Quantifier Elimination and Definability: Theory and Examples

Diego Alberto Barceló Nieves - PhD Student in Mathematics

Abstract:
Model Theory studies mathematical structures by considering which first-order sentences are true in them and which of their subsets are definable by first-order formulas. In this talk, we will introduce the general model-theoretic technique of Quantifier Elimination (QE), present the example of QE for the theory of algebraically closed fields—for which we will give a geometrical interpretation—and show a counterexample for the general model theory of modules, in which case we will see that a certain "approximation" of QE is always possible (Baur-Monk Theorem). As the main application, throughout we will note how a theory having QE facilitates the description of the definable subsets of its models.