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Duality and Definability in First Order Logic

Available Formats:
Electronic ISBN: 978-1-4704-0080-4
Product Code: MEMO/105/503.E
List Price: $38.00 MAA Member Price:$34.20
AMS Member Price: $22.80 Click above image for expanded view Duality and Definability in First Order Logic Available Formats:  Electronic ISBN: 978-1-4704-0080-4 Product Code: MEMO/105/503.E  List Price:$38.00 MAA Member Price: $34.20 AMS Member Price:$22.80
• Book Details

Memoirs of the American Mathematical Society
Volume: 1051993; 106 pp
MSC: Primary 03; 18;

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.

Mathematicians and philosophers interested in category theory and mathematical logic.

• Chapters
• 1. Beth’s theorem for propositional logic
• 2. Factorizations in 2-categories
• 3. Definable functors
• 4. Basic notions for duality
• 5. The Stone-type adjunction for Boolean pretoposes and ultragroupoids
• 6. The syntax of special ultramorphisms
• 7. The semantics of special ultramorphisms
• 8. The duality theorem
• 9. Preparing a functor specification
• 10. Lifting Zawadowski’s argument to ultra*morphisms
• 11. The operations in $\mathcal {BP}^*$ and $\mathtt {UG}$
• 12. Conclusion
• Requests

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Volume: 1051993; 106 pp
MSC: Primary 03; 18;

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.

Mathematicians and philosophers interested in category theory and mathematical logic.

• Chapters
• 1. Beth’s theorem for propositional logic
• 2. Factorizations in 2-categories
• 3. Definable functors
• 4. Basic notions for duality
• 5. The Stone-type adjunction for Boolean pretoposes and ultragroupoids
• 6. The syntax of special ultramorphisms
• 7. The semantics of special ultramorphisms
• 8. The duality theorem
• 9. Preparing a functor specification
• 10. Lifting Zawadowski’s argument to ultra*morphisms
• 11. The operations in $\mathcal {BP}^*$ and $\mathtt {UG}$
• 12. Conclusion
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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