  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$(\Omega,\leq)$.
