LEAPS-MPS: Fragments of Compactness

The field of model theory studies classes of structures: groups, fields, graphs. This is a broad field to study, so important distinctions are made based on how the class of structures is described (or axiomatized). If the class is describable in first-order logic, it is called an elementary class.

First-order logic has many powerful properties, especially the property of compactness. Compactness allows model theorists to build structures with exotic properties and has driven much of the model theory of elementary classes, most notably classification theory. However, many classes of structures are not describable in first-order logic (these are called nonelementary classes).

Lacking compactness, the development of nonelementary model theory and classification theory has proceeded much slower than its elementary counterpart. Recent work in nonelmentary classes has shown that various fragments of compactness can still hold in some nonelmentary classes and are still powerful enough to prove various results of elementary classification theory.

The PI will develop more of these fragments in nonelementary classes. Additionally, the PI will run a program to build research infrastructure at their home institution (an R2 institution and HSI). The first year of the program will bring in a series of logic speakers that focus on connections between logic and other fields. The second year will support both faculty and student research in logic inspired by these speakers.

More specifically, the PI will research compactness properties in nonelementary classes using techniques from model theory, set theory, and category theory. As stated above, compactness fails to hold in nonelementary classes, but the PI will explore a variety of methods to find to fragments of compactness in these classes. The first objective is to extend compactness and other concepts of model theory to higher algebra.

The second objective is to develop nonstandard ultraproducts in nonelementary classes using a category theoretic perspective. The third objective is to continue the PI’s exploration of connections between nonelementary compactness principles and large cardinal in set theory.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.