# Recognizing the real line

## TL;DR

This paper proves that any totally ordered set with an automorphism group sharing the same first-order properties as the real line's automorphism group must be order-isomorphic to the real line, extending previous results.

## Contribution

It improves a theorem by Gurevich and Holland by establishing isomorphism under first-order group-theoretic conditions, using analysis of centralizers in automorphism groups.

## Key findings

- Automorphism groups of the real line are characterized by their first-order properties.
- Transitivity and shared first-order sentences imply order-isomorphism to the real line.
- Centralizer analysis is key to establishing the main theorem.

## Abstract

Let $(\Omega, \leq)$ be a totally ordered set. We prove that if Aut$(\Omega,\leq)$ is transitive and satisfies the same first-order sentences as the automorphism group of the real line (in the language of groups) then $\Omega$ and and the real line are isomorphic ordered sets. This improvement of a theorem of Gurevich and Holland is obtained as a consequence of a study of centralizers associated with certain transitive subgroups of Aut$(\Omega,\leq)$.