Computable Structure Theory with Noncomputable Structures

Explore computable structure theory and its applications to noncomputable structures in this 56-minute Association for Symbolic Logic Invited Address. Delve into the basics of categoricity, computability, and effective categoricity. Examine alternative paths and definitions of computable categoricity for computable structures. Investigate hands-on examples, including W-computably categorical fields and noncomputable fields. Analyze the number of structures, isomorphism types of subfields, and their relationships to computable fields. Gain insights into theorems related to Rx and HTP(K), and uncover answers to complex mathematical questions in this comprehensive talk given by Russell Miller from Queen's College and CUNY Graduate Center at the Joint Mathematics Meetings.

Intro
An opening thought
Basics of categoricity
Computability and categoricity
Trying to define effective categoricity First stab at computable categoricity
An alternative path
Is this really much different? The definition of computable categoricity for a computable structure
Chronology
Two hands-on examples
Another W-computably categorical field K
Using noncomputable fields
New question: how many structures....?
Using computable fields
Using all subfields of
Isomorphism types of subfields of 0
Answers about Rx and HTP(K)
Summarizing the theorem
Answer to another question