Subject(s) 
Incompleteness theorems


Gödel's theorem


Mathematics  Philosophy

Subject 
Gödel, Kurt

Physical Description 
xx, 233 p. ; 24 cm 
Note 
Includes bibliographical references (p. [217]224) and index 
Contents 
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 nonobvious 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  Selfreference 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 omegaconsistency  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 
