On the Foundation of Numbers and Reflexive Set Theory: A Dual-Axis Construction and Its Topos-Theoretic Closure

Abstract Classical Cantor-ZFC set theory acts as the ultimate bedrock of essentialist mathematics, assuming that elements possess an inherent, independent existence (Svabhāva) and restricting the membership relation (∈) to flat, two-dimensional Boolean valuations. This paper presents a complete topological and algebraic reformulation of both set theory and thefoundational number system, establishing Reflexive Set Theory (RST) and its corresponding Reflexive Topos (RST). By redefining elements, belonging (∈∗), and inclusion (⊆∗) as dynamical projections over a non-Abelian Heyting sheaf Ω∗, we eliminate the pathological settheoretic singularities (e.g., Vitali non-measurable sets) that plague classical analysis. Underthis framework, we construct the holistic number system N∗ using a modified set of Peanoaxioms, where every number is represented as a dual-axis pair n = (A,B)—representing comoving objective manifestation and subjective orientation. We prove the Reflexive ClosureTheorem, demonstrating that the subobject classifier Ω∗ is recursively closed as an internalized object within RST, thus providing a pristine, non-regressive foundation for both settheory and the reflexive logic proposed in Paper 0.