Title:

On Galois correspondences in formal logic

This thesis examines two approaches to Galois correspondences in formal logic. A standard result of classical firstorder model theory is the observation that models of Ltheories with a weak form of elimination of imaginaries hold a correspondence between their substructures and automorphism groups defined on them. This work applies the resultant framework to explore the practical consequences of a modeltheoretic Galois theory with respect to certain firstorder Ltheories. The framework is also used to motivate an examination of its underlying modeltheoretic foundations. The modeltheoretic Galois theory of pure fields and valued fields is compared to the algebraic Galois theory of pure and valued fields to point out differences that may hold between them. The framework of this logical Galois correspondence is also applied to the theory of pseudoexponentiation to obtain a sketch of the Galois theory of exponential fields, where the fixed substructure of the complex pseudoexponential field B is an exponential field with the field Qrab as its algebraic subfield. This work obtains a partial exponential analogue to the KroneckerWeber theorem by describing the pure fieldtheoretic abelian extensions of Qrab, expanding upon work in the twelfth of Hilbertâ€™s problems. This result is then used to determine some of the modeltheoretic abelian extensions of the fixed substructure of B. This work also incorporates the principles required of this modeltheoretic framework in order to develop a model theory over substructural logics which is capable of expressing this Galois correspondence. A formal semantics is developed for quantified predicate substructural logics based on algebraic models for their propositional or nonquantified fragments. This semantics is then used to develop substructural forms of standard results in classical firstorder model theory. This work then uses this substructural model theory to demonstrate the Galois correspondence that substructural firstorder theories can carry in certain situations.
