A Simplified Explanation of Gödel’s Incompleteness Proof:
Introduction

Page last updated 11 May 2021

This simplified explanation of Gödel’s incompleteness proof (see online at English translation of Gödel’s original proof) 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 a 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 different languages:

• the formal languages that he is talking about, and
• the language that talks about those 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 section demonstrates just one such logical contradiction.