions that both bitter almonds and apricots kernels (and other plants in the rose family) contain amygdalin. Looks like it may be a case of apricot kernels also being referred to as bitter almonds. This should probably be noted, but a more rigorous examination of the nomenclature from a historical/scientific POV is definitely in order. Halogenated (talk) 16:59, 17 April 2008 (UTC)

investigation. Halogenated (talk) 16:52, 17 April 2008 (UTC)

They certainly both contain amygdalin. They are obviously closely related, both being drupe kernels. I did see somewhere a reference to apricot kernels being called "bitter almonds" in culinary usage, which is not implausible; perhaps a hatnote should be put at the top of the almond article, something like

But for the amygdalin article what we need to find out is what Robiquet actually used as a raw material. If it was almonds then the bitter almonds link, redirected to almond, is fine; if it was apricots, then in my opinion the text should say apricots, not just link to it. --Trovatore (talk) 17:05, 17 April 2008 (UTC)

[edit] Automaton group

I replied to your message.

Basically, singular sounds fine. I'll make the plural a redirect. It won't be until July at any event. JackSchmidt (talk) 00:09, 14 May 2008 (UTC)

Sounds great. Looking forward to finding out what we're talking about. --Trovatore (talk) 01:12, 14 May 2008 (UTC)

RUTH ELLIS Hi - Mr Bickford's first name was John. 81.131.59.220 (talk) 11:55, 31 May 2008 (UTC)