Incompleteness

Gödel’s proof of incompleteness:

A summary of information about Gödel’s proof and related matters

Gödel’s paper:

An online English translation of Gödel’s incompleteness proof with clickable cross-references

A guide to Gödel’s proof:

A step by step guide to Gödel’s proof for those wishing to investigate it in depth

Gödel’s proof simplified:

A simplified version of Gödel’s incompleteness proof that demonstrates the method used by Gödel

Gödel’s contradiction:

Why Gödel’s proof is inherently contradictory

Gödel’s flaw:

The crucial error in Gödel’s incompleteness proof

The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem:

A formal paper that gives a detailed analysis of the flaw in Gödel’s paper

Gödel’s Proposition V:

The crucial part of Gödel’s proof that he never actually proved; instead he simply assumed it to be correct

PDF Representability:

A formal paper that explains why a formal system cannot refer to itself in a logical and consistent manner

Gödel’s 1934 Undecidability lectures:

The confusion of language inherent in Gödel’s 1934 lectures on incompleteness

Errors in incompleteness proofs by Kleene and Rogers:

How Stephen Kleene and Hartley Rogers make the same erroneous assumption to generate an incompleteness proof

Gödel, Escher, Bach:

The flawed proof of incompleteness in the book Gödel, Escher, Bach

Peter Smith’s Gödel Without (Too Many) Tears:

The error in Peter Smith’s incompleteness proof in his Gödel Without (Too Many) Tears

PDF Peter Smith’s An Introduction To Gödel’s Theorems:

The error in Peter Smith’s incompleteness proof in his book An Introduction To Gödel’s Theorems

Nagel & Newman

The error in Nagel & Newman’s incompleteness proof

Other incompleteness proofs:

An overview of the analyses of several other incompleteness proofs

Common errors in incompleteness proofs:

An overview of some common errors in various attempts at incompleteness proofs

The “Diagonal Lemma”:

The key step in several incompleteness proofs and which contains an obvious error

Hilbert’s tenth problem:

How a proof regarding Hilbert’s tenth problem is erroneously claimed to lead to a proof of incompleteness

Good Math, Bad Math and Gödel:

An attempt at an explanation of Gödel’s incompleteness proof that falls down at the final hurdle

The Halting Problem and incompleteness proofs:

The naively simplistic assumptions used in attempts to prove incompleteness using Turing’s halting problem

Oh No ! Yet Another Flawed Incompleteness Proof:

A look at various simplistic attempts at incompleteness proofs and which have very obvious errors

Formal papers:

A list of formal papers detailing errors in published claims of incompleteness proofs, including computer proofs

David Pierce on Gödel:

How mathematicians avoid mentioning the unproven assumption that Gödel used to obtain his incompleteness result

Another intuitive error by Gödel:

An intuitive error by Gödel in another of his published papers, an error that was only discovered years later

A brief history of meta-mathematics:

How the field of meta-mathematics has made untenable and illogical assumptions

Some aspects of Gödel’s philosophy:

Some rather surprising statements by Gödel that reveal beliefs that have no logical nor evidential foundation

Gödel, Minds, and Machines:

A selection of claims for and against the notion that Gödel’s proof show that humans can deduce truths that machines cannot

Man versus Machine:

Roger Penrose claims that Gödel’s incompleteness proof indicates that human brains must be using quantum processes that no machine could replicate

Gödel’s substitution function:

How Gödel uses substitution in his proof

True but unprovable?

An examination of the claim that Gödel’s proof shows a formula to be true but unprovable in the given system

Intuitionism:

Intuition, Intuitionism and Gödel’s proof

Representability:

A key claim in many attempts at incompleteness proofs is the claim that a certain expression has a precise representation in the formal system

Church’s Theorem (An Unsolvable Problem):

How Church makes the same error as in many other incompleteness proofs

Jean-Yves Girard on Incompleteness:

A critical look at Jean-Yves Girard’s claims regarding Incompleteness from his book The Blind Spot

Flawed disproofs of Gödel’s proof:

Because Gödel’s proof appears paradoxical to many people, many people have attempted unsuccessfully to find the flaw in it

The Shackles of Conviction:

A novel based around the protagonist finding the flaw in Gödel’s proof

Logic and Language:

Logic is not independent of the language used to express it; this crucial fact is often ignored in attempts at incompleteness proofs

Mathematical Proofs:

Despite claims to the contrary, the reality is that much of mathematics is not supported by logical proof

Computer Proofs:

Are computer proofs as infallible as their programmers claim?