Constructivism in Mathematics, Vol 2

Klappentext 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. Inhaltsverzeichnis Constructivism in Mathematics: Contents Preliminaries 7. The Topology of Metric Spaces 1. Basic Definitions 2. Complete, Separable Metric Spaces 3. Located Sets 4. Complete, Totally Bounded Spaces 5. Locally Compact Spaces 6. Notes Exercises 8. Algebra 1. Identity, Apartness and Order 2. Groups 3. Rings and Modules 4. Linear Algebra 5. Polynomial Rings 6. Fields and Local Rings 7. The Fundamental Theorem of Algebra 8. Notes Exercises 9. Finite-Type Arithmetic and Theories of Operators 1. Intuitionistic Finite-Type Arithmetic 2. Normalization, and a Term Model For HA? 3. The Theory APP 4. Models for APP 5. Abstract Realizability in APP 6. Extensionality and Choice in APP and HA? 7. Some Metamathematical Applications 8. Theories of Operators and Classes 9. Notes Exercises 10. Proof Theory of Intuitionistic Logic 1. Preliminaries 2. Normalization 3. The Structure of Normal Derivations of N-IQCE 4. The Decidability of IPC 5. Other Applications of Normalization 6. Conservative Addition of Predicative Classes 7. Sequent Calculi 8. N-IQC as a Calculus of Terms 9. Notes Exercises 11. The Theory of Types and Constructive Set Theory 1. Towards a Theory of Types 2. The Theory MLi0 3. Some Alternative Formulations of MLi0 4. The Types Nk and Reformulation of the E-Rules 5. The Theory ML0 6. Embeddings into APP 7. Extensions of MLi0 and ML0 8. Constructive Set Theory 9. Notes Exercises 12. Choice Sequences 1. Introduction 2. Lawless Sequences 3. The Elimination Translation for the Theory LS 4. Other Notions of Choice Sequence 5. Notes Exercises 13. Semantical Completeness 1. Beth Models 2. Completeness for Intuitionistic Validity 3. Incompleteness Results 4. Lattices, Heyting Algebras and Complete Heyting Algebras 5. Algebraic Semantics for IPC 6. O-Sets and Structures 7. Validity as Forcing 8. Postscript on Realizability 9. Notes Exercises 14. Sheaves, Sites and Higher-Order Logic 1. Presheaves, Sheaves and Sheaf-Completion 2. O-Presheaf and O-Sheaf Structures 3. Some Notions from Category Theory 4. Forcing Over Sites 5. Sheaf Models for Higher-Order Logic 6. Notes Exercises 15. Applications of Sheaf Models 1. Interpretation of N, Q, Z, R,N in Sh(O(T)) 2. The Axiom of Countable Choice 3. Topologies in Sheaves Over a cHa 4. A Derived Rule of Local Continuity 5. The Monoid Model for CS 6. A Site Model for LS 7. Notes Exercises 16. Epilogue 1. The Role of Language and "Informal Rigour" 2. Intuitionistic Logic, For...

Inhalt
Constructivism in Mathematics: Contents Preliminaries
7. The Topology of Metric Spaces

1. Basic Definitions

2. Complete, Separable Metric Spaces

3. Located Sets

4. Complete, Totally Bounded Spaces

5. Locally Compact Spaces

6. Notes

Exercises

8. Algebra

1. Identity, Apartness and Order

2. Groups

3. Rings and Modules

4. Linear Algebra

5. Polynomial Rings

6. Fields and Local Rings

7. The Fundamental Theorem of Algebra

8. Notes

Exercises

9. Finite-Type Arithmetic and Theories of Operators

1. Intuitionistic Finite-Type Arithmetic

2. Normalization, and a Term Model For HA?

3. The Theory APP

4. Models for APP

5. Abstract Realizability in APP

6. Extensionality and Choice in APP and HA?

7. Some Metamathematical Applications

8. Theories of Operators and Classes

9. Notes

Exercises

10. Proof Theory of Intuitionistic Logic

1. Preliminaries

2. Normalization

3. The Structure of Normal Derivations of N-IQCE

4. The Decidability of IPC

5. Other Applications of Normalization

6. Conservative Addition of Predicative Classes

7. Sequent Calculi

8. N-IQC as a Calculus of Terms

9. Notes

Exercises

11. The Theory of Types and Constructive Set Theory

1. Towards a Theory of Types

2. The Theory MLi0

3. Some Alternative Formulations of MLi0

4. The Types Nk and Reformulation of the E-Rules

5. The Theory ML0

6. Embeddings into APP

7. Extensions of MLi0 and ML0

8. Constructive Set Theory

9. Notes

Exercises

12. Choice Sequences

1. Introduction

2. Lawless Sequences

3. The Elimination Translation for the Theory LS

4. Other Notions of Choice Sequence

5. Notes

Exercises

13. Semantical Completeness

1. Beth Models

2. Completeness for Intuitionistic Validity

3. Incompleteness Results

4. Lattices, Heyting Algebras and Complete Heyting Algebras

5. Algebraic Semantics for IPC

6. O-Sets and Structures

7. Validity as Forcing

8. Postscript on Realizability

9. Notes

Exercises

14. Sheaves, Sites and Higher-Order Logic

1. Presheaves, Sheaves and Sheaf-Completion

2. O-Presheaf and O-Sheaf Structures

3. Some Notions from Category Theory

4. Forcing Over Sites

5. Sheaf Models for Higher-Order Logic

6. Notes

Exercises

15. Applications of Sheaf Models

1. Interpretation of N, Q, Z, R,N in Sh(O(T))

2. The Axiom of Countable Choice

3. Topologies in Sheaves Over a cHa

4. A Derived Rule of Local Continuity

5. The Monoid Model for CS

6. A Site Model for LS

7. Notes

Exercises

16. Epilogue

1. The Role of Language and "Informal Rigour"

2. Intuitionistic Logic, Formalisms, and Equality

3. Brouwer's Theory of the Creative Subject

4. Dummett's Anti-Realist Argument

Bibliography

Index

Index of Names

List of Symbols

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