Responses to the demonstration of the flaw in Gödel’s Incompleteness proof
Page last updated 03 Mar 2023
A number of people have raised objections to the demonstration of the flaw in Gödel’s incompleteness proof. As yet there is no objection that shows any attempt to give a reasoned logical argument against my demonstration of the flaw.
There are several articles that claim to disprove Gödel’s proof, but for anyone who has a thorough understanding of Gödel’s original proof, it’s not actually very difficult to find the errors in these articles – as shown in Flawed Disproofs. It is most odd that persons who are supposedly well-versed in the application of logic are adamant that my demonstration of the flaw in Gödel’s proof is incorrect, while at the same time they refuse to give any logical argument to indicate why that demonstration might be incorrect.
In 2002, Leonid Rozenblit and Frank Keil invented the term, “the illusion of explanatory depth” (IOED), (Footnote: Leonid Rozenblit and Frank Keil, PDF The misunderstood limits of folk science: An illusion of explanatory depth, Cognitive science 26.5 (2002): pp 521-562.) stating that most people believe that they understand things in far greater detail, coherence, and depth than they really do. They devised a test where people are first asked to rate how well they understood something, and they are then asked to write a detailed explanation of their understanding of that thing. They are then asked to re-rate their understanding. Rozenblit and Keil found that the rating of understanding was much lower on the re-rating. My communications with several people have made it clear that most people greatly overestimate their knowledge of Gödel’s proof - they have an initial belief that they have a good understanding of Gödel’s proof, but as the communication proceeds it becomes evident that their knowledge is a sketchy knowledge derived from second hand sources rather than the proof itself.
A Mr Russell O’Connor has published a risible ‘review’ where he asserts that I misunderstand Gödel’s theorem. O’Connor thinks that he was qualified to review a paper on Gödel’s original proof, although by his own admission he wasn’t familiar with the details of that proof. He makes the ludicrous assertion that the actual sections where I detail the flaw in Gödel’s paper don’t make any controversial claims, but he claims that since he thinks that there are errors in the other sections that discuss the main text, he doesn’t need to make a detailed review of the actual text that details the flaw in Gödel’s proof. By doing this he thinks he can sidestep the need to actually identify any actual error in the demonstration of the flaw in Gödel’s proof. His review and my response are given at O’Connor Review.
Below I give the arguments that I have received according to type.
Common Objection 1
The most common objection is the claim that Gödel’s proof cannot be incorrect because there are numerous other ‘versions’ of proofs of Gödel’s theorem that logicians accept as being correct. (Update: there are detailed papers demonstrating the errors in many such proofs, see Errors in incompleteness proofs and Analysis of incompleteness proofs.)
This is simply a deflection of the argument from the main issue which is: Is the demonstration of the flaw in Gödel’s original paper correct? It introduces another matter of contention which is: How are we to define what constitutes a ‘version’ of Gödel’s original argument? One has to wonder why a logician would attempt to deflect the argument concerning Gödel’s proof in this way instead of simply pointing out an error in the demonstration of a flaw in Gödel’s original paper. If logicians believe that the argument is incorrect, why do they try to deflect attention from the main argument? That approach appears to suggest that logicians have not been able to find any error in the argument.
If they are refusing to even examine a demonstration of a flaw in Gödel’s original proof, then why would they ask to see a demonstration of a flaw in some other ‘version’ of Gödel’s proof? Clearly, if they refuse to examine one such demonstration, isn’t it likely that they would refuse to examine any other similar demonstration? But in any case, it has been shown that many other incompleteness proofs have been found to have fundamental errors that render these proofs mathematically invalid. See Errors in incompleteness proofs and Analysis of incompleteness proofs.
Common Objection 2
Another common objection is that the demonstration of the flaw in Gödel’s proof must be wrong since so many logicians have examined Gödel’s proof and have been unable to find any flaw in it. But if they have examined it in full detail then they must also have fleshed out Gödel’s perfunctory sketch of how one might prove his Proposition V, a key step in Gödel’s paper that Gödel actually stated in the paper that he wasn’t going to prove it since the assumed step seemed to be obvious, and besides he noted that it would require a lengthy proof (Footnote: “We are content here with indicating the proof of this proposition in outline, since it offers no difficulties in principle and is somewhat involved.” For the actual text in context, see Gödel’s Proposition V.)
Now, if so many mathematicians have actually done the hard slog of fully working out the outline that Gödel suggested, isn't it amazing that not one of them published the results of his hard work and time for others to see? Even Gödel never did so. Which makes the claim - that many mathematicians have examined Gödel’s proof in detail - rather questionable. In fact, it was precisely because there was no such detailed proof that I had to do the work and flesh out the perfunctory outline myself, and which is included in my paper PDF Paper: The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem.
But, in any case, simply because hundreds of people have believed something too be true for 80+ years isn’t conclusive evidence for that notion. Otherwise one might as well say that a religion that has been believed by millions over thousands of years must be correct. Some people respond to this by claiming that the case of Gödel’s proof is quite different since the people that believe it are highly intelligent. But if that is the case, these highly intelligent people should have no difficulty in pointing out the alleged errors in my work. But they don’t. Why not? It’s not as if such people don’t relish analyzing writings by people that they consider to be cranks and pointing out their errors for public consumption, see for example For every natural number N, there’s a Cantor Crank C(n) or A Crank among Cranks: Debating John Gabriel or Beware of Cranks.
It is illogical to claim that the mass of accumulated wisdom of hundreds of logicians cannot ever be mistaken. It is illogical to claim that one individual could never perceive anything of importance if hundreds of logicians have not already perceived it; that assumes that such examination by logicians is absolutely perfect.
But in any case, it is a deflection from the main issue: Is my demonstration of the flaw in Gödel’s original paper correct or not?
It might be noted that the incompleteness paper is not the only paper where Gödel made a completely unproven assertion which almost everyone assumed to be correct because of Gödel’s esteemed reputation. In that other paper Gödel made a similar intuitive assertion, but it was eventually proved (some 50 years later) that Gödel’s intuitive assumption stated precisely the opposite of what was actually the case. For details of this see Gödel’s Intuitive Error No.2. Not many mathematicians are aware of this - Gödel’s reputation is still so honored today that most discussions of Gödel’s work skip over this inconvenient detail.
Common Objection 3
Another common objection is the claim that Gödel did not actually prove in his original paper that for every consistent formal language, there will always be statements that are ‘true’ but unprovable in that formal language.
Several persons have, for some reason, assumed that the demonstration of the flaw in Gödel’s proof relies on disputing this aspect of the proof. Is it really too much to ask such people to actually examine the paper before they start criticizing it? If they had actually read the paper, they would find that the demonstration of the error in Gödel’s proof does not depend in any way on whether the proof produces a statement that might be said to be ‘true’ but unprovable in that formal language.
That said, there is an ongoing disagreement as to whether Gödel’s proof produces a statement that is ‘true’ but unprovable in that formal language - even among the most ardent supporters of the proof. This position that it does not actually produce a ‘true’ statement is currently fashionable in some quarters; it is a position which conveniently ignores the facts of the matter. It is the head in the sand approach - if Gödel’s theorem appears to lead to a contradiction, then - let’s deny that Gödel’s proof says what even Gödel thought it said - and ask everyone else to put their heads in the sand as well in the hope that the problem will disappear. More details regarding this are given at True but unprovable.
Gödel’s original paper produces a statement that Gödel referred to in his paper as “the proposition which is undecidable in the [formal language] turns out to be decided by [other] considerations”. Gödel also states that the statements that are unprovable in a formal language are not absolutely unprovable, which means that, according to Gödel, there is some way in every such statement can be proved to be ‘true’ or ‘false’.
And it is not the case that it is only those who believe that there are problems with Gödel’s proof that assert that he proved that “for every consistent formal language, there will always be statements that are ‘true’ but unprovable in that formal language.” In fact, most supporters of Gödel’s theorem also make such an assertion - this includes the most ardent supporters of Gödel’s proof, for example Gregory Chaitin, Douglas Hofstadter and John Dawson.
Again, this objection is a deflection of the argument from the main issue which is: Is the demonstration of the flaw in Gödel’s original paper correct? - and it is completely irrelevant to this question.
Common Objection 4
Another common objection is the claim that if I did not study pure mathematics in a university course, then I cannot have a sufficient grounding in mathematics to be able to understand and make pronouncements about Gödel’s proof.
These people assume, entirely without foundation, that I have no capacity for self-study, that I am unable to reach a sufficient level of knowledge and understanding by intensive study of a subject, unless I follow a university course. Where do people get these notions?
So, partly in response to those people who claim that I don’t understand Gödel’s proof at all, and partly to assist anyone who wishes to get to grips with Gödel’s original paper, I have created a walk-through guide to Gödel’s original incompleteness proof, and which is intended to be read alongside the paper. If you are speculating that I don’t understand Gödel’s proof, I suggest that you take a look at it; it may help you to decide whether it is reasonable to suggest that I do not understand the proof. When I decided to make that guide, I did an intensive search to see if there was anything similar already published anywhere, but I found no detailed guide at all - so I constructed my guide from scratch - which I would claim is hardly indicative of someone who does not understand Gödel’s proof.
And yes, I did not follow the traditional study route, but this was actually very beneficial, because I was not inculcated with the conventional viewpoint of all those who follow the conventional path. In a university course, there are so many things to learn within a short time that it is impossible to analyze everything in your course, so you have to assume that much of what you are being taught is correct. By the time you have finished the university course, you have reached the point where you have accepted the majority viewpoint to the extent that you will argue that a particular proof is correct, even though you have not actually verified it yourself.
And, again, this is a deflection from the main issue which is: Is the demonstration of the flaw in Gödel’s original paper correct?
The Platonist response
Some people appear to believe that by adopting a Platonist standpoint they can circumvent the absurdities that are inherent in Gödel’s proof. They argue that certain expressions in Gödel’s paper are not actually themselves number relationships, but that they correspond to number relationships that “exist” independently of any language. It is easily shown that this notion makes no difference whatsoever to the demonstration that there is an irredeemable flaw in Gödel’s proof, see The Platonist response to the flaw in Gödel’s proof, and also Platonism, The Myths of Platonism, Numbers, chairs and unicorns, Platonism’s Logical Blunder, and the posts Moderate Platonism and Descartes’ Platonism.
And the more bizarre responses
One person (Russell O’Connor) sent me a long criticism of the paper on the flaw in Gödel’s proof, saying that it was full of errors, and it had to be wrong because he had devised a formalized version of Gödel’s proof. When it was pointed out to him that he had failed to find any error whatsoever in my argument that demonstrated the flaw in Gödel’s theorem, his response was this:
“Even if Gödel did make an error in his proof, it’s only a historical curiosity.”
A truly astonishing response. This person thinks it unimportant that a proof that has been hailed as the most significant proof in the entire history of logic is wrong. He doesn’t even stop to consider that we might be able to learn something from the discovery that it is wrong. And in any case, his own ‘formalized version’ of an incompleteness proof contains an obvious error. See An Error in a Proof of Incompleteness by Russell O’Connor.
As Wilfrid Hodges (Footnote: Wilfrid Hodges, PDF An Editor Recalls Some Hopeless Papers, The Bulletin of Symbolic Logic, Vol. 4, No. 1, March 1998.) has pointed out, ‘to attack an argument, you must find something wrong in it. Several authors believed that you can avoid [that] by simply doing something else.’
As a point of interest, the reader might be interested in the various ways that people avoid actually giving a logical response to a demonstration of a flaw in Gödel’s proof - and various other results.
There are several good web pages on fallacious arguments at
- A List of Fallacious Arguments - Don Lindsay Archive
- Fallacies - The Internet Encyclopedia of Philosophy
- Wikipedia entry - List of Fallacies
An excellent book was written many years ago (in fact, 1930, a year before Gödel published his incompleteness theorem) that deals with fallacious reasoning and shows common ways in which arguments are deflected by those unwilling to engage in fair argument. It is still very relevant today. It is called “Straight and Crooked Thinking”, by Robert H. Thouless and I have made an online copy of the book on this site complete with all cross reference links on the page Straight and Crooked Thinking. In the book, Thouless gives a summary of the dishonest tricks used in arguments, see Dishonest tricks used in Arguments.
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.
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