Duality and Definability in First Order LogicUsing 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. |
Contents
1 Beths theorem for propositional logic | 1 |
2 Factorizations in 2categories | 8 |
3 Definable functors | 19 |
4 Basic notions for duality | 25 |
5 The Stonetype adjunction for Boolean pretoposes and ultragroupoids | 35 |
6 The syntax of special ultramorphisms | 41 |
7 The semantics of special ultramorphisms | 55 |
8 The duality theorem | 64 |
9 Preparing a functor specification | 72 |
10 Lifting Zawadowskis argument to ultramorphisms | 84 |
11 The operations in BP and UG | 91 |
12 Conclusion | 96 |
| 105 | |
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Common terms and phrases
2-arrows 2-regular arrows Aut(M automorphism B-valued Beth definability theorem bicategory bijection bilimits Boolean algebras Boolean pretoposes c-complex category theory category whose objects co-quotient codomain colimits commutes composition concrete 2-category construction coregular corresponding definable functor definition denoted descent theorem diagram duality theory e-mail E-relations elementarily equivalent equivalence of categories fact finite follows forgetful functor formula full and faithful given graph groupoid Hom(K Hom(T inverse covering isomorphism of categories L-AUT f L-formula lemma limit ultrapower loc.cit maps Mathematics Mod T/L models natural transformations node notation notion obtained order logic pair paper point of matching pretopos functor proof propositional logic quotient morphism regular factorization regular set relation-conservative Section special ultramorphisms Stone adjunction strict structure subobjects surjective T/L-IFS transition isomorphisms tuple ultra morphisms ultrafilter ultragroupoid ultralimits ultraproduct underlying universal property Zawadowski setup