Axiomatic Foundations of Fractal Analysis and Fractal Number Theory

Abstract: We develop an axiomatic framework for fractal analysis and fractal number theory grounded in hierarchies of definability. Central to this approach is a sequence of formal systems F_n, each corresponding to a definability level S_n contained in R of constructively accessible mathematical objects. This structure refines classical analysis by replacing uncountable global constructs with countable, syntactically constrained approximations. The axioms formalize: - A hierarchy of definability levels S_n, indexed by syntactic and ordinal complexity; - Fractal topologies and the induced notions of continuity, compactness, and differentiability; - Layered integration and differentiation with explicit convergence and definability bounds; - Arithmetic and function spaces over the stratified continuum R_{S_n}, which is a subset of R. This framework synthesizes constructive mathematics, proof-theoretic stratification, and fractal geometric intuition into a unified, finitistically structured model. Key results include the definability-based classification of real numbers (e.g., algebraic, computable, Liouville), a stratified fundamental theorem of calculus with syntactic error bounds, and compatibility with base systems such as RCA_0 and ACA_0. The framework enables constructive approximation and syntactic regularization of classical analysis, with applications to proof assistants, computable mathematics, and foundational studies of the continuum.