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A Simplified Explanation of Gödel’s proof - Introduction


 

 


 

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 (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

 

NEWS

Peter Smith’s ‘Proof’

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’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.

 

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

 

Links  

 

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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Comments

 

Comments on this site are welcome, please see the comment section.

 

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.

 

If you prefer to ask me directly about the material on this site, please send me an e-mail with your query, and I will attempt to reply promptly.

 

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