Uniform Computable Categoricity for Fields

Logic Seminar

Meeting Details

For more information about this meeting, contact Kristin Berrigan, Jan Reimann, Linda Westrick.

Speaker: Russell Miller, CUNY Graduate Center

Abstract: Computability theorists have learned not to take isomorphism for granted. It is quite possible for two computable structures (i.e., structures with decidable atomic diagrams) to be isomorphic to each other, yet for there to be no isomorphism between them that can be computed (as a function from one domain onto the other). This is important because, if the structures fail to be computably isomorphic, then one of them may have very nice computability-theoretic properties, while the other does not, despite being (non-computably) isomorphic to the first. For example, two computable fields can be isomorphic, but if they are not computably isomorphic, then in one it may be undecidable which field elements have square roots, while in the other the same question may be decidable. \emph{Computable categoricity} of a structure means that these pathologies cannot happen: for every pair of computable copies of the structure, there is a computable isomorphism between them. Fields were the first class of structures for which computable categoricity was studied, by Fr\"ohlich and Shepherdson in the 1950's. We will discuss a uniform version of computable categoricity for fields, which will be accessible with no background at all beyond a very basic knowledge of field theory. Even so, we will be able to understand several recent results, some by the speaker and (time permitting) some joint with Hans Schoutens.

Room Reservation Information

Room Number: 315 McAllister

Date: 10/01/2019

Time: 2:30pm - 4:00pm