Talk:Gödel's incompleteness theorems/Archive 9

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Archive 5

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Archive 7

Vile "we"-word

"The banner was hung on the article with care, in hopes that someone, caring, wouldn't despair [would quickly repair, would actually care]". Observe the egregious passive voice in the preceding sentence. The vile "we" or "you" etc is a writer's dodge to avoid the passive voice and the really affected, awful "one" (as in "One worries about writing 'we' in a sentence." One what? Infant reader? Platypus? Pet rabbit Emily?) I located all the offending "we"-words; there are quite a few in the proof-sketches and following paragraphs. Thoughts about rewording? Is this worth the time? (To avoid changes to the passive voice I have little choice but to substitute "one" for "we" (plus the verb-changes required). Or should I just delete the banner? Bill Wvbailey (talk) 22:09, 8 August 2011 (UTC)

I would go for the passive voice. We is fine in papers but not in an encyclopedia — it's not so much that it's informal as that it suggests an originality that is not claimed nor even allowed in this context. One does not have that flaw, but it's too pedagogical in tone — here we give "just the facts".

There is really nothing wrong with the passive voice. In some contexts it can come across as overly dry, but once we've excluded "original" and "pedagogical", "dry" is pretty much what we're left with. --Trovatore (talk) 22:32, 8 August 2011 (UTC)

I've gone through and eliminated the we-words and the banner. Someone should check the changes, though, especially in the "Proof sketch for the first theorem" section. Bill Wvbailey (talk) 15:10, 9 August 2011 (UTC)

Looks great, thanks. I did make one slight rewording, but the article sounds much better now (and doesn't have the hateful banner). CRGreathouse (t | c) 19:56, 9 August 2011 (UTC)

Expert Needed To Clean This Article!!

I attempted to fix the inaccuracies and vagueness of just the introduction but someone switched it back. But the article as a whole is just a giant, disorganized mess and is filled with much nonsense. There are lots of people out there with expertise in this area. But this topic also attracts lots of people who have no idea what they're talking about. Can an expert please step forward?? And then lock it!!! — Preceding unsigned comment added by 98.224.222.241 (talk) 22:53, 30 August 2011 (UTC)

What are you talking about? The only one who has recently reverted one of your edits is you.

I looked over your edits and at a quick glance had no objection to them and thought they were even a modest improvement, so I left them in place. I didn't consider them large enough changes to get that worked up about in either direction, though (arithmetic is a more precise word than mathematics, but from my POV mathematics that can't even do arithmetic isn't very much mathematics, so to me this is not a huge change). --Trovatore (talk) 23:03, 30 August 2011 (UTC)

Without any hocus-pocus

So, after having read the page let me see if I got it straight. Without any hocus-pocus, the Godel sentence is in essence no more than a formal statement about natural numbers. Is this correct? Southttext (talk) 04:45, 30 October 2011 (UTC)

That's what it is, in fact. The essence is something else again. — Arthur Rubin (talk) 07:41, 30 October 2011 (UTC)

Thanks, in fact sounds indeed better. And 'true' as in 'true but unprovable' is as much as, and no more than, that the arithmetical fact 1+1=2 is 'true'? Southttext (talk) 12:32, 30 October 2011 (UTC)

Pretty much. The only difference is that the incompleteness theorem looks at a sequence of equations of the form F(n) = 0, where F is a particular concrete function from the natural numbers to the natural numbers, and n ranges over the natural numbers. The incompletness theorem shows that if the theory of arithmetic at hand is consistent then each of these equations is true (in the same sense as the equation "1+1=2"), but the single sentence "For every n, F(n) = 0" cannot be proved in the theory of arithmetic at hand. — Carl (CBM · talk) 20:02, 30 October 2011 (UTC)

Polynomial

My first impression is that this edit is overly detailed for this article, and perhaps for Wikipedia in general. What do others think? Possibly something about it could be summarized in a sentence or two. --Trovatore (talk) 01:11, 9 December 2011 (UTC)

Yah, you think! I couldn't believe my eyes . . . no explanation at all, 36+/- symbols on the first line alone (one of which has an exponent to the 5^60)^2, let alone 10 more lines of the same yielding about 350-400 symbols in a string that ends in =0 [about 3000 bits of ASCII code]. If the intent is to be ironic, it's out of place. Our ultimate role here, (Bill's version), is to make the hard easy, the complex simple, the inscrutible scrutible. Yes, if someone can gracefully, elegantly show how a Diophantine equation can encode a machine-like process, and from this one can prove incompleteness or undecidability, this would be a worthy endeavor. Bill Wvbailey (talk) 02:45, 9 December 2011 (UTC)

Addendum: This sort of thing should remain in a textbook, or perhaps go into its own sub-article with significant development around it. As I noted above: can we use Diophantine equations to prove incompleteness or undecidability? Chaitin 2005 Metamath! makes a rather noble stab at this cf pages 37-45, and Franzen 2005:70-71 briefly discusses the MRDP theorem. An example of really big numbers in texts: In Penrose's 1989 Emporer's New Mind on pages 71-73 is the complete listing in binary of a universal Turing machine code in 75 +/- lines of 74 +/- 1's and 0's. That's a lot of bits (5500 +/-). On page 56-57 he writes down a number (base 10) that he claims is the "denary" equivalent number of his U. All well and good to provide some literary pungency or a bit of constructive evidence (it's unclear what Penrose's point is otherwise), but Penrose has spent over 20 pages developing the notion of a (Post-)Turing machine and showing one encoding of a U [O.R. alert, he could cut his bit-count in half.] Bill Wvbailey (talk) 16:15, 12 December 2011 (UTC)

I removed the explicit polynomial, and just mentioned that it was given in a paper. VladimirReshetnikov (talk) 19:33, 12 December 2011 (UTC)

2012-1-14

Concerning this edit [1], I agree with David Eppstein's revert:

The claim about Hilbert's program is in line with NPOV, and there is no problem with using primary sources there. There is a nice summary of the contemporary literature at the article Hilbert's program, and "widely but not universally" is a good NPOV summary of that. I think that removing the "but not universally" might give the false impression that the matter is resolved. For example, we say that the Church-Turing thesis is "widely" accepted in its article.
I don't see any need for an additional source for the literal statement of the halting problem, in a paragraph that already has several sources where authors use the halting problem. At some point, we have to accept that standard terms like "halting problem", "computable function", "group", etc. do not need an extra source every time they are mentioned, any more than common terms in any other field would.

— Carl (CBM · talk) 20:41, 14 January 2012 (UTC)

I agree with both of the above. Someone already fixed "halting problem". I restored "but not universally" in the lede since I think it's sufficiently discussed and referenced toward the end of the section "Meaning of the first incompleteness theorem". I don't think there's any actual controversy over the statement, so leaving it uncited in the lede seems fine to me. The existing reference "Hilbert's program then and now" (Zach 2006) also discusses the issue. 67.122.210.96 (talk) 19:37, 22 January 2012 (UTC)

Arithmetically Incorrect does not mean Inconsistent

The section on the first theorem says If G were provable under the axioms and rules of inference of T, then G would be false but derivable, and thus the theory T would be inconsistent. This is a bit misleading, since omega inconsistency does not entail inconsistency and I don't see how we can conclude that T would be inconsistent without using the Rosser construction.--141.70.13.41 (talk) 18:41, 21 February 2012 (UTC)

Hmm, well, it says it's an informal analysis (and as such I can accept that it might not capture every possible subtlety), but yeah, maybe we can tweak the wording a bit. Any suggestions? 67.117.145.9 (talk) 04:00, 27 February 2012 (UTC)

There had been an edit back on October that changed that section. There were a few other issues with the wording, so I changed it back to the original wording. Then I added another sentence to try to point out again that the analysis in that paragraph is informal. — Carl (CBM · talk) 11:44, 27 February 2012 (UTC)

Thanks.--141.70.13.75 (talk) 12:28, 7 March 2012 (UTC)

philosophy

I'm thinking of adding a mention of Neil Tennant if I can figure out what he is saying. He has a list of publications here and this book blurb looks relevant to the discussion of truth and knowability in the article. I just came across a link to his page and haven't looked at much of it yet. I wonder if anyone here has any thoughts. 67.117.145.9 (talk) 01:43, 27 February 2012 (UTC)

From your links, I don't see anything very directly relevant. He seems to have done a lot of work in the general area of things that people interested in the incompleteness theorems might also be interested in, but this article is about the incompleteness theorems, not about that general area of things. --Trovatore (talk) 16:48, 27 February 2012 (UTC)

Relationship with computability

Given the below, it must be clear that the halting problem occurs when non-meaningful input has been programmed or that the computer is running an inifinite set, one issue that should be calculated by the machine itself before running of the input happens! Remember that most testing of these things happen on "scientific" computer, the big mainframes, Tevafloppies and more, i.e., the supercomputers, and as such, qualifying the input by looking for inifinite input should be no problem! Because there is a significant difference in running input directly vs. checking for infinity input before running the input, i.e., the programming in the loose sense. 62.16.241.158 (talk) 20:31, 18 June 2012 (UTC)

modern developments

I can't edit the page due to lockdown, but I was interested to come across some recent (1962) developments that are not widely know in the popsci version of Godel. Could someone with edit access please add a link to this up to date textbook, which covers them and answers "what happened to godel's theorem"... http://www.amazon.co.uk/Inexhaustibility-Non-Exhaustive-Treatment-Lecture-Notes/dp/1568811756/ref=reg_hu-rd_add_1_dp (The book covers the work in Turing's PhD thesis and the 1962 work of Feferman, which show how to "grow" logics sequentially by adding Godel sentences as new axioms. These systems are then complete in the sense that for any theorem, there exists some extension of the system which can prove it. I'm not an expert logician though, so it would be facinating to hear any comments on this from people working in the field -- is that the last word in the Godel story or are there still open questions?) Charleswfox (talk) 12:34, 2 July 2012 (UTC)

Recent edits

An editor has recently been attempting to add a paragraph to the paper with the following reference:

Thompson (2012), "Arithimetic Proof and Open Sentences", Philosophy Study 2 (1) 43-50.

This seems very dodgy to me. The title is misspelled. The very few web sites I found mentioning it give the same volume and page number data but strangely differing publication years, and appear to all have been written by the same person. The web site for "Philosophy Study" [2] looks amateurish (e.g. no editorial board) and has no volume 2 (any attempt to go to a list of papers beyond the first issue gets a page not found error). This web site warns that the journal's publisher may not provide true peer review, although the comments give varying experiences. The author is not identified with a specific enough name to be able to verify that he or she exists. I don't like deleting the reference without being able to actually read it, but it seems in this case there is no alternative. Can someone give me evidence that this paper actually exists and has any value? —David Eppstein (talk) 07:19, 20 July 2012 (UTC)

The journal itself appears to be very dodgy. Google suggests the author's name is "Neil Thompson" but I cannot find him or tell what school he is at. I cannot find a preprint of the paper, and even if I wanted to buy it the publisher's web site is broken and won't show it. There is an abstract online, which says the following:

If the concept of proof (including arithmetic proof) is syntactically restricted to closed sentences (or their Gödel numbers), then the standard accounts of Gödel’s Incompleteness Theorems (and Löb’s Theorem) are blocked. In these standard accounts (Gödel’s own paper and the exposition in Boolos’ Computability and Logic are treated as exemplars), it is assumed that certain formulas (notably so called “Gödel sentences”) containing the Gödel number of an open sentence and an arithmetic proof predicate are closed sentences. Ordinary usage of the term “provable” (and indeed “unprovable”) favors their restriction to closed sentences which unlike so-called open sentences can be true or false. In this paper the restricted form of provability is called strong provability or unprovability. If this concept of proof is adopted, then there is no obvious alternative path to establishing those theorems.

Of course the strange thing about that is that the usual Goedel sentence is a closed sentence, although perhaps the paper itself has some explanation. I think that in this case the best option does seem to be to remove the reference from this article until the paper itself can be located. — Carl (CBM · talk) 11:48, 20 July 2012 (UTC)

The Neil Thompson paper (2012) is freely available online here, as a PDF. The ideas are not without interest, but it presumably it is too early for this work to have found recognition within the field. If you use Google Scholar to look for 'neil thompson gödel', it locates the paper but does not find any citations to it, or any other logic-related work by this Thompson. I wouldn't propose including it here unless there is some evidence that someone who knows mathematical logic has reviewed it. EdJohnston (talk) 15:55, 20 July 2012 (UTC)

Thank you for finding that - I couldn't. I agree with what you are saying. My impression, which is hard to see an argument against, is that the journal is serving as a press for hire which exerts minimal editorial control. The author is apparently a lawyer, for example, rather than a professional philosopher or mathematician. Now after having looked through the paper I agree with the removal of the section here. — Carl (CBM · talk) 17:35, 20 July 2012 (UTC)

Mathematics self-proves its own consistency

Carl Hewitt showed that mathematics self-proves its own consistency. See his article "What is Computation? Actor Model versus Turing's Model" in "A Computable Universe: Understanding and Exploring Nature as Computation" edited by Hector Zenil (University of Sheffield, UK). World Scientific. May 2012.

Hewitt recently presented the result at logic seminars at SRI and Stanford University.75.18.225.160 (talk) 20:46, 21 July 2012 (UTC)

Thompson's Article

I was one of the editors who made a reference to NeIl Thompson's article in the January Edition of Philosophy Study. This is a peer reviewed journal with an editorial board. The website is bit clunky but says

Philosophy Study is a new monthly peer-reviewed journal published across the United States by David Publishing Company, El Monte, CA, USA in print (ISSN 2159-5313) and online (ISSN 2159-5321), covers all sorts of research on Metaphysics, Epistemology, Ethics, History of Philosophy, Philosophy of Science, Philosophy of Language, Philosophy of Religion, Philosophy of Mind, Political Philosophy, and other relevant areas, and tries to provide a platform for scholars worldwide to exchange their latest findings. All papers considered suitable for the journal are reviewed anonymously by at least two referees who may be external to the Reviewers or Editorial Board (That is, many other editorial members and referees' personal identifying information and CV are not available at this site).

Those persons who have been removing the reference to it should read the article first before making irrelevant criticisms. Some of the people criticising it have not shown any signs of reading let alone coming to grips with it.

Explaining it simply as far as I can what is said in the article by Thompson follows. This can be easily confirmed by looking at the article. Hopefully this might allow those interested to come to gips with it rather than saying silly ad hominem things.

Only closed sentences can be made the subject of proof predicates. So brackets are neither provable or not provable in the sense that a sentence saying that a bracket is provable is unsyntactical. (Proof in this sense is always a series of closed sentences from an agreed sentence or axiom to a conclusion ie the (closed) sentence to be proved.)

An open sentence (a sentence with an unbounded free variable) is not provable (or unprovable) either. Such open sentences (as Quine says) are true of things but not true by themselves. Some schematic open certain sentences look like they are provable in the sense that all sentences in the schematic form are provable but are not strictly speaking provable. However Godel's sentence is certainly not a 'schematic' sentence of this kind.

That's 'provable' in the ordinary sense. How about 'arithmetic(al)' proof? (As for the spelling check OED 2nd ed Vol 1 p631 -either spelling as an adjective is correct.) Godel's arithmetisation proceeds on assigning certain numbers to all the symbols of an amalgam of Peano Arithmetic and Principia Mathematica. The resulting system of numerical symbols are meant to be in an isomorphic relationship with the sentences of that system itself.

Godel treats any formula as being potentially the subject of both ordinary or arithmetic proof. This is, according to Thompson, contrary to ordinary and justified usages of 'provable' or 'unprovable'.

Godel proceeds to construct his proof by constructing a sentence expressed in arthimetic terms. Go and read this in Godel's own article in Feferman's Complete Works Vol 1 at 151.

Godel points out the the open sentence 'x is unprovable' can be constructed in arithmetic terms. It therefore has its own godel number. So far so good according to Thompson.

The Godel sentence is next constructed by substituting that number for the free variable in the original sentence (or its arithmetic counterpart).

This looks like a closed sentence because the godel number of is qua number a constant. But if that number signifies anything it signifies the number of an open sentence containing an unbounded or free variable.

According to Thompson, arithmetic probability or unprovability is no different to ordinary proof. Saying ' "x is unprovable" is unprovable.' is unsyntactical but so is saying 'the godel number of 'x is unprovable' is arithmetically unprovable.' The Godel 'sentence' does say it itself is unprovable as a non sentence it says nothing. Its just a formula with limited significance.

If the Godel sentence is unsyntactical it is neither provable nor unprovable --- simply a formula. This does not disprove Godel's theses but shows that a different mode of proofwill be required of them.

The same analysis according to Thompson applies to Lob's Theorem (the Theorem based around Curry paradox ideas) that allow us, according to Boolos' Computability and Logic (p225) to prove Santa Claus exists.

Now I find Thompson's arguments compelling but certainly not yet established by any consensus. . I await comment from anyone who thinks he can decisively refute them. Nothing in the way of criticism on the site I have seen so far carries any weight in that direction.

Fernando Fernandodelucia (talk) 03:21, 23 July 2012 (UTC)

ps A link which should take you to the journal's editorial board & etc is http://www.davidpublishing.com/journals_info.asp?jId=680l

Fernando Fernandodelucia (talk) 04:21, 24 July 2012 (UTC)

Fernando,

Thanks for bringing Thompson's article to the attention of the community. It's a legitimate paper that ought to be referenced in the Wikipedia article.

The critical point is in the construction of the fixed point, a Gödel number is substituted for x in x is not probable to produced the "self-referential" sentence. However, Wittgenstein showed that the resulting system is inconsistent. Hewitt traced the source of inconsistency to allowing the use of fixed points to construct such "self-referential" sentences in "What is Computation? Actor Model versus Turing's Model" in "A Computable Universe: Understanding and Exploring Nature as Computation" edited by Hector Zenil (University of Sheffield, UK). World Scientific. May 2012.

13.7.64.123 (talk) 00:04, 27 July 2012 (UTC)

Do sentences have fixed points?

Hewitt addressed the issue discussed above in his article cited here in the section previous to this one. Gödel used the fixed point construction to create the sentence "This sentence is unprovable." Similarly, the fixed point construction is used in creating the paradoxical sentences of Curry and Löb. Hewitt has defined sentences in such a way that these monsters by Gödel, Curry, and Löb do not exist because in general fixed points do not exist for sentences, just as they do not exist for the simply typed lambda calculus.75.18.229.45 (talk) 23:31, 23 July 2012 (UTC)

Löb's sentences (use to derive Löb's theorem) and Curry sentences amount to much the same thing. It is very odd that the Löb sentence exactly parallels Godel's'.

Fernandodelucia (talk) 07:04, 31 July 2012 (UTC)

Thompson's Article

I think it is about time that the group who so persistently removed the reference to Thompson's article justify their position.

It is clear that Philosophy Study whilst new is a peer reviewed journal.

If the editor's position is justifiable they should be able to produce a critique showing that what is claimed is simply not credible or in any way tenable.

Otherwise it must go to mediation. Seven more days seems time enough.

Fernandodelucia (talk) 01:23, 30 July 2012 (UTC)

It's not clear that Philosophy Study is peer reviewed in any genuine sense. Even if it was peer reviewed, it is not clear that Thompson's article, being extremely new, is of any real interest to the field. If other papers were to comment on Thompson's work, that would strengthen the argument for including it here. Lacking that, several editors including me are unconvinced that the article is of sufficient interest to include here. That is an editorial judgement - every Wikipedia article has to choose among all the available references to decide which to include.

Separately, although this is somewhat less important, the diagonal lemma produces a sentence (with no free variables), and the Goedel number of that sentence is what is inserted into the negation of the provability predicate to form the Goedel sentence. No open formula with free variables is used. Perhaps Thompson is looking at some other proof of the incompleteness theorem, or perhaps he has misunderstood it. I think we need to wait to see whether anyone else finds Thompson's argument of interest. — Carl (CBM · talk) 01:58, 30 July 2012 (UTC)

In response, To claim its not peer reviewed youhave to have some basis to do so. Where is it?

As far as the diagonal lemma is concerned have a look at Thompson's treatment of that - see page 5 of my copy part 6: "The Boolos proof (Boolos 2007, 221) of the diagonal lemma depends on diagonalization producing a sentence in every case including those involving provability predicates. So if strong provability/unprovability is adopted, sentences containing provability predicates constitute exceptions to a measure of generality of the diagonal lemma. And if the diagonal lemma is not valid for provability predicates, Boolos and Smullyan’s accounts of Gödel’s incompleteness theorem results cannot be obtained any more than Gödel’s own.

It does seem to be a matter of reading the article and not skipping through it.

And how many peer reviewed articles to you need to validate an article. A strange idea that seems hard to apply.

Fernandodelucia (talk) 02:54, 30 July 2012 (UTC)

Was this ever taken to mediation? The statement "every Wikipedia article has to choose among all the available references to decide which to include" doesn't hold apply here, as this very long article devotes very little space at present to dissenting views. It is not for us to debate the merits of the viewpoints, as Carl does above - that is for the readers of wikipedia. Readers are not likely to do this if they are not exposed to the viewpoints in the first place. Paul1andrews (talk) 01:04, 14 September 2012 (UTC)

Note and for Pedagogics of the Article

The blunt point of Gödel therefore, for these two incompleteness theorems, is that Prime Numbers are ("most likely") infinite, but that there´s no mathematical formula for identifying them and that, as they are wholly, from a point "up there" unknown in the outset, Prime Numbers themselves constitute, at least in part and as phenomenon, these two incompleteness theorems and that, of course, consequently, Gödel claims that the theorems are definitive, truthfully obtained as shown by his paper, i.e., his use of "proof". LFOlsnes-Lea (talk) 05:30, 13 September 2012 (UTC)

I don't want to remove what appears to be an attempt to critique this article, as well as the theorems, but what do "Prime Numbers" have to do with the incompleteness theorems. Gödel may have used a prime number representation for the coding of strings (and then formulae) by numbers, but other representations are also acceptable. — Arthur Rubin (talk) 06:04, 13 September 2012 (UTC)

It proceeds very well from the article of Gödel´s paper what his considerations of prime numbers do as an example out of the mathematical reasoning by "Sätze 1 - 11"(?) and "Formeln". You do not appear to have read the paper very well (26 pages) and I also note the common practice of this by other famous people such as Schrödinger´s use of a cat in an experimental set-up to illustrate fundamental problems in his field, physics, as opposed to Gödel´s field in this paper, mathematics, in the "Monatsch. für Mathematik und Physik", i.e. "monthly journal for mathematics and physics". Alright? I hope you can carry the rest through yourselves as I have limited interest and I will (conseq.) not bother with more input/responses. Cheers! LFOlsnes-Lea (talk) 07:16, 14 September 2012 (UTC)

Not too many mathematicians really read that paper anymore, except for historical interest. I am confident there is no serious mathematical problem with the paper, though I have not personally verified that, but in terms of notation and presentation it's far from ideal. Those things have been tidied up considerably in the intervening years, and there is little reason for anyone now (at least, anyone interested in the math rather than the history) to struggle through Goedel's original notation. --Trovatore (talk) 10:04, 14 September 2012 (UTC)

One "unexplainable" smacker for Gödel

One last smacker for Godel: All axioms are needed to establish a (logical) system - Premise

All axioms - Premise

-----------------------

Logical System (Cond. Elim.) Conclusion

You can add the extra reiteration for classical premises, deduction and conclusion to obtain yourself. (Really LFO-L, but...) 62.16.241.158 (talk) 11:48, 24 June 2012 (UTC)

The section of "Limitations of Gödel's theorems" now seems entirely correct so objections are removed. Alright? LFOlsnes-Lea 17:30, 23 June 2012 (UTC)

The issue under "Limitations of Gödel's theorems" isn't whether "Gödel could use logics too", but whether

1. y is the Gödel number of a formula and x is the Gödel number of a proof of the formula encoded by y 2. y is the Gödel number of a formula THEN 3. Bew(y) = ∃x (x is the Gödel number of a proof of the formula encoded by y) AND therefore DEFEATS Gödel's two theorems because the above completely describes the disposition of the field given, i.e., the disposition of logics, the UoD, by Everything (in Mathematics), entities and so on, all the way up to "the whole of mathematics and so on", best seen by "Bew(y) = ∃ x ( y is the Gödel number of a formula and x is the Gödel number of a proof of the formula encoded by y)", being the defeated part, under "Construction of a statement about "provability". So the war is on between Gödel/Gödel followers and Gödel critics, whereof I am one. This is the reason that Gödel stands against Tarski in the intellectual World today! But criticism has to be met and we'll see. LFOlsnes-Lea 20:31, 23 June 2012 (UTC)

Remember also that Gödel's two theorems refer directly to Principia Mathematica by Whitehead and Russell, providing the start point for Gödel's work on this, as given by the article. LFOlsnes-Lea 20:38, 23 June 2012 (UTC)

(Removal to below, under meaninglessness of Liar's Paradox!)

Despite the impossibility to come up with a formula for all prime numbers (unifiedly), they can however, as they are named, be checked for whether they are prime numbers or not, by algorithms (starting with numbers 2 and 3) and that although the formula probably never can be reached, the whole thing will evidently marked off for the mystery it is, as for so many other natural phenomena. Good? LFOlsnes-Lea (talk) 04:21, 18 September 2012 (UTC)

On the Liar Paradox and more

Generally the liar paradox is shown to be meaningless (now). Next, Tarski and others hold powerful arguments against Gödel's incompleteness theorems. And there is a set theory that you may want to note, http://whatiswritten777.blogspot.no/2011/08/philosophical-notes-of-intellectual.html (a bit down on the page), that has this text: "For the time being, I have this to write. Out of 'I know nothing and my set is empty! Can you call illusions knowledge? I don't think so! What is it to know? I have absolutely no idea! To "know" has been assigned to me! Thanks, Russell, for pointing out the danger of having a single proposition of knowledge!' [Ed.], I think the set theory that breaks the Principia Mathematica can be solved by S = Ø (set of solution is empty [Ed. by mathematical mark for empty, Ø]). In case of protest, one should remember that one object/member lower down the hypothetical chain of sets (by categories) triggers necessary objects/members all the way up to the "first natural level where one would otherwise see an empty set right below it". "The first natural level" can also be seen as "the deepest level" before, if any at all, the empty set can occur." "You can add all the (meaningless) categories/set containers you want under a natural set/one set that contains members, but where do you get when the bottom container is empty? Clearly, it's just rubbish and thus it's not a serious argument against the project that Principia Mathematica represents." That is, by this explanation, that the maximal number of empty sets under the natural chain of sets, can only be 1, one, but usually is 0, zero, by the usual descriptions of commoners and non-mathematicians. This, thus, represents the final solution to set-theory for all time to come. Good? I am of the opinion that criticism should be presented on the same page under the header "Criticism of the Gödel's incompleteness theorems" because this is about presenting the truth. That is, you can't leave out the fact that his incompleteness theorems may be untrue! (I'm sorry this entry has been looking rather ugly.) 62.16.241.158 (talk) 20:37, 18 June 2012 (UTC)

For people who think that to make a title "This is not a title" on a book (Raymond Smullyan, fx.) matters, you do not do much other than positing a Austin statement, that is, you commit a speech act, NOT logics! So now, finally, us who belong to the group who are searching for "system completeness", we are to have successful days in the months and years to come, possibly even Gödel, from the "other" side, being a Christian. Cheers! LFOlsnes-Lea 03:30, 24 June 2012 (UTC)

However, Gödel still defeats these other "lunatics" who say that they have these axioms and that this system therefore has to generate these and other results, in that definite sense, so Gödel is a winner in these other respects! LFOlsnes-Lea 03:21, 24 June 2012 (UTC) LFOlsnes-Lea 03:30, 24 June 2012 (UTC)

Hi, 62.16.241.158. This page is not a forum for general discussion of Gödel's incompleteness theorems; it is a place to discuss how to improve the article. Do you have a concrete proposal? If not, you might consider asking a question at the mathematics reference desk. --Trovatore (talk) 19:29, 18 June 2012 (UTC)

Hi Trovatore. Thanks for asking. I'll consider it. See you! 62.16.241.158 (talk) 20:11, 18 June 2012 (UTC)

In the mean time, please refrain from publishing your research on this page. Please see WP:TALK. --Trovatore (talk) 20:54, 18 June 2012 (UTC)

To Trovatore. I hope you consider the notion of "significant views" and that still after, by disagreement, you can REMOVE whatever you like, as I totally submit to the policies of Wikipedia, all from the 5 pillars and onwards. Cheers! LFOlsnes-Lea 03:47, 24 June 2012 (UTC)

I've now suggested the line "One should note that the Liar Paradox as such may just as well prove meaningless and that, by this, Gödel's two incompleteness theorems are in jeopardy [by implications of comment literature if not his own paper too]." to the article. Is it alright? LFOlsnes-Lea (talk) 02:48, 19 September 2012 (UTC)

No, that's just completely incorrect. The Goedel methodology is not a formalization of the Liar. It's an analogue of the Liar, but with truth replaced by provability — this makes all the difference, because provability is expressible in the object language of the theory being studied, but truth is not. --Trovatore (talk) 03:37, 19 September 2012 (UTC)

Thanks. However, the Liar's Paradox is well cited with the secondary literature, I believe, and I do have the patience for this to mature as to discussion. As with so much else, let's see what turns out! Cheers! LFOlsnes-Lea (talk) 04:11, 19 September 2012 (UTC)

It's probably true that you can find good citations for connecting the theorems to the Liar paradox, and you can find other good citations for the view that the Liar is meaningless (that's probably the dominant view, I'd guess). You can't put them together to claim that the Goedel theorems might be wrong! That's original research by synthesis, and this is an excellent example of why this is disallowed, because that's a totally invalid inference. --Trovatore (talk) 04:22, 19 September 2012 (UTC)

To my knowledge, validity of inference isn't decided upon by King Trovatore. No, it is supposed to be backed by literature, as with conventions throughout this encyclopedia, the Wikipedia, and insofar as literature (which is for now uncertain) tell of the Liar Paradox as crucial for these incompleteness theorems then of course, the standing on truth and validity (numbers or not, "objects" or not) are then decided upon by plausibility (by sound judgment, i.e., from the experts!!! I think you're standing on the head today, I'm sorry! Over to "others". Trovatore, you can answer others. LFOlsnes-Lea (talk) 05:45, 19 September 2012 (UTC)

The fact that the inference is invalid, although true, is not really the point. You aren't supposed to make your own inferences at all. Well, I mean routine ones, yes, you have to be able to paraphrase. But you're not allowed to make your own conclusions by synthesizing the sources into something they didn't say. --Trovatore (talk) 07:09, 19 September 2012 (UTC)

To some extent, you're dead right here, but when it's so easy as to say "Liar's Paradox - meaningless?" why should we wait for these "important heads" to come around with a book for it? And as much as input or airing a thought, why leave the article potentially untrue and less intelligent than it can be? There are probably blogs now mentioning this already and I find it "picky"/counter to free speech to grant these people "exclusivity to truth", so to speak! Well, well, if you delete, you delete, I have no strong opinions, much less feelings for this! Cheers! LFOlsnes-Lea (talk) 11:50, 19 September 2012 (UTC)

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I can only assume from the above that LFO has not read the quotation from Goedel that I submitted on 16 September. Goedel did not shy away from discussing his theorems in the context of the paradoxes, so a thorough reading of his words, published at different times, is recommended, in particular the following: (1) the original paper where both Richard's and the liar paradoxes are mentioned, (2) his difficulties with Paul Finsler, trying to convince the man that Finsler's proof was rubbish (you'll need to get the appropriate volume of Collected Works; its either in the published letters or unpublished letters, I can't remember which, plus the commentary is excellent), (3) the following from the 1934 Princeton lectures. Keep in mind that the following is Goedel's own words and reflects his own thinking. And you can't beat that for background. I'll repeat the salient parts: from Goedel's 1934 "On Undecidable Propositions of Formal Mathematical Systems §7. Relation of the forgoing arguments to the paradoxes." Goedel is delivering a lecture, and speaking very clearly and carefully to an audience: "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. As I recall (I'm away from my books) Goedel is discussing this in the context of the Liar Paradox, i.e. right before this quotation. BillWvbailey (talk) 16:05, 19 September 2012 (UTC)

I see the unbeatable background, not to mention Gödel's own pedagogical skills (and from the pedagogics section a bit down on this page) it is very clear that he seeks his case in terms of prime numbers as the practical case out of his knowledge of mathematics, that the case he makes for prime numbers settle the issue definitely, he seems to think, in proving the 2 theorems. We've just reached a kind of anti-climax here and basically just waiting (at least I am, and probably for 2 months and more). But thanks for the recommendation on "Collected Works", that's brilliant! So I leave you to it! Cheers! PS: Of course, his original paper of 26 pages (per PDF document) has been read thoroughly and I think the case is clear. However, today's standing is most unclear as one of the two targets of "incompleteness" and "completeness" (by Tarski) can still be reached at the expense of the other, one achieved, the other demolished! LFOlsnes-Lea (talk) 16:54, 19 September 2012 (UTC)

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For future use either for this article or in the Finsler stub-article or both: cf Kurt Goedel Collected Works Volume IV Correspondence A-G, Oxford University Press. Two entries re Finsler's failed proof (This proof can be found in van Heijenoort, with commentary).

Draft letter (1970, not sent) to Yossef Balas, wherein Goedel discusses why Finsler's proof failed --

"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 the "truth" of the propositions of a language cannot be expressed in the same language, while provability (being an arithmetical relation), can. Hence true ≢ provable.

". . . he (Finsler) omits exactly the same point which makes such a proof possible, namely restriction to some well-defined formal system in which the proposition is undecidable. . . . If Finsler had confined himself to some well-defined formal system S, his proof (by replacing nonsensical section 11 with a proof that the proposition in question is expressible in S) could be made correct and applicable to any formal system." (p. 10-11).

Letters back and forth between Finsler and Goedel, with commentary by John W. Dawson, Jr starting at page 405ff.

After Turing's proof of 1936-1937 presented a formal, mechanical "syntactic" system (being just mechanical operations, no "truth" is found anywhere in it) and yet had enough structure to support arithmetic, Goedel would eventually embrace the TM as the best candidate of the three available (Church's lambda calculus, Goedel-Herbrand recursion, and Turing's machine). For reference, Goedel asserted this twice, and these can be found in addenda to articles in both The Undecidable and van Heijenoort. BillWvbailey (talk) 10:37, 20 September 2012 (UTC)

Undecidable statements provable in larger systems

This arose out of a discussion I had with other mathematicians. Realised there is a fair bit of confusion about Godel's "true but undecidable" sentence, with many assuming its negation must be a possible candidate for a true mathematical statement just because it can be added without formal inconsistency.

So I felt it improved the clarity of the article to put these sentences into a separate section easily found by readers of the article. They are good clear examples of statements that are formally undecidable, but also according to natural ways of thinking of things, "true" statements as well, which helps clarify the status of the Godel sentence.

It wasn't so easy to spot them on the page when mixed up with other types of undecidable statement in roughly historical order. Also is a distinct category of undecidable statement so deserves a section of their own for that reason too.

Hope this is acceptable. Robert Walker (talk) — Preceding undated comment added 10:43, June 30, 2012

Criteria for inclusion of critiques

I added a reference to a critique by James R Meyer that was removed by trovatore on the basis of it being "amateur". I don't think wikipedia rules prevent references to articles written by non-professionals? trovatore can you please provide further justification for the removal, or else reinstate the edit?

Paul1andrews (talk) 00:47, 14 September 2012 (UTC)

I agree with Trovatore that the article should not be mentioned in this article. The incompleteness theorems are somewhat unique among mathematical theorems in drawing lots of these sorts of papers. But the article itself should stick to the best possible references, which are professionally published works and well-regarded popularizations. By way of analogy, imagine if the article on Christianity tried to mention large numbers of amateur papers on that religion. We usually don't have that sot of problem with mathematical articles, but this one is an exception. There are indeed criticisms that are widely discussed in the literature and are worth mentioning - but there are also far too many amateur (and, too often, crank) papers on the incompleteness theorems.

There are many criteria that we can balance to decide what references to use here. Has Meyer's work been mentioned or cited anywhere? Was it published in a professional journal with a reputation for quality peer review? It Meyer's work self published instead? Has Meyer been invited to speak about his work at any conferences? In short, is there any evidence that Meyer's work is of interest to the community of incompleteness theorem scholars? — Carl (CBM · talk) 01:42, 14 September 2012 (UTC)

The article and its references do not need to be of interest to "the community of incompleteness theorem scholars", it needs to be of interest to a general readership. Naturally the community of incompleteness theorem scholars has a vested interest in supporting the theorem, and therefore should not set themselves up as gatekeepers of an independent reference source. Paul1andrews (talk) 02:48, 14 September 2012 (UTC)

See Wikipedia:Fringe theories. "A theory that is not broadly supported by scholarship in its field must not be given undue weight in an article about a mainstream idea." ... "A conjecture that has not received critical review from the scientific community or that has been rejected may be included in an article about a scientific subject only if other high-quality reliable sources discuss it as an alternative position." —David Eppstein (talk) 04:13, 14 September 2012 (UTC)

Thanks David for your clear answer - on the basis of this I agree that Meyer's work should not be referenced at this time. Paul1andrews (talk) 04:49, 14 September 2012 (UTC)

The point of the pages David Eppstein quoted is that sources we use here do need to be of interest to the community of incompleteness theorem scholars, just as sources cited in the article on Christianity need to be of interest to the community of academic scholars of Christianity. There are certainly mainstream scholars who criticize the theorems - the article mentions several of them - but it would not be appropriate for us to include every criticism with the idea that the reader can decide for themselves. The reader doesn't want us to barrage them with criticisms that nobody who knows what's going on would take seriously. — Carl (CBM · talk) 11:01, 14 September 2012 (UTC)

Yes, understood. Thanks. Paul1andrews (talk) 13:37, 14 September 2012 (UTC)

Another Published Critique

Professor Carl Hewitt published an important paper in the Turing memorial collection volume that made the following points:

The first theorem can be simply proved in a metatheory of mathematics using computational undecidability of inference. See Entscheidungsproblem
The second theorem is wrong: mathematics easily self-proves its own consistency.

“What is computation? Actor Model versus Turing's Model” in A Computable Universe: Understanding Computation & Exploring Nature as Computation. Edited by Hector Zenil. World Scientific Publishing Company. 2012. PDF available here. 50.131.244.2 (talk) 17:09, 16 September 2012 (UTC)

The above results were presented at professional academic seminars including the following:

SRI: April 24, 2012
Stanford: June 6, 2012 with slides here

75.144.241.166 (talk) 17:21, 16 September 2012 (UTC)

Thompson

This discussion is going nowhere, and will continue to go nowhere unless/until Fernandodelucia stops arguing with people and provides reliable secondary sources about this material (if they exist). By now it seems clear that he is unwilling to do either of those things, so in an attempt to preserve the civility of the discussion I'm closing this section of it. Please don't add to it unless your addition contains new information about article content or sources, and is neither about user behavior nor merely a repetition of what has already been said here. —David Eppstein (talk) 04:56, 15 May 2013 (UTC)

The following discussion has been closed. Please do not modify it.

Since a section on Neil Thompson, "Arithmetic Proof and Open Sentences" in Philosophy Study volume 2, no. 1, pg. 43-50 (2012) is being added repeatedly and putting emphasis on the fact that nobody has refuted it, I note that Philosophy Study is a journal found at http://www.davidpublishing.com/journals_info.asp?jId=680 , that the other articles posted in it are not mathematical articles, and that the publisher is not one of high repute; cf. http://chronicle.com/forums/index.php?topic=81342.0 . It's not a journal MIT gets, which brings up the question of whether anybody else does, and if anyone has seen it in the field to critique it.--Prosfilaes (talk) 06:50, 1 May 2013 (UTC)

The editors involved have shown remarkable restraint with this. It was only after months of abuse that a block warning has finally been placed at User_talk:Fernandodelucia. Hopefully that will suffice. Tkuvho (talk) 08:46, 1 May 2013 (UTC)

~~ince~~ a section on Neil Thompson, "Arithmetic Proof and Open Sentences" in Philosophy Study volume 2, no. 1, pg. 43-50 (2012) is being added repeatedly and putting emphasis on the fact that nobody has refuted it, I note that Philosophy Study is a journal found at http://www.davidpublishing.com/journals_info.asp?jId=680 , that the other articles posted in it are not mathematical articles, and that the publisher is not one of high repute; cf. http://chronicle.com/forums/index.php?topic=81342.0 . It's not a journal MIT gets, which brings up the question of whether anybody else does, and if anyone has seen it in the field to critique it.--Prosfilaes (talk) 06:50, 1 May 2013 (UTC)

The editors involved have shown remarkable restraint with this. It was only after months of abuse that a block warning has finally been placed at User_talk:Fernandodelucia. Hopefully that will suffice. Tkuvho (talk) 08:46, 1 May 2013 (UTC)

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What has been posted is a fact not a claim to its having been accepted. Not one of the posts has attempted to show that the arguments made are inherently untenable or are based on fallacious ideas. I would suggest that this group of editors +++read+++. Nothing they have said suggests they had.

Godel's Theorems are not the pure stuff of mathematicians and it would not be surprising that a critique would be found in a philosophy journal. I wouldn't think its critical that you couldn't find in the MIT library.Its a new journal. Its paywalled but you can find it (unpaywalled) at http://www.davidpublishing.com/davidpublishing/Upfile/2/29/2012/2012022981760545.pdf . The first step is to try and read before taking down things without any acquaintance with them.

A year or more ago a couple of editors tried to include something similar and it was suggested that it was too early to do so. Apart from Russell and Wittgenstein who were unhappy with the Theorems and who didn't seem to be be able to put a finger on what was wrong this is the first real challenge to the Theorems. Whilst the Theorems mights sound OK their counterpart Lob's Theorem is certainly hard to accept. As Boolos says it offerss a way of proving that Santa Claus exists.

I'm not trying to give an authoritative account of what Thompson says but the paper is very short and not hard for anyone competent in logic to understand. Its certainly easier than Godel's!

A rough outline:

In this context, a proof is a sequence of sentences using the standard rules of inferences and resulting in the conclusion which is also a sentence.

Godel introduces the idea of arithematisation which translates a symbolic system into a system of numbers which serve an indexical function. His arithmetisation is intended to be isomorphic to the original system.

He then introduces an arithmetic idea of proof which allows that any godel number of any formula is capable of proof including a single bracket. This is most odd and almost certainly wrong but doesn't matter that much. Boolos' text talks about sentences being proved. Boolos' text does not draw a distinction between open sentences which contain a free variable and closed sentences where all the variables are bound.

This doesn't seem surprising to mathematicians who tend to be focussed on formulas rather than sentences but in normal English its like using a sentence contain a pronoun where the person who is talked about is never identified. Quine, America's greatest logician point out that open sentences are true of things but not true or false in themselves. Sentences, properly so called must be true or false and open sentences are neither.

If you look at Godel's informal proof it quickly emerges that the sentence he talks about is an open sentence. As far as his formal proof is concerned arithmetic proof because it is intended to be isomorphic to ordinary proof can only be concerned with the proof of the godel numbers of closed sentences.

His famous sentence starts with 'x is arithmetically unprovable' ; that formula has its own godel number; that godel number (which is the godel number of an open sentence) is then used to to create a new 'sentence' saying the godel number of the original open sentence is arithmetically unprovable. But its not a valid sentence if both proof (including arithmetic proof) is restricted to closed sentences.

In theory, we could stick to Godel's idea of proof or something similar and allow open sentences into proof. But there's no good reason I can see for doing so.

If Thompson is right a lot of people will find it shocking but is that so important?

I am certainly sure there is nothing crazy about what he is saying and that a lot of logicians think that there is a problem here.

Lets ask all the snipping editors to get together and show (within say 7 days:

Thompson's thesis rests on some untenable assumption or mode of reasoning or say Godel's approach to proof of open sentences is right.

This shouldn't be too hard given their convictions about these things. Fernando — Preceding unsigned comment added by Fernandodelucia (talk • contribs) 03:00, 2 May 2013 (UTC)

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I responded at your talkpage, see there. Tkuvho (talk) 07:47, 2 May 2013 (UTC)

You ask people to "try and read" the section. Perhaps, first, you should try and make it more readable. Adding wikilinks to some of the things it mentions would help. You should also supply the one reference in it in <ref> format, rather than as plain text. Maproom (talk) 13:31, 2 May 2013 (UTC)

I would ask for evidence that a new journal, does, in fact, perform accurate fact-checking and has experts in the relevant field (mathematical logic); or a reliable source commenting on the paper. — Arthur Rubin (talk) 20:50, 2 May 2013 (UTC)

Part of the reason journal reputation matters is that it helps busy people avoid wasting their time reading borderline self-published nonsense. This post, and the extensive responses by the many people solicited to publish in Philosophy Studies, tells the story more than adequately. EEng (talk) 21:22, 2 May 2013 (UTC)

Clarification: All of the previous 4 posts respond to Fernandodelucia's post just above.

If a lot of logicians think there's a problem here, good! Cite some.

Having looked at that article, you have someone with a B.A. and a LLM who citing undergraduate textbooks against a theory that some of the greatest minds in mathematics have attacked and pronounced solid. I know who I'm betting on.--Prosfilaes (talk) 00:18, 3 May 2013 (UTC)

Arbitrary break

I have been told that I should post my proposed edit here and seek further discussion: Thompson ('Arithmetic Proof and Open Sentences' Philosophy Study 2 (1) 43-50 (2012)) claims that only closed and not 'open' sentences containing free variables are capable of proof. Similarly, only gödel numbers of open sentences are capable of arithmetic proof. Gödel's key sentence stating that the gödel number of a certain open sentence is unprovable becomes unsyntactical. On this approach to proof standard proofs of not only Gödel's incompleteness theorems, but of the Diagonal Lemma, Tarski's and Löb's Theorems are all claimed to be blocked. Thompson's critique does not preclude proof of these theorems by other means.

I have found some of the objections irrelevant: I make the following comments It is published in a peer reviewed journal which is cited in major reference citation services including Cambridge Scientific Abstracts (CSA) Chinese Database of CEPS, American Federal Computer Library Center (OCLC) Chinese Scientific Journals Database, VIP Corporation, Chongqing, P.R. China EBSCO Databases Norwegian Social Science Data Services (NSD) The Philosopher’s Index ProQuest Summon Serials Solutions Ulrich’s Periodicals Directory Universe Digital Library S/B and Phil Papers

A brief reference is to be included in a "Criticisms" section. Obviously inclusion in such a section is not an endorsement of the paper.

I have not been able to discern any substantial comment about its content so no-one is attempting to say that its claims are amde on the basis of some nonsense principle or it contains some obvious flaw.

What I'm really concerned about is whether anyone is claiming that its content is obviously untenable.

Some accounts are given on my talk page about the contents of the paper.

Fernandodelucia (talk) —Preceding undated comment added 09:02, 13 May 2013 (UTC)

Repeating what I said on your talk page:

You've been told very clearly why nothing on Thompson's ideas, and certainly not your interpretation of them and their implications or non-implications, is appropriate unless multiple secondary sources (or a single, extremely authoritative one) comment on it. In any event this is not the talkpage for the article; I suggest you repost there ... and see what others think. If you proceed straight to editing the article, you will be very close to another block, I assure you.

And by the way, I seem to recall someone being blocked for refusing to bother with WP:INDENT. You're angering a lot of people with your refusal to pay attention to WP norms -- big and small.

EEng (talk) 11:27, 13 May 2013 (UTC)

There's no evidence the journal is really peer-reviewed. There's no response; you interpret that as silence means consent, but I see an article that if it has got into the hands of mathematicians, it still hasn't been seen as worth the dignity of a response. There's thousands of articles like this, squaring the circle, proving the parallel postulate and generally attacking the most public targets of mathematics despite the fact that they are some of the most well-checked proofs by mathematicians. They aren't worth mention.--Prosfilaes (talk) 15:15, 13 May 2013 (UTC)

The Journal claims to be peer reviewed and gives a list of reviewers. To say it is not is just an assertion. Publication in a peer reviewed is not proof of excellence its just a necessary filter. I am really interested in what others say about its content. A whimsical dismissal based on some argument that many citations are necessary doen't get you far. I guess I am seeing that there's nothing you can offer about the content.Fernandodelucia (talk) —Preceding undated comment added 05:36, 14 May 2013 (UTC)

You cannot blackmail us into evaluating the content by saying that, if we don't, then you'll put it in the article. Doesn't work that way. I am not interested in discussing the content. --Trovatore (talk) 08:55, 14 May 2013 (UTC)

How can you claim I'm blackmailing you by asking you to think. I am challenging you to show that there's nothing substantial in the Thompson article. Why is such a challenge so uncomfortable?Fernandodelucia (talk) —Preceding undated comment added 10:28, 14 May 2013 (UTC)

And we are asking you to find reliable secondary sources that think there is something substantial in the Thompson article, and in the meantime to stop wasting our time and attention. One or two more rounds of editors telling you that, interspersed with your accusations of intellectual laziness, and you'll be blocked -- I guarantee it. Everyone's tired of your willful blindness. EEng (talk) 11:32, 14 May 2013 (UTC)

EEng, ad hominem attacks are out of place in a logic/maths talk site. I can't see how they improve your position. You have made your position clear that you have no intention of addressing the content of the article. However if there is anyone who wishes to show that Thompson's points are simply untenable and needs some pointers or explanations about what the article is about I'm happy to try and respond. I have already tried this on this site and my own talk site but am happy to take it further. The paper seems quite clear (and is certainly not 'crackpot' or crazy) but nothing about the Theorems is that easy.Fernandodelucia (talk) —Preceding undated comment added 01:42, 15 May 2013 (UTC)

Still playing dumb, huh? WP:ICANTHEARYOU, much? EEng (talk) 03:12, 15 May 2013 (UTC)

Thank you for once again making my case.Fernandodelucia (talk) —Preceding undated comment added 04:27, 15 May 2013 (UTC)

Should we add Kenneth Kunen's formulation?

This discussion was part of a longer thread that was recently moved to /Arguments. I am restoring the part of the thread that's directly relevant to improving the article. Tkuvho (talk) 09:17, 17 May 2013 (UTC)

I would be gratefull if you add a few very short remarks "true in the standard model of N", resp. "true in the standard model of N in ZFC" or so. "Disquotationally" is a quite complicated concept (though I don't want to remove it) that is, at least in my opinion, not sufficient to clerify it to most readers. If you even add some of the remarks that you wrote here, Carl, it would be definitely better (if you have time for this, of course).. Thx Franp9am (talk) 13:09, 10 April 2013 (UTC)

No, that would be completely wrong. It's a category error to say "true in the standard model in ZFC". It sounds like you're using "true" to mean "provable". But the most important takeaway from the Goedel theorems is precisely that provability is not strong enough to capture truth. --Trovatore (talk) 22:51, 10 April 2013 (UTC)

I think that's a little strong. "True in a model" is not the same as "true", but the standard model is a model and "true in the standard model" is a perfectly reasonable notion. — Carl (CBM · talk) 01:40, 11 April 2013 (UTC)

It's a perfectly reasonable notion; the problem is the addition of "in ZFC", which makes it appear that he's still identifying truth with provability, just provability in ZFC instead of in the object theory. --Trovatore (talk) 01:51, 11 April 2013 (UTC)

(Not that that's the only problem — the other problem is that bringing up the standard model, as far as I can see, only serves to add confusion to something that is really very simple.) --Trovatore (talk) 01:52, 11 April 2013 (UTC)

Just defining what is a number in ZF doesn't define addition and multiplication. I don't see how ZF helps with this. If anything it just gets in the way.

By the way, Davis, M. (1980). The mathematics of non-monotonic reasoning. Artificial Intelligence, 13(1), 73-80 presents a first-order theory which has a unique minimal model, which is, as he says, "the standard model of arithmetic". What's wrong with understanding this as a "definition" of the standard model? At least it's intelligible. Compulogger (talk) 13:55, 10 April 2013 (UTC)

Defining addition and multiplication of natural numbers in ZFC is easy. However, with or without ZFC; I just want to say that the word "standard" should appear somewhere. Franp9am (talk) 14:06, 10 April 2013 (UTC)

ZFC is a red herring, and the word "standard" is unnecessary weaseling. If T is consistent, then its Goedel sentence is true, period. There is no need to qualify or soften the word. --Trovatore (talk) 00:32, 11 April 2013 (UTC)

So, what do you mean by "true" if not "true in the standard model"? Franp9am (talk) 08:39, 11 April 2013 (UTC)

Given a formal theory T, its Goedel sentence, G_T, says that there is no natural number n with a certain property P(n). The property P is one that can be checked by a fixed computer program, one we can actually work out, and one that is guaranteed to terminate in an amount of time we can calculate in advance, given n.

Now, if T is consistent, then G_T is true. What does that mean? It means that no natural number has property P. What does that mean? It means that P(0) is false (that is, if you give 0 as an input to the program, it will respond "false"). And that P(1) is false, P(2) is false, P(3) is false, and so on.

So that's what it means. Can you word this as "true in the standard model"? Sure, you can. It isn't wrong. But it's unnecessary, and it makes it sound complicated when it's actually simple. If T is consistent, then G_T is just plain true, which means the thing I said above, which is the obvious thing it should mean. --Trovatore (talk) 13:27, 11 April 2013 (UTC)

I agree with you on this. Also see Emil J.'s comment below. Goedel's "satisfiability" criterion (Theorem X) is an iff, i.e. a logical equivalence so it defines what is meant by "true" in this context; as noted by Nagel and Newman 1958 this bridges the expanse between an arithmetical purely-mechanical computation, and the metamathematical "true". On pages 79 and 80 they state it this way: "We now ask the reader to observe that a meta-mathematical statement which says that a certain sequence of formulas is a proof for a given formula is true, if and only if, the Godel number of the alleged proof stands to the Goedel number of the conclusion in the arithmetical relation here designated by 'Dem'. Accordingly, to establish the truth or falsity of the meta-mathematical statement under discussion, we need concern ourselves only with the question whether the relation Dem holds between two numbers. Conversely, we can establish that the arithmetical elation holds between a pair of numbers by showing that meta-mathematical statement mirroed by this elation between the number is true". On page 91 a footnote describes pretty much what Trovatore wrote above about testing property P. Bill Wvbailey (talk) 14:41, 11 April 2013 (UTC)

But G_T can be chosen to be Cons(T) itself! So what you are claiming is that the sentence "Cons(T) implies Cons(T)" is just plain true. I would agree with that. Tkuvho (talk) 13:41, 11 April 2013 (UTC)

It seems to me that you are trying to prove from first principles that the Platonist view is better than the Pragmatic view (see my comments below). Perhaps we can try to avoid philosophical commitments altogether. Tkuvho (talk) 13:42, 11 April 2013 (UTC)

What I am doing is pointing out that it is very simple. I am deflating, if you like. What does it mean to say the Goedel sentence is true? The Goedel sentence says that no natural number has a certain property. Saying that it's true says only that no natural number really has that property. The only way that can be wrong is if some natural number does have that property, and if that were true, then by the arguments used to prove the theorems, from that number, you could recover a proof of a contradiction in T. There is really no way around this. --Trovatore (talk) 13:51, 11 April 2013 (UTC)

OK, well, perhaps the Platonist view is provably superior to the Pragmatic one, but perhaps User:Franp9am and I can still raise the issue whether Kunen's Pragmatic formulation can be mentioned in a subsection (perhaps subsubsection) toward the end of the article? Tkuvho (talk) 14:04, 11 April 2013 (UTC)

You keep trying to un-deflate. There is no need to talk about Platonists or pragmatists. The formulation I gave works for everybody. --Trovatore (talk) 14:07, 11 April 2013 (UTC)

Are we to assume that Kunen is provably wrong? Tkuvho (talk) 14:16, 11 April 2013 (UTC)

I am saying Kunen would agree with me. --Trovatore (talk) 15:13, 11 April 2013 (UTC)

But Kunen is the one who gave a non-Platonist formulation of Goedel's theorem on page 38 of his book Kunen, K.: Set theory. An introduction to independence proofs. Studies in Logic and the Foundations of Mathematics, 102. North-Holland Publishing Co., Amsterdam-New York, 1980. Do you think he changed his mind after reading your post? Tkuvho (talk) 15:20, 11 April 2013 (UTC)

Ah, found my copy (had to dig a little — had to box up some books a while back). I see nothing on page 38 that is in any way incompatible with what I say above. --Trovatore (talk) 16:11, 11 April 2013 (UTC)

I wonder if we are looking at the same edition. In my edition, Kunen certainly does not formulate the result as asserting that there are propositions that are true but not provable, which is the formulation you appear to favor. Tkuvho (talk) 16:17, 11 April 2013 (UTC)

He omits to discuss the truth of the proposition. It's an enormous leap to conclude he has no opinion about it, and an even bigger one to conclude that he would actively disagree with what I've said. If that's all you're going on, that's an awfully thin reed. --Trovatore (talk) 16:23, 11 April 2013 (UTC)

Again, what I am "going on" is not to disagree with what you've said (as I already mentioned we are not going to resolve the Platonist-Pragmatic discrepancies from "first principles"), but merely to see if editors support the inclusion of Kunen's type of formulation of the incompleteness result. To the extent that he does not formulate it in the way it is currently formulated in the article, the inclusion of a Kunen-type formulation may be justified on the grounds of notability. In other words, I am recommending a mitigation of an "all-or-nothing" attitude. Tkuvho (talk) 16:31, 11 April 2013 (UTC)

That does not require any change at all, as far as I can see. Kunen says that the Goedel sentence can neither be proved nor disproved from T, assuming T is consistent. We say that as well, I think (right?). We also provide the extra information that the sentence, understood as a sentence of arithmetic rather than necessarily as one talking about the objects of discourse of T, is true. --Trovatore (talk) 16:34, 11 April 2013 (UTC)

I just looked through the article again. The first three formulations I found all insist on "true but not provable" aspect. If it's there, you would need a microscope to find it. Also, Kunen prefaces the "true but not provable" comment by explicitly referring to a Platonist interpretation, which suggests that it is not the only one. I would propose creating a subsection where a non-Platonist formulation is clearly stated without the terminology "true but unprovable". Tkuvho (talk) 16:50, 11 April 2013 (UTC)

It is certainly true that the information that the sentence cannot be disproved by the theory is also important. Perhaps we should say "true but not provable (and also not disprovable)" or some such. --Trovatore (talk) 16:54, 11 April 2013 (UTC)

Sorry to belabor the point, but Kunen does not say that. You keep trying to get to the bottom of the "truth of the matter", whereas I keep trying to insist on sourceability. Tkuvho (talk) 16:57, 11 April 2013 (UTC)

It is sourceable, and well-sourced. It's not sourced to Kunen. Kunen says less; he doesn't in any way say anything that contradicts what we have. --Trovatore (talk) 16:59, 11 April 2013 (UTC)

Kunen certainly does not contradict the Platonist formulation, because it is not something that can be refuted mathematically. He does provide an alternative formulation, which is therefore sourceable, and yet not represented in our article. Tkuvho (talk) 17:09, 11 April 2013 (UTC)

So what do you want to write? "Some sources say A and also B, and some other sources say A but don't mention B (but also don't say anything contrary to B; they just don't mention it)"? What's the value in that? --Trovatore (talk) 17:12, 11 April 2013 (UTC)

No, I would suggest reproducing Kunen's "Pragmatic" formulation, which does not make Platonist assumptions, to complement the current formulation. There is no end to discussions at mathexchange, stackoverflow, and the like :-) pitting Platonist and Pragmatic viewpoints, with no clear consensus in sight. Similarly, as was already pointed out by User:Franp9am, other wikis follow Kunen's formulation and not ours. He checked it for the German wiki, and I obtained the same result for the French wiki. This in itself is not conclusive, but it is hard to understand why our page should be dominated by the Platonist formulation when there is a variety of views out there. Tkuvho (talk) 11:51, 15 April 2013 (UTC)

Kunen's formulation is a subset of ours (or, at least, of what ours probably should be). If you want to leave a part of it out, you should have a source that makes a point of leaving it out. There is no direct evidence that Kunen even disagrees with what you're calling the "Platonist" formulation (it doesn't even have to be Platonist, as explained by the footnote), and I would suggest that he almost certainly does agree with it — it just wasn't important to mention it at that point in the book. --Trovatore (talk) 18:05, 15 April 2013 (UTC)

Your interpretation of Kunen does not fit very well with what he wrote. After presenting a definition in the terminology of "neither provable or disprovable" at the beginning of the section, he makes a comment toward the end of the section that, from a Platonist perspective, this can be formulated as "true but unprovable". This clearly indicates that Kunen distinguishes between the two approaches. Tkuvho (talk) 09:21, 17 May 2013 (UTC)

I have Kunen's book on my shelf, so I re-read section I.14. One thing that stands out is that Kunen does not state the first incompleteness theorem at all - only the second. Now the formula in the second incompleteness theorem, CON^T, is already assumed to be true in the hypotheses, so it is trivially a true sentence unprovable in T. The question of truth is more interesting for the first incompleteness theorem, and the sentence "I am not provable in T", because the hypothesis of the first incompleteness theorem is only that T is consistent. But Kunen goes very quickly past the first incompleteness theorem, alluding to it rather than stating it as a theorem and using it only in one introductory paragraph to motivate the rest of the section.

So I do not see an alternate formulation in Kunen's book, much less anything that disagrees with the presentation already given here. The single paragraph that mentions the theorem (essentially as a side remark) is too short to serve as any sort of useful reference here.

I do think it would be worth adding a short section explaining the various senses in which the Gödel sentence of a consistent theory is true (I have said this before). Perhaps I will finally try to write one, now that Peter Smith's book is published and can provide better sourcing. — Carl (CBM · talk) 15:18, 17 May 2013 (UTC)

Thanks. Note that this page is about the incompleteness theoremS, not merely the first one. Tkuvho (talk) 09:42, 19 May 2013 (UTC)

The unself-evident within Gödel's theorem

A wall can be touched and its existence can therefore be considered self-evident with respect to the observer. A rectangular opening within this wall, that functions as a gateway, cannot be touched in order to demonstrate its existence because its essence is void, still it exists. Its form and function depend completely on the surrounding wall and it would therefore be more correct to refer to the existence of a gateway as unself-evident – existing but untouchable by the observer. By introducing the idea of unself-evident our thought process becomes enhanced with an imaginary component similarly to the way real numbers can be enhanced with an imaginary component. This underpinning makes it much easier to understand and to explain Gödel’s work.

If the formal system can give the Gödel sentence truth-value (i,e,. can prove it, can touch it) then the Gödel sentence is self-contradicting. Yet if the formal system cannot give the Gödel sentence truth-value (i,e,. cannot prove it, cannot touch it) then the Gödel sentence is unself-evident – true with respect to the formal system but untouchable by the formal system. The whole idea is that the Gödel sentence has both a real component (found in the realm of the self) and an imaginary component (found in the realm of the unself). The real component of the Gödel sentence is self-contradicting, whereas the imaginary component of the Gödel sentence is unself-evident. Gödel's work clearly points out the value of this imaginary component. Aristotelian logic, primarily the law of identity, only covers the real component of logic (i.e., the self-evident) and does not encompass this imaginary component (i.e., the unself-evident). Hence, Gödel's theorem ultimately alludes to the incompleteness of 'real' logic. EvitaTomjanovich (talk) 06:33, 9 August 2013 (UTC)

Here's some metatheory about Goedel's Thm that I think most will agree is self-evident: a lot of the discussion that takes place on this page is opaque and confusing. When you say unself-evident it's not evident wheterh you mean un–self-evident, or un-self–evident, or indeed unself-evident -- this creates a real problem, don't you imagine? EEng (talk) 07:58, 9 August 2013 (UTC)

Your sense of humour reflects your open-mindedness. Do you fear the unused 90 percent of your brain? Anyway, though the term 'unself-evident' is not commonly used, I came across this term in the book ”Some varieties of superparadox” p.9 by Christopher Ormell http://sammelpunkt.philo.at:8080/625/1/Ormell.pdf

“It [Principia Mathematica Vol. I] was supposed to "reduce mathematics to logic". If this meant anything at all it meant that it demonstrated clearly that the edifice of modern mathematics could be generated from a few wholly self- evident, wholly transparent principles. But this general, procedural, "philosophical" clarity was equally obviously missing, and the authors themselves admitted that they had had to introduce four unself-evident principles to pull the trick! The doctrine of Types The axiom of choice The axiom of infinity The axiom of reducibility”

I used the term 'unself-evident' as a means to complement the term 'self-evident' which is different from opposing the self-evident with not self-evident. Apart from this, mathematics has already been infiltrated by the unself-evident. The number zero and the empty set are clear examples of this infiltration, though mathematicians have carefully repositioned these concepts in the realm of the self-evident, especially with the use of axioms. The infinite also belongs to the realm of the unself-evident and Dedekind's realisation that an infinite set can fold back upon part of its self is a consequence of this.

Furthermore, this is not a meta-theory but a humble attempt to explain Gödel's theorem in layman terms by supplying a layman underpinning. As we all know, Gödel's incompleteness theorem is ultimately not about incompleteness per se, but about the impossible combination of completeness and consistency. In other words, the theorem proves that none of the formal systems to which G's theorem applies can be presented as a coherent whole. So there is something disturbed within these formal systems and according to the theorem, a balance cannot be established by leaning towards the self-evident. So have all exits been closed by Gödel's theorem, or is there still a gateway to resolution? EvitaTomjanovich (talk) 11:38, 14 August 2013 (UTC)

Thanks for the compliment on the humor. "its essence is void, still it exists." Was it Koestler that defined philosophy as the systematic abuse of concepts and terminology designed specifically for that purpose? I think you and User:Fernandodelucia should get together and talk about this, then get back to us when you have something to propose by way of improvements to the article. EEng (talk) 12:11, 14 August 2013 (UTC)

Yep, it was Koestler. Still, Coincidentia oppositorum or unity of opposites is a profound philosophical concept found in many cultures all over the world – this cannot be said of Koestler's work. This concept has also been fruitful in math (e.g., positive and negative numbers making up the integers; rational and irrational numbers making up the real numbers; real and imaginary numbers making up the complex numbers). Every time the boundaries of human thought are pushed a greater reality is revealed. If possible, let it sink in. Meanwhile, I will leave you guys in peace until I have something to propose by way of improvements to the article. Thanks for the interaction, EvitaTomjanovich (talk) 07:36, 23 August 2013 (UTC)

Personally I've found that even greater realities have been revealed by pushing the boundaries of non-human thought. EEng (talk) 11:19, 23 August 2013 (UTC)

Statement of the first incompleteness theorem

It seems to me there is no actual statement of the first incompleteness theorem in the article. The closest we come is:

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic).

and

Gödel's first incompleteness theorem states that: "Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory" (Kleene 1967, p. 250).

The former is an interpretation by a wikipedia writer.

The latter seems to be Kleene's interpretation of what the theorem states, not Gödel's actual statement of the theorem or an English translation thereof. Also I think the footnote [1] should not be included after the word "true", because it is not part of Kleene's interpretation. As it stands, the stated "theorem" is an interpretation of an interpretation of what Gödel meant.

I think the article would be improved by inclusion of an exact statement of the theorem in addition to these interpretations, and by a clear indication that these are interpretations, not statements of the theorem.

Paul1andrews (talk) 05:06, 14 September 2012 (UTC)

Can you give an example of what you think would be a "statement" of the theorem rather than an "interpretation" of it? If you mean "Gödel's actual statement of the theorem", that is a separate matter. There is a great deal of secondary literature on the theorem, so the literal statement that Gödel used is primarily of historical, rather than mathematical, interest. — Carl (CBM · talk) 10:54, 14 September 2012 (UTC)

Yes I mean "Gödel's actual statement of the theorem". I think the article would be improved by the inclusion of it, regardless of whether one believes it to be of historical or mathematical interest. Paul1andrews (talk) 13:35, 14 September 2012 (UTC)

Is this what you're talking about? I added these after some dissent (these appeared in Nov 2009) but sometime this year, as I recall, the article went through a revision and we decided (and I agreed) that the article had developed enough to remove them. When you read them, and you will see why. There may be other verbatim Goedel statements e.g. in the Princeton lectures, but these original statements are opaque to anyone but an academic professional or a very-proficient student:

Original statement [first incompleteness theorem]

The first incompleteness theorem first appeared as "Theorem VI" in his 1931 paper On Formally Undecidable Propositions in Principia Mathematica and Related Systems I. In Gödel's original notation, it states:

"The general result about the existence of undecidable propositions reads as follows:

"Theorem VI. For every ω-consistent recursive class κ of FORMULAS there are 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).^[1](van Heijenoort 1967:607.)

Original statement of Gödel’s Theorem XI

While contemporary usage calls it the “Second incompleteness Theorem”, in the original Gödel presented it as his “Theorem XI”. It is stated thus (in the following, “Section 2” is where his Theorem VI appears, and P is Gödel’s abbreviation for the system obtained by adding the Peano axioms to the logical system of Principia Mathematica.

”The results of Section 2 have a surprising consequence concerning a consistency proof for the system P (and its extensions), which can be stated as follows:

”Theorem XI. Let κ be any recursive consistent⁶³ class of FORMULAS; then the SENTENTIAL FORMULA stating that κ is consistent is not κ-PROVABLE; in particular, the consistency of P is not provable in P,⁶⁴ provided P is consistent (in the opposite case, of course, every proposition is provable [in P])". (Brackets in original added by Gödel “to help the reader”, translation and typography in van Heijenoort 1967:614)

”⁶³ “κ is consistent” (abbreviated by “Wid(κ)”) is defined as thus: Wid(κ)≡ (Ex)(Form(x) & ~Bew_κ(x))."

(Note: In the original "Bew" has a negation-“bar” written over it, indicated here by ~. “Wid” abbreviates “Widerspruchfreiheit = consistency”, “Form” abbreviates “Formel = formula”, “Bew” abbreviates “Beweisbar = provable” (translations from Meltzer and Braithwaite 1962, 1996 edition:33-34) )

”⁶⁴ This follows if we substitute the empty class of FORMULAS for κ.”

^ Here Flg(κ) represents the theory generated by κ and "v Gen r" is a particular formula in the language of arithmetic. "Flg" is from "Folgerungsmenge = set of consequences" and "Gen" is from "Generalisation = generalization" (cf Meltzer and Braithwaite 1962, 1992 edition:33-34).

Wvbailey (talk) 18:20, 14 September 2012 (UTC)Bill

What was the reason for taking this out? It seems to me that if an article can show the actual thing it is discussing then it should. For example, the article on "circle" shows a picture of a circle. It would be silly not to, and it seems no less silly in the case of these theorems. Many parts of Wikipedia are opaque to non-experts but this poses no problem to the reader - he/she merely skims over these parts, noting on the way that, yes, this stuff is complicated. As it stands the article is inaccurate because it presents secondary and tertiary interpretations as being the actual theorems, which they are not. It would be more accurate to show the theorems in their original state, then present one or two interpretations that others in the field have placed upon them.

Paul1andrews (talk) 05:17, 15 September 2012 (UTC)

The statement in the article is the actual thing that it is discussing - it is a genuine statement of the first incompleteness theorem. The original statement by Goedel is neither better nor worse in that respect, although it uses outdated terminology and bizarre capitalization. Can you give any example of any modern textbook that states the first incompleteness theorem in the way the Goedel originally did? I don't believe I have ever seen one. The focus of this article is not "Goedel's original paper on the incompleteness theorem", it is "Goedel's incompleteness theorem". — Carl (CBM · talk) 11:56, 15 September 2012 (UTC)

I don't think that's a tenable argument. There are two (different) statements that purport to be the theorem but they are both interpretations, not the actual theorem. That is clear, but it appears you will not be convinced. You seem to be using rather loose definitions of is and genuine. Would anyone else care to add their opinions? I'd be interested to know what people believe is being achieved by leaving it out, and what would be lost by putting it back in.

Paul1andrews (talk) 11:28, 16 September 2012 (UTC)

I'm about to travel so I can't do justice to a search. But if you can get your hands on a cc of Goedel's 1934 "On Undecidable Propositions of Formal Mathematical Systems" there is some interesting stuff at "§7. Relation of the forgoing arguments to the paradoxes." Goedel is delivering a lecture, and speaking very clearly and carefully to an audience, and he does a nice job of discussing the results of his research: "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. Whether this is germane I leave to better minds. Bill Wvbailey (talk) 23:39, 16 September 2012 (UTC)

As a first step towards inclusion of the actual theorem, I've split the existing cut down "Original Statement" section and moved to the top of the respective sections for theorems 1 and 2. (This sub-section was under the wrong section - dealing with theorem 1 - in any case). The reader is now gently introduced via a reference to the original statement, with a segue into the broadly used natural language interpretations.

Paul1andrews (talk) 08:58, 18 September 2012 (UTC)

I do not think it is helpful to include the actual text of theorems VI and XI of Godel (1931), as much of the notation is specific to the paper in question, and hence meaningless even to educated persons familiar with modern statements of the same results. I would recommend using Gödel’s restatement of theorem VI from van Heijenoort 1967:610 that reads: "...(I)n every formal system that satisfies (certain basic definability) assumptions... and is ω-consistent there are undecidable propositions of the form (x)F(x), where F is a recursively defined property of the natural numbers, and likewise in every extension of such a system by a recursively definable ω-consistent class of axioms." Parenthetical content and ellipsis mine. This is much closer to what is meant by "the first incompleteness theorem" than theorem VI is in the absence of the additional material that follows it's proof (ie: the remainder of the section [2] in which it is proved). Theorem VI says, in the absence of notation (paraphrase) "For every arithmetizable, recursive class of formulas K, either elementary or resulting from a sequence of other formulas by means of disjunction, negation and generalization, which is not only syntactically consistent (ie: does not have any contradictory formula as an immediate consequence), but also avoids, as consequences, certain infinite combinations of sentences that are intuitively contradictory, there are recursive properties F s.t. neither '(x)F(x)' nor '~(x)F(x)' are immediate consequences of the formulas contained in K; notions of negation, generalization, "immediate consequence", etc. all being arithmetizable." It is not clear that this is superior to the passage quoted from van Heijenoort 1967:610. I do agree, however, with the inclusion of the original theorem in some form, and not just in these (more familiar) forms strengthened or generalized by additional theoretical work due to Rosser, Kleene, Tarski, Church, Turing, Post, etc. This applies to the second incompleteness theorem as well, where numerous published descriptions include unnecessary and/or subjective epistemological content.

I would also recommend the following as the natural language description of the first incompleteness theorem:

"If L is a consistent and adequate artithmetical logic, then L is incomplete: there are correctly formed statements (about the natural numbers) for which neither the statement, nor its negation are theorems of L."

The natural language description in the article is in a generalized form that assumes the Church-Turing thesis. MixedMartialArtsHistory (talk) 21:27, 26 October 2013 (UTC)

Rather than "arithmetical logic" I'd prefer "formal theory" or "formal system", where this is previously defined to include the [core, sufficient] postulates of the propositional and predicate calculus [formal logical system] and the postulates of number theory [ala Peano axioms, aka "a certain amount of arithmetic"]" (cf Kleene 1952:81-82). This necessary addition of the Peano axioms to the formal logic of PM is probably why Goedel used the symbol P rather than L. Bill Wvbailey (talk) 14:22, 29 October 2013 (UTC)

I don't know why you would say such a thing; Gödel clearly says that the Peano Axioms are completely extraneous to the demonstration (Gödel 1931, footnote 16): "The addition of the Peano axioms, as well as all other modifications introduced in the system PM, merely serves to simplify the proof and is dispensable in principle." The system developed in Principia Mathematica is fully capable of expressing all the information in the Peano axioms (as theorems).66.30.202.15 MixedMartialArtsHistory (talk) 19:04, 8 December 2013 (UTC)

Natural language description of the first incompleteness theorem is incomplete

I cannot make the edit due to the page being semi-protected:

Gödel's first incompleteness theorem shows that any consistent effective formal system that includes enough of the theory of the natural numbers is incomplete: there are true statements expressible in its language that are unprovable.

should be changed to

Gödel's first incompleteness theorem shows that any consistent effective formal system that includes enough of the theory of the natural numbers is incomplete: there are true statements expressible in its language that are unprovable within the system.

--Klemensbaum (talk) 19:40, 12 February 2013 (UTC)

Hmm? That is what it says — word for word, I think, though I haven't checked it word for word. And the last edit to the article was almost a month ago. Maybe your browser did a partial load, or something?

(By the way, I think "system" should be replaced by "theory" — as it stands, readers might wonder if a system is something different.) --Trovatore (talk) 19:54, 12 February 2013 (UTC)

Doesn't "system" also refers to the proof calculus? I really hate all the inconsistent terminology in logic. Hans Adler 19:58, 12 February 2013 (UTC)

A "deductive system" is a set of axioms plus a set of rules, but there is no mention of the rules here. I don't think "system" is being used in a technical sense. --Trovatore (talk) 20:00, 12 February 2013 (UTC)

Thanks, done. Hans Adler 19:58, 12 February 2013 (UTC)

Creativity as a solution to Incompleteness of human thought

One obvious implication of Godel's theory is that the human mind has a capacity to create axioms, and therefore is not confined to the arbitrary set of first order logic. This has been offered from numerous sources, and should be included somewhere in the article. — Preceding unsigned comment added by 72.238.115.40 (talk) 04:23, 10 December 2013 (UTC)

I generally prefer to keep this article focused on the mathematics. Certainly the less-mathematical aspects of the topic have their place, but perhaps pointers to them, plus a brief discussion, is sufficient for this article. And those we already have: See the "Discussion and implications" section. In particular, the "Minds and machines" subsection mentions the topic you're interested in, and points to mechanism (philosophy)#Gödelian arguments for a more in-depth treatment. --Trovatore (talk) 05:02, 10 December 2013 (UTC)

Example section

The section of the article labeled 'Examples of undecidable statements' doesn't actually have any examples of undecidable statements, I feel like some could greatly help explaining the nature / scope of the theorem to non-mathematicians. — Preceding unsigned comment added by 129.170.194.128 (talk) 17:23, 19 December 2013 (UTC)

But it does, like "In 1973, the Whitehead problem in group theory was shown to be undecidable, in the first sense of the term, in standard set theory." In any case, the examples of undecidable statements are not simple, and I double a non-mathematician can do anything with them; the Whitehead problem is "Is every abelian group A with Ext¹(A, Z) = 0 a free abelian group?", a statement my undergraduate class in group theory did not prepare me to understand. The Continuum hypothesis is slightly easier, maybe, but concrete examples are not the easy way to understand this theory.--Prosfilaes (talk) 01:05, 4 March 2014 (UTC)

Possible example of Godel's inconsistence theorem?

For those who understand better with examples, perhaps this might work:

Mathemeticians manipulate rules, which command numbers, spaces, or vectors to follow an order. These results, in turn, may command other math terms. Eventually, a result circles back to order the original rule, with no certainty that it will conform to the original. For example, a rule orders up a set of all odd numbers. That set's rule, in turn, orders every possible pair of odd numbers to add up, to make a set of all even numbers. That in turns orders its complement, all odd numbers, to form. But this set is larger than the original set of all odd numbers, because it's based on the original numbers added together. Whatever the biggest original odd number was, when something was added to it, it got even larger.

The term "order" may be unorthodox, but it's not misleading. Unless heirarchy exists, another relational dimension, orders end in circularity, and may not conform consistently with the original meaning.

Brian Coyle — Preceding unsigned comment added by 208.80.117.214 (talk) 09:56, 3 March 2014 (UTC)

That doesn't make any sense; all odd numbers, no matter how you figure it, are countably infinite. There is no biggest original odd number. The term "order" is deeply misleading; it does not pertain to order as it is commonly known to mathematicians.--Prosfilaes (talk) 01:16, 4 March 2014 (UTC)

Hi Brian,

This is an article talk page, and its purpose is exclusively to discuss what should appear in the article on Gödel's incompleteness theorems. It is not supposed to be used to ask questions about the theorems, except in the context of what should appear in the article. I confess I don't really understand your question, but if you like, you can ask it at Wikipedia:Reference desk/Mathematics, and someone (heck, maybe even me) will take a stab at addressing it. --Trovatore (talk) 01:27, 4 March 2014 (UTC)

Just a suggestion

I just read this on the Wiktionary page for 'doctrine of necessity', and this page came to mind. Perhaps this is of some quotable value? It notably predates the existence of these proofs, even if it's not really a proof itself.

"1803, John Dawson, The Doctrine of Philosophical Necessity Briefly Invalidated, page 7,

In examining the doctrine of necessity, I shall endeavour to bring what I have to say about the subject into as narrow a compass as possible, and for that purpose the following axioms are premised; as in all sciences, some principles must be taken for granted, else nothing can be proved."

67.171.222.203 (talk) 20:41, 10 May 2014 (UTC)

Hawking

Godel has been endorsed by Hawking. Hawking draws far-reaching conclusions from Godel's theorems of 1931. See http://www.hawking.org.uk/godel-and-the-end-of-physics.html — Preceding unsigned comment added by 88.150.234.8 (talk) 08:47, 27 June 2014 (UTC)

Hawking says that "The Godel number of 2+2=4 is *." The asterisk is apparently a permanent part of Hawking's text, not a computer error. — Preceding unsigned comment added by 88.150.234.8 (talk) 08:52, 27 June 2014 (UTC)

Hi, 88.150.234.8. Please be aware that article talk pages are not for general discussion of the subject matter, but for discussing what the article should say and how it should say it. Is there something from the page you linked that you think could be used to improve this article? If so, could you be more specific? Thanks, --Trovatore (talk) 09:10, 27 June 2014 (UTC)

Hawking's article of 2002 is already mentioned three times in Wikipedia. See Quantum mechanics, Theory of everything and History of electromagnetic theory. — Preceding unsigned comment added by 88.150.234.8 (talk) 09:40, 27 June 2014 (UTC)

Apocryphal quote?

Dear all,

There's a quote by Stephen Kleene (Kleene 1967, p. 250) on Gödel's incompleteness theorem which supposedly clarifies Gödel's VI (so called "first") theorem. Kleene's more "accessible" statement is now quoted left, right and center throughout the interwebs (It's even in published books!). The quote is below:

“Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory” (Kleene 1967, p. 250).

I have Kleene's book (Mathematical Logic, 1967), and the quote is nowhere to be found!

Can anyone please verify this? — Preceding unsigned comment added by Akineton (talk • contribs) 03:16, 6 July 2014 (UTC)

I am not interested in fake maths from Godel and Kleene. — Preceding unsigned comment added by 2.98.54.17 (talk) 15:11, 13 December 2014 (UTC)

This does not seem to be on p. 250 of Kleene 1967. I am not a mathematician, but Gödel's 1931 paper had a note added to it in 1963, which states:

"...it can be proved rigourously that in every consistent formal system that contains a certain amount of finitary number theory there exist undecidable propositions and that, moreover, the consistency of any such system cannot be proven in the system." [Kurt Gödel: Collected Works: Volume 1: Publications 1929-1936, p. 195]

Hope this helps.(Edited to fix bungled indentation). Astrakhan4 (talk) 00:38, 15 January 2015 (UTC)

Yes, the same quote can also be found at van Heijenoort 1967:616 (addendum added by Gödel in 1963). I hunted through Turing 1936-7 but didn't find it there. A hint in the quote is "effectively generated", a notion that was defined after Gödel 1931 and Turing 1936-7 in the work by Kleene and Rosser (1938 and after). I'm still searching. Bill Wvbailey (talk) 23:07, 16 January 2015 (UTC)

There's a theorem in Kleene 1952:208 that states that " . . .if the system is consistent then it is (simply incomplete) with [a particular formula A_q(q)] as an undecidable formula". Kleene names this "Rosser's form of Gödel's theorem". The formula is a diagonalization of something, but I don't know what that something is. The search continues. BillWvbailey (talk) 23:44, 16 January 2015 (UTC)

Here are some "Informal" statements of Gödel's Incompleteness Theorems, quoted from Franzen; it's clear that the quote-in-question is an amalgam of Gödel and Rosser:

First Incompleteness Theorem (Gödel-Rosser): Any consistent formal system S within which a certain amount of elementary arithmetic can be carried out is incomplete with regards to statements of elementary arithmetic: there are statements which can neither be proved, nor disproved in S. (Franzen 2005:16)

Second Incompleteness Theorem (Gödel): For any consistent formal system S within which a certain amount of elementary arithmetic can be carried out, the consistency of S cannot be proved in S itself. (Franzen 2005:34)

Franzen observes that Gödel never formally proved his "second" incompleteness theorem, and this was left to David Hilbert and Paul Bernays in their 1939 Grundlagen der Mathematik (Foundations of mathematics). If someone has a cc of this in English this would be an interesting place to search for the quote. [Note to self -- have searched early Rosser and Kleene a few times, Kleene 1952, also Nagel and Newman. Tarski? not in my Tarski. ]. BillWvbailey (talk) 17:50, 17 January 2015 (UTC)

---

Here is the reason we cannot find the quote. Read on:

So I went back (this took a while) through the history, starting at the beginning, dividing and conquering, until I lit on how this "quote" came to be referenced. Here is a selection of versions that show its evolution over the years. By middle of 2009 editor CBM was in the midst of a major re-write, and that when the paragraph was first tagged as needing a reference, and then referenced by CBM to Kleene 1967:250:

end Dec 2007

Gödel's first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that:

For any consistent formal, computably enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed.(refactored from 1) That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

end June 2008:

Gödel's first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that:

For any consistent formal, recursively enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed.(refactored from 1) That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

end Dec 2008:

Gödel's first incompleteness theorem, states that:

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory.

13:05 20 July 2009

Gödel's first incompleteness theorem states that:

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory.[citation needed]

13:07 20 July 2009, CBM edit: (→‎First incompleteness theorem: ref)

Gödel's first incompleteness theorem states that:

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250).

The reference did not appear before this time, and it has been there ever since. I suspect that the reference is being used "disquotationally", i.e. as a "cf" or "see, for example Kleene 1967:250). However, Kleene 1967:250 is not a very good place to land. Propose we change the "quote" to that of Franzen quote, above:

First Incompleteness Theorem (Gödel-Rosser): Any consistent formal system S within which a certain amount of elementary arithmetic can be carried out is incomplete with regards to statements of elementary arithmetic: there are statements which can neither be proved, nor disproved in S. (Franzen 2005:16)

The use of "proved" here makes me queasy (it's not defined: does it mean "derived from" using modus ponens and substitution?) but I prefer this because it does make it very, very clear that this statement of the theorem is the result of work by Rosser as well as Gödel. I am not sure what to do with the "disquotational" footnote [1] however; the word "true" complicates the matter. I have a cc of Goldstein and cannot find the quote there either (CBM added both Kleene 1967 and Goldstein as references in an earlier edit).

Any thoughts out there about this? I'm going to remove the reference to Kleene, but I'm not going to do anything else for a little while. BillWvbailey (talk) 17:29, 18 January 2015 (UTC)

My key sticking point is that, whatever we do, we must continue to make clear that the Goedel (or Rosser) sentence of a consistent theory (considered as a statement about arithmetic, not about the objects of discourse of the theory) is true. There is far too much obfuscation on this point in the popularizations. We need to put this right out front, and let there be no confusion about it.

The bit about which version is due to Goedel and which to Rosser strikes me as more of a historical detail, and the distinction between them more of a technical detail. Both are certainly important enough to deal with in the article, and we ought to get them right, which I am grateful to Will for on his attention to these details. But I don't care as much about exactly how or where they are presented. --Trovatore (talk) 21:54, 18 January 2015 (UTC)

I hear you about true, and Gödel-Rosser. Are you okay with what's there now in the article? If so let's just leave the wording as it is (as a précis, but not a quote). What about the footnote, is that satisfactory? (BTW: When I googled the quote I was surprised at how widely travelled the spurious Kleene reference has become. As long as the "précis" is okay technically I don't think there's much to worry about here.) BillWvbailey (talk) 22:20, 18 January 2015 (UTC)

Yes, I think it's OK, though I'm not sure "précis" is the exact word I would have chosen, partly because I'm not sure exactly what it means. In school it seems to me that a précis was a summary of an essay, sort of like an abstract. Is that what you were trying to get at? --Trovatore (talk) 22:42, 19 January 2015 (UTC)

Yes, that would be my intent. I admit "précis" is an unusual word. My trusty Webster's 9th defines précis as "a concise summary of essential points, statements, or facts" (first usage approx 1760). I suppose the other word (you used it, too) would be plain-vanilla, simple old "summary". Bill Wvbailey (talk) 23:34, 19 January 2015 (UTC)

Disquotationally??

In the Note, it is asserted that the use "here" of "true" is disquotational. First, the link fails to provide any clear insight about wtf the editor is attempting to say. Second, my Shorter Oxford English dictionary has no entry for that word. Speak English, please! Third, the link is to the Redundancy Theory and no longer to Disquotationalism. Fourth, what does "here" mean? Does it mean the editor will on a minute by minute basis check that any future change in this article will comply with his (its gotta be a guy, I think) demands on how "true" must be used in all cases? Or does it mean "in this section"? paragraph? moment? Fifth, How does this "note" contribute to this article? It doesn't as far as I can see. Sixth, it would be nice for the various types of "truth" to be distinguished...I don't think its possible to have only one kind of "true" when discussing a theory of true statements - am I wrong? (There is true, meta-true, meta-meta-true, etc.). My general comment (and based on the arguments page, this is going to be lost on the fan-boys here) is that if a topic is presented in needlessly obfuscatory language, then most of its utility is lost. And this article is clearly needlessly obfuscatory. I suggest the editors, prior to starting their editing session, look in the mirror and repeat "Oh, what a clever, intelligent, witty boy I am!" until its out of their system. Then try to write clearly for the audience that Wikipedia is intended for. Like, by always distinguishing statements (propositions) IN a system from statements ABOUT a system and clearly indicating when a statement is one, the other, or both. Oh, what a smart boy I am!173.189.79.42 (talk) 22:31, 17 May 2015 (UTC)

Just taking one point: You say "I don't think its possible to have only one kind of "true" when discussing a theory of true statements - am I wrong? (There is true, meta-true, meta-meta-true, etc.)".

Answer: Yes, you are wrong. Hope this helps. --Trovatore (talk) 18:09, 18 May 2015 (UTC)

Severely underreferenced

Unfreakingbelievable! Such a fundemental concept with footnotes close to none. Am I supposed to believe every utterance of an anonymous Wikipedian At Large? - üser:Altenmann >t 15:16, 31 August 2015 (UTC)

I see what happened here. You came to this article thinking that sourcing is always done with footnotes. That is not so. This article uses Harvard-style parenthetical referencing. Look for parentheses with a name and a date, like this: (Goodauthor, 2027). --Trovatore (talk) 19:02, 31 August 2015 (UTC)

Don't patronize me and show me Harvard, e.g., in section "Background". If are you telling me that "(simply)" is a Harvard-style ref, then it is malformed (year missing). - üser:Altenmann >t 03:05, 1 September 2015 (UTC)

OK, look, you made a mistake in not noticing it was Harvard. Admit that and quit asking for "footnotes" or complaining about "patronizing".

That said, looking over it, there aren't very many Harvard refs either, and that probably does need to change. But with more Harvard refs.

Some of them are probably malformed, but they're there. Search for "(Hellman 1981, p. 451–468)" for example. --Trovatore (talk) 03:33, 1 September 2015 (UTC)

My mistake was my insufficient English language discrimination. I meant no say "citations close to none". Admit that the article is severely underreferenced and quit nitpicking on technicalities. - üser:Altenmann >t 15:12, 1 September 2015 (UTC)

A significant problem, here, seems to be that the references appear manually at the end of the article, disconnected from their citations, so I'm thinking a solution to this might be to, for each manual citation, move the matching manual reference to the manual citation's location, formatting its contained information as an equivalent "ref" tag, then suffix a "ref-list" so as to auto-generate the references (hope I've explained that clearly enough :-). It'll be a big change. but should (IMHO :-) greatly improve the article. Before I try to make such a change, however: what does everybody think?

Rpot2 (talk) 07:49, 10 March 2016 (UTC)

In any case, there is no lack of experts who have edited this article, or who have this article on their watchlist. I see Trovatore and David Eppstein in the recent edit history, who both have PhDs in mathematics; Wvbailey has done a lot of reading on the subject and knows quite a bit; and if you count Carl Hewitt through his sockpuppets that is another. I have some experience with the incompleteness theorems, as well. As always, my advice for those who want to see more sources is to add them – just adding templates will not lead to the outcome you are looking for, based on experience with many articles on Wikipedia. — Carl (CBM · talk) 02:06, 2 September 2015 (UTC)

One correction: I have a BS in mathematics but my PhD is in CS. —David Eppstein (talk) 04:14, 2 September 2015 (UTC)

May be you have PhD but it seems you cannot read plain English. The template says "This article needs attention from an expert in mathematics". And it is lack of attention to the state of the article I am complaining about. Please don't delete the template until the explained issue is resolved, i.e, until experts in maths take seriously the wikipedia rule about references. CBM: your advice to add them myself is ridiculous. How in freaking hell I can add them if I am not familiar with the subject? This is not a pokemon or pornstar article, it is MATHEMATICS, for God's sake! So, either you start adding references, or I start deleting original exegesis. - üser:Altenmann >t 14:56, 2 September 2015 (UTC)

You should read Wikipedia:Scientific citation guidelines as well, if you are interested in the Wikipedia policies that apply. Remember that there is no rule that requires a citation for each sentence. There are some unreferened sections, which do need some improvement, but remember WP:DEADLINE. I also want to warn you about the three revert rule, which you are in danger of violating. — Carl (CBM · talk) 16:49, 2 September 2015 (UTC)

I have no problems with uncontroversial knowledge. Fact is, I have issues with the text of the article. I am not going to read 15 books listed at the bottom to verify the article. I need citations in situ. - üser:Altenmann >t 02:37, 3 September 2015 (UTC)

In regard the "expert" tag; there is a potential reason for inclusion, as an expert in whatever field Hewitt is expert in might have a different take on the article. We do not have agreement as to whether the theorems are in that field, so I'm not sure consensus could be reached for inclusion. — Arthur Rubin (talk) 18:22, 2 September 2015 (UTC)

remove "background" section or move it to later in article if found to be meaningful...

that first "background" section is all over the place/not sure coherent in whatever it's trying to say/but certain doesn't fit the role of "background" for this topic...article would flow far better if just went right into the following "first incompleteness theorem" section....and whatever is in that 'background' section could be moved to later on, if it's deemed relevant or coherent... 68.48.241.158 (talk) 22:22, 29 February 2016 (UTC)

"meaning of first incompleteness theorem" section would better fit as "background" imo....as it's more general in nature....that first section is trying to get too technical too fast or something...it just feels very, very odd as the first thing you start reading for this topic in article proper...68.48.241.158 (talk) 22:33, 29 February 2016 (UTC)

The background section introduces the terminology that is required for the theorem. On the other hand, the section on "meaning" shouldn't come before the section that actually states the theorem whose meaning is being discussed. — Carl (CBM · talk) 21:27, 10 March 2016 (UTC)

"formal axiomatic system"/"formal system" vs "formal theory"

terminology is inconsistent in the article and jumps back and forth abruptly....most accurately, I think "formal axiomatic system" should be used but could be shortened in places to "formal system"....I've never really heard "formal theory" used in place of the former in this context...but there are many times throughout where "system" and "theory" are used to mean the same thing, and this is very confusing, particularly to someone new to the topic...Hilbert, Russell, Gödel, etc would refer to Principia Mathematica as a formal axiomatic system (or formal system) for arithmetic, I think, and not as a formal theory for arithmetic...(or even more confusing just "theory for arithmetic"...which pops up in this article...).. 68.48.241.158 (talk) 14:39, 14 March 2016 (UTC)

"Formal system" everywhere is fine with me, although "theory" would be the more common word in contemporary logic. They are essentially synonymous. — Carl (CBM · talk) 17:01, 14 March 2016 (UTC)

I would prefer "theory" everywhere. "System" is kind of vague; "theory" is the precise technical term. It does have the disadvantage that its meaning in this context is quite different from its ordinary-English meaning, but that happens a lot in math. --Trovatore (talk) 18:29, 14 March 2016 (UTC)

idk as long as consistent throughout though to avoid confusion..think 'theory' causes more problems in the context of this topic though...as there's usage of 'deductive system' and 'axiomatic system' and would be odd to say 'deductive theory' or 'axiomatic theory'.... 68.48.241.158 (talk) 19:18, 14 March 2016 (UTC)

It's a difference in emphasis. Roughly, a deductive system is axioms plus rules, whereas a theory is axioms plus consequences. --Trovatore (talk) 19:49, 14 March 2016 (UTC)

Sections

just thinking the article has a bit too much of the everything but the kitchen sink feel to it (too much clutter)... SECTIONS: PROOF VIA BERRY'S PARADOX: probably delete or briefly mention in liar paradox section...?? FORMALIZED PROOFS: delete or expand (but probably delete) what does this actually mean?? or actually add to the article?? probably vanity thing for the guy... UNDECIDABLE STATEMENTS PROVABLE IN LARGER SYSTEMS: planted right in middle of the section...seems distracting from actual topic and largely tangential.. EXAMPLES UNDECIDABLE STATEMENTS: same thoughts more or less as previous (at least made more aimed specifically at topic at hand in some way).. PARACONSISTIC LOGIC: idk seems way tangential to merit own section ROLE OF SELF-REFERENCE: I think clearly delete or briefly mention in other section... PROOF SKETCH SECOND THEOREM: seems thrown in at a random place..and not sure qualifies as a proof sketch or adds anything not already discussed..??

obviously these things could be worked on slowly and carefully...but if get rid of some of the clutter would make easier to work on accuracy of what should definitely be included in the article...68.48.241.158 (talk) 20:22, 17 March 2016 (UTC)

The article should be comprehensive, within reasonable space limits. I don't favor removing large parts of the article. Also, it might help discussion if you avoid using the word "accuracy" as if the article is patently inaccurate. Everyone has a preferred way of saying things, but the overall the article is quite accurate as it stands. — Carl (CBM · talk) 10:31, 18 March 2016 (UTC)

the article isn't largely inaccurate but there are a lot of assertions throughout that are vague to the point of being unhelpful/potentially misleading imo..but it would take much time to tighten them all up...not an overnight thing...but a couple sections should be dropped as they stand now, like FORMALIZED PROOFS...simply not worthy of own section within such an important topic...simply vanity reference for the professors...like if every professor who did something to do with Darwin's theory of natural selection created their own three sentence section in that article.....(comprehensive good though...some sections should be expanded..like the HISTORY section and others).. 68.48.241.158 (talk) 12:51, 18 March 2016 (UTC)

I don't see mention of formalized proofs as vanity references - in fact I suspect I added at least a majority of those references, and I am not one of the authors. The point of that text is that, even if someone has doubts about the informal proofs, there are also two fully formalized proofs, which may provide some reassurance to skeptics. This is particularly relevant for the second incompleteness theorem . Moreover, the number of nontrivial theorems for which a fully formalized proof has been produced is very small - I can think of only the prime number theorem, the Kepler conjecture/Hales' theorem, and the incompleteness theorems off the top of my head, and I try to pay attention to such things when they are announced. So the fact that there is a fully formalized proof of the incompleteness theorem also worth noting because it can only be said of a small number of results. — Carl (CBM · talk) 13:18, 18 March 2016 (UTC)

okay perhaps what you just explained above should be explained in the section to make it more worthy of a stand alone section..again, none of this has to be done overnight....68.48.241.158 (talk) 13:38, 18 March 2016 (UTC)

The trouble, of course, is that when we want to add (correct) explanation, someone else will come along and complain that the explanation is not directly quoted from a cited source. So, to limit the amount of commentary that we include in the article, sometimes it is better to say less than we know. If we do find a good source discussing the formalized proofs, then adding some additional, sourced commentary would be great. — Carl (CBM · talk) 13:41, 18 March 2016 (UTC)

first sentence..

"Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic." this is a tricky topic in general that requires very precise wording to be accurate...anyway, think "establish" should be replaced with "demonstrate" as establish kind of suggests Gödel invented the truths of the theorems as opposed to discovered/demonstrated them (ie their truths are independent of Gödel discovering them).. think "all but most trivial" is confusing/not necessary...think reference to a proof of the completeness and consistency of a system capable of addition/subtraction but not multiplication/division...but arithmetic in the context of this topic has to include (Hilbert's program)addition/subtraction/multiplication/division...so the trivial system referenced isn't capable of arithmetic at all but something else..and not sure this 'trivial system' is referenced/explained later in piece so why go off on the tangent at all..particularly in the very first sentence..and since the incompleteness theorems don't address this kind of 'trivial' system anyway?? how's this: Gödel's incompleteness theorems are two proofs in mathematical logic that demonstrate inherent limitations of any formal axiomatic system capable of (expressing) arithmetic. ...don't like the 'doing' either...just sounds funny...maybe 'expressing...' and did away with the repetition of 'theorems' right of the bat.. 68.48.241.158 (talk) 16:49, 14 March 2016 (UTC)

Try just making the change. I will not revert blindly, but I will try to improve on edits (can't speak for others, but hopefully they will do the same). I don't like calling the theorems "proofs". I edited the first sentence with something like what you suggested. — Carl (CBM · talk) 17:09, 14 March 2016 (UTC)

No serious objections to the changes so far (currently, it says Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate inherent limitations of any formal axiomatic system capable of expressing basic arithmetic). I think the "all but the most trivial" part was meant to emphasize that you don't even need all that much of arithmetic (Robinson arithmetic suffices), but we probably don't need to cut things that fine in the opening sentence anyway.

I understand 68's objection to saying "...theorems are two theorems..." but I don't think we have a lot of choice in this case; see Fowler on elegant variation here. When there's a single word that's the most precise, it does not aid understanding to avoid it simply in the name of avoiding repetition. I suppose it wouldn't be totally implausible to replace the second "theorems" by "results", but on the whole I would vote no. --Trovatore (talk) 20:49, 14 March 2016 (UTC)

Ah, I actually do have one issue to raise. There was a change from "capable of doing arithmetic" to "capable of expressing arithmetic". But expressive power lies in the language (and intended interpretation), not the theory or deductive system. I do kind of see the point; it's not clear that "doing" really captures the idea we want. Any thoughts on a better word? --Trovatore (talk) 23:16, 14 March 2016 (UTC)

yeah, if the system is symbolically or typographically complex enough to 'represent' or 'express' or 'do' or 'mirror' whole number arithmetic than it is subject to the theorems...don't know exactly how to best state that in a brief introduction...one solution is to drop it and just state "system capable of arithmetic" or "system capable of whole number arithmetic." 68.48.241.158 (talk) 00:40, 15 March 2016 (UTC)

The use of "arithmetic" to mean the theory of the natural numbers -- rather than just "arithmetic" in the normal sense of addition and multiplication facts -- is confusing for a first sentence in a general article whose readers are not likely to read the term as jargon. I have changed it to "basic properties of the natural numbers" which is less likely to be misinterpreted. — Carl (CBM · talk) 13:53, 15 March 2016 (UTC)

think 'basic properties of natural numbers' is too vague and probably inaccurate...as some systems are capable of expressing basic properties of the natural numbers (addition/subtraction) and have been proven complete and consistent...these theorems specifically relate to systems capable of natural number arithmetic in the classic sense (add/sub/mult/div)...this was crucial to Hilbert's program as every relationship between the natural numbers can in theory be built up from these basic operations....so think the word arithmetic (or what the word technically means in the context of these theorems) must be used somehow....perhaps something like "capable of expressing the arithmetic of the natural numbers."68.48.241.158 (talk) 14:57, 15 March 2016 (UTC)

in that second paragraph the phrase, " is capable of proving all truths about the relations of the natural numbers (arithmetic). " is excellent imo with the parenthetical (arithmetic) as this is exactly what arithmetic meant in the context of these theorems...ie the relations of the natural numbers...to the big brains working on this stuff back then arithmetic technically meant the study of the relationships between the natural numbers...and they understood that the tools needed to study any conceivable relationship were the 4 basic operations... 68.48.241.158 (talk) 15:28, 15 March 2016 (UTC)

The four operations are not enough, though. For the second theorem, a significant amount of strength is required for the theory to prove the Hilbert-Bernays condition, and of course this strength goes well beyond the axioms of

PA^{-}

that determine the basic operations. Separately, comments about the "big brains" in the past won't help us improve the article. — Carl (CBM · talk) 10:35, 18 March 2016 (UTC)

In particular Robinson arithmetic Q proves every true quantifier-free sentence in the language of PA, including every true fact about the arithmetical operations on standard numbers, but Q is not strong enough for the second incompleteness theorem. The article arithmetic seems to be written in a way that will not be very helpful for clarifying this. Because of most people's association of the term with the four basic operations only, I think it is more likely to confuse a naive reader than to clarify matters, in the first sentence. — Carl (CBM · talk) 10:40, 18 March 2016 (UTC)

"capable of expressing sufficient facts about the natural numbers" this may not be wrong but it's so vague that it's sort of meaningless so think not how we'd ultimately want to leave the fist sentence..all that is actually explicitly necessary of a formal system to qualify is addition and multiplication as their inverses (subtraction/division) are derivable from these..B Meltzer in the Dover preface writes, "any formal logical system that disposes of sufficient means to compass the addition and multiplication of positive integers and zero is subject to this limitation."

the second incompleteness theorem is inseperable from the first (naming it 'second' is sort of artificial..Gödel certainly never did) and it relates to the exact same systems as the first...so not sure what you're getting at....

so that last clause might be best if just got quite specific with something like ..system capable of compassing the addition and multiplication of the natural numbers

a to do type thing too is to better explain the requirements of the system in the article proper/explain what arithmetic meant in the context of this topic...68.48.241.158 (talk) 13:31, 18 March 2016 (UTC)

The first incompleteness theorem (rather, its proof) applies to every true theory of arithmetic - no matter how weak - along with theories that may have false axioms but are able to represent all the computable functions. The second incompleteness theorem, in its explicit hypotheses, requires the system to be able to prove various facts about the Provability predicate (these are the so-called Hilbert-Bernays conditions). So, for example, the first incompleteness theorem applies to Robinson's Q, and shows that Q does not prove the Gödel sentence of Q, but Robinson's Q does not satisfy the hypotheses of the second incompleteness theorem, and the second incompleteness theorem cannot be used to show that Q does not prove Con(Q). So it is not the case that the two theorems apply to exactly the same theories.

Now, Q is indeed able to prove every true sentence of the form $n + m =k$ and every true sentence of the form $n * m = k$, but it does not satisfy the hypotheses of the second incompleteness theorem, so I am not especially content with a claim that "capable of compassing the addition and multiplication of the natural numbers" is sufficient for the two theorems. I think it is better to just say "enough facts" or "is sufficiently strong" than to try to detail the Hilbert-Bernays conditions in the first sentence of the article.

Finally, please see the section of this article "Translations, during his lifetime, of Gödel's paper into English" for sourced commentary on the quality of the Meltzer translation. We should cite it, but we should not use it for many serious purposes when there are much better sources available. — Carl (CBM · talk) 13:35, 18 March 2016 (UTC)

perhaps, but the statement I quote is uncontroversial and well put by him...what you're speaking of is beyond me technically but my main concern is vague language within an encyclopedia...like vague technical jargon is a big danger in an article like this...like above when you say "every true theory of arithmetic -no matter how weak..." this is vague and confusing as exactly how weak/strong the system must be is easily and specifically defined...and that Presburger system is too weak for these theorems to apply.. (this is just an example of how vagueness (especially masked in technical jargon) potentially misleads...) 68.48.241.158 (talk) 14:13, 18 March 2016 (UTC)

or to clarify..I've never seen or read anything anywhere to suggest there's any controversy about what kind of systems are subject to the theorems or that defining such is of any difficulty...you seem to be suggesting otherwise in technical jargon that is beyond me personally..so others would have to weigh in...68.48.241.158 (talk) 14:25, 18 March 2016 (UTC)

I think the first sentence will need to be somewhat vague - the one you proposed was at least as vague, but I thought it gave an incorrect impression.

Avoiding any jargon: the first incompleteness theorem applies to Robinson's Q, but the second incompleteness theorem does not, because Q is strong enough to satisfy the hypotheses of the first theorem but not strong enough to satisfy the hypotheses of the second theorem. — Carl (CBM · talk) 15:33, 18 March 2016 (UTC)

this is getting off topic and probably better for the arguments page that was created for this topic but..are you absolutely certain of this?? my clear understanding is if a system is incomplete in this realm then it is by definition subject to what the 'second theorem' states about consistency because the second theorem is entirely demonstrated via what is demonstrated in the 'first theorem.' so if this Q indeed falls prey to the 'first theorem' it will automatically fall prey to the 'second theorem,' no matter what... 68.48.241.158 (talk) 16:21, 18 March 2016 (UTC)

but back to first sentence..perhaps best to go full vague in a sense then, like: ...demonstrate the inherent limitations of any formal axiomatic system of a certain expressive power. and just get out of the arithmetic/number/facts/etc etc business altogether in that first sentence as this will all be explained later on (beginning in the second paragraph of the intro)....68.48.241.158 (talk) 16:37, 18 March 2016 (UTC)

I think that "certain expressive power" is fine, and I made that change. For Q, the first thing I recommend is getting a copy of Peter Smith's "An introduction to Gödel's theorems". Then see Chapter 26. Now, if you won't take my word for anything, I don't know why you would take anyone else's, but you could see e.g. [3] or [4] also. The specific hypotheses needed for both the canonical proofs of the first and second incompleteness theorems (with Rosser's method incorporated) have been thoroughly studied, and the second requires more than the first. Some low-level books either never say clearly what is needed, or just assume the stronger hypotheses throughout. — Carl (CBM · talk) 17:53, 18 March 2016 (UTC)

that's interesting..question though: is Q a kind of "weakened system" for arithmetic not unlike Presburger Arithmetic (though less weakened) that just happens to be susceptible to the first incompleteness theorem..and would therefore not be considered sufficient to potentially satisfy what is known as Hilbert's Program? (ie not fully capable of what was meant by "arithmetic" and therefore of "trivial" interest to Hilbert's program..?)..and not the kind of system addressed by KG...? 68.48.241.158 (talk) 18:27, 18 March 2016 (UTC)

[1] Here Flg(κ) represents the theory generated by κ and "v Gen r" is a particular formula in the language of arithmetic. "Flg" is from "Folgerungsmenge = set of consequences" and "Gen" is from "Generalisation = generalization" (cf Meltzer and Braithwaite 1962, 1992 edition:33-34).

[1]