From Sets and Types to Topology and Analysis Towards Practicable Foundations for Constructive Mathematics,9780198566519

From Sets and Types to Topology and Analysis Towards Practicable Foundations for Constructive Mathematics

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ISBN13: 9780198566519
ISBN10: 0198566514
Format: Hardcover
Pub. Date: 2005-12-08
Publisher(s): Clarendon Press
List Price: $261.33

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Summary

This edited collection bridges the foundations and practice of constructive mathematics and focusses on the contrast between the theoretical developments, which have been most useful for computer science (eg constructive set and type theories), and more specific efforts on constructiveanalysis, algebra and topology. Aimed at academic logicians, mathematicians, philosophers and computer scientists Including, with contributions from leading researchers, it is up-to-date, highly topical and broad in scope. This is the latest volume in the Oxford Logic Guides, which also includes: 41. J.M. Dunn and G. Hardegree: Algebraic Methods in Philosophical Logic 42. H. Rott: Change, Choice and Inference: A study of belief revision and nonmonotoic reasoning 43. Johnstone: Sketches of an Elephant: A topos theory compendium, volume 1 44. Johnstone: Sketches of an Elephant: A topos theory compendium, volume 2 45. David J. Pym and Eike Ritter: Reductive Logic and Proof Search: Proof theory, semantics and control 46. D.M. Gabbay and L. Maksimova: Interpolation and Definability: Modal and Intuitionistic Logics 47. John L. Bell: Set Theory: Boolean-valued models and independence proofs, third edition

Table of Contents

List of Contributors
xviii
Introduction
1(20)
Laura Crosilla
Peter Schuster
Foundations for Constructive Mathematics
1(3)
Bridging Foundations and Practice
4(1)
Different but Related Foundations
5(1)
Practicable and Weak Foundations
6(1)
Martin-Lof Type Theory
7(2)
Constructive Set Theory
9(2)
Bishop-Style Constructive Mathematics
11(3)
Formal Topology
14(7)
References
17(4)
Part I Foundations
21(142)
Generalized Inductive Definitions in Constructive Set Theory
23(18)
Michael Rathjen
Introduction
23(1)
Inductive Definitions in CZF
24(5)
The system CZF
24(1)
Inductively defined classes in CZF
25(1)
Inductively defined sets in CZF+REA
26(2)
General inductive definitions
28(1)
Lower Bounds
29(7)
The μ-calculus
29(3)
Fragments of second order arithmetic
32(1)
A first lower bound
33(2)
Better lower bounds
35(1)
An Upper Bound
36(5)
Acknowledgements
39(1)
References
39(2)
Constructive Set Theories and their Category-Theoretic Models
41(21)
Alex Simpson
Introduction
41(3)
Constructive set theories
41(1)
Category-theoretic models
42(2)
Sets in Constructive Mathematics
44(2)
Axioms for Constructive Set Theory
46(4)
Infinity and Induction
50(2)
Constructive Set Theories
52(1)
Categories of Classes
53(4)
Categories of Sets
57(5)
Acknowledgements
59(1)
References
59(3)
Presheaf Models for Constructive Set Theories
62(16)
Nicola Gambino
Variable Sets in Foundations and Practice
62(2)
Classes and Sets
64(7)
Set-theoretic axioms
64(1)
Categories of classes
65(2)
Axioms for small maps
67(4)
Presheaves
71(3)
Basic definitions
71(1)
Presheaf models
72(2)
Kripke-Joyal Semantics
74(1)
Conclusions
75(3)
Acknowledgements
76(1)
References
76(2)
Universes in Toposes
78(13)
Thomas Streicher
Background and Motivation
78(2)
Universes in Toposes
80(5)
Hierarchies of universes
84(1)
Existence of Universes in Toposes
85(2)
Further Properties and Generalizations
87(2)
Conclusions and Open Questions
89(2)
Acknowledgements
90(1)
References
90(1)
Toward a Minimalist Foundation for Constructive Mathematics
91(24)
Maria Emilia Maietti
Giovanni Sambin
Arguments for a Minimal Constructive Type Theory
93(7)
The computational view
93(3)
Intrinsic reasons: intensionality vs. extensionality
96(1)
Extrinsic reasons: compatibility
97(2)
Conceptual reasons: against reductionism
99(1)
A Proposal for a Minimal Type Theory
100(8)
Logic explained via the principle of reflection
100(2)
A formal system for minimal type theory
102(6)
Some Visible Benefits
108(1)
Open Questions and Further Work
108(3)
Acknowledgements
109(1)
References
109(2)
Appendix: The System mTT
111(4)
Interactive Programs and Weakly Final Coalgebras in Dependent Type Theory
115(22)
Peter Hancock
Anton Setzer
Introduction
115(4)
Other approaches to interactive programs in dependent type theory
116(1)
Related work
117(1)
Notations and type theory used
117(2)
Non-Dependent Interactive Programs
119(4)
The set IO
120(1)
Introduction of elements of IO
121(1)
Bisimilarity
121(1)
Equalities and weakly final coalgebras
122(1)
Dependent Interactive Programs
123(2)
Dependent interactive programs
123(1)
Programs for dependent interfaces
123(1)
Weakly final coalgebras on S → Set
124(1)
Bisimilarity
124(1)
Server-Side Programs and Generalization
125(3)
Server-side programs
125(1)
Generalisation: polynomial functors on families of sets
125(1)
Equivalents of polynomial functors
126(1)
The natural numbers, conatural numbers, iteration, and coiteration
127(1)
Coiteration in Dependent Type Theory
128(2)
Non-dependent version
129(1)
Inductive data types vs. coalgebras
130(1)
Guarded Induction
130(4)
Coiter and guarded induction
130(3)
Bisimilarity as a state-dependent coalgebra
133(1)
Normalization
133(1)
Conclusion
134(3)
Acknowledgements
134(1)
References
134(3)
Applications of Inductive Definitions and Choice Principles to Program Synthesis
137(12)
Ulrich Berger
Monika Seisenberger
Introduction
137(2)
Inductive Definitions
139(3)
Formalization and program extraction
141(1)
Classical Dependent Choice
142(3)
Proof-theoretic and computational optimizations
144(1)
Conclusion
145(4)
References
146(3)
The Duality of Classical and Constructive Notions and Proofs
149(14)
Sara Negri
Jan von Plato
Introduction
149(1)
From Mathematical Axioms to Mathematical Rules
150(3)
Derivations in Left and Right Rule Systems
153(1)
Geometric and Cogeometric Axioms and Rules
154(6)
Duality of Dependent Types and Degenerate Cases
160(3)
References
161(2)
Part II Practice
163(172)
Continuity on the Real Line and in Formal Spaces
165(11)
Erik Palmgren
Introduction
165(1)
Point-Free Topology
166(1)
Continuous Mappings
166(2)
Functions on Real Numbers
168(5)
Open Subspaces and the Reciprocal Map
173(3)
Acknowledgements
174(1)
References
174(2)
Separation Properties in Constructive Topology
176(17)
Peter Aczel
Christopher Fox
Introduction
176(1)
Constructive Topological Spaces (ct-spaces)
177(2)
The Space of Formal Points of a Formal Topology
179(1)
Local collections
179(1)
The notion of a formal topology
179(1)
The ct-space of formal points
180(1)
Constructive Separation Properties
180(5)
The separation properties for i = 0, 1, 2
180(1)
Notions of regular space
181(1)
Some implications
182(1)
Summary
182(1)
Some counterexamples
183(2)
Separation Relative to an Inequality
185(2)
Ru-Metric Spaces
187(2)
Regular Formal Topologies
189(1)
Sober Spaces
189(4)
References
192(1)
Spaces as comonoids
193(9)
A. Bucalo
G. Rosolini
Introduction
193(1)
A Categorical View of Basic Pairs
193(4)
The Characterization
197(2)
Comparison with the Presentations of Formal Topologies
199(3)
Acknowledgements
200(1)
References
200(2)
Predicative Exponentiation of Locally Compact Formal Topologies over Inductively Generated Topologies
202(21)
Maria Emilia Maietti
Introduction
202(1)
Preliminaries on Formal Covers
203(5)
Join formal covers
207(1)
Locally Compact Formal Covers
208(2)
Exponentiation
210(10)
The exponential formal cover
211(3)
Application and abstraction
214(6)
Any Exponentiable Formal Cover is Locally Compact
220(3)
Acknowledgements
221(1)
References
221(2)
Some Constructive Roads to Tychonoff
223(16)
Steven Vickers
Introduction
223(6)
Point-free topology
224(1)
Locales
224(2)
Formal topologies
226(3)
Compactness
229(3)
Preframes
229(3)
Tychonoff
232(4)
Synthetic Locale Theory
236(3)
References
237(2)
An Elementary Characterisation of Krull Dimension
239(6)
Thierry Coquand Henri Lombardi
Marie-Francoise Roy
Introduction
239(1)
Boundaries of an Element in a Distributive Lattice
239(2)
Krull Dimension of a Distributive Lattice
241(1)
The Two Boundaries of an Element in a Commutative Ring
242(3)
Acknowledgements
244(1)
References
244(1)
Constructive Reverse Mathematics: Compactness Properties
245(23)
Hajime Ishihara
Introduction
245(2)
A Formal System
247(4)
CSM-Spaces and CTB-Spaces
251(3)
The Heine-Borel Theorem
254(2)
The Cantor Intersection Theorem
256(2)
The Bolzano--Weierstraß Theorem
258(3)
Sequential Compactness
261(7)
Acknowledgements
265(1)
References
266(2)
Approximating Integrable Sets by Compacts Constructively
268(12)
Bas Spitters
Introduction
268(1)
Preliminaries
269(2)
Ishihara's trick
270(1)
Integration Theory
271(2)
Regular Measures and Ulam's Theorem
273(3)
Intuitionistic Theorems
276(2)
Conclusions
278(2)
Acknowledgements
278(1)
References
278(2)
An Introduction to the Theory of C*-Algebras in Constructive Mathematics
280(13)
Hiroki Takamura
Introduction
280(1)
Constructive C*-Algebras
281(1)
Positive Elements
281(4)
Positive Linear Functionals and States
285(2)
Representations
287(2)
The GNS Construction Theorem
289(4)
Acknowledgements
291(1)
References
292(1)
Approximations to the Numerical Range of an Element of a Banach Algebra
293(11)
Douglas Bridges
Robin Havea
Introduction
293(2)
Approximating the Numerical Range
295(4)
Sinclair's Theorem
299(5)
Acknowledgements
302(1)
References
302(2)
The Constructive Uniqueness of the Locally Convex Topology on Rn
304(12)
Douglas Bridges
Luminita Vita
Introduction
304(1)
Some Convex Geometry
305(7)
The Locally Convex Topology is Unique
312(4)
Acknowledgements
314(1)
References
314(2)
Computability on Non-Separable Banach Spaces and Landau's Theorem
316(19)
Vasco Brattka
Introduction
316(1)
Preliminaries from Computable Analysis
317(2)
Computable Metric and Normed Spaces
319(2)
General Computable Normed Spaces
321(5)
Topology on General Computable Spaces
326(2)
Closure Properties of General Computable Spaces
328(2)
A Computable Version of Landau's Theorem
330(1)
Conclusion
331(4)
Acknowledgements
332(1)
References
332(3)
Index 335

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