Talk:Gödel's incompleteness theorems: Difference between revisions

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Revision as of 00:44, 29 January 2007 edit68.114.185.27 (talk) →is the incompleteness theorem complete?← Previous edit		Latest revision as of 10:16, 29 November 2024 edit undoJochen Burghardt (talk \| contribs)Extended confirmed users23,706 edits →Provability: ReplyTag: Reply
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			== Second theorem in terms of the first ==
	{{DelistedGA}}

			The first theorem states that there exists a true statement ''F'' cannot prove. A few months ago I added an stating that the second theorem in particular provides a concrete example of one of the true statements that ''F'' cannot prove, namely Cons(''F''). It was reverted, with edit summary "how do you know it is true?" But since one of the premises of the theorem is that ''F'' is consistent, Cons(''F'') is true, so I believe Cons(''F'') is a concrete example of a true statement not provable from ''F''. If this reasoning is correct, should the specification be re-added to the article? I think it may be helpful to see the second theorem as providing a specific example of the first theorem, even if the first theorem already involves the Godel sentence. ] (]) 08:02, 27 October 2022 (UTC)
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			: I have a source that I think claims the second theorem provides an example of the first:
	== Re: ==

			:: "He followed this with his First and Second Incompleteness Theorems. The first one asserts that every sufficiently extensive, consistent formal system (and almost all formal systems are sufficiently extensive) is incomplete in the sense that there exist sentences expressed within the system that cannot be decided within it. The second one provides additional information that consistency of such a system is a sentence of this kind."
	Hey, ], I am curious what fault you found with the article at http://www.mind-crafts.com/godels_incompleteness_theorems.html that made you remove the link to it I put in External Links. I have to confess that IMHO it's the best non-technical introduction to Gödel's theorems on the Internet. ] 16:55, 16 November 2006 (UTC)
	:Paul, actually, I took a brief look and (at least at that level of scrutiny) it does look pretty good. We don't want to encourage link farming but this one really might be worthwhile. --] 03:52, 27 November 2006 (UTC)
	::OK, I restored the link - since no arguments were given against it. Will be happy to listen and discuss if anybody comes up with cogent args. ] 22:45, 3 December 2006 (UTC)

			: From ''Harvey Friedman's Research on the Foundations of Mathematics'' (1985), Studies in Logic and the Foundations of Mathematics vol. 117. p.viii. ] (]) 09:45, 8 November 2022 (UTC)
	== Removal of Godel's statement of theorem ==

			== Image ==
	I removed Godel's original statement of the theorem, but it's been reverted back. Godel's statement:
	:To every ''ω''-consistent recursive class ''κ'' of ''formulae'' there correspond recursive ''class signs r'', such that neither ''v'' Gen ''r'' nor Neg(''v'' Gen ''r'') belongs to Flg(''κ'') (where ''v'' is the ''free variable'' of ''r'').
	is only meaningful if you explain Godel's notation (e.g., class signs, Gen, Neg, and Flg), which we don't really need to do in this article. Perhaps for historical interest, we can have all this in its own section of the article, but it's not useful where it stands now and will just mystify most people. If there are no objections, I'll remove it again soon. -] 09:55, 27 November 2006 (UTC)

			Regarding Jochen's addition of an image, .
	:Obviously, I disagree or I would not have reverted you. You took out a section header which is inappropriate. And the formal statement of the theorem may be of interest to some people, such as those who have read the paper. If you think that it is too advanced a topic, the appropriate course is to move it to the end of the article and add more explanatory material. ] 10:29, 27 November 2006 (UTC)
	::Thanks for the explanation, JRSpriggs. The section header was removed because it was introducing the deleted content and didn't make sense without it. As for the content itself, I don't think it's a matter of the topic being "too advanced"; it's that it's unreasonable for us to expect our audience to have read Godel's paper (with the German abbreviations intact, at that). As stated right now, I feel it does more harm than good, especially given the recent remarks about the article's accessibility. Like you suggest, a possible solution would be to move it to the end with more explanation, which I would be happy with and may attempt when I get a chance. -] 16:12, 27 November 2006 (UTC)
	:::With respect to JR, I tend to agree with Spurious here. Gödel's proof was brilliant; his notation, like almost all version-1.0 notation, was less than ideal from the point of view of long-after-the-fact exposition. And even if the notation were nicer, we still should not want to get into the details of a specific coding scheme early in the article, as it's kind of beside the point. --] 16:27, 27 November 2006 (UTC)

			Jochen says in the edit summary "no idea how to illustrate the theorem itself", and I basically agree. Oh, someone could probably come up with ''something'', maybe some sort of block diagram, but I doubt it would actually be helpful.
	::::Word. ] 21:26, 27 November 2006 (UTC)

			But to be honest I don't think the proffered image is particularly helpful either.
	== The bizarre proof sketch ==

			So my vote would be just not to ''have'' an image. I don't think there's any great value in having an image purely ''pro forma''. If there's no image that genuinely helps direct the reader to the point of the article, then why have one at all? --] (]) 20:25, 20 February 2023 (UTC)
	<s>How can you say that P(x) can be proved if x is the Gödel number of a provable statement explicitly explaining in the previous paragraph that the Gödel numbers encode single-parameter formulas, which are not statements and, therefore, cannot be proved or disproved? In other words, the concept of "provability" is inappropriate to F when we rewrite P(x) as P(G(F)). --] 02:17, 6 January 2007 (UTC)</s>

			==Confusing part about truth of Goedel sentence?==
	:A given Gödel number might encode a single-parameter formula, or any other formula, or a statement. I've made an edit that I think might clarify a bit; does it address the problem you see? If not, can you be more specific about what text you see the problem in? Thanks in advance! —]<sub><small><i>]</i></small></sub> 06:49, 6 January 2007 (UTC)
			: However, since the Gödel sentence cannot itself formally specify its intended interpretation, the truth of the sentence {{math\|''G''<sub>''F''</sub>}} may only be arrived at via a meta-analysis from outside the system. In general, this meta-analysis can be carried out within the weak formal system known as ], which proves the implication {{math\|''Con''(''F'')→''G''<sub>F</sub>}}, where {{math\|''Con''(''F'')}} is a canonical sentence asserting the consistency of {{mvar\|F}}.

			Could this part confuse the reader? I am not sure what it would mean for a sentence to "specify its intended interpretation" (perhaps meaning that no sentence can single out the standard/true natural numbers?) For the second sentence it may help to mention that T proving Con(PA) -> (Goedel sentence for PA) and T proving (Goedel sentence for PA) are different phenomena, and while PRA is an example of such a theory T in the former case, much stronger theories need to be considered in the latter case (theories stronger than PA). ] (]) 22:39, 19 April 2024 (UTC)
	::<s>At first, reading this article and the ], I have got the affirmation that a given Gödel number encodes a formula. The mapping one-to-one at that. Therefore, it seems that you are wrong when telling that a number (''a'' means ''one'' in english) encodes a form or another form. It encodes <u>only one form</u> but not another. Secondly, since x is a natural number, a form generates an infinite number of statements. That is, infinitely many statements F(x) conform to a form F. Taking into account that the notion of proof does not apply to the forms, we are precluded from defining the decider P(G(F)). The decider must be provided with a statement; i.e., both the number of F and x. --] 13:15, 6 January 2007 (UTC)</s>

			== bew redirects to here ==
	:::Sorry, you misunderstood me. I didn't mean that a Gödel number might ''simultaneously'' encode all of those things; I meant that a given Gödel number could be a Gödel number that encodes a single-parameter formula, or a Gödel number that encodes a formula of some other type, or a Gödel number that encodes a statement.

			Please see ] and reply there, if desired. I am proposing that the redirect ] be pointed to ] instead of to ]. - ] (]) 02:17, 24 April 2024 (UTC)
	:::Also, the article does not use the term ''decider'' and does not make reference to ''P''('''G'''(''F'')), so I don't know what you're referring to. Can you quote a specific sentence from the article so I can see what you're objecting to?

			== I think this phrasing is inaccurate ==
	:::—]<sub><small><i>]</i></small></sub> 19:41, 6 January 2007 (UTC)

			"For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system."
	:To Javalenok: Please do not remove the messages of other editors (unless they are vandalism, libel, obscenity, etc.). Also do not remove your own prior messages if they provide context to another user's messages. If you like, you may strike-out the parts of your messages with which you no longer agree using <nowiki><s> and </s></nowiki> (having this effect <s> and </s>). Consequently, I have restored the messages by Ruakh and yours which provide context. ] 07:35, 9 January 2007 (UTC)
			I know it's typically quoted like this but that gets corrected in a lot of books on incompleteness. If there are statements which are "true", or that we can find are true, then incompleteness wouldn't apply to it. I think it's thinking of truth in terms of consistency but the validity is being checked through provability in formalism. The correct phrasing is "if there are true statements within a (formalist) system then they are unprovable." The idea is to see if terms are semantically true through logical relations (or checking soundness through validity in a sense). ] (]) 04:48, 16 October 2024 (UTC)

			== Provability ==
	Sorry my fussiness, I have overlooked in the beginning that statements can also have Gödel numbers. Before this discussion will be deleted, please explain why do you substitute the Godel number ''x'' by formula ''F''(''G''(''F'')) in ''P''(''x'')? The formula is not a number. The grammatically correct statement would be P(<b>G(</b>F(G(F))<b>)</b>). Furthermore, the substitution of variable ''x'' in ''P''('x') by the number of statement ''p'' turns the form into statement and its number changes accordingly. But the resulting statement is p, so its number 1)must be fixed; and 2) known in advance!!! Where do we get the number from? --] 20:33, 11 January 2007 (UTC)

			I can't find in Misplaced Pages a definition of "provability" adequate for the context of this article. Can someone with knowledge of thia topic please help? Thanks, ] (]) 22:55, 28 November 2024 (UTC)
	Also, I suggest to incorporate the the '#' notation for 'G()' as in so that P(#F(#F)) looks better than P(G(F(G(F)))). --] 16:10, 8 January 2007 (UTC)

			:{{done}} I tried. I'm not sure whether we should state in a footnote that "limits of provability" means, more precisely, that it is impossible to find a proof for every truth (even about natural numbers' arithmetic). - ] (]) 10:16, 29 November 2024 (UTC)
	In addition, let me note that the formalism p = SU(#SU) corresponds to the sentence 'p is equal to "this statement is not provable"' rather than the 'desired' "''p'' cannot be proved". Peahaps, the difference between two sentences of lair paradox does not really matter. Nevertheless, the neccessety for the redundant 'p' confuses me. --] 23:02, 8 January 2007 (UTC)

	== What is the need for self-unprovable formulas? ==
	Let me give a shorter sketch. I see no need to define SU. The statement p equivalent to "p cannot be proven" can be easily formalized as p = ~P(p). Now, if p is provable true then the argument p would be not provable (a contradiction). Alternatively, if p is false (that is, not provable true) then P(p) would be true meaning the argument p is provable true (again contradiction). --] 13:44, 10 January 2007 (UTC)

	:You said "...if p is false (that is, not provable true)...". Falsity is the same as not BEING true. That is different from not PROVABLE. This distinction is essential to understanding Gödel's theorem. There is no inconsistency in the existence of a sentence ''p'' which is simultaneously true and not provable. Gödel never proved ''p'', i.e. that ''p'' cannot be proved. What he proved was ''ω-con→p'', i.e. that IF arithmetic is ω-consistent, THEN ''p'' cannot be proved. ] 06:39, 11 January 2007 (UTC)

	::OK, may be I should tell: IF p = false THEN P(p) = true HENCE p is provable true HENCE contradiction (p and ~p) = true. The purpose of the incompleteness theorem is to show that a consistent theory cannot be complete. I'm showing that it is possible to do without introducing the self-unprovable forms. Any objections? --] 08:47, 11 January 2007 (UTC)

	:::Your first step, going from ''Neg(p)'' to ''P(p)'', is changing context which is logically invalid. You are going from an arithmetic statement to a meta-arithmetic statement. Although Gödel showed that some meta-arithmetic statements can be pushed down into arithmetic, it does NOT go the other way (as you would have it). Also you say "I'm showing that it is possible to do without introducing the self-unprovable forms.". I do not see what you are getting at here. You are still using ''p'' in your argument, and ''p'' is the "self-unprovable form" to which you object, is it not? ] 08:36, 12 January 2007 (UTC)

	::: Look, SU(#F) is defined as ~P(#F(#F)), so SU(#SU) = ~P(#SU(#SU)). Given SU(#SU) = p, we can rewrite the equality as p = ~P(#p). This equality states self-unprovability explicitly and, furthermore, eliminates the needles self-unprovable form SU(#F). The Neg(p) is ~~P(#p). But you deny the logical conclusion ~~P(#p) => P(#p). Instead, you prefer to go to english language. But stating that some self-unprovable can be proved is an outright nonsense. The dares to state: "Suppose that ~(~P(p)) is a theorem. Hence P(p) is a theorem." --] 23:32, 13 January 2007 (UTC)

	::::You have completely misunderstood my last message. There is no meaningful difference between "P(p)" and "~~P(p)"; and I did not say otherwise. The vital distinction which you are ignoring is between statements in English about arithmetic and statements in the language of arithmetic which might be regarded as (in some sense) encoding the English statements. Although the encoding is chosen with the intention of making them equivalent, they are not (known to be) PROVABLY equivalent either from the axioms of arithmetic or from meta-arithmetic considerations. ] 09:24, 17 January 2007 (UTC)

	==How About==
	We split out the philosophical ruminations into a separate article and make this article as clear and concise a description of the theorem(s) and their context in mathematics and metamathematics (no easy feat, I'm sure).
	If nothing else this article is becoming very long and I think it'd benefit.

	That said, I confess I'm not sure how this would be done -- do we need to have a discussion/vote? should one '''be bold''' and just start the other article?
	Thoughts?
	] 21:31, 10 January 2007 (UTC)

	:Let's talk about it first. The ruminations here are quite short, which seems like the ideal situation to me. Are you saying that you would like to expand them?
	:If you have time and would like to improve this article, you might start by going through and changing the tone so that it is more encyclopedic, without the "we" statements. ] · <small>]</small> 12:22, 11 January 2007 (UTC)
	::Well, the ] section may be better fit in its own article, or merged over to ]. Actually, there are a bunch of disparate treatments of this scattered over Misplaced Pages, e.g., here, in ], ], and ]. We should pick a primary place for the content to go and the rest can link to it. -] 12:27, 12 January 2007 (UTC)

	:::I am all in favor of reducing redundancy. I am fearful about a separate article because I am not a philosopher. I am afraid it might grow into a giant mess of speculation, OR, and non-notable opinions, and it would be hard to fix. But I wouldn't revert the move if it happened. You might leave a one-paragraph summary here, and point to the main article with the {{tl\|main}} template. ] · <small>]</small> 14:23, 12 January 2007 (UTC)

	I think we need to distinguish between two sorts of philosophical issue. At least something ought to be said in this article about implications for philosopy ''of mathematics'' -- particularly, the difficulties that Gödel's results have presented for the formalist and logicist programs. That's part of the historical background in which the results were announced.

	The stuff about philosophy ''of mind'', on the other hand, should not be here in my opinion, with the exception of a link somewhere and a very brief summary. This is a mathematics article. --] 20:59, 12 January 2007 (UTC)
	:OK, I merged the "Minds and Machines" section over to ]. -] 01:14, 17 January 2007 (UTC)

	== Gödel sentence is an objection to AI. Is it true? ==
	''Moved to ].''
	: This question is not an "argument over the validity of the incompeteness theorem". Why was it moved there? --] 20:36, 11 January 2007 (UTC)
	::It's not directly related to improving the article. --] 21:24, 11 January 2007 (UTC)


	== Is the incompleteness theorem complete??? ==
	''Moved to ].''
	=="Publication" cat==
	There's been a bit of reverting going on over the category ]. My take: first of all, the cat itself should be renamed (categories of this sort take plural names). Second, no, this article doesn't belong in that category, though <s> a category</s> an article named after Gödel's famous paper, "On Formally Undecidable Propositions...", would belong in the category. This article is not (primarily) about that paper, and therefore doesn't. --] 20:43, 17 January 2007 (UTC)

	:I strongly agree that the category should be pluralized, and weakly agree that it isn't suitable here. —]<sub><small><i>]</i></small></sub> 22:44, 17 January 2007 (UTC)

	The category tag seems to be added and removed over and over again. I think Trovatore's contention is that this article is about the theorems, not the publication. I agree. How about I create a short article on the actual publication, which is certainly notable because it introduced these theorems. Then that page can get added to the category, and everyone wins. ] · <small>]</small> 18:55, 24 January 2007 (UTC)

	:Yes, please! And if the article is out of copyright, we can put its full text on Wikisource and both articles can link to it. :-D —]<sub><small><i>]</i></small></sub> 19:17, 24 January 2007 (UTC)

	::The article itself was originally published in German in 1931; I have no idea what its copyright status is. The later English translations are almost certainly still under copyright in the United States. ] · <small>]</small> 19:27, 24 January 2007 (UTC)

	:: Done: ]. ] · <small>]</small> 02:02, 25 January 2007 (UTC)
	:::Nice work, Carl. I might quibble on the link to the problematic article ], which I think should probably be deleted or redirected somewhere. But it looks very good, and you finished it quickly! --] 02:51, 25 January 2007 (UTC)
	::::My general philosophy is that a link should be provided if it theoretically helps the linking article. Otherwise, when the destination article is improved, how will anyone know which articles should link to it? I have the same belief about red links. ] · <small>]</small> 03:29, 25 January 2007 (UTC)
	:::::Well, that makes sense, if there ought to be an article called "Gödel numbering". Should there? I'm not convinced. A ''redirect'' for sure, but an article? And the argument doesn't strike me as being quite as strong when talking about redirects. --] 03:52, 25 January 2007 (UTC)
	::::::The concept of Godel numbering the sentences of a first-order theory deserves some article. When the current article on Godel numbering is fixed, at least "what links here" will be accurate. The person making the redirect, or a bot in some cases, should resolve the link in the original article to bypass the redirect.
	::::::My thought about the Godel numbering article is that it should be an article, but it should just give a catalog of the various senses of the phrase, possibly with links to larger articles on each of these senses. ] · <small>]</small> 04:12, 25 January 2007 (UTC)

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

	: I agree that Peano arithmetic is a formal system that adequately formalizes Peano arithmetic. I added a caveat to the lead - the well known decidable formal systems for geometry and the real numbers are simple, but not "trivial". ] · <small>]</small> 13:36, 26 January 2007 (UTC)


	==Commercial links==

	The link to "Askanas, Malgosia, 2006, "Gödel Incompleteness Theorems - A Brief Introduction." A lucid and enjoyable exposition of both theorems in the spirit of Gödel's original proof. "is not a bad site for explaining this theorem--well above the high school student's level though--but its "lucid and enjoyable" presents a value judgement which is not at all consistent with the purpose of the article and Misplaced Pages. It is a commercial site selling services in math education. It should be retained, in my view, it does a fairly good job of explaining the concepts, but the evaluation needs to be reconsidered. ] 03:31, 27 January 2007 (UTC)

	:I removed the annotation. You can (and, I would suggest, should) have moved it here along with your comment. In cases where a phrase is clearly not neutral POV, seems to advertise for a commercial service, or both, there is no leave the statement while inquiring about it. ] · <small>]</small> 03:54, 27 January 2007 (UTC)

	==is the incompleteness theorem complete?==
	This is not an argument ''against'' the incompleteness theorem. This is a ''question'' ''about'' the incompleteness theorem, what it ''means'', and how inclusive it is. Does the incompleteness theorem include itself, and if so, does that make it complete? Secondly, if the incompleteness theorem is complete, does that make it consistent, or inconsistent? How do we really know, and ''can'' we decide? If we ''can'' decide, isn't it the same thing as the ]? I would appreciate it if somebody could give me a straight answer. "I really don't know" is acceptable and all I'm asking for is honesty. -{{unsigned\|68.114.185.27}}
	:Two things: first, please use the preview button instead of constantly editing this comment. Second, you have already stated you are ] and as such should not be editing here. The proper course of action for you is to convince the admins that you should be unblocked, and to do that I would recommend demonstrating an understanding of why you were blocked in the first place. -] 23:29, 28 January 2007 (UTC)

	:I never said I was Germanium, and besides which, you're dodging the issue. I'm asking is a simple question that deserves a simple answer. Again, I don't know is fine and appreciated if you have no real answer to the question. THanks

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Second theorem in terms of the first

The first theorem states that there exists a true statement F cannot prove. A few months ago I added an edit stating that the second theorem in particular provides a concrete example of one of the true statements that F cannot prove, namely Cons(F). It was reverted, with edit summary "how do you know it is true?" But since one of the premises of the theorem is that F is consistent, Cons(F) is true, so I believe Cons(F) is a concrete example of a true statement not provable from F. If this reasoning is correct, should the specification be re-added to the article? I think it may be helpful to see the second theorem as providing a specific example of the first theorem, even if the first theorem already involves the Godel sentence. C7XWiki (talk) 08:02, 27 October 2022 (UTC)

I have a source that I think claims the second theorem provides an example of the first:

"He followed this with his First and Second Incompleteness Theorems. The first one asserts that every sufficiently extensive, consistent formal system (and almost all formal systems are sufficiently extensive) is incomplete in the sense that there exist sentences expressed within the system that cannot be decided within it. The second one provides additional information that consistency of such a system is a sentence of this kind."

From Harvey Friedman's Research on the Foundations of Mathematics (1985), Studies in Logic and the Foundations of Mathematics vol. 117. p.viii. C7XWiki (talk) 09:45, 8 November 2022 (UTC)

Image

Regarding Jochen's addition of an image, here.

Jochen says in the edit summary "no idea how to illustrate the theorem itself", and I basically agree. Oh, someone could probably come up with something, maybe some sort of block diagram, but I doubt it would actually be helpful.

But to be honest I don't think the proffered image is particularly helpful either.

So my vote would be just not to have an image. I don't think there's any great value in having an image purely pro forma. If there's no image that genuinely helps direct the reader to the point of the article, then why have one at all? --Trovatore (talk) 20:25, 20 February 2023 (UTC)

Confusing part about truth of Goedel sentence?

However, since the Gödel sentence cannot itself formally specify its intended interpretation, the truth of the sentence G_F may only be arrived at via a meta-analysis from outside the system. In general, this meta-analysis can be carried out within the weak formal system known as primitive recursive arithmetic, which proves the implication Con(F)→G_F, where Con(F) is a canonical sentence asserting the consistency of F.

Could this part confuse the reader? I am not sure what it would mean for a sentence to "specify its intended interpretation" (perhaps meaning that no sentence can single out the standard/true natural numbers?) For the second sentence it may help to mention that T proving Con(PA) -> (Goedel sentence for PA) and T proving (Goedel sentence for PA) are different phenomena, and while PRA is an example of such a theory T in the former case, much stronger theories need to be considered in the latter case (theories stronger than PA). C7XWiki (talk) 22:39, 19 April 2024 (UTC)

bew redirects to here

Please see Talk:BEW#BEW, Bew, bew and reply there, if desired. I am proposing that the redirect Bew be pointed to BEW instead of to Gödel's incompleteness theorems#Bew. - dcljr (talk) 02:17, 24 April 2024 (UTC)

I think this phrasing is inaccurate

"For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system." I know it's typically quoted like this but that gets corrected in a lot of books on incompleteness. If there are statements which are "true", or that we can find are true, then incompleteness wouldn't apply to it. I think it's thinking of truth in terms of consistency but the validity is being checked through provability in formalism. The correct phrasing is "if there are true statements within a (formalist) system then they are unprovable." The idea is to see if terms are semantically true through logical relations (or checking soundness through validity in a sense). 2407:4D00:AC00:8A6D:4283:C8DB:423F:37D7 (talk) 04:48, 16 October 2024 (UTC)

Provability

I can't find in Misplaced Pages a definition of "provability" adequate for the context of this article. Can someone with knowledge of thia topic please help? Thanks, DPdH (talk) 22:55, 28 November 2024 (UTC)

Done I tried. I'm not sure whether we should state in a footnote that "limits of provability" means, more precisely, that it is impossible to find a proof for every truth (even about natural numbers' arithmetic). - Jochen Burghardt (talk) 10:16, 29 November 2024 (UTC)

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