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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.

The simple reason why a brief summary cannot be given is because Gödel did not actually give a fully detailed proof of a crucial part of his incompleteness paper. He simply assumed that this crucial part was correct, and instead he just gave a rough outline of how a proof of that crucial part might proceed, saying in his paper: “We content ourselves here with indicating the proof of this proposition in outline, since it offers no difficulties of principle and is somewhat involved.” (Footnote: Peter Smith, although he is a staunch advocate of Gödel’s proof, acknowledges in his paper, Expounding the First Incompleteness Theorem (PDF), that:
“Gödel only sketches a proof by induction on the complexity of the definition of the (characteristic function of the) property or relation in terms of definitions by composition and recursion grounding out in the trivial initial functions. The crucial step is just asserted - ‘the processes of definition … (substitution and recursion) can both be formally reproduced in the system P’ … it certainly isn’t spelt out.”
See also Gödel Without Tears - Or Not? and A Fundamental Flaw in an Incompleteness Proof by Peter Smith PDF.)
This failure to give a fully detailed proof is even more surprising in view of Gödel’s insistence elsewhere on clarity and precision. And today almost every mathematician/logician seems content to make the same intuitive assumption, which is exceedingly strange - in mathematics, one should never let intuition triumph over strict logic. Stranger still is that you might expect that several mathematicians/logicians would have tried to do a fully detailed proof of this crucial part, and published their results - but where are such results? This is why I give in my paper (Footnote: See The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem (PDF)) a fully detailed account of Gödel’s suggested outline of the crucial part - and the flaw in that crucial part.

It is easy to show that Gödel’s Proposition V leads to a blatant contradiction, see the page Gödel’s Proposition V.

It might be noted here that his 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 (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.

A brief summary would inevitably mean 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 site-map.

The Flaw and the Substitution function in Gödel’s proof

The web-page 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.

Footnotes:

Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked. 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 - any address will do, it does not require verification. Your e-mail will only be used to notify you of replies to your comments - it will never be used for any other purpose and will not be displayed. If you cannot see any comments below, see Why isn’t the comment box loading?.

NEWS

Lebesgue Measure

There is now a new page on a contradiction in Lebesgue measure theory.

Illogical Assumptions

There is now a new page Halbach and Zhang’s Yablo without Gödel which analyzes the illogical assumptions used by Halbach and Zhang.

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.

Previous Blog Posts

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:

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

Gödel, Minds, and Machines

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