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There's something about Gödel : the complete guide to the incompleteness theorem

Auteur : Francesco Berto
Éditeur : Malden, MA : Wiley-Blackwell, 2009.
Édition/format :   Livre : AnglaisVoir toutes les éditions et tous les formats
Base de données :WorldCat
Résumé :
There's Something About Gödel is a lucid and accessible guide to Gödel's revolutionary Incompleteness Theorem, considered one of the most astounding argumentative sequences in the history of human thought. It is also an exploration of the most controversial alleged philosophical outcomes of the Theorem. Divided into two parts, the first section introduces the reader to the Incompleteness Theorem--the argument that  Lire la suite...
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Détails

Genre/forme : manuel
Personne nommée : Kurt Gödel; Kurt Gödel
Format : Livre
Tous les auteurs / collaborateurs : Francesco Berto
ISBN : 9781405197663 1405197668 9781405197670 1405197676
Numéro OCLC : 361965956
Description : xx, 233 pages : illustrations ; 24 cm
Contenu : 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 TNI (is not recursive!) --
Arithmetizing substitution --
How can a TNT sentence refer to itself? --
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 undecidable 2 --
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.
Autres titres : Tutti pazzi per Gödel!
Responsabilité : Francesco Berto.

Résumé :

There's Something About Godel is a lucid and accessible guide to Godel's revolutionary Incompleteness Theorem , considered one of the most astounding argumentative sequences in the history of human  Lire la suite...

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"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 Lire la suite...

 
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schema:description"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 TNI (is not recursive!) -- Arithmetizing substitution -- How can a TNT sentence refer to itself? -- 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 undecidable 2 -- 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."@en
schema:description"There's Something About Gödel is a lucid and accessible guide to Gödel's revolutionary Incompleteness Theorem, considered one of the most astounding argumentative sequences in the history of human thought. It is also an exploration of the most controversial alleged philosophical outcomes of the Theorem. Divided into two parts, the first section introduces the reader to the Incompleteness Theorem--the argument that all mathematical systems contain statements which are true, yet which cannot be proved within the system. Berto describes the historical context surrounding Gödel's accomplishment, explains step-by-step the key aspects of the Theorem, and explores the technical issues of incompleteness in formal logical systems. The second half, The World After Gödel, considers some of the most famous--and infamous--claims arising from Gödel's theorem in the areas of the philosophy of mathematics, metaphysics, the philosophy of mind, Artificial Intelligence, and even sociology and politics. This book requires only minimal knowledge of aspects of elementary logic, and is written in a user-friendly style that enables it to be read by those outside the academic field, as well as students of philosophy, logic, and computing."@en
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