Title: On Decidability of the Ordered Structures of Numbers,
Abstract: The ordered structures of natural, integer, rational and real numbers are
studied here. It is known that the theories of these numbers in the language of
order are decidable and finitely axiomatizable. Also, their theories in the
language of order and addition are decidable and infinitely axiomatizable. For
the language of order and multiplication, it is known that the theories of
$\mathbb{N}$ and $\mathbb{Z}$ are not decidable (and so not axiomatizable by
any computably enumerable set of sentences). By Tarski's theorem, the
multiplicative ordered structure of $\mathbb{R}$ is decidable also; here we
prove this result directly and present an axiomatization. The structure of
$\mathbb{Q}$ in the language of order and multiplication seems to be missing in
the literature; here we show the decidability of its theory by the technique of
quantifier elimination and after presenting an infinite axiomatization for this
structure we prove that it is not finitely axiomatizable.