Talk:Gödel's incompleteness theorems/Archive 2

message: "This easy corollary of the second incompleteness theorem shows that there is no hope of proving e.g. the consistency of first order arithmetic using finitistic means provided we accept that finitistic means are correctly formalized in a theory the consistency of which is provable in PA."
What?
Is this what is meant?:
"This easy corollary of the second incompleteness theorem shows that if we accept that the finite means are correct in a consistent theory proved by PA then there is no hope of proving the consistency of PA using these finite means."  129.186.18.149 22:00, 13 June 2006 (UTC)
replies: text: Although a single author (who was paid for the substantial amount of work involved) might be able to rewrite this to make it flow better, it would never be easy to understand. A lot of qualifications and careful wording are necessary because this is an inherently difficult topic. A sloppy presentation would leave you with something like "it is not what it is" which is impossible, of course. JRSpriggs 08:56, 3 January 2007 (UTC)
thread_title: More Understanding

message: If "the human mind is inconsistent", would it be an 'adaptive' trait?
IMHO If it is inconsistent it is because it would not be possible for a mind to be consistent and it would be in a way not relevant for the evolution. --Mandelum 20:50, 21 May 2006 (UTC)
thread_title:  Evolution 

message: When I first took a look at what Wikipedia has to say about Gödel's incompleteness theorems I recoiled in horror; the entry was full of non-sense and confusion. I then took the time to rewrite the first three sections, which resulted in something approximately like http://en.wikipedia.org/w/index.php?title=G%C3%B6del%27s_incompleteness_theorem&diff=18326634&oldid=18326416". I didn't do anything to the rest of the sections apart from explaining the two different senses the term "undecidable" is used in logic. I also wrote a section "A major overhaul needed?" in the discussion section, pointing out various errors and inaccuracies and offered to rework the whole article. This received no comments. Since then I have mostly watched the article slowly detoriate, making a few corrections now and then. The detoriation has not been only because new inaccuracies have been introduced, but also because people have introduced various clarifications that in reality just add confuse, reworded awkward sentences subtly changing their meaning and seem to have felt the need to introduce more technical details than I originally deemed appropriate. Of course, there's nothing wrong in introducing technicalities per se, but when it is done in an ad hoc fashion in middle of explanations on a more general level, I feel the result is confusing to potential readers. And the sections on implications and signifance of Gödel's theorem are still, on the whole, a haphazard collection of more or less relevant, accurate or comprehensible musings. 
Now, this being Wikipedia there's no reason for me to expect anyone to hold my contributions sacrosant. But if I undertake the effort to completely rework this article - as I suggested in the now archived discussion entry "A major overhaul needed?" - I would find it encouraging if upon noticing an apparent inaccuracy or an awkwardly worded phrase instead of editing it right off, people would discuss it here. But, being new to Wikipedia, I've no idea whether this is a reasonable expectation or not. If not, then I'll just crawl back into my hole and leave the article as it is. Perhaps there are some notes or instructions on writing and editing a Wikipedia article on a highly technical topic such as Gödel's incompleteness theorems?
A simple example of detoriation is the evolution of the statement of the first incompleteness theorem. The way I originally worded the first incompletness theorem was not merely a result of my poor English, but because of an important technical distinction. Consider this wording
:Gödel's 1st incompleteness theorem gives us a method to produce from any theory T proving basic arithmetical facts a sentence G_T such that G_T is true and undecidable in T iff T is consistent. 
Now, numerous times the statement of the first incompleteness theorem essentially like the above was reformulated into something like:
:Gödel's 1st incompleteness theorem gives us a method to produce from any consistent theory T proving basic arithmetical facts a sentence G_T which is true and undecidable in T.
A seemingly innocent change improving readability. However, this rewording considerably weakens the theorem to the point of being actually incorrect. For if T is inconsistent, the latter version gives us no information about the truth or falsity of G_T while the former does. Similarly, a well-intending person added a clarification to the note that the sentence G_T produced by the mechanism provided by Gödel's 1st incompleteness theorem is called "the" Gödel sentence of the theory, explaining that there are infinitely many formulas having the stated property of being true but undecidable in the theory. But this property is not at all what entitles a statement to be called a Gödel statement of T! The relevant property is that of the statement's being provably equivalent to its own unprovability in T or a suitable variation thereof (e.g. of Con(T)-->G being provable in T). 
Also, someone apparently thinks that "provable in T" should be replaced by "included in T". If a theory is defined to be a deductively closed set of sentences, this is of course technically correct. A statement being included in a theory is not a concept people in general would be familiar with and the people who know that for a deductively closed set of sentences it means same as provability wouldn't use that phrase in any case. So I think "included in T" is in no way an improvement over "provable in T".
Someone has added a statement explaining the scope of Gödel's theorems
: Gödel demonstrated the incompleteness of a theory of arithmetic, but it is clear that the demonstration could be given for any theory and language of a certain expressiveness.
Why is this necessary? How does e.g. "for any formal theory, it's possible to construct an arithmetical statement..." lead a potential reader to assume the theorem only applies to theories of arithmetic? Of course, if Gödel's original result was discussed it would be in place to mention that it generalized, but as the article is, the remark comes out of nowhere. Etc. Etc. But enough of this whining for now.
-- Aatu
:Some comments:
:#Major rewrites of highly-visible and much-edited articles need to be undertaken more delicately and transparently than ordinary edits. It would be good to put an outline of your proposed changes here first. Then you might do the rewrite as a subpage of your user page, ask for comments, and copy it over once it's accepted. (That's what I did at definable number, except I didn't actually get any comments on the rewrite, so I just copied it after a decent interval. Article still has problems, BTW, but at least it's talking about a well-defined concept now.)
:#Your point on "G_T iff T is consistent" is certainly correct, but I'm not convinced it's the better way to say it. From the point of view of the foundational implications, such as the near-refutation of Hilbert's program and the consequent difficulties for the formalist school, it's not the interesting half of the theorem. If our foundational theories are inconsistent, who cares what happens to their Gödel sentences?
:#I still don't think the "maximal consistent subset" stuff is useful (don't know if that was yours or not). Independently of what sorts of subsets we can talk about, I just can't see any reason anyone would think a formalization of "the maximal consistent subset of T is consistent" is even a candidate for a formalization of "T is consistent".
:#Why don't you sign in? Have you forgotten your password? You could try the "e-mail new password" button at the login screen. --Trovatore 16:50, 1 December 2005 (UTC)
thread_title:  A question about common practice and some whining 

message: Essay moved to User:BenCawaling/Essay. Gandalf61 08:28, 14 April 2006 (UTC)
thread_title:  Self-contradiction in one-to-one correspondence (About the incomplete totality of the set of all prime natural numbers) 

message: The article is supposed to be about the Gödel theorems; I don't understand why there's a long section on Gentzen's result. This should be split off to a separate article. By the way, Gentzen's consistency proof has been on requested articles for a while. One solution would be to just cut-and-paste the current section to create that article, but when I added it to requested articles I was hoping for a bit deeper treatment than is given here. But in any case the section doesn't belong in this article, though the result should certainly be mentioned and linked to. --Trovatore 23:15, 14 February 2006 (UTC)
replies: text: Went ahead and did the cut-and-paste. Someone (Chalst?) needs to give Gentzen's consistency proof some serious TLC. --Trovatore 04:37, 15 February 2006 (UTC)
thread_title: Why is there a section on Gentzen?

message: I propose that that part of the content of Gödel's incompleteness theorems that is specific to the first or the second theorem should be split off to Gödel's first incompleteness theorem or Gödel's second incompleteness theorem, respectively (currently these are redirects to the main article). The remainder of the article, the part that's common to both theorems, would stay here, and the individual theorems would retain short sections with a brief exposition and a  pointer to the "first" and "second" articles.
Reasons:
*The article is currently too long. Splitting off Gentzen's consistency proof helped, but not enough. This is not just a technical issue (that issue is fading as old browser versions fall into disuse); the length is a problem for readers trying to understand, and editors trying to keep the article in good global shape.
*The theorems are quite different from one another in terms of application. The second theorem can almost be viewed as a corollary of the first, but is the one that is usually more important in actual practice in mathematical logic. On the other hand the first theorem is the one to discuss in an elementary discourse on the limitations of formal deductive systems. --Trovatore 19:58, 28 February 2006 (UTC)
thread_title: Proposed split

message: The topic of Gödel's incompleteness theorems is something I have come across occasionally, and as a Major Theory in the History of Mathematicts it seems something I should have a basic handle on.  That said, are there any examples that can be given to speak about?  In truth, the article as written doesn't let me get it.  My background includes advanced calculus from college where we used a textbook by Spivak, a course in basic number theory, and a course where we were studied the Cantor set and Lebesgue integration.  So I'm not exactly a dolt.  However, I think I would need a major leap of sophistication to really understand the article as written.  Given that this is a general encyclopedia, any idea on how to bridge this?  In the spirit of Wikipedia I'd do something myself, but in this case I KNOW I could only damage the article by my efforts.  Steve Kd4ttc 01:46, 8 March 2006 (UTC)
replies: text: You have a good beef, I think. The article should be reorganized to give some context before leaping into the technicalities (in addition to the changes I mention above about splitting out the articles on the two individual theorems). I hope someone will take this on (I'm afraid it isn't going to be me, at the moment). --Trovatore 22:13, 8 March 2006 (UTC)
thread_title:  Mere mortal has a request 

message: I've changed the  template to ; I don't really think this is a "good article" as it stands. (I'm not terribly optimistic that it ever can be, using WP methodology; I'm afraid it's always going to look like a giraffe, which is a horse designed by committee. But maybe it can. It just isn't now.)
I'm supposed to explain which criterion or criteria I think it fails. The most important criterion that it fails is the very first one; the article is not well written. Specifically:
#It doesn't have a good lead section (this is the easiest thing to fix)
#It doesn't follow a clear logical structure.
##Rather than starting with (at least a little) motivation and historical context (say, Hilbert's programs), it jumps straight into a quote from Gödel's paper, using notation specific to that paper and not standardly used.
##The "meaning", "misconceptions", and "discussion" sections all treat roughly the same topic, but in a disjointed manner; they need to be unified into a single section and mercilessly edited for flow.
##The "minds and machines" section does not belong in this article at all; it should be a separate article.
##The first and second theorems have their own sections, followed by their own proof sections further down the article; these need to be moved to Gödel's first incompleteness theorem and Gödel's second incompleteness theorem.
##The "examples" section is irrelevant and should probably just be removed. These are examples of logical independence, not examples of the Gödel theorems.
I think the article does meet most of the other criteria for being a good article. I think it's mostly factually accurate and verifiable (haven't read the whole thing through recently), has broad coverage (too broad, really), is relatively NPOV and stable, and it's hard to see what images could really be added. But the writing is just so bad that it can't possibly be a GA. This is no one person's fault, which is maybe the heart of the problem. --Trovatore 20:22, 8 May 2006 (UTC)
thread_title: delisting as "good article"

message: I'm a statistics undergraduate student who took a philosophy of mathematics course in which we covered the godel incompleteness theorem.  During this course I remember the professor clearly explaining the incompleteness theorem in a way that was relatively easy to understand.  (I don't remember what tactic he used, be it simple examples etc or I'd remember the theorem)
And I STILL can't make heads nor tails of this article having forgotten what the theorem means exactly.  The language is way more overspecialized then it needs to be.
:Since you don't seem to remember how the professor made Gödel simple, may we have his contact address?  I'd love to see this article, and all mathematics articles, greatly improved.  --DryaUnda 19:51, 28 June 2006 (UTC)
thread_title:  Gobblygook 

message: I would like to ask an expert on Gödel's Incompleteness Theorem in what manner or in what way it regards (respects) and deals with Temporal logic? -- PCE 22:46, 21 May 2006 (UTC)
replies: text: These theorems have nothing to do with Temporal logic. Paul August ☎ 23:12, 21 May 2006 (UTC)
replies: text: What form or type of logic then would you classify these theorems as having something to do with? -- PCE 23:52, 21 May 2006 (UTC)
replies: text: Mathematical logic. Paul August ☎ 00:08, 22 May 2006 (UTC)
replies: text: As relating to Mathematical logic then would the theorems have anything to do with or cover such issues as the incorrectness or inability or unavailability of constructs such as x=x+1 or y(x(i)) being expressed using conventional mathematical notation? -- PCE 00:30, 22 May 2006 (UTC)

text:  No. Paul August ☎ 00:51, 22 May 2006 (UTC)
thread_title:  Incompleteness vs Temporal logic 

message: This section has been moved to the arguments page; please continue there. --Trovatore 20:31, 27 July 2006 (UTC)
thread_title:  Reals and Complexes are sets with complete Axiomatizations ? 

message: The article as it stands does not give any complete statement of the hypotheses. The previous section of the talk page shows why this is misleading.   Moreover, the article states that the key property necessary is that the theory be recursively enumerable; but the theory of real closed ordered fields is complete, and thus both decidable and recursively enumerable.  This ought to be fixed.
Later, the article says
:The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics: If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.
This rephrasing may be unsettling for foundations as Hilbert envisioned them, but it is not unsettling to foundations as they are generally considered nowadays (to the extent that they are considered at all).  The sentence confuses foundations in general with Hilbert style foundations. 
CMummert 02:33, 24 July 2006 (UTC)
thread_title:  Suggestions for improvement