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

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

ISBN/ASIN: 1405197668 | 2009 | English | pdf | 257/257 pages | 1.15 Mb
Publisher: John Wiley & Sons | Author: Francesco Berto | Edition: 1

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.

Contents
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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 “standardmodel” 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

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