There's something about Gödel! the complete guide to the incompleteness theorem
| Autor principal: | |
|---|---|
| Formato: | Libro |
| Idioma: | Inglés |
| Publicado: |
Malden (Massachusetts) :
Wiley-Blackwell
2009
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| Materias: | |
| Ver en Universidad de Navarra: | https://innopac.unav.es/record=b20083117*spi |
Tabla de Contenidos:
- 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
- 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
