University Seminar- Computability, Complexity, and Algebraic Structure-Types, existential closures, and cohesive powers in the study of arithmetic.

Time: Friday, September 29, 12:00noon–1:00pm

Place: Rome Hall, Room 352

Speaker:  Henry Klatt, GWU

Title: Types, existential closures, and cohesive powers in the study of arithmetic.

Abstract:  Models of Arithmetic are both mysterious and alluring.  One the one hand, Tennenbaum's theorem asserts that no non-standard model of Peano Arithmetic can be computable, meaning every such model must be difficult to describe.  On the other hand, arithmetic is one of the central focuses of mathematical logic, which compels us to study these structures anyway.  Some time in the 70s, Joram Hirschfeld studied existential closure, a generalization of the notion of algebraic closure from number theory, in the context of arithmetic.  Seemingly unrelated to this, many people have studied Nerode semirings, (a.k.a. recursively enumerable prime powers, a.k.a. cohesive powers of arithmetic, a.k.a. simple models of arithmetic) over the past half century.  The convergence of these two topics has recently shed light on the richness of the space of prime filters on the lattice of computably enumerable sets, and their connection to types in models of arithmetic. In this talk, we will cover some of the recent insights and headaches and demonstrate the potential for future research in this direction.