Talk:Gödel's incompleteness theorems/Arguments

message: I'd like to inform you that the two sections of 3 and 4 missing in Hirzel's paper are now copy and paste ready for people who are interested in translating it completely. It can be found here, avoiding the difficulties in moving text from the original pdf-document to a translation-ready document: http://whatiswritten777.blogspot.no/2012/09/a-part-of-godels-paper-on-two.html . Cheers! 46.9.87.116 (talk) 03:35, 11 January 2013 (UTC)
thread_title:  The Two Last Sections of the Gödel Paper, Sections 3 and 4 - Copy-and-Paste-Ready 

message: I'm going out on a limb here and present a historical context RE what Goedel meant by "true", "provable" and "formal system". Observe that it has nothing whatever to do with ZFC, except that ZFC is one of many "formal systems"; in fact his standard model appears to be "arithmetic" computed by Turing machine:
I remember [altho I now think my memory was planted there by space aliens] challenging Trovatore [and/or Carl] to find the word "true" in Goedel’s 1931 and indeed [this was confirmed . . . at least this is how I remember the dialog going, caution: aliens at work. Indeed] much to my chagrin [the world "true" does appear] in at least three places (cf van Heijoort p. 598, 599 (twice), 612). There are two places of particular interest:
(1)In the intro (page 599), Goedel says outright that he's going to "replace the second of the assumptions" -- this being "that every provable formula is true in the interpretaton considered" -- "by a purely formal and much weaker one", and then where he actually does this: 
(2) Theorem X where Goedel asserts that:
:"Theorem X. Every problem of the form (x)F(x ) (with recursive F) can be reduced to the question whether a certain formula of the restricted functional calculus is satisfiable (that is, for every recursive F we can find a formula of the restricted functional calculus that is satisfiable if and only if (x)F(x) is true." (cf van Heijenoort p. 612)
So I wondered what kind of "true" this was and Trovatore assured me it means true. So what is this true, what does it mean -- as a philosopher I know precisely what it means to me, but what did it mean to Goedel? Although reputed to be a Platonist, we know that the young, tentative Goedel at the time (1931) was quite sensitive to Formalism and Intuitionism, indeed he states in a discussion of Theorem VI that it is “constructive . . . [that it] has been proved in an intuitionistically unobjectionable manner”. Back then this would mean that he can exhibit the objects that he's constructed, without use of the LoEM over the infinite, inside his formal system. 
In his Princeton lectures of 1934 he uses the words "true" and "provable" freely: cf "On Undecidable Propositions of Formal Mathematical Systems" at "§7: "So we see that the class α of number of true formulas cannot be expressed by a propositional function of our system, whereas the class β of provable formulas can. Hence α ≠ β and if we assume β ⊆ α (i.e. every provable formula is true) we have β ⊂ α., i.e. there is a proposition A which is true but not provable. ~A then is not true and therefore not provable either, i.e. A is undecidable." (page 64-65) in The Undecidable.
By 1946 (cf Undecidable p. 84) he states that "with this concept [of computability by general recursion or "Turing's computability"] . . . one has for the first time succeeded in giving an absolute defintion of an interesting epistemological notion, i.e., one not depending on the formalism chosen.*" He added this footnote in ca 1965: *"To be more precise: a function of integers is computable in any formal system containing arithmetic, if and only if it is computable in arithmetic, where a function f is called computable in S if there is in S a computable term representing f"].  
Furthermore, that by 1963 we know that Goedel came to accept that “due to A. M. Turing’s work a precise and unquestionably adequate definition of a formal system [Footnote here: “ . . .formal systems in the proper sense of the term, whose characteristic property is that reasoning in them, in principle can be completely replaced by mechanical devices”] can now be given . . . that is, it can be proved rigorously that in every consistent formal system that contains a certain amount of finitary number theory there exist undecideable arithmetic propositons and that, moreover, the consistency of any such system cannot be proved in the system” [Note added 28 Aug 1963 to his 1931; cf van Heijenoort p. 616].
In Vol II of Collected Works, Gregory H Moore observes: "The Platonist views put forward by Goedel in 1947 were strengthened in 1964 . . . where he described himself as "someone who considers mathematical objects to exist independently of our constructions" ". Moore then quotes from the supplement to the 2nd edition (1964) where Goedel "pursued at some length the analogy between mathematics and physical theories" in which "sense preception . . . induces us to build up physical theories and to expect that future sense perceptions will agree with them . . .." Goedel's philosophy can be found on page 268 of vol. II. It's too involved to quote in full here, but here is a snippet: "Evidently the "given" underlying mathematics is closely related to the abstract elements contained in our empirical ideas. It no means follows, however, that this data of the second kind, because they cannot be associated with actions of certain things upon our sense organs are purely subjective, as Kant asserted. Rather they, too, may represent an aspect of objective reality, but, as opposed to the sensations, their presence in us may be due to another kind of relationship between ourselves and reality." (p. 268) [emended] Bill Wvbailey (talk) 20:17, 15 April 2013 (UTC)
replies: text: Just a little note:  It wasn't I who did the literature search Will is talking about.  I have never read Goedel's 1931 paper.  My guess is, it was Carl, but I'm not sure of that either. --Trovatore (talk) 21:58, 13 April 2013 (UTC)
replies: text: Actually I miswrote: it was I who followed up by finding the four places mentioned (there may be more). This is at least the third round of argument concerning this issue of "truth". There's one on 8 September 2010 that Tkuvho started off with a question about truth and numbers, then you, Carl, and I did indeed have a dialog around this issue of "truth". Then there's another long dialog on 12-16 July 2011 around challenges from editor Drgao; it's actually very interesting to read after reading the above, which is pretty much a repeat. I'm not sure how many more times we can beat Theorem X. Bill Wvbailey (talk) 20:17, 15 April 2013 (UTC)

text: It is well-known that Goedel was a Platonist. User:Franp9am's question concerns how this page could be presented from a non-Platonist viewpoint. Tkuvho (talk) 15:39, 10 April 2013 (UTC)
thread_title: historical context

message: Gregory Chaitin [“Dangerous Knowledge” BBC4 documentary. 2007.] famously complained about the triviality of basing something as important as incompleteness on such a ridiculous solipsistic sentence saying
:"[Gödel’s argument for incompleteness] was too superficial. It didn't get at the real heart of what was going on. It was more tantalizing than anything else. It was not a good reason for something so devastating and fundamental. It was too clever by half. It was too superficial. [It was based on the clever construction] “I'm unprovable.” So what? This doesn't give any insight how serious the problem is." 
Carl (talk) 16:07, 23 April 2016 (UTC)
replies: text: Gödel's proof is what it is. And it's correct. this is indisputable. it can clearly and unambiguously be explained and described. whether or not it's superficial or important or anything else is a matter of pure opinion. the term "incompleteness" was never even used in his paper, by the way. if someone's notion of "incompleteness" needs to be based on something other or beyond Gödel's paper, so be it. It's irrelevant.68.48.241.158 (talk) 21:54, 4 May 2016 (UTC)
thread_title:  Chaitin's critique of Gödel's approach to incompleteness theorems 

message: As the article currently points out: No doubts about the consistency of mathematics are resolved by the proof of the formal consistency of mathematics.
As Professor Meyer pointed out in his review of Vol. 52 of Studies in Logic, the existence of such a formal proof is primarily of philosophical interest.  Also, as pointed out by Meyer, the existence of a proof of formal consistency means that there is no reason to suppose that it is impossible to prove constructive consistency.  Unfortunately, such a proof seems to be very difficult.
Carl (talk) 16:25, 23 April 2016 (UTC)
thread_title:  No doubts about the consistency of mathematics are resolved by the proof of the formal consistency of mathematics. 

message: I am not a Wikipedia editor, so I will not correct this myself since I do not know the guidelines. However, I do have expertise in mathematical logic. Most Wikipedia articles on mathematical theorems have a section titled "precise statement" which gives an accurate statement of the theorem. This one does not. The statement of the first incompleteness theorem in the opening paragraph is fine for lay readers but is not a precise statement of the theorem for two reasons:
First, "whose theorems can be listed by an effective procedure (i.e., an algorithm)" should be replaced with "whose axioms are decidable" or "with a general recursive set of axioms." Second, "proving all truths about the arithmetic of the natural numbers" should be replaced with "proving all true statements of first order Peano arithmetic". I can basically tell what is intended by the statement that is there now, but it is too sloppy to be considered a precise statement of the theorem in my professional opinion.
I do not necessarily think the statement in the opening paragraph should be changed, but I do think a section should be added with a precise statement of the theorems, as seems to be standard Wikipedia practice.
:Of course, a theory can be axiomatized with a decidable set of axioms if and only if the theory can be axiomatized with a recursively enumerable set of axioms. So there is no need for the first replacement. Also, no theory with a recursively enumerable set of axioms can prove all true statements of Peano arithmetic - the set of true sentences of arithmetic is not r.e. - so the second suggestion would not be an improvement, and would be incorrect.  I do think that "proving all truths about the arithmetic of the naturals" is not very clear, but the suggestion goes too far. Of course, there is a better statement already under the section "First incompleteness theorem" for those who read beyond the introduction. — Carl (CBM · talk) 23:57, 24 September 2017 (UTC)
thread_title:  Article does not give a precise statement of the theorem.