Talk:Gödel's incompleteness theorems/Archive 3

message: Hey, User:Paul_August, I am curious what fault you found with the article at http://www.mind-crafts.com/godels_incompleteness_theorems.html that made you remove the link to it I put in External Links.  I have to confess that IMHO it's the best non-technical introduction to Gödel's theorems on the Internet.   66.217.176.211 16:55, 16 November 2006 (UTC)
replies: text: Paul, actually, I took a brief look and (at least at that level of scrutiny) it does look pretty good. We don't want to encourage link farming but this one really might be worthwhile. --Trovatore 03:52, 27 November 2006 (UTC)
replies: text: OK, I restored the link - since no arguments were given against it.  Will be happy to listen and discuss if anybody comes up with cogent args.  66.217.179.50 22:45, 3 December 2006 (UTC)
thread_title:  Re: [http://www.mind-crafts.com/godels_incompleteness_theorems.html Gödel's Incompleteness Theorems - A Brief Introduction] 

message: I removed Godel's original statement of the theorem, but it's been reverted back.  Godel's statement:
:To every ω-consistent recursive class κ of formulae there correspond recursive class signs r, such that neither v Gen r nor Neg(v Gen r) belongs to Flg(κ) (where v is the free variable of r).
is only meaningful if you explain Godel's notation (e.g., class signs, Gen, Neg, and Flg), which we don't really need to do in this article.  Perhaps for historical interest, we can have all this in its own section of the article, but it's not useful where it stands now and will just mystify most people.  If there are no objections, I'll remove it again soon. -SpuriousQ 09:55, 27 November 2006 (UTC)
replies: text: Obviously, I disagree or I would not have reverted you. You took out a section header which is inappropriate. And the formal statement of the theorem may be of interest to some people, such as those who have read the paper. If you think that it is too advanced a topic, the appropriate course is to move it to the end of the article and add more explanatory material. JRSpriggs 10:29, 27 November 2006 (UTC)
replies: text: Thanks for the explanation, JRSpriggs. The section header was removed because it was introducing the deleted content and didn't make sense without it.  As for the content itself, I don't think it's a matter of the topic being "too advanced"; it's that it's unreasonable for us to expect our audience to have read Godel's paper (with the German abbreviations intact, at that).  As stated right now, I feel it does more harm than good, especially given the recent remarks about the article's accessibility.  Like you suggest, a possible solution would be to move it to the end with more explanation, which I would be happy with and may attempt when I get a chance. -SpuriousQ 16:12, 27 November 2006 (UTC)
replies: text: With respect to JR, I tend to agree with Spurious here. Gödel's proof was brilliant; his notation, like almost all version-1.0 notation, was less than ideal from the point of view of long-after-the-fact exposition. And even if the notation were nicer, we still should not want to get into the details of a specific coding scheme early in the article, as it's kind of beside the point. --Trovatore 16:27, 27 November 2006 (UTC)
thread_title:  Removal of Godel's statement of theorem 

message: Let me give a shorter sketch. I see no need to define SU. The statement p equivalent to "p cannot be proven" can be easily formalized as p = ~P(p). Now, if p is provable true then the argument p would be not provable (a contradiction). Alternatively, if p is false (that is, not provable true) then P(p) would be true meaning the argument p is provable true (again contradiction). --Javalenok 13:44, 10 January 2007 (UTC)
replies: text: You said "...if p is false (that is, not provable true)...". Falsity is the same as not BEING true. That is different from not PROVABLE. This distinction is essential to understanding Gödel's theorem. There is no inconsistency in the existence of a sentence p which is simultaneously true and not provable. Gödel never proved p, i.e. that p cannot be proved. What he proved was ω-con→p, i.e. that IF arithmetic is ω-consistent, THEN p cannot be proved. JRSpriggs 06:39, 11 January 2007 (UTC)
replies: text: OK, may be I should tell: IF p = false THEN P(p) = true HENCE p is provable true HENCE contradiction (p and ~p) = true. The purpose of the incompleteness theorem is to show that a consistent theory cannot be complete. I'm showing that it is possible to do without introducing the self-unprovable forms. Any objections? --Javalenok 08:47, 11 January 2007 (UTC)
replies: text: Your first step, going from Neg(p) to P(p), is changing context which is logically invalid. You are going from an arithmetic statement to a meta-arithmetic statement. Although Gödel showed that some meta-arithmetic statements can be pushed down into arithmetic, it does NOT go the other way (as you would have it). Also you say "I'm showing that it is possible to do without introducing the self-unprovable forms.". I do not see what you are getting at here. You are still using p in your argument, and p is the "self-unprovable form" to which you object, is it not? JRSpriggs 08:36, 12 January 2007 (UTC)

text:  Look, SU(#F) is defined as ~P(#F(#F)), so SU(#SU) = ~P(#SU(#SU)). Given SU(#SU) = p, we can rewrite the equality as p = ~P(#p). This equality states self-unprovability explicitly and, furthermore, eliminates the needles self-unprovable form SU(#F). The Neg(p) is ~~P(#p). But you deny the logical conclusion ~~P(#p) => P(#p). Instead, you prefer to go to english language. But stating that some self-unprovable can be proved is an outright nonsense. The  dares to state: "Suppose that ~(~P(p)) is a theorem. Hence P(p) is a theorem." --Javalenok 23:32, 13 January 2007 (UTC)
replies: text: You have completely misunderstood my last message. There is no meaningful difference between "P(p)" and "~~P(p)"; and I did not say otherwise. The vital distinction which you are ignoring is between statements in English about arithmetic and statements in the language of arithmetic which might be regarded as (in some sense) encoding the English statements. Although the encoding is chosen with the intention of making them equivalent, they are not (known to be) PROVABLY equivalent either from the axioms of arithmetic or from meta-arithmetic considerations. JRSpriggs 09:24, 17 January 2007 (UTC)
thread_title:  What is the need for self-unprovable formulas? 

message: Moved to Talk:Gödel's incompleteness theorems/Arguments.
: This question is not an "argument over the validity of the incompeteness theorem". Why was it moved there? --Javalenok 20:36, 11 January 2007 (UTC)
replies: text: It's not directly related to improving the article. --Trovatore 21:24, 11 January 2007 (UTC)
thread_title:  Gödel sentence is an objection to AI. Is it true? 

message: While I generally feel that anyone wanting to make such extensive revisions to a high-profile article ought to log in, I think the recent spate of edits by 132.181.160.42 have generally been of high quality. However I can't agree with a couple of the changes to the intro.  Explaining myself here. The anon wrote:
:These theorems show that no consistent formal system can completely describe Peano arithmetic (addition and multiplication over the natural numbers), and that no system capable of representing that arithmetic can prove its own consistency.
This statement is insufficiently precise to be either right or wrong, but it has reasonable readings that are just wrong. Peano arithmetic, for example, is a syntactic rather than semantic construct (as opposed to the structure of the naturals, which is semantic); it's certainly possible for a consistent formal system to completely describe the syntactic manipulations of Peano arithmetic. Moreover there needs to be something said about what sort of formal system (say, that its axioms are computable), otherwise no such conclusion is possible. The anon also says:
:The theorems also have extensive implications for philosophy and cognitive science.
A great deal of bad philosophy and cognitive science has been based on the theorems (so say I, a hardcore metaphysical libertarian and substance dualist who likes the conclusion of the Lucas arguments, just not the arguments themselves). In general any claimed philosophical implications outside of mathematics should be viewed with suspicion; it's not ruled out a priori that no such implications can be drawn, but I have never seen one that worked. I think there's a good bit of possibly accurate stuff that's suggested by the theorems, by analogy as it were, and these may even be a reasonable heuristic guide to truth. But the word "implication" is too strong. These are theorems of mathematics, and we should keep the focus firmly there. --Trovatore 07:34, 26 January 2007 (UTC)
replies: text: I agree that Peano arithmetic is a formal system that adequately formalizes Peano arithmetic.  I added a caveat to the lead - the well known decidable formal systems for geometry and the real numbers are simple, but not "trivial". CMummert · talk 13:36, 26 January 2007 (UTC)
thread_title: Recent anon edits

message: The link to "Askanas, Malgosia, 2006, "Gödel Incompleteness Theorems - A Brief Introduction." A lucid and enjoyable exposition of both theorems in the spirit of Gödel's original proof. "is not a bad site for explaining this theorem--well above the high school student's level though--but its "lucid and enjoyable" presents a value judgement which is not at all consistent with the purpose of the article and Wikipedia. It is a commercial site selling services in math education. It should be retained, in my view, it does a fairly good job of explaining the concepts, but the evaluation needs to be reconsidered. Malangthon 03:31, 27 January 2007 (UTC)
replies: text: I removed the annotation.  You could (and, I would suggest, should) have moved it here along with your comment.  In cases where a phrase is clearly not neutral POV, seems to advertise for a commercial service, or both, there is no reason to leave the statement while inquiring about it. CMummert · talk 03:54, 27 January 2007 (UTC)
thread_title: Commercial links

message: :This section discussed the underlying mathematics issues rather than the article.  It has been moved to the arguments page and can be found here. CMummert · talk 02:16, 29 January 2007 (UTC)
thread_title: is the incompleteness theorem complete?

message: retitled Wvbailey 00:18, 10 October 2007 (UTC)
thread_title: Ademh: Your post "Paradox in Godel's incompleteness theorem that invalidates it" is now at [[Talk:Gödel's incompleteness theorems/Arguments]] 

message: Let A be a set of axioms and U=\{u_1,u_2,\dots\} be the set of sentences undecidable over A.  
Question: is U recursively enumerable?  I.e. is there an effective algorithm for writing down all the undecidable statements of number theory, set theory, etc.?  Does it depend on the particular axiom set?  If the answer is known maybe it can be added to the article.  Obviously you can write down infinitely many undecidable statements by iterating Gödel's diagonalization scheme but I don't know if you can get them all by such methods.  Thanks.
:I assume that by "u is decidable over A" you mean that either u or its negation is provable from A. If I knew that u was either provable or disprovable, then by searching all proofs I could determine which of these two options holds; but if a particular u is neither provable nor disprovable then the search would go on forever.  If the set of undecidable sentences was   alsou recursively enumerable, then I could enumerate that at the same time, preventing my search from going on forever. So if the set of undecidable sentences was enumerable, I could decide, given a sentence, whether it is provable, disprovable, or undecidable, which means that the set of provable sentences would be recursive. But for almost every first-order theory the set of provable sentences is not recursive, but only recursively enumerable. Tracing back, this means that the set of undecidable sentences must not be r.e. — Carl (CBM · talk) 12:28, 30 October 2007 (UTC)
replies: text: Ah yes, of course.  Thanks.  —Preceding unsigned comment added by 75.62.4.229 (talk) 16:11, 30 October 2007 (UTC)
thread_title:  recursive enumerability 

message: (outdent) I think I have a dead-tree source for this, I can't access the whole article tho (not a member) 
:: A Proof of Godel's Theorem in Terms of Computer Programs, Arthur Charlesworth, Mathematics Magazine, Vol. 54, No. 3 (May, 1981), pp. 109-121.  http://links.jstor.org/sici?sici=0025-570X%28198105%2954%3A3%3C109%3AAPOGTI%3E2.0.CO%3B2-1&size=LARGE&origin=JSTOR-enlargePage
CeilingCrash 03:14, 2 November 2007 (UTC)
thread_title:  Rewrite 

message: Please, everyone, quit writing "Godel", even in the talk pages; it's a flat misspelling. You don't have to go to the effort of putting in the ö character -- just write "Goedel", which is correct. The umlaut is not a diacritic in the ordinary sense, but a marker for the missing e. --Trovatore 00:00, 6 November 2007 (UTC)
thread_title: Spelling nit

message: Since the question of original research came up, I looked up a reference for this proof. It dates back at least to Kleene, 1943, "Recursive predicates and quantifiers". Theorem VIII states:
:"There is no complete formal deductive theory for the predicate (x)\bar{T}_1(a,a,x)."
As the T is Kleene's T predicate, and the bar indicates negation, this is exactly a statement about the halting problem. Kleene explicitly says: "This is the famous theorem of Gödel on formally undecidable propositions, in a generalized form."  This paper is available in the book The Undecidable as well. — Carl (CBM · talk) 12:49, 6 November 2007 (UTC)
replies: text: Dude, this argument was never about content. It was about language.Likebox 19:15, 7 November 2007 (UTC)
thread_title:  Reference - Kleene 1943 

message: The talk page was getting very long, so I have moved the discussion
prior to the current active one (about the new 'proof') to Archive03
and added a link in the Archive infobox. Zero sharp 16:17, 6 November 2007 (UTC)
thread_title:  Created Archive 03 

message: We now have three proof sketches in the article: the main sketch of the syntactic proof, the section on computability, and the section on Boolos' proof. I pulled Boolos' article today,  and it does seem worth mentioning briefly. I think we should rearrange things so that all the proofs are in one section. Within that, the first subsection should concern arithmetization, then the syntactical proof sketch, then a section on semantic proofs including the computability proof and Boolos' proof.   Boolos points out in his paper that Goedel gives another semantic proof in the intro to his paper on the incompleteness theorems. The current organization is less than ideal. — Carl (CBM · talk) 16:42, 8 November 2007 (UTC)
thread_title:  Semantic proofs 

message: This article and the separate article On Formally Undecidable Propositions of Principia Mathematica and Related Systems both say that the translation in van Heijenoort's book was done by van Heijenoort himself, but I think that is wrong.  Could someone who has the book please check?
:On p. 595, V.H. says "The translation of the paper is by the editor,...". — Carl (CBM · talk) 22:17, 15 November 2007 (UTC)
replies: text: Hmm, thanks.  I must have been thinking of one of the other translations in the book, probably the Begriffsschrift.  —Preceding unsigned comment added by 75.62.4.229 (talk) 22:26, 15 November 2007 (UTC)
thread_title:  translation in van Heijenoort book