Timeline for Did Russell understand Gödel's incompleteness theorems?
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| Jul 30, 2020 at 12:04 | comment | added | Ace shinigami | PM is not safe from Godel, and in fact no useful axiomatic system is. Type Theory and ZF Set Theory both set out to solve the sizing problems in Cantor's Set Theory, which lead to the Liar's paradox style issues you talked about. They do not protect by the style of diagonalization proof used by Godel. PM gives Godel more than enough room to prove PM's incompleteness. | |
| Sep 6, 2017 at 14:06 | comment | added | user13627 | Feel free to join the discussion we are having here (and which in my opinion is related to the question). | |
| Aug 9, 2017 at 19:07 | comment | added | Conifold | @user170039 See references in What sources discuss Russell's response to Gödel's incompleteness theorems? Russell is quoted there:"I was puzzled by it. It made me glad that I was no longer working at mathematical logic. If a given set of axioms leads to a contradiction, it is clear that at least one of the axioms must be false." But unlike the Liar sentence, or Russell's paradox modeled on it, "I am unprovable" does not lead to contradictory conclusions (there is no contradiction with it being unprovable rather than "false"). | |
| Jul 29, 2017 at 6:58 | comment | added | user13627 | @LuísHenrique: Can you explain a bit more what you meant by, "[t]he problem with this is that it supposes that the "meaning" of a sentence does not imply the whole language in which it is expressed" (especially the bold part)? | |
| Jul 29, 2017 at 6:56 | comment | added | user13627 | @Conifold: I think that paradox is not the exact word that Russell. So far as I recall, in My Philosophical Development he used the word "puzzle" (maybe George Chen also used the word in this sense). Anyway, can you give some references/arguments which supports your saying that "..."paradox presented by Gödel sentence was nothing new; it was the same old vicious circle paradox, which had been abundantly dispelled by Russell's Theory of Types" is not the case."? | |
| Jul 27, 2017 at 13:57 | comment | added | user13627 | I don't understand how (as Conifold asked) the "paradox presented by Gödel sentence was nothing new; it was the same old vicious circle paradox". Can you elaborate that a bit? | |
| Apr 13, 2017 at 12:42 | history | edited | CommunityBot |
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| Oct 15, 2016 at 14:03 | comment | added | George Chen | Russell is appealing to one's own sense. Quoting texts from authoritative figures is not going to help, although quite a few academics are doing swimmingly well by just that. This sense, Russell admitted, is not possessed by everyone. Descriptions in olfactory terms are definitely moot to people who have no sense of smell, but walking around with a strong BO does tell something about one's olfactory nerve. | |
| Oct 14, 2016 at 0:06 | comment | added | George Chen | I suspect Russell's meekness was some sort of mischief, a practical joke of some sort. People don't know how mischievous Russell was. | |
| Oct 13, 2016 at 21:56 | comment | added | Conifold | First, "paradox presented by Gödel sentence was nothing new; it was the same old vicious circle paradox, which had been abundantly dispelled by Russell's Theory of Types" is not the case. Gödel sentence involves no paradox, and that is what critics charged Russell (and Wittgenstein) with not understanding. Second, "G had no meanings and thus did not belong to the body of propositions" is moot for Gödel's reasoning. But these two views are exactly the mistakes attributed to Russell and Wittgenstein, rightly or wrongly wab.uib.no/agora/tools/alws/… | |
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| Oct 13, 2016 at 15:27 | history | edited | George Chen | CC BY-SA 3.0 |
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| Oct 13, 2016 at 13:28 | history | edited | George Chen | CC BY-SA 3.0 |
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| Oct 13, 2016 at 12:03 | comment | added | George Chen | @NieldeBeaudrap: Thanks, but I think your answer is more relevant to the question. In spite of everything I say, my answer as to Russell's understanding is indeed speculative in nature. I just shine some light on Russell's background knowledge and let the reader decide how likely it is that Russell did not understand G. | |
| Oct 13, 2016 at 10:12 | comment | added | Niel de Beaudrap | Your answer contains interesting and plausible claims; not to mention technical assertions with which I broadly agree, in respect to self-referentiality vis-a-vis Gödel's theorem. Properly substantiated, this answer should be voted more highly than mine, and 'accepted'. But your answer also contains quite a bit of your own opinion beyond what an answer on a philosophical question must unavoidably have, which makes it more difficult to separate answer from editorial. Would you be willing to revise your answer, to make clearer the correspondance of assertions to citeable sources? | |
| Oct 12, 2016 at 19:58 | comment | added | Luís Henrique | "A self-referential sentence G's meaning cannot be determined until each of its constituent's meaning is determined; one of G's constituent is G itself, thus G's meaning cannot be determined because G contains a vicious circle." - The problem with this is that it supposes that the "meaning" of a sentence does not imply the whole language in which it is expressed. But it does; a sentence in English implies the whole language, and so there is no possibility that it is not, on some level, metalinguistic. | |
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| Oct 11, 2016 at 16:06 | comment | added | Artem Kaznatcheev | Thank you. This is incredibly insightful and gives me an extra respect for Russell. Do you know of sources other than your answer that go further into this bit of history? | |
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| Oct 9, 2016 at 18:28 | history | answered | George Chen | CC BY-SA 3.0 |