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9781405197663

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

by
  • ISBN13:

    9781405197663

  • ISBN10:

    1405197668

  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2009-11-09
  • Publisher: Wiley-Blackwell
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Summary

Berto's highly readable and lucid guide introduces students and the interested reader to G_del's celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from G_del'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 G_del'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
Acknowledgments
The Gödelian Symphony:
Foundations and Paradoxes
"This sentence is false"
The Liar and Gödel
Language and metalanguage
The axiomatic method, or how to get the non-obvious out of the obvious
Peano's axioms ...
... and the unsatisfied logicists, Frege and Russell
Bits of set theory
The Abstraction Principle
Bytes of set theory
Properties, relations, functions, that is, sets again
Calculating, computing, enumerating, that is, the notion of algorithm
Taking numbers as sets of sets
It's raining paradoxes
Cantor's diagonal argument
Self-reference and paradoxes
Hilbert
Strings of symbols
"... in mathematics there is no ignorabimus"
Gödel on stage
Our first encounter with the Incompleteness Theorem ...
... and some provisos
Gödelization, or Say It with Numbers!
TNT
The arithmetical axioms of TNT and the "standard model" N
The Fundamental Property of formal systems
The Gödel numbering ...
... and the arithmetization of syntax
Bits of Recursive Arithmetic ...
Making algorithms precise
Bits of recursion theory
Church's Thesis
The recursiveness of predicates, sets, properties, and relations
... and How It Is Represented in Typographical Number Theory
Introspection and representation
The representability of properties, relations, and functions ...
... and the Gödelian loop
"I Am Not Provable"
Proof pairs
The property of being a theorem of TNT (is not recursive!)
Arithmetizing substitution
How can a TNT sentence refer to itself?
g
Fixed point
Consistency and omega-consistency
Proving G1
Rosser's proof
The Unprovability of Consistency and the "Immediate Consequences" of G1 and G2
G2
Technical interlude
"Immediate consequences" of G1 and G2
Undecidable1 and undecidable2
Essential incompleteness, or the syndicate of mathematicians
Robinson Arithmetic
How general are Gödel's results?
Bits of Turing machine
G1 and G2 in general
Unexpected fish in the formal net
Supernatural numbers
The culpability of the induction scheme
Bits of truth (not too much of it, though)
The World after Gödel:
Bourgeois Mathematicians! The Postmodern Interpretations
What is postmodernism?
From Gödel to Lenin
Is "Biblical proof" decidable?
Speaking of the totality
Bourgeois teachers!
(Un)interesting bifurcations
A Footnote to Plato
Explorers in the realm of numbers
The essence of a life
"The philosophical prejudices of our times"
From Gödel to Tarski
Human, too human
Mathematical Faith
"I'm not crazy!"
Qualified doubts
From Gentzen to the Dialectica interpretation
Mathematicians are people of faith
Mind versus Computer: Gödel and Artificial Intelligence
Is mind (just) a program?
"Seeing the truth" and "going outside the system"
The basic mistake
In the haze of the transfinite
"Know thyself": Socrates and the inexhaustibility of mathematics
Gödel versus Wittgenstein and the Paraconsistent Interpretation
When geniuses meet ...
The implausible Wittgenstein
"There is no metamathematics"
Proof and prose
The single argument
But how can arithmetic be inconsistent?
The costs and benefits of making Wittgenstein plausible
Epilogue
References
Index
Table of Contents provided by Publisher. All Rights Reserved.

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