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Realizability [[electronic resource] ] : an introduction to its categorical side / / Jaap van Oosten
| Realizability [[electronic resource] ] : an introduction to its categorical side / / Jaap van Oosten |
| Autore | Oosten Jaap van |
| Edizione | [1st ed.] |
| Pubbl/distr/stampa | Oxford, : Elsevier, 2008 |
| Descrizione fisica | 1 online resource (327 p.) |
| Disciplina |
511.3
612.843 |
| Collana | Studies in logic and the foundations of mathematics |
| Soggetto topico |
Logic, Symbolic and mathematical
Model theory |
| ISBN |
1-281-16508-5
9786611165086 0-08-056006-7 1-4356-2874-8 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Front Cover; Realizability: An Introduction to its Categorical Side; Copyright Page; Preface; Introduction; Table of Contents; Chapter 1 Partial Combinatory Algebras; 1.1 Basic definitions; 1.1.1 Pairing, Booleans and Definition by Cases; 1.2 P(A)-valued predicates; 1.3 Further properties; recursion theory; 1.3.1 Recursion theory in pcas; 1.4 Examples of pcas; 1.4.1 Kleene's first model; 1.4.2 Relativized recursion; 1.4.3 Kleene's second model; 1.4.4 K2 generalized; 1.4.5 Sequential computations; 1.4.6 The graph model P(ω); 1.4.7 Graph models; 1.4.8 Domain models; 1.4.9 Relativized models
1.4.10 Term models1.4.11 Pitts' construction; 1.4.12 Models of Arithmetic; 1.5 Morphisms and Assemblies; 1.6 Applicative morphisms and S-functors; 1.7 Decidable applicative morphisms; 1.8 Order-pcas; Chapter 2 Realizability triposes and toposes; 2.1 Triposes; 2.1.1 Preorder-enriched categories; 2.1.2 Triposes: definition and basic properties; 2.1.3 Interpretation of languages in triposes; 2.1.4 A few useful facts; 2.2 The tripos-to-topos construction; 2.3 Internal logic of C[P] reduced to the logic of P; 2.4 The 'constant objects' functor; 2.5 Geometric morphisms Chapter 3 The Effective Topos3.1 Recapitulation and arithmetic in εff; 3.1.1 Second-order arithmetic in εff; 3.1.2 Third-order arithmetic in εff; 3.2 Some special objects and arrows in εff; 3.2.1 Closed and dense subobjects; 3.2.2 Infinite coproducts and products; 3.2.3 Projective and internally projective objects, and choice principles; 3.2.4 εff as a universal construction; 3.2.5 Real numbers in εff; 3.2.6 Discrete and modest objects; 3.2.7 Decidable and semidecidable subobjects; 3.3 Some analysis in εff; 3.3.1 General facts about R; 3.3.2 Specker sequences and singular coverings 3.3.3 Real-valued functions3.4 Discrete families and Uniform maps; 3.4.1 Weakly complete internal categories in εff; 3.5 Set Theory in εff; 3.5.1 The McCarty model for IZF; 3.5.2 The Lubarsky-Streicher-Van den Berg model for CZF; 3.5.3 Well-founded trees and W-Types in εff; 3.6 Synthetic Domain Theory in εff; 3.6.1 Complete partial orders; 3.6.2 The synthetic approach; 3.6.3 Elements of Synthetic Domain Theory; 3.6.4 Models for SDT in εff; 3.7 Synthetic Computability Theory in εff; 3.8 General Comments about the Effective Topos; 3.8.1 Analogy between ▿ and the Yoneda embedding 3.8.2 Small dense subcategories in εff |
| Record Nr. | UNINA-9910777303903321 |
Oosten Jaap van
|
||
| Oxford, : Elsevier, 2008 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Realizability [[electronic resource] ] : an introduction to its categorical side / / Jaap van Oosten
| Realizability [[electronic resource] ] : an introduction to its categorical side / / Jaap van Oosten |
| Autore | Oosten Jaap van |
| Edizione | [1st ed.] |
| Pubbl/distr/stampa | Oxford, : Elsevier, 2008 |
| Descrizione fisica | 1 online resource (327 p.) |
| Disciplina |
511.3
612.843 |
| Collana | Studies in logic and the foundations of mathematics |
| Soggetto topico |
Logic, Symbolic and mathematical
Model theory |
| ISBN |
1-281-16508-5
9786611165086 0-08-056006-7 1-4356-2874-8 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Front Cover; Realizability: An Introduction to its Categorical Side; Copyright Page; Preface; Introduction; Table of Contents; Chapter 1 Partial Combinatory Algebras; 1.1 Basic definitions; 1.1.1 Pairing, Booleans and Definition by Cases; 1.2 P(A)-valued predicates; 1.3 Further properties; recursion theory; 1.3.1 Recursion theory in pcas; 1.4 Examples of pcas; 1.4.1 Kleene's first model; 1.4.2 Relativized recursion; 1.4.3 Kleene's second model; 1.4.4 K2 generalized; 1.4.5 Sequential computations; 1.4.6 The graph model P(ω); 1.4.7 Graph models; 1.4.8 Domain models; 1.4.9 Relativized models
1.4.10 Term models1.4.11 Pitts' construction; 1.4.12 Models of Arithmetic; 1.5 Morphisms and Assemblies; 1.6 Applicative morphisms and S-functors; 1.7 Decidable applicative morphisms; 1.8 Order-pcas; Chapter 2 Realizability triposes and toposes; 2.1 Triposes; 2.1.1 Preorder-enriched categories; 2.1.2 Triposes: definition and basic properties; 2.1.3 Interpretation of languages in triposes; 2.1.4 A few useful facts; 2.2 The tripos-to-topos construction; 2.3 Internal logic of C[P] reduced to the logic of P; 2.4 The 'constant objects' functor; 2.5 Geometric morphisms Chapter 3 The Effective Topos3.1 Recapitulation and arithmetic in εff; 3.1.1 Second-order arithmetic in εff; 3.1.2 Third-order arithmetic in εff; 3.2 Some special objects and arrows in εff; 3.2.1 Closed and dense subobjects; 3.2.2 Infinite coproducts and products; 3.2.3 Projective and internally projective objects, and choice principles; 3.2.4 εff as a universal construction; 3.2.5 Real numbers in εff; 3.2.6 Discrete and modest objects; 3.2.7 Decidable and semidecidable subobjects; 3.3 Some analysis in εff; 3.3.1 General facts about R; 3.3.2 Specker sequences and singular coverings 3.3.3 Real-valued functions3.4 Discrete families and Uniform maps; 3.4.1 Weakly complete internal categories in εff; 3.5 Set Theory in εff; 3.5.1 The McCarty model for IZF; 3.5.2 The Lubarsky-Streicher-Van den Berg model for CZF; 3.5.3 Well-founded trees and W-Types in εff; 3.6 Synthetic Domain Theory in εff; 3.6.1 Complete partial orders; 3.6.2 The synthetic approach; 3.6.3 Elements of Synthetic Domain Theory; 3.6.4 Models for SDT in εff; 3.7 Synthetic Computability Theory in εff; 3.8 General Comments about the Effective Topos; 3.8.1 Analogy between ▿ and the Yoneda embedding 3.8.2 Small dense subcategories in εff |
| Record Nr. | UNINA-9910813535403321 |
|
Oosten Jaap van
|
||
| Oxford, : Elsevier, 2008 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Realizability : an introduction to its categoricl side / Jaap van Oosten
| Realizability : an introduction to its categoricl side / Jaap van Oosten |
| Autore | Oosten, Jaap : van |
| Pubbl/distr/stampa | Amsterdam : Elsevier, c2008 |
| Descrizione fisica | XVI, 310 p. ; 24 cm |
| Disciplina | 511.33 |
| Collana | Studies in logic and the foundations of mathematics |
| Soggetto non controllato |
Logica categoriale
Logica algebrica |
| ISBN | 978-0-444-51584-1 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNINA-990008820350403321 |
|
Oosten, Jaap : van
|
||
| Amsterdam : Elsevier, c2008 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Sets, models and proofs / Ieke Moerdijk, Jaap van Oosten
| Sets, models and proofs / Ieke Moerdijk, Jaap van Oosten |
| Autore | Moerdijk, Ieke |
| Pubbl/distr/stampa | Cham, : Springer, 2018 |
| Descrizione fisica | xiv, 141 p. : ill. ; 24 cm |
| Altri autori (Persone) | Oosten, Jaap van |
| Soggetto topico |
03-XX - Mathematical logic and foundations [MSC 2020]
03Exx - Set theory [MSC 2020] 03C07 - Basic properties of first-order languages and structures [MSC 2020] 03B10 - Classical first-order logic [MSC 2020] |
| Soggetto non controllato |
Axiom of choice
Completeness theorem First-order logic Mathematical logic Model theory Proof tree Quantifier elimination Set Theory |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Titolo uniforme | |
| Record Nr. | UNICAMPANIA-VAN0124991 |
|
Moerdijk, Ieke
|
||
| Cham, : Springer, 2018 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Vanvitelli | ||
| ||
Sets, models and proofs / Ieke Moerdijk, Jaap van Oosten
| Sets, models and proofs / Ieke Moerdijk, Jaap van Oosten |
| Autore | Moerdijk, Ieke |
| Edizione | [Cham : Springer, 2018] |
| Pubbl/distr/stampa | xiv, 141 p., : ill. ; 24 cm |
| Descrizione fisica | Pubblicazione in formato elettronico |
| Altri autori (Persone) | Oosten, Jaap van |
| Soggetto topico |
03-XX - Mathematical logic and foundations [MSC 2020]
03Exx - Set theory [MSC 2020] 03C07 - Basic properties of first-order languages and structures [MSC 2020] 03B10 - Classical first-order logic [MSC 2020] |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNICAMPANIA-SUN0124991 |
|
Moerdijk, Ieke
|
||
| xiv, 141 p., : ill. ; 24 cm | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Vanvitelli | ||
| ||