Isomorphism and classification for countable structures

TL;DR

This paper develops a new topological framework for classifying countable structures' isomorphism types, blending effective descriptive set theory with computable structure theory, and applies it to fields and trees.

Contribution

Introduces a topology on the space of isomorphism types for countable models, extending classification methods beyond computable structures.

Findings

Classifications reveal differences between similar classes of structures.

A measure on algebraic fields' isomorphism types is defined and analyzed.

Prevalence of relative computable categoricity is examined under the new measure.

Abstract

We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and computable structure theory, extending concepts from the latter beyond computable structures to examine the isomorphism problem on arbitrary countable structures. We give examples using specific classes of fields and of trees, illustrating how the new concepts can yield classifications that reveal differences between seemingly similar classes. Finally, we use a computable homeomorphism to define a measure on the space of isomorphism types of algebraic fields, and examine the prevalence of relative computable categoricity under this measure.