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In his book My Philosophical Development Russell writes,

In my introduction to the Tractatus, I suggested that, although in any given language there are things which that language cannot express, it is yet always possible to construct a language of higher order in which these things can be said. There will, in the new language, still be things which it cannot say, but which can be said in the next language, and so on, ad infinitum. This suggestion, which was then new, has now become an accepted commonplace of logic. It disposes of Wittgenstein's mysticism and, I think, also of the newer puzzles presented by Gödel.

One of the most relevant references I have found is Drucker's Perspectives on the History of Mathematical Logic. In it I have found the following words,

Is Russell recalling his bewilderment at the time he first became acquainted with Gödel's theorems, or is he expressing his continuing puzzlement? Is he saying that, intuitively, he had recognized the futility of Hilbert's scheme for proving the consistency of arithmetic, but failed to consider the possibility of rigorously proving that futility? Or is he revealing a belief that Gödel had in fact shown arithmetic to be inconsistent? Henkin, at least, assumed the latter; in response to Russell's closing request, he attempted to explain the import of Gödel's second theorem, stressing the distinction between incompleteness and inconsistency. Eventually a copy of Russell's letter made its way to Gödel, who remarked that "Russell evidently misinterprets my result; however he does so in a very interesting manner .... " (Gödel to Abraham Robinson, 2 July 1973.)

My questions are as follows,

  1. "Is Russell recalling his bewilderment at the time he first became acquainted with Gödel's theorems, or is he expressing his continuing puzzlement? Is he saying that, intuitively, he had recognized the futility of Hilbert's scheme for proving the consistency of arithmetic but had failed to consider the possibility of rigorously proving that futility? Or is he revealing a belief that Gödel had in fact shown arithmetic to be inconsistent?"-Is(are) there any justification(s) for assuming "the latter" as Henking did?

  2. Gödel remarked that "Russell evidently misinterprets my result; however he does so in a very interesting manner .... ". How did Gödel came to the conclusion that Russell indeed "misinterpreted" his result? Why was Russell's "misinterpretation" interesting?


Is(are) there any detailed and critical account(s) of Russell's response to Gödel's Incompleteness Theorems which (at least partially) tries to answer these questions?

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Irvine in Bertrand Russell also writes that "Russell did not understand Gödel's celebrated results, which he interpreted as implying that arithmetic is inconsistent". Berto in Gödel Paradox and Wittgenstein's Reasons notes that both Russell and Zermelo mistook Gödel's theorem for a paradox due to not distinguishing between theory and meta-theory, syntax and semantics; but argues that Wittgenstein rejected the distinction consciously. Wittgenstein himself famously objected to Gödel's theorem, and Gödel responded similarly that "Wittgenstein did not understand it (or pretended not to understand it)". But recent scholarship is much more nuanced and sympathetic to Wittgenstein's objections, see Matthíasson's Interpretations of Wittgenstein’s Remarks on Gödel, Lampert's Wittgenstein’s “Notorious Paragraph” about the Gödel Theorem, Shanker's Gödel's Theorem in Focus, and Rodych's Wittgenstein on Gödel.

We have much less to go on with Russell, but there is also a more charitable way to read his quotes, including the ones from answers to the question linked by Niel de Beaudrap. And it is directly suggested by the mention of Wittgenstein. Wittgenstein argued in the Tractatus, contra Frege, that there can be no rules for logic, or in modern terms that language should be its own meta-language, as is indeed the case with natural languages. "And what can not be said should be passed over in silence", which is probably "Wittgenstein's mysticism" that Russell mentions. However, Russell himself was well aware of Wittgenstein's position, but suggested a hierarchy of languages instead. So perhaps he is referring to inconsistency on Wittgenstein's reading rather than his own.

On Wittgenstein's conception "true but unprovable" sentence is indeed perplexing. Russell mused:"I realized, of course, that Godel’s work is of fundamental importance, but 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. Does this apply to school-boy's arithmetic, and if so, can we believe anything that we were taught in our youth? Are we to think that 2 + 2 is not 4, but 4.001? Obviously, this is not what is intended". If we add the negation of the Godel sentence to Peano arithmetic Godel's proof presumably produces a contradiction, yet this can not be the case since then the sentence would be provable. So one of the axioms that "must be false" would be one from "school-boy's arithmetic", and clearly "this is not what is intended". Therefore, one needs a "language of higher order in which these things can be said", according to Russell. If so, Russell is fully on board with Gödel's separation of truth (in meta-language) and provability (in object language).

Obviously, this reading is not straightforward, and Godel probably took Russell to mean something like a paradox, but found the idea of (what is now called) Tarski hierarchy "interesting".

  • Can you cite some reference for your third paragraph? – user 170039 Nov 4 '15 at 3:06
  • @user 170039 The passage is quoted in Niel de Beaudrap's answer in the thread he linked (he quotes from Irvine), the interpretation is similar to those suggested in connection with Wittgenstein, see Matthíasson's thesis. – Conifold Nov 4 '15 at 20:26
  • Like most of others I have also thought Russell as someone who failed to understand Gödel's Incompleteness Theorems. The third paragraph showed me that my judgement was too hasty. Indeed it seems to me that if the interpretation provided in the third paragraph was what Russell actually had in mind then Russell completely understood Gödel's theorems and that too in a very "interesting" manner, contrary to Gödel's comment. Thank you very much. By the way, do you know of any other author(s) who argue in support of the view you put forward in the third paragraph. If so, can you mention some? – user 170039 Nov 5 '15 at 5:01
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    Doesn't Russell's claiming that his suggestion of hierarchy of languages disposing of "newer puzzles presented by Gödel" show that Russell didn't understand the theorem properly? – user 170039 Nov 20 '15 at 11:44
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    @user170039 The contradiction of the Godel sentence being both true and false if we accept Godel's meta-argument as a valid proof that it is true. – Conifold Jul 2 '18 at 20:20

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