On Axiomatizability of the Multiplicative Theory of Numbers

TL;DR

This paper investigates the axiomatizability of the multiplicative theories of various number sets, providing explicit axiomatizations and proving their non-finite axiomatizability, while also establishing quantifier elimination for these theories.

Contribution

It offers explicit axiomatizations for the multiplicative theories of complex, real, and positive rational numbers and proves they are not finitely axiomatizable.

Findings

Theories are decidable with explicit axiomatizations.

Theories are not finitely axiomatizable.

Quantifier elimination is achieved for each set.

Abstract

The multiplicative theory of a set of numbers (which could be natural, integer, rational, real or complex numbers) is the first-order theory of the structure of that set with (solely) the multiplication operation (that set is taken to be multiplicative, i.e., closed under multiplication). In this paper we study the multiplicative theories of the complex, real and (positive) rational numbers. These theories (and also the multiplicative theories of natural and integer numbers) are known to be decidable (i.e., there exists an algorithm that decides whether a given sentence is derivable form the theory); here we present explicit axiomatizations for them and show that they are not finitely axiomatizable. For each of these sets (of complex, real and [positive] rational numbers) a language, including the multiplication operation, is introduced in a way that it allows quantifier elimination (for the theory of that set).