Talk:Kurt Gödel

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statement of theorem[edit]

"any self-consistent axiomatic system powerful enough to describe integer arithmetic will allow for propositions about integers that can neither be proven nor disproven from the axioms. "

integers, or natural numbers? Gödel's incompleteness theorem says natural numbers.

-- John Joseph Bachir 23 sept 2004 (after taking a formal language and automata exam...)


Was Kurt really Austrian-born? I think this will always be problematic. You could probably also say he was Czech-born or Austro-Hungarian-born. As far as I know Brunn was at that time part of Austria which was in turn part of the Austrian-Hungarian empire. I'm not sure how Czech or Austrian the parents of Kurt were (or considered themselves) but the fact that he was sent to a German-speaking school may be a hint. Anyway, if you think you have good arguments to change this, please do. :-)

-- Jan Hidders July 10 2001

Der Herr Warum[edit]

I don't know German very well but is Der Herr Warum correct ?

I speak a little German, (I watched a lot of Sesamstrasse as a child :-)) and it is certainly correct. You can check for yourself:

-- JanHidders

Goedel Number[edit]

What is Goedel Number ? Taw

A Goedel numbering is a scheme (with certain nice properties) which associates logical formulas with numbers, so that instead of talking about strings like "(phi or psi) -> tau " you could talk about numbers that prepresent them instead. Once a particular Goedel numbering is fixed, a Goedel number of a particular logical formula/statement is the natural number that represents it according to the numbering. Why? --AV

Because some page on wiki (Light Bulb Jokes) has link named Goedel Number that points to Kurt Godel page. Taw

The links would best point to Gödel's incompleteness theorem where the concept is explained. Or we could write a separate article. --AxelBoldt

I'd support a separate article, it will make linking easier, and sometime somebody may want to talk about Godel numbers without getting into the whole incompleteness theorem. Perhaps the Godel Number page could just be a semi-short definition with links to Kurt Godel, and to the Incompleteness theorem, that way if there are other uses for Godel numbers than the proof of the incompleteness theorem we could have links to those pages as well. It certianly seems like there should be other uses for Godel numbers, but this is not my area and I don't really know anything about them... MRC

You're right, there're other uses, although they may be too advanced for Wikipedia. I agree that it should be a (short) article on its own. --AV

Computer vs Computable[edit]

It also implies that a computer can never be programmed to answer all mathematical questions.

I'm not sure this is the case. It implies that you cannot choose a formal system and then simply work out all its consequences, and as a result get the answer to all mathematical questions -- thus it proves that one potential way of a computer answering all mathematical questions doesn't work. But in the general case it's an open question whether computers are in principle capable of more or less intelligence than humans, and so this can only be said conclusively if either the AI question is resolved, or it is shown that it is in principle impossible to answer all mathematical questions (whether the answering is done by a human, computer, or something else). Delirium 04:07 1 Jul 2003 (UTC)

It's misleading. 'Answering all mathematical questions' is like running through a recursively enumerable set - can be done if you have an infinite supply of CPU cycles and don't mind waiting infinitely long.

Charles Matthews 04:37 1 Jul 2003 (UTC)

I have removed the sentence Charles Matthews objects to but not because it is wrong. The far more general point is true. The theorem does not only imply that computers cannot answer all mathematical questions; it implies people cannot either and, more than that, it implies that some mathematical questions are unanswerable. The sentence I have removed was written by someone who does not fully understand this theorem. Godel is often trotted out to support an anti-AI point of view, I suspect that that is what has happened here.

Psb777 09:44, 10 Feb 2004 (UTC)

I respectfully disagree. The remark is IMO correct and relevant and therefore should stay. What the motives were of the one who wrote it is simply irrelevant. In fact, it could very well have been me that put it there, and I hold no such view. Removing correct information from an article in Wikipedia requires more justification than that. -- Jan Hidders 17:30, 12 Feb 2004 (UTC)

You would be right if the "correct" thing I removed was not just a small part of the truth. But I said it was not wrong which is not quite the same thing as saying correct. There are a lot of consequences of Godel's theorems, the interpretation I removed was not wrong but it was misleading. Why are not all the consequences of the theorem listed? [Because there are pages for the theorems!] Why this one (sub-)consequence? If the comment goes back then the general point must be what is replaced, not one that is needlessly computer specific. Paul Beardsell 07:16, 13 Feb 2004 (UTC)

I don't agree that it is needlessly computer specific, and I would argue that it is the most important consequence from which almost all other consequences follow. In fact, it is essentially equivalent with the first theorem, so calling it "a small part of the truth" is, well, a bit misleading :-). Moreover, it illustrates why this is such an interesting theorem, so it certainly has its place there. If you don't like how it is worded, then by all means reword it, if you think it is too specific then make it more general, but removing statements from Wikipedia should always be done with the greatest care. So, since we have to stick to NPOV I will put it back and reword it a little so it reflects a bit more your point of view, even though I in fact disagree. Let me know if you find this unacceptable. -- Jan Hidders 10:28, 13 Feb 2004 (UTC)

I like your new wording. What part of it do you disagree with? And I'm being needlessly argumentative, now that you have crafted wording with which I agree, but in what way is the statement "It also implies that a computer can never be programmed to answer all mathematical questions" not computer specific? And, this quesion from interest only, do you think that the brain is capable of evaluating a super set of the algorithms which a computer can evaluate? Paul Beardsell 14:23, 13 Feb 2004 (UTC)

Good, I'm happy you like the new wording. What I myself don't like about it, is that it now is a bit academic and abstract. The answer to your last question is "extremely unlikely and without any evidence whatsoever". However, I don't think there is a definitive proof that shows that a human brain or all humanity as a collective cannot do noncomputable things. -- Jan Hidders 13:05, 14 Feb 2004 (UTC)

There isn't a definitive proof that there isn't reincarnation either. Paul Beardsell 01:04, 16 Feb 2004 (UTC)

There is however definitive proof that this discussion is over, if ever it started. ;-) Remember that Wikipedia is not a discussion forum and contributions should be made in the spirit of cooperation. Trying to lure people into little debates is usually not very productive. Good luck with your other contributions to Wikipedia. -- Jan Hidders 23:45, 16 Feb 2004 (UTC)

This page is the discussion forum for the article. Paul Beardsell 23:11, 17 Feb 2004 (UTC)

I am disappointed that the discussion has not continued. Upon reflection I agree with Jan Hidders that his new wording is academic and abstract. We have gone from something which was partially correct and perfectly understandable albeit misleading to something which is correct but jargon. I intend to replace the current text as follows. This removes the perceived anti-AI slant whilst maintaining readability. I propose we define computable and to do so elsewhere - and I hope the link I have used is considered adequate.

Comment to Aleph4 march 21 Thanks for your prompt reaction. I was prepared to wait four weeks for the first reader. The word ‘specific’ in my text seems to be misleading. So please omit it. The n in Gödels Z(n), itself not a symbol of System P, stands there for any positiv whole number out of the infinite sequence 0, f0, ff0, fff0, ...... etc. Another error that I just see in my text lies in my description of the number representations: evidently, the symbols f have to be put in front of the symbol 0 (zero) and not x ! Sorry, I must have slept! The symbol y in Gödels Z(y) however is a symbol of the System P, it stands there quite for itself, not for anything else, and, again I have to correct myself, its Gödel-number in Gödels paper is 19, my 13 comes from the Nagel-Newman booklet, from where I anyway assumed the way of writing the formulae to get them on a single line of typing. Yours Ginomadeira

Then: It also implies that a computer can never be programmed to answer all mathematical questions
Currently: It also implies that the set of truths about natural numbers is not recursively enumerable, which means that there is no algorithm that can enumerate all these mathematical truths.
New: It also implies that not all mathematical questions are computable.

Paul Beardsell 03:34, 22 Feb 2004 (UTC)

I do not see any meaning in It also implies that not all mathematical questions are computable. How can a question be computable?

There are two ways (well ... infinitely many really) of phrasing the incompleteness theorem:

  1. There is no axiom system that generates all mathematical truths.
  2. There is no computer program that lists all mathematical truths.

Of course the two are equivalent, but the equivalence is itself an interesting fact. The first of these is already in the article: These theorems ended a hundred years of attempts to establish a definitive set of axioms to put the whole of mathematics on an axiomatic basis... Why not use the sentence It also implies that a computer can never be programmed to answer all mathematical questions for the second? Aleph4 00:30, 11 Feb 2005 (UTC)


This so-called English pronunciation IS nonsense. RickK | Talk 07:32, 19 Mar 2004 (UTC)

OK, but how does one pronounce Goedel? Is it more gurdel than girdel? Paul Beardsell 08:18, 19 Mar 2004 (UTC)

There is no "r" sound in Gödel. There may be none in "gurdel" or "girdel" either for Paul Beardsell but there will be for many English speakers. Informal pronunciation guides like this are problematic, though I understand the need for conveying the sound of the "ö". BrendanH 11:37, 31 Mar 2004 (UTC)
In the absence of a change, I have edited the pronunciation guide. However, I feel my version and the previous version are both worse than nothing. Mine is too fussy, the previous version is simply wrong (because it does not work for many English speakers). Someone should delete both. BrendanH 11:15, Apr 8, 2004 (UTC)
Ok, I deleted both :-) linking to the 'rhotic' page was a nice idea, but apart from still not yielding the correct pronounciation it's also quite unwieldy (AC, 23:04, 9 Apr 2004)

This is all very interesting but I am at a loss: How does one pronounce "Go:dl"? Is that supposed to be SAMPA? Paul Beardsell 15:13, 10 Apr 2004 (UTC)

The correct pronounciation can be heard here. Curiously enough, there are two different pronounciations of which only the one labeled '...godels01.wav' (watch your browser's status bar) comes reasonably close (AC, 16:59, 18 Apr 2004)

That's an awful IPA transcription. The problem is that the template IPAc-en takes the transcriptions from Help:IPA for English, and it doesn't include the phoneme /ɜː/, just /ɜːr/, because Wikipedia IPA reduces the set to the diaphonemes of the English language, and the phoneme /ɜː/ happens to be one that only occurs in non rhotic English variants, and is not considered a diaphoneme, but just a phonetic transcription of a non rhotic English variant. When you try to represent the phoneme /ɜː/ using IPAc-en, it transforms it to the diaphoneme /ɜːr/, making it impossible to render the referenced phonetic transcription (/ˈɡɜːdəl/). Am I right? Any idea? --Gradebo (talk) 15:24, 2 February 2016 (UTC)

Let's just not use IPAc-en then.--Anareth (talk) 17:58, 1 August 2016 (UTC)

If it were provable it would be wrong, so one could prove wrong statements in this system.

Is some punctuation missing here ?

Shyamal 11:12, 8 Apr 2004 (UTC)

There is an often-repeated story about Godel's US citizenship interview, during which he began to describe the loophole he had found in the US Constitution, whereby the USA could be (legally) transformed into a dictatorship.

See for example this post from sci.math

1. Is it worth making some mention of this curious biographical detail in the article?

2. Is it recorded anywhere just what this "loophole" was? I have seen the anecdote in a number of different versions, but never any indication of how Godel's discovery was supposed to work.

User:Stuart Presnell

It's very important to view Godel's fears within a historical context, which the author of the entry failed to provide. Godel had just witnessed Nazi Germany be transformed from a functional democracy into a hated dictatorship; and they gained their power partly because of a loophole in the German Constitution that made the Nazi takeover legal on paper, if not in practice.

Godel may have been an eccentric person, but he wasn't just being an eccentric. He'd just seen one country (legally) transformed into a dictatorship. He had no reason to assume it couldn't happen again. Warning the judge about it probably sounded like his civic duty as a potential citizen.

Brno or Brünn? Or both?[edit]

Where was Gödel born?

  1. Brno
  2. Brünn, now Brno
  3. In a city which is now (2005) known as Brno in the English-speaking world, but which in his time (at least by him and his family) was called "Brünn".

I think that (1) is misleading, and (3) is too verbose, so I prefer (2), which really is an abbreviation for (3). Please do not remove "Brünn" without explaining it here. -- Aleph4 23:40, 9 Apr 2005 (UTC)

In fact the only misleading proposal is (2) suggesting nonexistent renaming from Brünn to Brno. He was born in the city called Brno in Czech and Brünn in German. Since his family was German-speaking he most likely called his hometown "Brünn", but it doesn't make the Czech name (from which the German version was once derived) less valid or less English. My proposal (4) is therefore "Brno (Brünn)" with the Czech and present-day English name in the first place and the German name with which he is also associated in parentheses. Qertis 10:09, 18 Apr 2005 (UTC)
I think it should remain in the current form (X in A, now Y in B). This is standard all over the English Wikipedia. We have to include A (here Austro-Hungary) to give information on his nationality at birth. If X (Brünn) was the then-official name of the place, we should give that too, as a matter of record. Charles Matthews 18:37, 18 Apr 2005 (UTC)

I am sure that it was an official name, but perhaps not the (i.e., the only) official one. According to Meyer's 1886 encyclopedia, the city had in 1880 "82660 inhabitants, among them 60% Germans, 40% Czechs, and 5498 Jews". (I see, so Gödel was really German after all, just like Mozart... :-)

But I am sure (again without being able to prove it) that Gödel's certificate said Brünn, not Brno. -- Aleph4 23:41, 18 Apr 2005 (UTC)

proved v proven[edit]

Is this a WP:ENGVAR issue or a WP:COMMONALITY issue? Or just a general don't-make-useless-changes consideration?

In the US, "proved" and "proven" are about equally good when used for the passive or for the present perfect tense ("proven" is required when used as an adjective). If UK readers have a strong preference for "proven" and US readers are neutral, then maybe we should consider going with "proven" on COMMONALITY grounds. --Trovatore (talk) 19:27, 28 March 2016 (UTC)

Ngrams suggests 'proved' is several times more common, both in American and British English. — Carl (CBM · talk) 23:53, 28 March 2016 (UTC)
What query did you use? "Proved" by itself would be confounded by hits for the simple past. I saw strange things when I searched for "has proved"/"has proven" or "is proved"/"is proven" (for example, I sometimes got a yellow warning box saying there was only one hit, which struck me as dubious). --Trovatore (talk) 20:45, 29 March 2016 (UTC)
That's a good point; I just looked at 'proved' versus 'proven', which probably is a meaningless query. — Carl (CBM · talk) 23:31, 29 March 2016 (UTC)

External links modified[edit]

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In the article we have

In 1934 Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey, entitled On undecidable propositions of formal mathematical systems. Stephen Kleene, who had just completed his PhD at Princeton, took notes of these lectures that have been subsequently published.

But in A to Z of Mathematicians by Tucker McElroy, Infobase Publishing, May 14, 2014 page 120

In 1933 he visited the Institute for Advanced Study at Princeton (a mathematical think tank affiliated with Princeton University), where he would spend increasing amounts of time as the political situation in Europe deteriorated. Godel also suffered from mental depression, and he stayed in a sanatorium in Europe in 1934 after a nervous breakdown

Where the article statement comes from?--Vujkovica brdo (talk) 12:50, 30 June 2016 (UTC)

So certainly it's completely in order to ask for a source for the statement, but I don't see too much relevance to the other item you present. There is no contradiction between the two things. 1934 was an entire year; he could have both spent time in a sanatorium and also given the series of lectures. --Trovatore (talk) 22:20, 30 June 2016 (UTC)
I do not understand your comment here. The source I've cited mentioned two years: 1933 and 1934. You are talking about 1934 only and about some relevance.--Vujkovica brdo (talk) 05:51, 1 July 2016 (UTC)
I don't understand your comment. Why do you think the statement by Tucker McElroy contradicts what is in the article? It appears to me that it does not. --Trovatore (talk) 08:49, 1 July 2016 (UTC)
I'm not talking about a contradiction. It's more about completeness and the source saying the same as other sources. Please compare:

Kurt Gödel: Collected Works: Volume I: Publications 1929-1936 by Kurt Gödel, Solomon Feferman, OUP USA, 1986 page 8
Godel visit in 1933-1934 was the first of three that he was to make to the Institute before taking up permanent residence there in 1940. He lectured on the incompleteness results in Princeton in the spring of 1934. Apparently he had already begun to work with some intensity on problems in set theory at the same time he felt rather lonely and depressed during this period in Princeton. Following his return to Europe, he had a nervous breakdown and entered a sanatorium for a time. In the following years there were to be recurrent bouts of mental depression and exhaustion. A scheduled return visit to Princeton had to be postponed to the fall of 1935 and then was unexpectedly cut short after two months, again on account of mental illness. More time was spent in a sanatorium in 1936, and Godel was unable to carry on at the University of Vienna until the spring of 1937
The most obvious reason for this situation was Gödel’s increasing association with the Institute for Advanced Study at Princeton, of which he was a member for three years during the 1930’s; a less obvious reason was the state of his mental health. When the institute first began operation in the fall of 1933, he was a visiting member for the academic year, thanks to the efforts of Oswald Veblen, and from February to May 1934, he lectured on the incompleteness results. There he made Einstein’s acquaintance, but came to know him well only a decade later. Lonely and depressed while at Princeton, he had a nervous breakdown after re turning to Europe in June 1934, and was treated for this condition by the eminent psychiatrist Julius von Wagner-jauregg. that fall he again stayed briefly in a sanatorium (the first stay had been in 1931, for suicidal depression), postponing an invitation to re turn to the Institute for Advanced Study for the spring term of 1935 and informing Veblen that the delay was due to an inflammation of the jawbone. At Vienna, during the summer semester of 1935, he gave a course on topics in mathematical logic, then traveled to Princeton in September, suffering from depression and overwork, he resigned suddenly from the institute in mid-November, returned to Europe in early December, and spent the winter and spring of 1936 in a sanatorium. Veblen who had seen him to the boat, wrote to Paul Heegaard (who was on the organising committee for the 1936 International congress of Mathematicians), urging that Gödel “be invited to give one of the principal addresses. There is no doubt that his work on the foundations of mathematics is the most important which has been done in this field in our time.”
In 1935 Goumldel had made the first breakthrough in his new area of research: set theory. During May and June 1937 he lectured at Vienna on his striking result that the axiom of choice is relatively consistent. That summer he obtained the much stronger result that the generalized continuum hypothesis is relatively consistent; and in September 1937 John von Neumann, an editor of the Princeton journal Annals of Mathematics, urged him to publish his new discoveries there. Yet Gödel did not announce them until November 1938, and then not in the Annals but in a brief summary communicated to the Pro ceedings of the National Academy of Sciences.

Gödel's Theorem in Focus by Stuart Shanker, Psychology Press, 1990 page 7
He first came in 1933-1934 to lecture on his incompleteness theorems at the Institute for Advanced Study, where he spent the academic year. etc etc--Vujkovica brdo (talk) 12:56, 4 July 2016 (UTC)