A Theoretical Study on Axiomatizability of Some First-Order Mathematical Structures

Abstract

Decidability and undecidability are central challenges in mathematical logic, particularly in the axioma-tization of first-order structures. This study investigated the axiomatizability of various mathematical structures, emphasizing the roles of completeness, consistency, and compactness. We provide an explicit axiomatization for ⟨Z; |⟩, where u | v indicates that u divides v (i.e., ∃t (u · t = v)), demonstrating its decidability through quantifier elimination. This work extends these findings to ⟨Q; |⟩ and explores the multiplicative theory of integers, ⟨Z; ×⟩, highlighting computable axiomatizations in decidable theories.