Axioms as Derived Structures: Generating and Switching Mathematical Foundations from World Axioms

Can axioms themselves be derived rather than assumed? This paper demonstrates that Peano arithmetic emerges as a constrained special case of more general structures—and argues this points toward a system capable of generating and switching between arbitrary mathematical foundations.The foundation rests on three frameworks we call "world axioms": Reduction Network (structural transformations), Relational Emergence Systems (relational ontology), and Description without Causation (covariation over causation). Formalized through information geometry, graph theory, and dynamical systems, these world axioms yield Peano arithmetic when a multiplicity parameter M is constrained to zero.This derivation suggests a broader vision: a meta-axiomatic system that (a) generates foundations by specifying constraints, (b) switches between them while tracking what is preserved, and (c) translates theorems across foundational frameworks. We identify six open problems toward realizing this vision.