Louise Hay Logic Seminar
Rishi Banerjee
UIC
Axiomatizing countable Borel equivalence relations in infinitary logic
Abstract: Descriptive set theorists study the properties of “definable” (i.e., Borel or analytic) subsets of “nice” (i.e., Polish) topological spaces. The study of these subsets is closely linked to the model theory of the infinitary logic $L_{\omega_1\omega}$, which extends first order logic by allowing countably infinite conjunctions and disjunctions. Of particular interest is the Borel reducibility hierarchy for Borel equivalence relations, which can be thought of as a way of comparing the complexity of classification problems. A common technique is to look at the “local structure” of an equivalence relation, i.e., what types of first order structures can be placed on all the equivalence classes in a uniform Borel manner. A countable Borel equivalence relation (CBER) is a Borel equivalence relation in which every equivalence class is countable. We show that there is an Lw1w-theory precisely axiomatizing the local structure of CBERs. Background in descriptive set theory and infinitary logic will not be required. (Talk based on joint work with Ronnie Chen, University of Michigan)
Wednesday September 25, 2024 at 2:00 PM in 427 SEO