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

by Berto, Francesco

• ISBN13: 9781405197670
• ISBN10: 1405197676
• Edition: 1st
• Format: Paperback
• Copyright: 2009-11-09
• Publisher: Wiley-Blackwell

Summary

Berto's highly readable and lucid guide introduces students and the interested reader to Godel's celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Godel'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 Godel's theories Written in an accessible, non-technical style

Author Biography

Francesco Berto, Ph.D., is a lecturer in Philosophy at the University Venice - Ca'Foscari, Italy, and is a Chaire d'Excellence Fellow at the École Normale Supérieure (Sorbonne), Paris. He has written numerous articles on logic, metaphysics and philosophy of language. He is the author of How to Sell a Contradiction: The Logic and Metaphysics of Inconsistency (2005), for which he won the Castiglioncello prize for the best philosophical book by a young philosopher in 2007.

Table of Contents

Prologue	p. xi
Acknowledgments	p. xix
The Gödelian Symphony	p. 1
Foundations and Paradoxes	p. 3
"This sentence is false"	p. 6
The Liar and Gödel	p. 8
Language and metalanguage	p. 10
The axiomatic method, or how to get the non-obvious out of the obvious	p. 13
Peano's axioms...	p. 14
...and the unsatisfied logicists, Frege and Russell	p. 15
Bits of set theory	p. 17
The Abstraction Principle	p. 20
Bytes of set theory	p. 21
Properties, relations, functions, that is, sets again	p. 22
Calculating, computing, enumerating, that is, the notion of algorithm	p. 25
Taking numbers as sets of sets	p. 29
It's raining paradoxes	p. 30
Cantor's diagonal argument	p. 32
Self-reference and paradoxes	p. 36
Hilbert	p. 39
Strings of symbols	p. 39
"...in mathematics there is no ignorabimus"	p. 42
Gödel on stage	p. 46
Our first encounter with the Incompleteness Theorem...	p. 47
...and some provisos	p. 51
Gödelization, or Say It with Numbers!	p. 54
TNT	p. 55
The arithmetical axioms of TNT and the "standard model" N	p. 57
The Fundamental Property of formal systems	p. 61
The Gödel numbering...	p. 65
...and the arithmetization of syntax	p. 69
Bits of Recursive Arithmetic...	p. 71
Making algorithms precise	p. 71
Bits of recursion theory	p. 72
Church's Thesis	p. 76
The recursiveness of predicates, sets, properties, and relations	p. 77
...And How It Is Represented in Typographical Number Theory	p. 79
Introspection and representation	p. 79
The representability of properties, relations, and functions...	p. 81
...and the Gödelian loop	p. 84
"I Am Not Provable"	p. 86
Proof pairs	p. 86
The property of being a theorem of TNT (is not recursive!)	p. 87
Arithmetizing substitution	p. 89
How can a TNT sentence refer to itself?	p. 90
¿	p. 93
Fixed point	p. 95
Consistency and omega-consistency	p. 97
Proving G1	p. 98
Rosser's proof	p. 100
The Unprovability of Consistency and the "Immediate Consequences" of G1 and G2	p. 102
G2	p. 102
Technical interlude	p. 105
"Immediate consequences" of G1 and G2	p. 106
Undecidable1 and undecidable2	p. 107
Essential incompleteness, or the syndicate of mathematicians	p. 109
Robinson Arithmetic	p. 111
How general are Gödel's results?	p. 112
Bits of Turing machine	p. 113
Gl and G2 in general	p. 116
Unexpected fish in the formal net	p. 118
Supernatural numbers	p. 121
The culpability of the induction scheme	p. 123
Bits of truth (not too much of it, though)	p. 125
The World after Gödel	p. 129
Bourgeois Mathematicians! The Postmodern Interpretations	p. 131
What is postmodernism?	p. 132
From Gödel to Lenin	p. 133
Is "Biblical proof" decidable?	p. 135
Speaking of the totality	p. 137
Bourgeois teachers!	p. 139
(Un)interesting bifurcations	p. 141
A Footnote to Plato	p. 146
Explorers in the realm of numbers	p. 146
The essence of a life	p. 148
"The philosophical prejudices of our times"	p. 151
From Gödel toTarski	p. 153
Human, too human	p. 157
Mathematical Faith	p. 162
"I'm not crazy!"	p. 163
Qualified doubts	p. 166
From Gentzen to the Dialectica interpretation	p. 168
Mathematicians are people of faith	p. 170
Mind versus Computer: Gödel and Artificial Intelligence	p. 174
Is mind (just) a program?	p. 174
"Seeing the truth" and "going outside the system"	p. 176
The basic mistake	p. 179
In the haze of the transfinite	p. 181
"Know thyself": Socrates and the inexhaustibility of mathematics	p. 185
Gödel versus Wittgenstein and the Paraconsistent Interpretation	p. 189
"When geniuses meet...	p. 190
The implausible Wittgenstein	p. 191
"There is no metamathematics"	p. 194
Proof and prose	p. 196
The single argument	p. 201
But how can arithmetic be inconsistent?	p. 206
The costs and benefits of making Wittgenstein plausible	p. 213
Epilogue	p. 214
References	p. 217
Index	p. 225
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