Talk: Gödel's incompleteness theorems

The first theorem states that there exists a true statement F cannot prove. A few months ago I added an [https://en.wikipedia.org/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&diff=next&oldid=1089149326 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 [Cons(F)] 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 [Godel] 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)

Regarding Jochen's addition of an image, [https://en.wikipedia.org/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&diff=1140576680&oldid=1140561191 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)

: However, since the Gödel sentence cannot itself formally specify its intended interpretation, the truth of the sentence {{math|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 {{math|Con(F)→G_F}}, 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). C7XWiki (talk) 22:39, 19 April 2024 (UTC)

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)

"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)

I can't find in Wikipedia 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)

What is the rationale for having a section entitled "Translations, during his lifetime, of Gödel’s paper into English", rather than a section simply entitled "Translations of Gödel's paper into English"? Does such a translation date cutoff appear anywhere else on the entire WikiPedia website? Furthermore, the cutoff makes no sense when there is another section "Articles by others" that includes a reference to an English translation by Martin Hirzel dated 2000, long after Gödel’s death.

It would appear that the only reason it is done this way is in order to exclude my online freely accessible translation (https://www.jamesrmeyer.com/ffgit/godel-original-english), a HTML translation that is well-formatted and has clickable cross-references, and which includes a colored background for certain words which enables a clear distinction between a word that refers directly to a relation and the same word when it is intended to refer to the corresponding Gödel number for that relation. The translation is the first result in a search for "English translation of Gödel's incompleteness proof" in nearly all Internet search engines.

The exclusion of this translation has all the signs of implying an underlying bias against either myself or my website, or both, and which is completely contrary to the Wikipedia ethos.

This exclusion should be remedied as soon as possible. I suggest that the heading "Translations, during his lifetime, of Gödel's paper into English" be replaced by "English translations of Gödel's paper", and that the section should include details of all translations of the paper. If it is considered appropriate, one could have sub-headings "During Gödel's lifetime" and "Later translations". Jamesrmeyer (talk) 21:54, 27 May 2025 (UTC)