Minimal Generating Axiom Method (MGAM): Foundations, Principles, and Mathematical Implications

Abstract

The pursuit of foundational economy has long been a guiding principle in mathematical logic and set theory. This paper introduces the Minimal Generating Axiom Method (MGAM), a rigorous theoretical framework designed to systematically identify, isolate, and evaluate the absolute minimum set of axioms required to generate a given mathematical theory or object. By modeling theories as deductive closures over foundational spaces, MGAM provides a structural approach to analyze the trade-off between expressive power and axiomatic economy. We formalize the core mapping from a theory to its minimal generating axioms, denoted as Theory -> Axiom_min, and explore the philosophical, structural, and foundational consequences of this reduction process. Furthermore, we demonstrate how MGAM can be applied across algebraic, topological, and set-theoretic structures, paving the way for the discovery of novel mathematical foundations and unmasking deep symmetries hidden beneath redundant axioms.