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Set Theory, Arithmetic, and Foundations of Mathematics : Theorems, Philosophies.
A collection of remarkable papers from various areas of mathematical logic, written by outstanding members of the field.
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2011.
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| Colección: | Lecture Notes in Logic, 36.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Title; Copyright; Dedication; Contents; Introduction; Historical remarks on Suslin's problem; 1. Suslin's problem.; 2. Consistency of ƠSH.; 3. Consistency of SH.; 4. Envoi.; REFERENCES; The continuum hypothesis, the generic-multiverse of sets, and the O conjecture; 1. A tale of two problems; 2. The generic-multiverse of sets.; 3. O-log; 4. The O conjecture.; 5. The complexity of O-logic.; 6. The weak multiverse laws and H(c+).; 7. Conclusions.; 8. Appendix.; REFERENCES;?-models of finite set theory; 1. Introduction.; 2. Preliminaries.; 3. Building?-models.
- 4. Models with special properties. 5. ZFfin and PA are not bi-interpretable.; 6. Concluding remarks and open questions.; REFERENCES; Tennenbaum's theorem for models of arithmetic; 1. Some historical background.; 2. Tennenbaum's theorem.; 3. Diophantine problems.; REFERENCES; Hierarchies of subsystems of weak arithmetic; 1. Introduction.; 2. Preliminaries.; 3. The main results.; 3.1. Proof of Theorem A; 3.2. Proof of Theorem B.; 3.3. Proofs of Theorems C and D.; REFERENCES; Diophantine correct open induction; Introduction.; Background.
- Wilkie's theorems and the models of Berarducci and Otero. 1. Axioms for DOI.; 2. Diophantine correct rings of Puiseux polynomials; 3. Generalized polynomials.; Special sequences of polynomials.; Theorems on generalized polynomials.; 4. A Class of Diophantine correct ordered rings.; REFERENCES; Tennenbaum's theorem and recursive reducts; 0. Conventions.; 1. Rich theories.; 2. Thin theories.; 3. Examples.; 4. Some 1-thin theories.; 5. More about LO.; REFERENCES; History of constructivism in the 20th century; 1. Introduction.; 2. Finitism.; 2.1. Finitist mathematics.; 2.2. Actualism.
- 3. Predicativism and semi-intuitionism. 3.1. Poincaré.; 3.2. The semi-intuitionists.; 3.3. Borel and the continuum.; 3.4. Weyl.; 4. Brouwerian intuitionism.; 4.1. Early period.; 4.2. Weak counterexamples and the creative subject.; 4.3. Brouwer's programme.; 5. Intuitionistic logic and arithmetic.; 5.1. L.E.J. Brouwer and intuitionistic logic.; 5.2. The Brouwer-Heyting-Kolmogorov interpretation.; 5.3. Formal intuitionistic logic and arithmetic through 1940.; 5.4. Metamathematics of intuitionistic logic and arithmetic after 1940.; 5.5. Formulas-as-types.
- 6. Intuitionistic analysis and stronger theories. 6.1. Choice sequences in Brouwer's writings.; 6.2. Axiomatization of intuitionistic analysis.; 6.3. The model theory of intuitionistic analysis.; 7. Constructive recursive mathematics.; 7.1. Classical recursive mathematics.; 7.2. Constructive recursive mathematics.; 8. Bishop's constructivism.; 8.1. Bishop's constructive mathematics.; 8.2. The relation of BCM to INT and CRM.; 9. Concluding remarks.; REFERENCES; A very short history of ultrafinitism; 1. Introduction:; 2. Short history and prehistory of ultrafinitism.; 2.1. Murios.