// Gödel (for all): Table of contents

Translated by J. C. Kelly

 

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …13

 

PART ONE

 

Chapter one

A general overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …21

The true and the provable. Formal axiomatic systems.

Completeness and axioms. Infinity: The bête noire of the

foundations of mathematics. The Incompleteness Theorem.

Gödel’s original proof. The theorem of consistency.

Extension and scope of Gödel’s theorem. Warnings.

Gödel, computers and artificial intelligence. Philosophical

applications. Examples and exercises.

 

Chapter two

Hilbert and the problem of foundations . . . . . . . . . . . . . . . ....51

Hilbert’s program. Discussion: What Gödel’s theorems do

and do not say. Examples and exercises.

 

Chapter three

Language for arithmetic and the definition of truth . . . . . . …..71

Formal language. Statements. Axioms and inference

rules in first order logic. Proofs and theories. Truth in

mathematics: a formal definition. Completeness and

consistency in our formal theory. The solution to a

dilemma. Exercises.

 

Chapter four

Gödel’s theorem beyond mathematics . . . . . . . . . . . . . . . . .103

Julia Kristeva: Gödel and semiotics. The development of a

formal theory for poetic language. Paul Virilio:

Gödel and the new technologies. Régis Debray and Michel

Serres: Gödel and politics. Deleuze and Guattari: Gödel and

philosophy. Jacques Lacan: Gödel and psychoanalysis. Jean-

François Lyotard: Gödel and the postmodern condition.

Exercises.

 

 

PART TWO

 

Proof of the theorems

 

Roadmap

Concatenation and the Incompleteness Theorem . . . . . . . . ..147

If there exists an expressible concatenation, Gödel’s

theorems hold.

 

Chapter five

The semantic version of the Incompleteness Theorem . . . . . 151

Concatenation using Morse code. The Self-referential

method. “To be true” is not expressible.

 

Chapter six

General (syntactic) version of the Incompleteness Theorem.

The theorem of consistency . . . . . . . . . . . . . . . . . . . . . . . . 165

Exercises.

 

Chapter seven

There exists an expressible concatenation in arithmetic . . . . .189

 

Chapter eight

Any recursive property can be expressed using

concatenation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  197

 

 

PART THREE

 

Incompleteness in a general and abstract context

 

Chapter nine

Incompleteness in a general and abstract context . . . . . . . . . 217

An intrinsic proof of Gödel’s theorem.

Concatenation and Gödel’s argument. Conclusions

and unsolved problems. Exercises.

 

Appendix I

Examples of complete and incomplete theories . . . . . . . . . . .261

 

Appendix II

Historical appendix: Landmarks in the history of the

Incompleteness Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . .277

 

Appendix III

Kurt Gödel, Mr Why . . . . . . . . . . . .  . . . . . . . . . . . . . . . . ..289

Chronology of his life   . . . . . . . . . . . . . . . . . . . . . . . . . . . .293

 

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .297

 

Recommended reading . . . . . . . . . . . . . . . . . . . . . . . . . . . ..301

 

Index of names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .303

 

Index of terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..305