Title: Scott Complexity and Torsion Abelian Groups
Speaker: Rachael Alvir (University of Waterloo)
Abstract: The logic $L_{\omega_1 \omega}$ admits sentences that are infinitely long by allowing countably many conjunctions and disjunctions. In this logic, we can describe countable structures - such as the natural numbers - up to isomorphism (among countable structures) via a single sentence known as the Scott sentence of that structure. The syntactic complexity of a Scott sentence for a structure tells us a host of information about the structure. We consider a finer notion of this complexity than that historically considered in the literature known as the Scott complexity.
Next, we discuss computing the Scott complexity for torsion groups. In particular, we will focus on computing the Scott complexity for reduced abelian groups. To do this, we give a characterization of the back-and-forth relations on such groups. This gives a new proof of the fact that reduced abelian groups attain arbitrarily high Scott complexity. Moreover, we can give an explicit example of a sequence of groups which exhibit this behavior, with the Scott complexity strictly increasing with the length of the group.
URL associated with Seminar: https://research.math.osu.edu/logicseminar/
