Axiomatizing Mathematical Theories: Multiplication

TL;DR

This paper investigates the axiomatizability of multiplication theories across various number domains, reviewing classical results, providing new proofs, and addressing open questions in the field.

Contribution

It offers new proofs for existing theorems and resolves an open question regarding the axiomatizability of multiplication in certain mathematical structures.

Findings

Some structures' theories are proven to be axiomatizable.

Identification of structures with unknown axiomatizability status.

Resolution of an open question about multiplication theories.

Abstract

Axiomatizing mathematical structures is a goal of Mathematical Logic. Axiomatizability of the theories of some structures have turned out to be quite difficult and challenging, and some remain open. However axiomatization of some mathematical structures are now classical theorems in Logic, Algebra and Geometry. In this paper we will study the axiomatizability of the theories of multiplication in the domains of natural, integer, rational, real, and complex numbers. We will review some classical theorems, and will give some new proofs for old results. We will see that some structures are missing in the literature, thus leaving it open whether the theories of that structures are axiomatizable (decidable) or not. We will answer one of those open questions in this paper.