Constructivism in Mathematics, Vol 2

Inbunden, Engelska, 1988

Del 123 i serien Studies in Logic and the Foundations of Mathematics

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Beskrivning

Studies in Logic and the Foundations of Mathematics, Volume 123: Constructivism in Mathematics: An Introduction, Vol. II focuses on various studies in mathematics and logic, including metric spaces, polynomial rings, and Heyting algebras.

The publication first takes a look at the topology of metric spaces, algebra, and finite-type arithmetic and theories of operators. Discussions focus on intuitionistic finite-type arithmetic, theories of operators and classes, rings and modules, linear algebra, polynomial rings, fields and local rings, complete separable metric spaces, and located sets. The text then examines proof theory of intuitionistic logic, theory of types and constructive set theory, and choice sequences.

The book elaborates on semantical completeness, sheaves, sites, and higher-order logic, and applications of sheaf models. Topics include a derived rule of local continuity, axiom of countable choice, forcing over sites, sheaf models for higher-order logic, and complete Heyting algebras.

The publication is a valuable reference for mathematicians and researchers interested in mathematics and logic.

Produktinformation

Utgivningsdatum:1988-11-01
Mått:156 x 234 x 37 mm
Vikt:1 140 g
Format:Inbunden
Språk:Engelska
Serie:Studies in Logic and the Foundations of Mathematics
Antal sidor:140
Förlag:Elsevier Science
ISBN:9780444703583

Utforska kategorier

Matematikens grunder inom Naturvetenskap och teknik

Innehållsförteckning

Constructivism in Mathematics: Contents Preliminaries7. The Topology of Metric Spaces1. Basic Definitions2. Complete, Separable Metric Spaces3. Located Sets4. Complete, Totally Bounded Spaces5. Locally Compact Spaces6. NotesExercises8. Algebra1. Identity, Apartness and Order2. Groups3. Rings and Modules4. Linear Algebra5. Polynomial Rings6. Fields and Local Rings7. The Fundamental Theorem of Algebra8. NotesExercises9. Finite-Type Arithmetic and Theories of Operators1. Intuitionistic Finite-Type Arithmetic2. Normalization, and a Term Model For HAω3. The Theory APP4. Models for APP5. Abstract Realizability in APP6. Extensionality and Choice in APP and HAω7. Some Metamathematical Applications8. Theories of Operators and Classes9. NotesExercises10. Proof Theory of Intuitionistic Logic1. Preliminaries2. Normalization3. The Structure of Normal Derivations of N-IQCE4. The Decidability of IPC5. Other Applications of Normalization6. Conservative Addition of Predicative Classes7. Sequent Calculi8. N-IQC as a Calculus of Terms9. NotesExercises11. The Theory of Types and Constructive Set Theory1. Towards a Theory of Types2. The Theory MLi03. Some Alternative Formulations of MLi04. The Types Nk and Reformulation of the E-Rules5. The Theory ML06. Embeddings into APP7. Extensions of MLi0 and ML08. Constructive Set Theory9. NotesExercises12. Choice Sequences1. Introduction2. Lawless Sequences3. The Elimination Translation for the Theory LS4. Other Notions of Choice Sequence5. NotesExercises13. Semantical Completeness1. Beth Models2. Completeness for Intuitionistic Validity3. Incompleteness Results4. Lattices, Heyting Algebras and Complete Heyting Algebras5. Algebraic Semantics for IPC6. Ω-Sets and Structures7. Validity as Forcing8. Postscript on Realizability9. NotesExercises14. Sheaves, Sites and Higher-Order Logic1. Presheaves, Sheaves and Sheaf-Completion2. Ω-Presheaf and Ω-Sheaf Structures3. Some Notions from Category Theory4. Forcing Over Sites5. Sheaf Models for Higher-Order Logic6. NotesExercises15. Applications of Sheaf Models1. Interpretation of N, Q, Z, R,N in Sh(O(T))2. The Axiom of Countable Choice3. Topologies in Sheaves Over a cHa4. A Derived Rule of Local Continuity5. The Monoid Model for CS6. A Site Model for LS7. NotesExercises16. Epilogue1. The Role of Language and "Informal Rigour"2. Intuitionistic Logic, Formalisms, and Equality3. Brouwer's Theory of the Creative Subject4. Dummett's Anti-Realist ArgumentBibliographyIndexIndex of NamesList of Symbols