Talk:Gödel's incompleteness theorems
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Second theorem in terms of the first
editThe 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 [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)
Image
editRegarding 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?
edit- However, since the Gödel sentence cannot itself formally specify its intended interpretation, the truth of the sentence GF 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)→GF, 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
editPlease 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
edit"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)