Classification over a predicate: non-structure using forcing

Classification theory, pioneered by Shelah in the 1980`s, is the study of "dividing lines" and dichotomy theorems. A classical example of such a dichotomy theorem is Shelah`s celebrated Main Gap Theorem that states that, given a class of structures (also called "models") axiomatized by a countable theory (set of axioms), it either has the largest possible number of non-isomorphic models of any uncountable cardinality, or every model can be "classified" using a small number of invariants (dimensions). The first option in the above dichotomy is called "non-structure", and the second one "structure". The proof of the theorem relies on identifying the correct dividing lines - properties that divide theories into good (whose properties lead to structure) and bad (that lead to non-structure). Main Gap-type results are conjectured in other contexts in model theory, but none have been fully established. In this project, we study one specific such context: models of countable theories over a fixed predicate. Specifically, we have a countable collection of axioms T and a predicate P (which can be thought of as a distinguished subset of every model), and we consider classes of models axiomatized by T with a fixed P-part. A prototypical example of such a class is the class of vector spaces over a fixed field. A much more interesting and less understood example is the class of exponentially closed fields with a fixed kernel of the exponential function. Although the context of model theory over a predicate sounds pretty close to the classical one, experience shows that it exhibits various phenomena that do not occur in classical model theory, which suggests that it requires more sophisticated methods and techniques, The main goal of this project is to advance the understanding of the non-structure side in this context. Specifically, we hope to show, using set theoretic techniques such as forcing, that classes that exhibit ``complicated combinatorial behavior, have (potentially in some extension of the set theoretic universe) ``non-structure over P. Our hope is that such results will help us to identify true dividing lines in this context, and bring us closer to a Main Gap- type dichotomy theorem. In addition, we hope to deduce the so-called Generalized Gaifmans Conjecture, proposed recently by Shelah and Usvyatsov: ``structure implies the existence property (which means that every model of P occurs as the P-part of some model of T). We hope to prove this by showing that the failure of the existence property leads to non-structure.