Duality And Definability In First Order Logic
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Duality and Definability in First Order Logic
Author: Michael Makkai
language: en
Publisher: American Mathematical Soc.
Release Date: 1993
We develop a duality theory for small Boolean pretoposes in which the dual of the [italic capital]T is the groupoid of models of a Boolean pretopos [italic capital]T equipped with additional structure derived from ultraproducts. The duality theorem states that any small Boolean pretopos is canonically equivalent to its double dual. We use a strong version of the duality theorem to prove the so-called descent theorem for Boolean pretoposes which says that category of descent data derived from a conservative pretopos morphism between Boolean pretoposes is canonically equivalent to the domain-pretopos. The descent theorem contains the Beth definability theorem for classical first order logic. Moreover, it gives, via the standard translation from the language of categories to symbolic logic, a new definability theorem for classical first order logic concerning set-valued functors on models, expressible in purely syntactical (arithmetical) terms.
Duality and Definability in First Order Logic
Author: Mihály Makkai
language: en
Publisher: Oxford University Press, USA
Release Date: 2014-08-31
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, the author 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 carefuly written book shows an attention to both organization and detail and will appeal to mathematicians and philosophers interested in category theory.
Topological Duality for Distributive Lattices
Author: Mai Gehrke
language: en
Publisher: Cambridge University Press
Release Date: 2024-03-07
Introduces lattice-theoretic and topological methods in logic and computer science, with applications in domain theory and automata theory.