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A Simplified Explanation of Gödel’s proof - Introduction





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:


Part 1: Basic Logical Concepts

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.


Part 2: Formal Language Systems

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.


Part 3: Gödel’s numbering system

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.


Part 4: Creating a special number relationship

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


Part 5: Number relationships and the concept of substitution

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.


Part 6: Matching number relationships and formal sentences

In this part Gödel shows how number relationships can have corresponding formal sentences.


Part 7: Another Numbering System

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


Part 8: The Crux of the Proof

Here we show how Gödel creates his special ‘true but unprovable’ sentence, using the previous steps in the proof.


Part 9: The Flaw 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.


Part 10: A Demonstration of a Contradiction Arising from Gödel’s Proof

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.






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?.

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The Lighter Side


Paper on ‘indefinable’ real numbers

There is now a paper that deals with a common misconception regarding real numbers, see On the Reality of the Continuum and Russell’s Moment of Candour. Also see the associated paper that deals with the matter of language and the diagonal proof, see On Considerations of Language in the Diagonal Proof.

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Other recently added pages

Proof of more Real numbers than Natural numbers


The Myths of Platonism


Goodman’s Paradox


The Platonist Rod paradox


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Lebesgue Measure

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

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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).

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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 Posts

A Failure to Understand

Random selection of numbers

Moderate Platonism

Descartes’ Platonism

The duplicity of Mark Chu-Carroll

A John Searle Inanity

Man versus Machine

Fake News and Fake Mathematics

Ned Block’s Blockhead

Are we alone in the Universe?

Good Math, Bad Math?

Bishops Dancing with Pixies?

Artificial Intelligence

Cranks and Crackpots

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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