Did Russell understand Gödel's incompleteness theorems?



Russell was active in philosophy (although no longer in math) for many years after the Gödel's 1931 publication. Gödel's paper were not obscure, and Russell would have been aware of their effect on the Principia and his logicism (and Hilbert's formalism). Logicomix (a partially fictionalized accounts of Russell's life) and common sense suggest that Russell would have grasped Gödel's theorem and its drastic effect to his philosophy. On the other hand writers such as Hofstadter (in "I am a Strange Loop") suggest that Russell never understood Gödel's theorem: going as far as to rudely compare Russell to a dog staring blankly at a TV screen.

Is there any writing of Russell's thoughts on Gödel's incompleteness theorem? Is there any reliable historic/biographic source on Russell's understanding of Gödel? Did Russell understand Gödel's incompleteness theorems?

Artem Kaznatcheev

Posted 2012-10-12T05:25:15.530

Reputation: 1 884

Artem, I cross-posted this question for you, I hope you don't mind (I see you don't have a Quora account) This excellent answer to your question might encourage you to start posting your questions there :) http://www.quora.com/Bertrand-Russell/Did-Russell-understand-Godels-incompleteness-theorems/answer/Sridhar-Ramesh

– None – 2012-10-13T06:00:50.987


@AnonymousCoward I think my posts here count as creative commons, so you may do with them what you like. Although copy-and-pasting my question into an unattributed question with no back-links is hardly cross-posting, so I would appreciate more if you either attributed correctly or acknowledged that you simply copied not cross-posted. The atmosphere at quora does not appeal to me (for reasons like this), and I doubt I will be interested in making an account. Thank you for the invite, though.

– Artem Kaznatcheev – 2012-10-14T01:35:20.983


Russell is indeed rather sloppy for a logician. For example: "[in] Principia Mathematica [...] the syntax is never precisely described, and the axioms and rules of inference are presented in a way that mixes together the syntax with its intended meaning. The formalism appears to be inextricably tied to its informal interpretation. [...] it is this last feature of Russell’s logic that seems to have led to some misunderstandings on his part." − (Russell and Godel).

– user21820 – 2018-12-13T07:14:55.740

2Russell himself admitted as much in a postscript to a 1943 article by Godel: "His great ability, as shown in his previous work, makes me think it highly probable that many of his criticisms of me are justified. The writing of Principia Mathematica was completed thirty-three years ago, and obviously, in view of subsequent advances in the subject, it needs amending in various ways. [...] I must therefore ask the reader to give Dr. Gödel’s work the attention that it deserves, and to form his own critical judgment on it." – user21820 – 2018-12-13T07:21:58.243

Thank you for those comments @user21820, I feel like they could be put together into an answer (which would be easier to find and read than the comments). – Artem Kaznatcheev – 2019-01-03T14:58:06.340

@ArtemKaznatcheev: I lost the comment I was typing. Basically, you must go through a rigorous proof of Godel's theorems yourself, and then you would know how little Russell understood. If you have basic knowledge of classical logic and programming, this (first half) should give a self-contained proof. For a conventional proof read Peter Smith's "Godel without tears". Hence questions about how much Russell understood do not need to be answered by citing anyone.

– user21820 – 2019-01-04T10:30:06.193

In just the same way Russell did not grasp the meaning of the calculus of Spencer Brown, which has a direct relevance to Godel's work and to the foundations of set theory. When I asked Brown about this he replied, in a kindly and wistful tone of voice, "Oh, Bertie was a fool". This seems to sum it up. Great man in many ways but very dense in certain respects. . . – None – 2019-11-28T14:54:57.493

3In fact, stack exchange questions are licensed under creative commons with attribution, @user2539 must link back here – Max – 2014-04-11T06:37:09.003

Not entirely off topic, I hope. Russell greatly admired the "philosophical" stance of Frege when Russell's paradox dashed his hopes for completion. Russell wearied of the logical technicalities of philosophy and felt beaten down by Wittgenstein on one side and Godel on the other. He became a humanist, and perhaps, in some sense, a greater or more "timeless" philosopher. – Nelson Alexander – 2015-11-04T03:13:12.953

1Your comments on Boolean algebra are well taken especially on light of complex adaptive systems.Russell could be a very inconsistent thinker. For example his book "Why I am not a Christian" is silly and poorly expressed. I myself am not a Christian but not because of his book! Unfortunately some of this heuristic nonsense populates some of his "deeper" thoughts as well. Unfortunately there is little understanding among logicians and mathematicians of the subjective nature of the mind from which supposedly objective ideas spring. What it appears we are left with in the name of clarity and insig – Kent Brosveen – 2016-10-29T20:52:17.543



After a bit of searching, I found some promising leads (and quite a few consistent descriptions) which suggest that Russell thought Gödel's results were of cardinal importance, but misunderstood their implications. In particular, he thought that Gödel's result essentially entailed that Peano Arithmetic was inconsistent rather than incomplete; but also realized that this is not something which Gödel was likely to be claiming.

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.

(From Russel, Gödel, and Logicism.) It would be interesting to have a more complete record of how Russell came to this misconception: was it that something was being established as a Theorem, whose content was to establish as true a statement which was provably not a theorem (of another formal system)? Of course, Russell may not have exponded at much length why he interpreted the Incompleteness Theorem as he did; he had, after all, stopped working in mathematical logic. As a prodigious writer and an obvious person to ask about Gödel's results, it does seem plausible that my cursory search has revealed only the top tenth of the iceberg. This hypothesis is supported by the record of Gödel's reaction to Russell's reaction to his Incompleteness Theorem:

Russell evidently misinterprets my result; however he does so in a very interesting manner.

(From Information and Randomness: An Algorithmic Perspective.) Perhaps Gödel simply found Russell's confused concern a refreshing change of reaction from that of others'; perhaps he said this to heighten the contrast against Wittgenstein's reaction to the Incompleteness Theorem (which was trivializing, but what more should one expect from someone who considers set theory akin to a childhood disease?); or perhaps Gödel was simply being polite to an elder statesman. But if he genuinely found Russell's reaction interesting, this would suggest a meatier misinterpretation.

Niel de Beaudrap

Posted 2012-10-12T05:25:15.530

Reputation: 9 640

3Godel's theorem strictly speaking states that no axiomatic system of arithmetic can be both complete and consistent. Maybe Godel sees Russell as rephrasing Godel's theorems as "Inconsistency" theorems, with the interpretation that sees strict arithmetic consistency as less important a feature of logical axiomatisation of mathematics than theoretical completeness? Either way, great answer. – Paul Ross – 2012-10-13T15:29:41.357

1I have accepted this answer, but if someone has further information then I would be very happy to know! – Artem Kaznatcheev – 2012-10-14T01:40:41.663

@Dennis: That cannot be a valid response, whether or not Russell made it. The reason is that every computable formal system S that interprets arithmetic is subject to Godel's incompleteness theorem. It is irrelevant whether S is a set theory, a type theory, or something else, and whether S has classical truth values, multiple truth values or not even the concept of truth values. S can even be some crazy formal system that makes no sense. Still, S is essentially incomplete (this is a technical term, not an English phrase). See this for proof.

– user21820 – 2017-09-06T12:11:06.413

@Dennis: In other words, whatever kind of logical system humans can ever think of will never succeed in Hilbert's original goal. Any hierarchy either would be computably describable (and hence any system built on such would be subject to the incompleteness theorems) or would be computably indescribable (and hence useless for human reasoning). – user21820 – 2017-09-06T12:14:12.673

@user170039: See my above comments, and also this interesting blog post about iterating Godel's theorem.

– user21820 – 2017-09-06T12:14:27.487

@user21820 I'm a bit confused at what you're responding to. I never claimed I was told Russell never made any mistake here, just that I was told he didn't make the mistake quoted in the answer. I'm also not sure what view you're ascribing to Russell on the basis of that very short second-hand reporting. As in the link you provide, in a hierarchy of theories extending PA, no theory can prove its consistency, but there's nothing to stop it from proving the consistency of the theory it extends.... – Dennis – 2017-09-06T13:59:37.593

@user21820 ....This might have been what Russell (allegedly) had in mind. Note "hierarchy of languages" vs. "hierarchical language". – Dennis – 2017-09-06T14:00:17.400

@user21820 Also, see the answer by Monad, since I think it's likely that is the quote my Professor was recalling. – Dennis – 2017-09-06T15:12:29.770

@Dennis: I did note the use of "hierarchy of languages", not necessarily a single language, but the problem remains the same; either the hierarchy is computably describable or it is not. I also did read the quote given by Monad. If accurate, then I do think it's fair to say Russell was wrong or at least highly misleading in his writing. It can potentially be argued that he had in mind indescribable languages, but I then fail to see the point of stating such a claim, since the theory of the natural numbers is precisely one such language, and we didn't even need multiple languages... – user21820 – 2017-09-06T16:04:18.637

@Dennis: I'm sorry my initial comment was ambiguous. I didn't mean that your response is invalid. I meant that the response that is attribute to Russell is invalid, whether or not he actually gave such a response. – user21820 – 2017-09-06T16:05:21.683

@user21820 I got that much, but I'm failing to see how any of what you've pointed to would show the falsity of Russell's claim. Each step up in the hierarchy of theories might be a recursively enumerable theory, and be able to prove the truths of the theory it extends, but the new theory will of course be incomplete. This can be carried on ad infinitum but at no stage will you reach a theory which is itself complete, course -- hence "logically incapable of completion" (as a single, complete theory). He doesn't seem to claim that the hierarchy of theories itself is describable, nor should he. – Dennis – 2017-09-06T16:10:49.473

@user21820 All of this is to say that each theory in the hierarchy might be "computably describable", but there is no commitment to a complete and "computably describable" theory of the hierarchy of theories. I take this all to be in line with fairly standard Tarskian ideas of semantic ascent and a hierarchy of truth predicates. What is it that you're finding fault with? Is it that you think such a hierarchy was claimed (by Russell) to counter Gödel's theorems? – Dennis – 2017-09-06T16:14:10.507

@Dennis: My main point is that there is no need for a hierarchy if we are unable to fully describe it. It is far easier and of lower complexity to just use the theory of the natural numbers as a single language. It is not philosophically meaningful to claim that there is some unknown hierarchy of languages that suffices for mathematics. The fact that it cannot be described fully means that it is useless as a whole. Only the parts which have been described are useful. Also, feel free to come to Philosophy of Math Chat! =)

– user21820 – 2017-09-06T16:18:20.793

@Dennis: Note that the typical way of creating such hierarchies that actually reach some kind of completeness relies on transfinite recursion. What if I don't buy the meaningfulness of transfinite recursion? Do you see why I say it's somewhat philosophically objectionable? – user21820 – 2017-09-06T16:19:35.157

@user21820 Ok, I see, but then your actual claim is significantly more qualified than the initial one (to no detriment, imho). Russell is only clearly wrong insofar as you reject transfinite recursion or hold something like a "pragmatic constructivism" according to which the only meaningful/useful mathematical "concepts" (for lack of a better term) are those that admit of a computable description. I agree that, given all of that, Russell's response to the theorems doesn't get off the ground. – Dennis – 2017-09-06T16:31:43.810

1@Dennis: Okay then I think we pretty much agree. I'd just like to say that it's not that I explicitly reject transfinite recursion; rather I don't agree to accept it just like that, as it can't be justified non-circularly. I've sort of a leaning towards predicative systems as being far more philosophically justifiable than impredicative ones. Thanks for sharing your views too! – user21820 – 2017-09-06T16:53:00.360

Can you give a link to the article Russell, Gödel, and Logicism from which it can be downloaded? – None – 2017-10-04T07:51:37.340

5@ArtemKaznatcheev I contacted one of my professors who is a Russell scholar, I'll let you know if anything comes of it. He was fairly certain Russell didn't make the mistake quoted in this answer. He did say this much: "There isn't much, however. I do recall his saying that Goedel's results show there must be a hierarchy of languages, which is a much more reasonable conclusion." – Dennis – 2013-06-19T18:17:39.993

@Dennis:...and essentially right conclusion, contrary to Gödel's accusation of evident "misunderstanding" to Russell. – None – 2015-11-04T04:33:42.203


The following is from a late paper of Russell's titled "Logical Positivism". It can be found in "Logic and Knowledge"

It appeared that, given any language, it must have a certain incompleteness, in the sense that there are things to be said about the language which cannot be said in the language. This is connected with the paradoxes - the liar, the class of classes that are not members of themselves, etc. These paradoxes had appeared to me to demand a hierarchy of 'logical types' for their solution, and the doctrine of hierarchy of languages belongs to the same order of ideas. For example, if I say 'all sentences in the language L are either true or false', this is not a sentence in the language L. It is possible, as Carnap has shown, to construct a language in which many of the things about the language can be said, but never all the things that might be said: some of them will always belong to the 'metalanguage'. For example, there is mathematics, but however mathematics may be defined, there will be statements about mathematics which will belong to 'metamathematics', and must be excluded from mathematics on pain of contradiction.
There has been a vast technical development of logic, logical syntax, and semantics. In this subject, Carnap has done the most work. Tarski's "Der Begriff der Wahrheit in den formalisierten Sprachen" is a very important book, and if it compared with the attempts of philosophers in the past to define 'truth' it shows the increase of power derived from a wholly modern technique. Not that difficulties are at an end. A new set of puzzles has resulted from the work of Godel, especially his article "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme" (1931), in which he proved that in any formal system it is possible to construct sentences of which the truth or falsehood cannot be decided within the system. Here again we are faced with the essential necessity of a hierarchy, extending upwards ad infinitum, and logically incapable of completion.

It looks like here he came a little closer to understanding it, but still pretty far off. He seems to be confusing Turing's decidability, the Tarski definability theorem, and incompleteness into one homogeneous lump. His statement of Gödel's theorem is either trivially false or interestingly true depending on what he means by "decidable in a formal system": the man does have a knack for statements which skirt the line between the two.


Posted 2012-10-12T05:25:15.530

Reputation: 146

Ah! This might be the quote I was told about! – Dennis – 2017-09-06T15:10:52.680

3"His statement of Gödel's theorem is either trivially false or interestingly true depending on what he means by "decidable in a formal system": the man does have a knack for statements which skirt the line between the two."-if it is not certain that what he actually meant, then how can you say that he was still "pretty far off" in understanding it? – None – 2015-11-05T04:45:37.403


As mentioned in a comment, Alasdair Urquhart has written a paper, Russell and Gödel (Bull. Symb. Logic 22 (2016), 504–520), that discusses a number of different topics, including Russell’s view of Gödel’s results. He provides many of the Russell quotes that other respondents here have given, as well as the following quote from an “Addendum” that was written in 1965 but published only posthumously in 1971, in the fourth edition of The Philosophy of Bertrand Russell.

Not long after the appearance of Principia Mathematica, Gödel propounded a new difficulty. He proved that, in any systematic logical language, there are propositions which can be stated, but cannot be either proved or disproved. This has been taken by many (not, I think, by Gödel) as a fatal objection to mathematical logic in the form which I and others had given to it. I have never been able to adopt this view. It is maintained by those who hold this view that no systematic logical theory can be true of everything. Oddly enough, they never apply this opinion to elementary everyday arithmetic. Until they do so, I consider that they may be ignored. I had always supposed that there are propositions in mathematical logic which can be stated, but neither proved nor disproved. Two of these had a fairly prominent place in Principia Mathematica—namely, the axiom of choice and the axiom of infinity. To many mathematical logicians, however, the destructive influence of Gödel’s work appears much greater than it does to me and has been thought to require a great restriction in the scope of mathematical logic. … I adhere to the view that one should make the best set of axioms that one can think of and believe in it unless and until actual contradictions appear.

This quote seems to demonstrate a reasonably good understanding of what Gödel proved. On the other hand, as already noted, there are other remarks by Russell that seem to misunderstand Gödel. Urquhart concludes, “In the end, it is probably impossible to interpret Russell’s comments on the incompleteness theorem in a fully consistent way. His remarks combine correct summaries of Gödel’s work with what appear to be quite confused and muddled ideas.”

Timothy Chow

Posted 2012-10-12T05:25:15.530

Reputation: 161

Thanks for converting my comment into an answer! =) I only have a tiny quibble; the quote does not demonstrate proper understanding of what Godel proved. Specifically, "Oddly enough, they never apply this opinion to elementary everyday arithmetic." shows that Russell failed to understand that Godel's results indeed applied to PA− (which is as elementary arithmetic as one can get, with just the finitely many discrete ordered semi-ring axioms and no induction), and produces an arithmetical sentence (even Π1) independent of any theory that extends PA−. Russell's vague grasp led to confusion. – user21820 – 2021-01-13T07:27:38.707

@user21820 : I prefer a more charitable interpretation of Russell. I see him as saying, "Some people think that because Goedel's theorem applies to the logicist program, it follows that the logicist program is fatally flawed. But Goedel's theorem applies to PA too, yet those critics don't conclude that PA is fatally flawed. Incompleteness isn't a bug; it's just a feature." – Timothy Chow – 2021-01-13T13:11:09.177

The logicist program aims to reduce mathematics to purely logical grounds. It failed. Nobody is saying PA is flawed; just the notion that everything can be given purely logical justification is fatally flawed, and incompleteness is a very concrete reason for that judgement, since most mathematicians believe that arithmetical sentences are meaningful but their truth values cannot be acquired via any possible purely logical justification, even justifications discovered in the future. – user21820 – 2021-01-13T15:27:12.610

Russell had the intellectual ability to understand Godel if he had wished, but for whatever reason he decided not to put in the necessary (non-trivial) effort to do so, hence his confusion and later admission that he did not understand Godel's work enough to be able to comment on it. There is no reason to give a so-called 'charitable interpretation' when Russell himself admitted that he could not dispute Godel's criticism of him. – user21820 – 2021-01-13T15:29:20.670

@user21820 : When did Russell admit that he could not dispute Goedel's criticism of him? Not here. Parenthetically, he says, "(not, I think, by Goedel)". Furthermore, I don't find your criticism of logicism on the grounds of incompleteness to be convincing. Logicism just means that arithmetical sentences can be defined in logical terms, not necessarily that we have an algorithm for determining their truth. So just because Russell didn't find that criticism convincing either doesn't mean that Russell didn't understand incompleteness. – Timothy Chow – 2021-01-14T02:28:46.543

I quoted Russell's admission here. Your quote does come later, but why assume that Russell understood incompleteness in the presence of clear evidence that he didn't (as evident from his conflation of syntax and semantics)?

– user21820 – 2021-01-14T03:20:06.293

And your interpretation of "logicism" is simply not what the original proponents had in mind: "THE present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles" (Russell 1903). Changing the interpretation of "logicism" to cater for Godel's theorems is just moving goalposts; the original logicist program truly failed. – user21820 – 2021-01-14T03:22:03.247

By the way, I'm not saying that the "many" in the quote you cited are correct. I have no idea who Russell is referring to by that, but certainly all who understood Godel didn't make any silly claims like "no systematic logical theory can be true of everything", so either Russell misunderstood "many" or was just responding to nonsense (which doesn't need response). – user21820 – 2021-01-14T03:26:21.820

@user21820 : Your quote from Russell is precisely the definition of logicism that I had in mind. He uses the same word "definable" that I did, and I interpret "deducible" in the same way. To interpret "deducible" algorithmically is, I maintain, anachronistic. As for evidence that Russell conflated syntax and semantics, that was earlier in his life. He might have come to a clearer understanding later. – Timothy Chow – 2021-01-14T03:53:40.047

Furthermore, the failure of logicism has more to do with mathematicians' sense that axioms of set theory and even arithmetic aren't "purely logical." The failure doesn't have much, if anything, to do with incompleteness. – Timothy Chow – 2021-01-14T03:57:09.853

No, I did not interpret "deducible" anachronistically, since that is the only way one can perform rigorous logical reasoning regardless of the kind of formal system one uses. Leaving aside the question of whether or not Russell came to a clearer understanding later, I think our disagreement arose from our different interpretation of "all pure mathematics". Originally, logicism's goal was to not only reduce "all pure mathematics" to purely logical principles, but also to show its consistency from purely logical principles. The problem is, the question of consistency itself is mathematics... – user21820 – 2021-01-14T04:00:29.133

Now if you insist that consistency is not part of "all pure mathematics", then do please tell me what exactly you interpret by "all pure mathematics", because I truly don't see any simple way to interpret it that is completely unaffected by incompleteness in some fashion. – user21820 – 2021-01-14T04:08:29.640


Russell's comments on Gödel were scanty, but it was very unlikely that Russell did not understand what Gödel was talking about. The 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[source 1]. Russell discovered the Theory of Types in 1906. The Theory of Types provided no shelter for vicious circles.[source 3] On the other hand, Gödel's raising this paradox anew in 1931 indicated that Gödel probably never understood Russell's Theory of Types.

In Russell's Theory of Types, meaning is fundamental. 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.

No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the "whole theory of types").

Wittegenstein, Tractatus 3.332

Gödel, from formalist's point of view, regarded symbols in PM as meaningless empty signs[source 9], and, by means of numbering, Gödel managed to show that G belonged to the body of propositions for which PM was supposed to be a foundation - this was the point of contention: by Russell's type theory, G had no meanings and thus did not belong to the body of propositions: all self-referential sentences were specifically weeded out by theory of types as nonsensical. Gödel's attack on Principia was similar to planting a bag of weed in one's roommate’s car then accusing him of illegal possession of drugs, or to accusing one's room-mate, who was actually speaking another language on the phone, of making threats about the President of the United States

The root of the problem was formalists' disregard for meanings. In Russell's 1937's "Introduction To The Second Edition" of The Principles of Mathematics, Russell categorically dismissed formalists:

The formalists are like a watchmaker who is so absorbed in making his watches look pretty that he has forgotten their purpose of telling the time, and has therefore omitted to insert any works.

Was Gödel's work below contempt? There are, in this world, lunatics, crooks and dupes. The crooks are below contempt; the lunatics and dupes are not. Russell was very generous; he lavished praises on Wittgenstein and Ramsey who ruthlessly attacked his Principia but at the same time unmistakingly demonstrated their understanding of his theory of types. On the other hand, nothing in Gödel's work indicated that Gödel even had a clue about Russell's theory of types. Although Russell was eager to say something nice, there was really not much he could say; towards fledging philosophers, Russell was tender and caring and was very careful to avoid raising his voices.[source 6] Russell actually came face to face with Gödel in early 1940s'; the following excerpt was Russell's recount of his encounter with Gödel:

While in Princeton, I came to know Einstein fairly well. I used to go to his house once a week to discuss with him and Gödel and Pauli. These discussions were in some ways disappointing, for, although all three of them were Jews and exiles and, in intention, cosmopolitans, I found that they all had a German bias towards metaphysics, and in spite of our utmost endeavours we never arrived at common premises from which to argue. Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal 'not' was laid up in heaven where virtuous logicians might hope to meet it in hereafter.

Russell, Bertrand. "America. 1938-1944." Autobiography. 1967. London and New York: Routledge, 2000. Print. 466.

Gödel's theorems were probably more devastating to formalists because they showed what absurdities were admissible in formalism. Nevertheless, it is possible that PM is incomplete or inconsistent, but proving complete and proving consistent were not PM's concerns. Both completeness and consistency involve all, exact what propositions constitute all is the point of contention.[source 4] PM did not aim at all; it aimed at deducing arithmetics, which was the starting point of ordinary mathematics; PM was on target, and PM made several serendipitous discoveries along the way. Regarding consistency, all that PM could say was probably something like this: "as of today, no contraction has been discovered within PM" - more than that was beyond what inductive reasoning could warrant [source 2]. By deducing arithmetics from logical principles, PM demonstrated that mathematics and logic are identical - this thesis, first proposed in his 1900's Principles, Russell had never seen any reason to modify.[source 5]

If Gödel sentence were true, PM would be inconsistent because, in virtue of PM ✳2.02, which says a true proposition is implied by any proposition, PM implies G. Any body of premises that can deduce G is not a valid foundation because it contains a contradiction; an inconsistent body of premises contains false premises, and a false premise implies any conclusion (PM ✳2.21) - that is why Russell asked "are we to think that 2 + 2 is not 4, but 4.001?" Then again, if principles in PM were asserted, Gödel sentence implicated no one because it was either nonsense or non-contradiction by the Theory of Types.[source 3]

Can you know that a proposition is true before it is proven? Yes, you can. But those propositions are what Wittgenstein called tautologies, none of which is Gödel sentence G. Basically, tautologies are different ways of saying the same thing. Tautology

Like all theories whose justifications are inductive, PM by its nature should be tentative, subject to revisions based new evidence; the second edition of PM demonstrated Russell's attitude more than proved a point: Russell learned a lesson from Newton-Leibniz quarrel, and he had no desire to take the place of Aristotle to establish himself as a towering authority - Russell actually took pains to avoid playing the authority role.7[8]

Sources: 1. The Vicious Circle Principle

2.The Inductive Nature of Principia Mathematica

3. If Gödel sentence is taken literally, it is nonsense; if it is interpreted in type theory, it is a simultaneous assertion of multiple statements - this is what Russell means by "Here again we are faced with the essential necessity of a hierarchy, extending upwards ad infinitum, and logically incapable of completion. " See Liar's paradox. Note that every time I point to the Liar's paradox, people automatically say I mistook true for provable. Actually, this distinction is irrelevant; what the liar's paradox and G in common is that they are all self-referential. If a sentence can't comment itself, then it is commenting its counterpart one order below itself, thus a hierarchy rises from the second order ad infinitum. There is no first order G, because a proposition about a proposition is at least 2nd order. First order propositions are all about individuals, not propositions. Theory of Logical Types

4. If consistency means there is no contradiction, then one needs to examine all propositions, then again what goes into all? Russell somewhere said Whitehead and himself believed it was impossible to prove that a formal system was consistent.


The foundamental thesis of the following pages, that mathematics and logic are identical, is one which I have never since seen any reason to modify. Russell, Bertrand. Introduction to the second Edition. Second paragraph. Principles of Mathematics, 1937.

6. Russell in several writings mentioned that extremely intelligent people are also emotional unstable; instead of calling for "mental toughness" he advocated separating sensitive children from the crowd. One source I can tell for sure is Education and the good Life. Russell was definitely aware of the fragile mental state in philosophical community. This awareness permeates almost all of his non-technical writings.

7. Somewhere Russell blamed Newton of retarding British Mathematics for 150 years, and he did not know how far British mathematics had fall behind Germany until he visited the US. I can't come up with the source off the top my head, but somewhere he definitely said something like this

8. Russell tried to make do without Axiom of Reducibility in the 2nd. Ramsey accused the AOR of being a matter of brute fact, not a tautology (see Ramey's Foundations of Mathematics); Russell admitted that AOR lacked self-evidence, and was willing to show what it was like without AOR in the 2nd.

  1. The following interpretation of PM by formalists is the major misunderstanding of PM by formalists:

The symbols of PM are, however, fully devoid of meanings in the sense that derivation of theorems depends only on following the formal rules of PM.

Source: Nagel & Newman. Gödel's Proof. New York and London: New York University Press, 2001. Print 71.

To have a sense of how the Theory of Types works, take for example the sentence "a set is not a member of itself"; this sentence is neither true nor false; it is meaningless and thus has no membership in the body of propositions for which PM is supposed to be a foundation. Obviously sentences like this are not only permissible to formalists, who do not consider meaning as a guarding criterion that bars the entrance of nonsense, they are even fundamental to ZFC.

George Chen

Posted 2012-10-12T05:25:15.530

Reputation: 2 102

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?

– None – 2017-07-27T13:57:37.367


@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."?

– None – 2017-07-29T06:56:09.163

@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)? – None – 2017-07-29T06:58:09.173


@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").

– Conifold – 2017-08-09T19:07:58.960

Feel free to join the discussion we are having here (and which in my opinion is related to the question).

– None – 2017-09-06T14:06:15.180

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. – Ace shinigami – 2020-07-30T12:04:39.107

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? – Artem Kaznatcheev – 2016-10-11T16:06:45.943

"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. – Luís Henrique – 2016-10-12T19:58:58.267

2Your 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? – Niel de Beaudrap – 2016-10-13T10:12:02.683

@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. – George Chen – 2016-10-13T12:03:15.153


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 http://wab.uib.no/agora/tools/alws/collection-6-issue-1-article-6.annotate

– Conifold – 2016-10-13T21:56:27.463

I suspect Russell's meekness was some sort of mischief, a practical joke of some sort. People don't know how mischievous Russell was. – George Chen – 2016-10-14T00:06:46.587

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. – George Chen – 2016-10-15T14:03:06.320