The complexity of classification problems for models of arithmetic

TL;DR

This paper explores the complexity of classifying various models of arithmetic, revealing some are Borel and others are Borel complete, with implications for understanding their structural intricacies.

Contribution

It precisely characterizes the Borel complexity of classification problems for different types of models of arithmetic, including countable, finitely generated, and recursively saturated models.

Findings

Countable models of arithmetic have Borel complete classification problems.

Finitely generated and recursively saturated models have Borel classification problems.

Classification problems for pairs of recursively saturated models and automorphisms are Borel complete.

Abstract

We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.