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The Flaw in Gödel’s proof of his Incompleteness theorem

You might be expecting to find here a brief summary of the flaw in Gödel’s proof of incompleteness (which you can see online at English translation of Gödel’s original proof or as a PDF file at English translation of Gödel’s original proof, PDF file). However, it is impossible to give a proper explanation of the flaw in the proof in a few short sentences. This should not be surprising; if it was that simple to explain the flaw, one would have expected that the flaw in the proof would be relatively simple to discover.


A brief summary inevitably means making shortcuts and simplifications. Because of that, there would necessarily be deficiencies in the explanation - if that was not the case, then a brief summary would contain all of the information of the full article! Experience shows that, regardless of whatever warnings one makes about such deficiencies that a brief summary might contain, they are ignored by a multitude of irrational individuals. These persons take delight in spreading misinformation, quoting material from the brief summary as though it was material from the full explanation. Because of that, no attempt will be made here to give a brief summary.



A Simplified explanation of Gödel’s proof and the flaw in the proof

If you are not familiar with the details of Gödel’s proof, I suggest that you either read the simplified explanation of Gödel’s proof on this website, or read the novel The Shackles of Conviction . Both include an explanation of the theorem and the flaw in Gödel’s proof of it in a simplified way that makes it accessible to the general reader.



A step by step walk-through guide to Gödel’s Proof

For 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. Having looked for such a guide and could find none, I thought it odd that there were no such guides, considering how much other material has been written about Gödel’s proof. So I decided to create this guide to assist anyone else who wishes to become familiar with the details of Gödel’s argument.



A Detailed explanation of the flaw in Gödel’s proof

On the other hand, if you are very familiar with the details of Gödel’s theorem, you will probably also want to see the details of the flaw in Gödel’s proof as covered in the paper:


The Fundamental Flaw in Gödel’s proof of his Incompleteness Theorem PDF


The paper gives a comprehensive demonstration of the flaw in Gödel’s proof of his incompleteness theorem. The link above is a link to the latest version. Please note that the central argument of the paper remains unchanged throughout the revisions. The revisions have been to make the principles easier to understand. If you want to see the previous versions please see the sitemap.



The Flaw and the Substitution function in Gödel’s proof

The webpage Gödel’s Substitution Function describes how Gödel’s incorrect use of a ‘substitution’ function confuses the systems that he is dealing with.



Responses to the demonstration of the flaw in Gödel’s proof

If you would like to see some of the responses that I have received regarding the demonstration of the flaw in Gödel’s proof, please see Responses to the demonstration of the flaw: here.



Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked (comments will be checked before appearing on this site). Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - this will only be used to notify you of replies to your comments - it will never be used for any other purpose, will never be displayed and does not require verification. Comments are common to the entire website, so please indicate what section of the site you are commenting on.


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The Lighter Side



There’s something about Gödel by Francesco Berto

There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book There’s something about Gödel comes under scrutiny.



Easy Footnotes

I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).



O’Connor’s “computer checked” proof

I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.



New page on Chaitin’s Constant

There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.



Updated: Diagonal Lemma

Flawed proofs of the Diagonal Lemma by Panu Raatikainen and Vann McGee have been added to the Diagonal Lemma web page.

Previous Blog Posts  


16th Mar 2015 Bishops Dancing with Pixies?


23rd Feb 2015 Artificial Intelligence


31 Mar 2015 Cranks and Crackpots


30 Apr 2015 The Chinese Room




For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem


– a page with general links:

Gödel Links


– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines


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Please note that this web site, like any other is a collection of various statements. Not all of this web site is intended to be factual. Some of it is personal opinion or interpretation.


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Copyright © James R Meyer 2012 - 2016