Computability for tree presentations of continuum-size structures

TL;DR

This paper develops a computability-theoretic framework for analyzing continuum-sized structures via tree presentations, focusing on their first-order properties and introducing the concept of tree-decidability.

Contribution

It introduces a new approach to study continuum-size structures through tree presentations, emphasizing first-order properties and decidability concepts.

Findings

Decidability results for additive and multiplicative groups of p-adic integers

Decidability of products like the profinite completion of Z

Decidability of the field of real numbers

Abstract

We formalize an existing computability-theoretic method of presenting first-order structures whose domains have the cardinality of the continuum. Work using these methods until now has emphasized their topological properties. We shift the focus to first-order properties, using computable structure theory (on countable structures) as a guide. We present three basic questions to be asked when a structure is presented as the set of paths through a computable tree, as in our definition, and also propose the concept of tree-decidability as an analogue to the notion of decidability for a countable structure. As examples, we prove decidability results for certain additive and multiplicative groups of $p$-adic integers, products of these (such as the profinite completion of $\mathbb Z$), and the field of real numbers.