Talk:Gödel's incompleteness theorems

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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)\to 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)[reply]

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)[reply]

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)[reply]

Provability

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)[reply]

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)[reply]

Exclusion of English translation of Gödel’s paper

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)[reply]

Discussion moved

The discussion entitled "What Do We Call 'Truth' Here?" has been moved to Talk:Gödel's incompleteness theorems/Arguments#What Do We Call 'Truth' Here?. D.Lazard (talk) 10:00, 29 May 2025 (UTC)[reply]