Talk:Löb's theorem

message: Can someone add an example? --Michalchik (talk) 09:43, 8 April 2009 (UTC)
replies: text: Sad to say, Martin Loeb died a couple of months ago.  I have just done an article based on obituaries. Robin Gandy is mentioned in the obits, but I left him out as I was not sure how important he was. -- 15:40, 3 October 2006 (UTC)

message: Most of the content I know of appropriate either for this article or for provability logic could go in either:
#The case for putting the overlap in provability logic is that that article is most easy to situate within the wider context of modal logic, so the content is eaasier to fit in context;
#On the other hand this overlap is a nice entry point into provability logic for the majority of logic students who are more familiar with the formalisation of consistency proofs than they are with modal logic.
I'm undecided, but I lean towards giving a reasonable short introduction on this page, and a full treatment on the provability logic page. ---- Charles Stewart 06:38, 6 Oct 2004 (UTC)
replies: text: The questions are (i) where should the main body of information appear, and (ii) how much overlap gives the best reader experience ---- Charles Stewart 11:39, 6 Oct 2004 (UTC)
thread_title:  Dividing up content between here and provability logic 

message: I have removed the following text:
"This theorem is problematic for a number of reasons, and for this reason, has been referred to by Barwise & Etchemendy as "Löb's paradox."  For example, consider the statement: "Santa Claus exists."
If it is necessarily true that Santa Claus exists, it seems that we ought to infer that Santa Claus does, in fact, exist---indeed, that this inference must be drawn necessarily.
But by Löb's theorem, this implies that Santa Claus necessarily exists, from which we infer that Santa Claus exists! Any statement could be exchanged for "Santa Claus exists," and so a paradox results."''
There is misunderstanding of what Lob's theorem proves by the author who inserted this erroneous discussion. Lob's theorem states that if T proves ""exists proof of A in T" -> A" then T proves A. There is no link to necessary existence, where Lob's theorem is hardly to be applied. Indeed Lob's theorem is valid only in theories T that can express the concept of "provability" at first place, otherwise it is not clear how you can formulate the Lob's theorem at first place. Danko Georgiev MD 06:11, 25 June 2007 (UTC)
thread_title:  Removed wrong text

message: Can't we argue in the following way:
Assume T \vdash ((T \vdash \varphi) \Rarr \varphi).
Then T + \neg\varphi \vdash \neg(T \vdash \varphi).
But T \vdash \varphi iff T \vdash (\neg\varphi \Rarr \varphi) iff (T + \neg\varphi) \vdash \varphi, and moreover these equivalences are provable in T.
Therefore T + \neg\varphi \vdash \neg((T + \neg\varphi) \vdash \varphi).
Therefore T + \neg\varphi proves its own consistency, so by Gödel's second theorem T + \neg\varphi is inconsistent.
Therefore T \vdash \varphi.
(One can deduce Gödel's second theorem from Löb's theorem even more easily: If T \vdash \neg(T \vdash \bot) then T \vdash ((T \vdash \bot) \Rarr \bot). Then by Löb's theorem, T \vdash \bot.)
: I believe this is essentially Kripke's proof of Löb's theorem.  In the introduction to his book _The Logic of Provability_, Boolos writes "Around 1966, Kripke realized that Löb's theorem is a direct consequence of Gödel's second incompleteness theorem for single-sentence extensions of PA."  Then he quickly runs through a proof that I think is essentially this one (though I found this one easier to understand!).  John Baez (talk) 05:27, 16 December 2013 (UTC)
thread_title: Löb's theorem as a corollary of Gödel's second incompleteness theorem

message: "If there is a proof of the Goldbach conjecture, then the Goldbach conjecture is true" - I can accept that.
As a consequence of Löb's theorem, "Goldbach conjecture is true"??
--88.153.45.144 (talk) 19:19, 8 April 2013 (UTC)
replies: text: "I can accept that." You can accept, but can you describe this acception in PA? --Davidsusu (talk) 22:27, 18 June 2021 (UTC)
thread_title:  I don't get it. 

message: The theorem effectively says: "If statement P cannot be proven, then (within the same theory) it cannot be proven that the statement P cannot be proven" (i.e. it can't be proven that there doesn't exist a natural number N that encodes a proof of statement P). (Taking P to be a contradiction, e.g. x!=x, we get Gödel's result that a consistent theory can't prove its own consistency.) - Mike Rosoft (talk) 20:22, 19 January 2016 (UTC)
thread_title: Alternate formulation

message: In the formula, '\vdash' and 'Prov()', and also '\to' and 'if-then' ('\dfrac{A}{B}') are mixed. Also, is the two variant necessary? --Davidsusu (talk) 00:34, 19 June 2021 (UTC)
replies: text: The string of "and"s make it hard to tell which symbols you consider mixing.  Only the if-then and \dfrac{A}{B} are equivalent.   I think the second is formula is slightly preferable since I think the use of if-then is often taken as \rightarrow, which it should not be in this case. AquitaneHungerForce (talk) 09:01, 19 June 2021 (UTC)
thread_title: More consistent notation of the formula