This is the title of a talk given at the CMU graduate student seminar on February 21, 2017. A video of the talk is on Youtube.
Classically, model theorists have studied classes of models satisfying a fixed set of axioms, where the axioms can be described using only quantification over elements (not sets), and the disjunctions and conjunctions can only be finite. For some obscure reason, such classes are now called elementary classes.
In this talk, I will explain why studying elementary classes may not be enough, and will discuss frameworks that allow one to study classes axiomatized by axioms with infinite conjunctions and disjunctions. This includes the so-called abstract elementary classes (AECs), developed by Shelah in the mid seventies, which are a kind of concrete category closed under directed colimits. I will present some of the results of my thesis on AECs: the eventual categoricity conjecture for universal classes and generalizations of linear independence from linear algebra to certain nice kind of AECs.
John T. Baldwin, Categoricity, University Lecture Series, vol. 50, American Mathematical Society, 2009.
Saharon Shelah, Classification Theory for Abstract Elementary Classes, Studies in Logic: Mathematical Logic and foundations, vol. 18 and 20, College Publications, 2009.
Sebastien Vasey, Shelah's eventual categoricity conjecture in universal classes: part I, Preprint: pdf arXiv. See a short outline of the proof.
Sebastien Vasey, Shelah's eventual categoricity conjecture in universal classes: part II, Preprint: pdf arXiv.
Will Boney and Sebastien Vasey, A survey on tame abstract elementary classes, Preprint: pdf arXiv.