There's Something About Gödel

The Complete Guide to the Incompleteness Theorem
1. Auflage 2009
ISBN: 978-1-4051-9766-3
Verlag: Wiley

Buch, Englisch, 256 Seiten, Format (B × H): 160 mm x 239 mm, Gewicht: 526 g

Berto's highly readable and lucid guide introduces students and the interested reader to Gödel's celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments.
- Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters
- Discusses interpretations of the Theorem made by celebrated contemporary thinkers
- Sheds light on the wider extra-mathematical and philosophical implications of Gödel's theories
- Written in an accessible, non-technical style

Autoren/Hrsg.

Berto, Francesco

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Mathematische Logik
• Geisteswissenschaften Philosophie Philosophie der Mathematik, Philosophie der Physik
• Mathematik | Informatik Mathematik Mathematik Allgemein Philosophie der Mathematik

Inhaltsverzeichnis

Prologue xi

Acknowledgments xix

Part I: The Gödelian Symphony 1

1 Foundations and Paradoxes 3

1 “This sentence is false” 6

2 The Liar and Gödel 8

3 Language and metalanguage 10

4 The axiomatic method, or how to get the non-obvious out of the obvious 13

5 Peano’s axioms … 14

6 … and the unsatisfied logicists, Frege and Russell 15

7 Bits of set theory 17

8 The Abstraction Principle 20

9 Bytes of set theory 21

10 Properties, relations, functions, that is, sets again 22

11 Calculating, computing, enumerating, that is, the notion of algorithm 25

12 Taking numbers as sets of sets 29

13 It’s raining paradoxes 30

14 Cantor’s diagonal argument 32

15 Self-reference and paradoxes 36

2 Hilbert 39

1 Strings of symbols 39

2 “… in mathematics there is no ignorabimus” 42

3 Gödel on stage 46

4 Our first encounter with the Incompleteness Theorem … 47

5 … and some provisos 51

3 Gödelization, or Say It with Numbers! 54

1 TNT 55

2 The arithmetical axioms of TNT and the “standard model” N 57

3 The Fundamental Property of formal systems 61

4 The Gödel numbering … 65

5 … and the arithmetization of syntax 69

4 Bits of Recursive Arithmetic … 71

1 Making algorithms precise 71

2 Bits of recursion theory 72

3 Church’s Thesis 76

4 The recursiveness of predicates, sets, properties, and relations 77

5 … And How It Is Represented in Typographical Number Theory 79

1 Introspection and representation 79

2 The representability of properties, relations, and functions … 81

3 … and the Gödelian loop 84

6 “I Am Not Provable” 86

1 Proof pairs 86

2 The property of being a theorem of TNT (is not recursive!) 87

3 Arithmetizing substitution 89

4 How can a TNT sentence refer to

Über Autor(innen)

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.