On Decidability of the Ordered Structures of Numbers

TL;DR

This paper investigates the decidability of ordered structures of various number systems, providing new proofs and axiomatizations, especially for the rational numbers, and clarifying their logical properties.

Contribution

It offers a direct proof of decidability for the multiplicative ordered structure of rationals and presents an infinite axiomatization, filling a gap in existing literature.

Findings

Decidability of the ordered structure of rationals established.

Direct proof of decidability for the multiplicative ordered structure of reals.

Infinite, non-finitely axiomatizable characterization of the rational structure.

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.