Theres Something About Gödel: The Complete Guide to the Incompleteness Theorem

There s Something About Gödel is a lucid and accessible guide to Gödel s revolutionary Incompleteness Theorem, considered one of the most astounding argumentative sequences in the history of human thought. It is also an exploration of the most controversial alleged philosophical outcomes of the Theorem.

Divided into two parts, the first section introduces the reader to the Incompleteness Theorem the argument that all mathematical systems contain statements which are true, yet which cannot be proved within the system. Berto describes the historical context surrounding Gödel′s accomplishment, explains step–by–step the key aspects of the Theorem, and explores the technical issues of incompleteness in formal logical systems. The second half, The World After Gödel, considers some of the most famous and infamous claims arising from Gödel′s theorem in the areas of the philosophy of mathematics, metaphysics, the philosophy of mind, Artificial Intelligence, and even sociology and politics.

This book requires only minimal knowledge of aspects of elementary logic, and is written in a user–friendly style that enables it to be read by those outside the academic field, as well as students of philosophy, logic, and computing.

Prologue.

Acknowledgments.

Part I: The Gödelian Symphony.

1 Foundations and Paradoxes.

1 "This sentence is false".

2 The Liar and Gödel.

3 Language and metalanguage.

4 The axiomatic method, or how to get the non–obvious out of the obvious.

5 Peano′s axioms .

6 and the unsatisfied logicists, Frege and Russell.

7 Bits of set theory.

8 The Abstraction Principle.

9 Bytes of set theory.

10 Properties, relations, functions, that is, sets again.

11 Calculating, computing, enumerating, that is, the notion of algorithm.

12 Taking numbers as sets of sets.

13 It′s raining paradoxes.

14 Cantor′s diagonal argument.

15 Self–reference and paradoxes.

2 Hilbert.

1 Strings of symbols.

2 " in mathematics there is no ignorabimus".

3 Gödel on stage.

4 Our first encounter with the Incompleteness Theorem .

5 and some provisos.

3 Gödelization, or Say It with Numbers!

1 TNT.

2 The arithmetical axioms of TNT and the "standard model" N.

3 The Fundamental Property of formal systems.

4 The Gödel numbering .

5 and the arithmetization of syntax.

4 Bits of Recursive Arithmetic .

1 Making algorithms precise.

2 Bits of recursion theory.

3 Church′s Thesis.

4 The recursiveness of predicates, sets, properties, and relations.

5 And How It Is Represented in Typographical Number Theory.

1 Introspection and representation.

2 The representability of properties, relations, and functions .

3 and the Gödelian loop.

6 "I Am Not Provable".

1 Proof pairs.

2 The property of being a theorem of TNT (is not recursive!)

3 Arithmetizing substitution.

4 How can a TNT sentence refer to itself?

5

6 Fixed point.

7 Consistency and omega–consistency.

8 Proving G1.

9 Rosser′s proof.

7 The Unprovability of Consistency and the "Immediate Consequences" of G1 and G2.

1 G2.

2 Technical interlude.

3 "Immediate consequences" of G1 and G2.

4 Undecidable1 and undecidable2.

5 Essential incompleteness, or the syndicate of mathematicians.

6 Robinson Arithmetic.

7 How general are Gödel′s results?

8 Bits of Turing machine.

9 G1 and G2 in general.

10 Unexpected fish in the formal net.

11 Supernatural numbers.

12 The culpability of the induction scheme.

13 Bits of truth (not too much of it, though).

Part II: The World after Gödel.

8 Bourgeois Mathematicians! The Postmodern Interpretations.

1 What is postmodernism?

2 From Gödel to Lenin.

3 Is "Biblical proof" decidable?

4 Speaking of the totality.

5 Bourgeois teachers!

6 (Un)interesting bifurcations.

9 A Footnote to Plato.

1 Explorers in the realm of numbers.

2 The essence of a life.

3 "The philosophical prejudices of our times".

4 From Gödel to Tarski.

5 Human, too human.

10 Mathematical Faith.

1 "I′m not crazy!"

2 Qualified doubts.

3 From Gentzen to the Dialectica interpretation.

4 Mathematicians are people of faith.

11 Mind versus Computer: Gödel and Artificial Intelligence.

1 Is mind (just) a program?

2 "Seeing the truth" and "going outside the system".

3 The basic mistake.

4 In the haze of the transfinite.

5 "Know thyself": Socrates and the inexhaustibility of mathematics.

12 Gödel versus Wittgenstein and the Paraconsistent Interpretation.

1 When geniuses meet .

2 The implausible Wittgenstein.

3 "There is no metamathematics".

4 Proof and prose.

5 The single argument.

6 But how can arithmetic be inconsistent?

7 The costs and benefits of making Wittgenstein plausible.

Epilogue.

References.

Index.

Francesco Berto teaches logic, ontology, and philosophy of mathematics at the universities of Aberdeen in Scotland, and Venice and Milan–San Raffaele in Italy. He holds a Chaire d′Excellence fellowship at CNRS in Paris, where he has taught ontology at the École Normale Supérieure, and he is a visiting professor at the Institut Wiener Kreis of the University of Vienna. He has written papers for American Philosophical Quarterly, Dialectica, The Philosophical Quarterly, the Australasian Journal of Philosophy, the European Journal of Philosophy, Philosophia Mathematica, Logique et Analyse, and Metaphysica, and runs the entries Dialetheism and Impossible Worlds in the Stanford Encyclopedia of Philosophy. His book How to Sell a Contradiction has won the 2007 Castiglioncello prize for the best philosophical book by a young philosopher.

"There′s Something about G¨odel is a bargain: two books in one. The first half is a gentle but rigorous introduction to the incompleteness theorems for the mathematically uninitiated. The second is a survey of the philosophical, psychological, and sociological consequences people have attempted to derive from the theorems, some of them quite fantastical." (Philosophia Mathematica, 2011)

There is a story that in 1930 the great mathematician John von Neumann emerged from a seminar delivered by Kurt Gödel saying: It′s all over. Gödel had just proved the two theorems about the logical foundations of mathematics that are the subject of this valuable new book by Francesco Berto. Berto′s clear exposition and his strategy of dividing the proof into short, easily digestible chunks make it pleasant reading ... .Berto is lucid and witty in exposing mistaken applications of Gödel′s results ... [and] has provided a thoroughly recommendable guide to Gödel′s theorems and their current status within, and outside, mathematical logic. (Times Higher Education Supplement, February 2010)