A Theoretical Study on Axiomatizability of First-Order Mathematical Structures
DOI:
https://doi.org/10.22105/kmisj.v2i1.82Keywords:
Axiomatizability, Boolean algebras, Decidability, Incompleteness, Number structure, First-order logicAbstract
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.
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