**Memoirs of the American Mathematical Society**

1993;
106 pp;
Softcover

MSC: Primary 03; 18;

Print ISBN: 978-0-8218-2565-5

Product Code: MEMO/105/503

List Price: $38.00

Individual Member Price: $22.80

**Electronic ISBN: 978-1-4704-0080-4
Product Code: MEMO/105/503.E**

List Price: $38.00

Individual Member Price: $22.80

# Duality and Definability in First Order Logic

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*Michael Makkai*

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

# Table of Contents

## Duality and Definability in First Order Logic

- Table Of Contents v6 free
- Abstract vi7 free
- Introduction vii8 free
- 1. Beth's theorem for propositional logic 112 free
- 2. Factorizations in 2-categories 819
- 3. Definable functors 1930
- 4. Basic notions for duality 2536
- 5. The Stone-type adjunction for Boolean pretoposes and ultragroupoids 3546
- 6. The syntax of special ultramorphisms 4152
- 7. The semantics of special ultramorphisms 5566
- 8. The duality theorem 6475
- 9. Preparing a functor specification 7283
- 10. Lifting Zawadowski's argument to ultra*morphisms 8495
- 11. The operations in BP* and UG 91102
- 12. Conclusion 96107
- References 105116