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Duality and Definability in First Order Logic
Michael Makkai
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Memoirs of the American Mathematical Society
1993; 106 pp; softcover
Volume: 105
ISBN-10: 0-8218-2565-8
ISBN-13: 978-0-8218-2565-5
List Price: US$36
Individual Members: US$21.60
Institutional Members: US$28.80
Order Code: MEMO/105/503
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Using the theory of categories as a framework, this book develops a duality theory for theories in first order logic in which the dual of a theory is the category of its models with suitable additional structure. This duality theory resembles and generalizes M. H. Stone's famous duality theory for Boolean algebras. As an application, Makkai derives a result akin to the well-known definability theorem of E. W. Beth. This new definability theorem is related to theorems of descent in category theory and algebra and can also be stated as a result in pure logic without reference to category theory. Containing novel techniques as well as applications of classical methods, this carefully written book shows attention to both organization and detail and will appeal to mathematicians and philosophers interested in category theory.

Readership

Mathematicians and philosophers interested in category theory and mathematical logic.

Table of Contents

  • Beth's theorem in propositional logic
  • Factorizations in \(2\)-categories
  • Definable functors
  • Basic notions for duality
  • The Stone-type adjunction for Boolean pretoposes and ultragroupoids
  • The syntax of special ultramorphisms
  • The semantics of special ultramorphisms
  • The duality theorem
  • Preparing a functor specification
  • Lifting Zawadowski's argument to ultra\(^\ast\) morphisms
  • The operations on \({\mathcal B}{\mathcal P}^\ast\) and UG
  • Conclusion
  • References
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