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There's Something About Gödel: The Complete Guide to the Incompleteness Theorem

There's Something About Gödel: The Complete Guide to the Incompleteness Theorem

Francesco Berto

ISBN: 978-1-444-31502-8

Jul 2010

256 pages


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


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 γ

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.




""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)