Decidability of the Multiplicative and Order Theory of Numbers

TL;DR

This thesis explores the decidability of multiplicative and ordered theories of various number systems, establishing new results for the rational numbers and providing explicit axiomatizations.

Contribution

It proves the decidability of the multiplicative ordered structure of the rationals and offers explicit axiomatizations, filling gaps in existing literature.

Findings

Ordered structures of natural, integer, rational, and real numbers are decidable in order language.

Theories in order and addition are decidable and infinitely axiomatizable.

The multiplicative ordered structure of the rationals is decidable and not finitely axiomatizable.

Abstract

The ordered structures of natural, integer, rational and real numbers are studied in this thesis. 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. In this thesis we prove this result directly by quantifier elimination and present an explicit infinite axiomatization. The structure of $\mathbb{Q}$ in the language of order and multiplication seems to be missing in the literature. 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.