Introduction of the book Gödel (para todos) translated by J. C. Kelly (all rights reserved)
Gödel’s Incompleteness Theorem is one of mathematical logic’s most profound and paradoxical results. It is also perhaps the theorem which has been of most interest in areas far removed from the exact sciences. It has been cited in disciplines as diverse as semiotics, psychoanalysis, philosophy, and the political sciences. Authors such as Kristeva, Lacan, Debray, Deleuze, and Lyotard, to name just a few, have invoked the name of Gödel and his Incompleteness Theorem in daring analogies. Alongside other buzzwords from the postmodern period, words such as “chaos”, “fractal”, “indeterminism” and “randomness”, the phenomenon of incompleteness has also been associated with alleged defeats of reason and the end of certainty in the most exclusive domain of human thought: the kingdom of exact formulas. Gödel’s theorem also crops up in heated epistemological controversies within the world of science itself, such as in the discussions on artificial intelligence. Appearing with a ubiquity which rivals the Theory of Relativity, and in a stealthier manner, Gödel’s theorem has become a cornerstone and unavoidable reference point of contemporary thought.
Gödel’s Incompleteness Theorem is one of mathematical logic’s most profound and paradoxical results. It is also perhaps the theorem which has been of most interest in areas far removed from the exact sciences. It has been cited in disciplines as diverse as semiotics, psychoanalysis, philosophy, and the political sciences. Authors such as Kristeva, Lacan, Debray, Deleuze, and Lyotard, to name just a few, have invoked the name of Gödel and his Incompleteness Theorem in daring analogies. Alongside other buzzwords from the postmodern period, words such as “chaos”, “fractal”, “indeterminism” and “randomness”, the phenomenon of incompleteness has also been associated with alleged defeats of reason and the end of certainty in the most exclusive domain of human thought: the kingdom of exact formulas. Gödel’s theorem also crops up in heated epistemological controversies within the world of science itself, such as in the discussions on artificial intelligence. Appearing with a ubiquity which rivals the Theory of Relativity, and in a stealthier manner, Gödel’s theorem has become a cornerstone and unavoidable reference point of contemporary thought.
However, while the sophistication of the equations in Einstein’s theory means even the best attempts to make it accessible are confined to examples with clocks and people who do not age in journeys through space, in the case of Gödel’s Incompleteness Theorem there is good news: it is possible to provide a rigorous and accessible explanation without any prior mathematical knowledge beyond addition and multiplication, as taught at school.
Indeed, this is precisely what we have set out to do in this book: to provide an explanation that is at once detailed whilst being simple and self-contained, one which allows people from all disciplines, armed only with an indispensible “spirit of curiosity” to challenge themselves to experience in depth one of the most extraordinary intellectual feats of our time.
Gödel
A word regarding the book’s title: when the suffix “for all” is added to the title of books which explain a subject—and even more so when the book deals with questions or authors regarded as “difficult”— it is implied that the “for all” is really just a euphemistic way of saying, somewhere in between condescension and compassion, that it is indeed written for people “who don’t know anything”. This is not the case with this book: when we use the term “for all”, we are talking, rather, about the true meaning of the expression, with all its implications. We haven’t just written this book for people “who know don’t know anything”, but also for readers who have already read partial explanations of Gödel’s theorem, and even those who have studied his theorems and their proofs in some detail. Even if our book does start from zero, it goes much further than some of the best known attempts to make the theory accessible. In particular, we provide a rigorous proof with the complete details of all the theorems, making use of an approach that varies slightly from the norm, one that is novel on account of its simplicity and in which the use of technical mathematical jargon is kept to a bare minimum. We have also included a final chapter with our own investigation of the phenomenon of incompleteness in a general context, and a discussion of problems which remain unsolved in order to show the longevity of the ideas and questions which Gödel’s thought continues to provoke.
The material has been organized along the following lines:
- In the first chapter, we provide a general overview and a first, informal approximation both of the statements of Gödel’s theorems and some of their philosophical applications.
- In chapter 2, we describe the historical context and the state of the debate surrounding the foundations of mathematics at the time in which Gödel’s idea came forth. The chapter concludes by discussing some of the most frequent distortions and errors when explaining the statements.
- In chapter 3, we introduce the formal language required to state the theorems with all the required precision, and pave the way for the proofs.
Each of the three chapters seems to conclude in the same way: by stating Gödel’s theorems. However, our intention and hope is that they are read each time with a deeper understanding and with the new meaning and greater precision we incorporate at each stage.
- In chapter 4, we introduce some analogies and attempts that have been made in a number of social sciences to apply Gödel’s theorem in areas which lie beyond mathematics. In particular we consider texts by Julia Kristeva, Paul Virilio, Régis Debray, Gilles Deleuze, Félix Guattari, Jacques Lacan, and Jean-Françios Lyotard.
This concludes the first part of the book.
The second part is dedicated to the proofs of the theorems. We believe our proof makes the least possible use of technical mathematical jargon. Essentially, we show that all Gödel’s arguments can be developed from just one mathematical fact: the existence in arithmetic of an operation which reflects the way in which the letters of a language are juxtaposed one after the other to make up words.
The third and final part of the book is dedicated to our own exploration of the phenomenon of incompleteness in a more general and abstract context. We ask what mathematical fact can be found reflected in other objects and which property draws the boundary between complete and incomplete theories.
A selection of exercises is included at the end of most of the chapters and, in spite of some initial doubts, we also decided to include the solutions. We hope the exercises will provide the reader with an additional stimulus, firstly trying to come up with their own solution “without help”, and then comparing it with the one we have proposed in each case.
The book is completed by three appendices: the first is to be consulted while reading and brings together a variety of theories which serve as examples and counterexamples to various assertions. The second is a selection of texts written by the protagonists themselves, such as Cantor, Russell and Hilbert, on key milestones in phenomenon of incompleteness, and when read together, these make up a small history on the subject. The third is a brief biography of Kurt Gödel with a chronology of his life.
In the final chapter, we have left the reader with some of the unsolved problems, and perhaps some readers will also set themselves the challenge of solving them. Others may wish to make suggestions or criticisms about different parts of our explanation, or point out any mistakes which might have slipped past us. For this reason, we have set up a blog in order to receive your comments:
We will also publish there, in their complete form, some of the texts that had to be summarized to fit the format of this book, as well as various articles from the bibliography which we have found particularly interesting.
Finally, we would like to thank Xavier Caicedo for various conversations and explanations which helped clarify the more delicate points of the theorem and also the generous and attentive final reading of the book by Pablo Coll .
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