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Roads to infinity : the mathematics of truth and proof /
Offers an introduction to modern ideas about infinity and their implications for mathematics. It unifies ideas from set theory and mathematical logic, and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics. From publisher description.
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Natick, Mass. :
A K Peters,
©2010.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- The diagonal argument : Counting and countability ; Does one infinite size fit all? ; Cantor's diagonal argument ; Transcendental numbers ; Other uncountability proofs ; Rates of growth ; The cardinality of the continuum ; Historical background
- Ordinals : Counting past infinity ; The countable ordinals ; The axiom of choice ; The continuum hypothesis ; Induction ; Cantor normal form ; Goodstein's Theorem ; Hercules and the Hydra ; Historical background
- Computability and proof : Formal systems ; Post's approach to incompleteness ; Gödel's first incompleteness theorem ; Gödel's second incompleteness theorem ; Formalization of computability ; The halting problem ; The entscheidungsproblem ; Historical background
- Logic : Propositional logic ; A classical system ; A cut-free system for propositional logic ; Happy endings ; Predicate logic ; Completeness, consistency, happy endings ; Historical background
- Arithmetic : How might we prove consistency? ; Formal arithmetic ; The systems PA and PA ; Embedding PA and PA; Cut elimination in PA ; The height of this great argument ; Roads to infinity ; Historical background
- Natural unprovable sentences : A generalized Goodstein Theorem ; Countable ordinals via natural numbers ; From generalized Goodstein to well-ordering ; Generalized and ordinary Goodstein ; Provably computable functions ; Complete disorder is impossible ; The hardest theorem in graph theory ; Historical background
- Axioms of infinity : Set theory without infinity ; Inaccessible cardinals ; The axiom of determinacy ; Largeness axioms for arithmetic ; Large cardinals and finite mathematics ; Historical background.