Axiomatic Foundations of Fractal Analysis and Fractal Number Theory
Stanislav Semenov
TL;DR
This work introduces a stratified axiomatic framework for fractal analysis and fractal number theory by building a hierarchy of formal systems $\\mathcal{F}_n$ that define corresponding definability levels $S_n$, culminating in the constructive continuum $\\mathbb{R}_{S_\\omega} = \\bigcup_n \\mathbb{R}_{S_n}$. Each level supports a definable topology, arithmetic, and calculus, with an explicit focus on convergence bounds, computational complexity, and a stratified measure-theoretic landscape. The paper develops a constructive Hahn–Banach extension within this hierarchy, a Fractal PCP Gap Principle, and reverse mathematics correspondences, providing a unified framework that integrates constructive analysis, proof theory, and fractal geometry. Applications span approximation theory, computable analysis, and algorithmic mathematics, all within a definability-controlled setting that remains consistent with established subsystems such as $\\mathsf{RCA}_0$, $\\mathsf{ACA}_0$, and beyond. The framework thus offers a flexible, countable yet topologically rich model of the continuum that clarifies the boundary between definability and nonconstructivity and opens avenues for new constructive theorems and formal tools in proof assistants.
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.