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Introduction to Higher-Order Categorical Logic

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ISBN-10: 0521356539

ISBN-13: 9780521356534

Edition: N/A

Authors: J. Lambek, P. J. Scott, B. Bollobas, W. Fulton, A. Katok

List price: $69.99
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In this volume, Lambek and Scott reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory. In Part I, they show that typed lambda-calculi, a formulation of higher-order logic, and cartesian closed categories, are essentially the same. Part II demonstrates that another formulation of higher-order logic, (intuitionistic) type theories, is closely related to topos theory. Part III is devoted to recursive functions. Numerous applications of the close relationship between traditional logic and the algebraic language of category theory are given. The authors have included an introduction to category theory and develop the necessary logic…    
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Book details

List price: $69.99
Publisher: Cambridge University Press
Publication date: 3/25/1988
Binding: Paperback
Pages: 304
Size: 6.25" wide x 9.00" long x 0.75" tall
Weight: 0.792
Language: English

Introduction to category theory
Introduction to Part 0
Categories and functors
Natural transformations
Adjoint functors
Equivalence of categories
Limits in categories
Examples of cartesian closed categories
Cartesian closed categories and [lambda]-calculus
Introduction to Part I
Historical perspective on Part I
Propositional calculus as a deductive system
The deduction theorem
Cartesian closed categories equationally presented
Free cartesian closed categories generated by graphs
Polynomial categories
Functional completeness of cartesian closed categories
Polynomials and Kleisli categories
Cartesian closed categories with coproducts
Natural numbers objects in cartesian closed categories
Typed [lambda]-calculi
The cartesian closed category generated by a typed [lambda]-calculus
The decision problem for equality
The Church-Rosser theorem for bounded terms
All terms are bounded
C-monoids and cartesian closed categories
C-monoids and untyped [lambda]-calculus
A construction
Historical comments on Part I
Type theory and toposes
Introduction to Part II
Historical perspective on Part II
Intuitionistic type theory
Type theory based on equality
The internal language of a topos
Peano's rules in a topos
The internal language at work
The internal language at work II
Choice and the Boolean axiom
Topos semantics
Topos semantics in functor categories
Sheaf categories and their semantics
Three categories associated with a type theory
The topos generated by a type theory
The topos generated by the internal language
The internal language of the topos generated
Toposes with canonical subobjects
Applications of the adjoint functors between toposes and type theories
Completeness of higher order logic with choice rule
Sheaf representation of toposes
Completeness without assuming the rule of choice
Some basic intuitionistic principles
Further intuitionistic principles
The Freyd cover of a topos
Historical comments on Part II
Supplement to Section 17
Representing numerical functions in various categories
Introduction to Part III
Recursive functions
Representing numerical functions in cartesian closed categories
Representing numerical functions in toposes
Representing numerical functions in C-monoids
Historical comments on Part III
Author index
Subject index