Valentin Goranko and Ruaan Kellerman: Towards model theory of trees generated by elementary classes of linear orders

Abstract: This is an overview talk on an ongoing research project studying logical theories of classes of trees (as partial orders), also joint with Alberto Zanardo (University of Padova).

Given a class of linear order types C, several different classes of trees are naturally associated with C in terms of how the paths in those trees are related to the order types in C. In a previous work we have studied and completely determined the set-theoretic relationships between these classes of trees and between their corresponding first-order theories. We have also obtained some general and some specific results on axiomatizations of the first-order theories of some such classes of trees defined in terms of the generating class of linear orders C.

In this talk we present an overview of that work and will outline its extension to a more general model-theoretic study of classes of trees generated by elementary classes of linear orders. We will focus on the axiomatizations of their first-order theories, first-order definability and non-definability of paths and other special sets in such trees, and some structural properties of certain natural classes of trees. Besides some known results, we will present a number of open problems.

For this talk we will not presume any essential background beyond some basic knowledge on model theory of first-order logic.