Copyright © James R Meyer 2012 - 2017 www.jamesrmeyer.com
This simplified explanation of Gödel’s incompleteness proof (see online at English translation of Gödel’s original proof or as a PDF file at English translation of Gödel’s original proof, PDF) has been written in an attempt to make the proof as accessible as possible to as many people as possible, using straightforward English language with a few simple symbols. I have deliberately avoided any need for knowledge on the part of the reader of any complex mathematics. Please note that this simplified explanation is not intended to be a substitute for my detailed explanation of the flaw in Gödel’s proof, which can be found at The Flaw in Gödel’s proof
The simplified explanation is broken down into ten parts:
This is a brief survey of variables, what is meant by free and bound variables, and the concept of the substitution of a variable, and the concepts of propositions and functions.
This is a brief explanation of what is meant by a formal language system, its axioms and its rules, and what we mean by a proof in a formal language system – and what we mean by saying that a formal system is consistent, complete or incomplete.
This part explains how Gödel was able to use numbers to capture all the information of any expression of formal language system. This was essential for his proof.
This part explains how Gödel came up with the idea of creating a special number relationship that would correspond to the concept that one expression in a formal language system can be a proof of another expression in that formal language system
This part explains how Gödel came up with the idea of creating number relationship that would correspond to the concept of the substitution of a variable in a formal language expression.
In this part Gödel shows how number relationships can have corresponding formal sentences.
Here we explain how Gödel defined another numbering system that is similar but not quite the same as his special numbering system as seen in Part 3
Here we show how Gödel creates his special ‘true but unprovable’ sentence, using the previous steps in the proof.
This part shows how Gödel confuses the languages that he is talking about and the language that is talking about these languages.
Once you see that Gödel’s can only get the result that he does by confusing languages, there are several ways you can show that the proof is logically absurd. This part shows just one such logical contradiction.
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It has come to my notice that, when asked about the demonstration of the flaw in his proof (see A Fundamental Flaw in an Incompleteness Proof by Peter Smith PDF), Smith refuses to engage in any logical discussion, and instead attempts to deflect attention away from any such discussion. If any other reader has tried to engage with Smith regarding my demonstration of the flaw, I would be interested to know what the outcome was.
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.
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).
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.
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.
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:
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Copyright © James R Meyer 2012 - 2017