Fractal Analysis on the Real Interval: A Constructive Approach via Fractal Countability
Stanislav Semenov
TL;DR
This paper introduces fractal countability, a constructive framework that treats the real interval as a layered union of definable point-sets $S_n$, each associated with a formal system $\mathcal{F}_n$. It develops a full cascade of layer-relative notions—topology, continuity, differentiation, integration, and measure—where objects and operations may shift definability level (e.g., $S_n$ to $S_{n+k}$) and where compactness and dimension become syntactically constrained. By redefining classical analysis within this stratified hierarchy, it offers a constructive, syntax-aware reformulation of analysis, including a fractal calculus, definable covers, and a stratified measure theory, while reinterpreting pathologies and paradoxes as layer-dependent phenomena. The framework has implications for constructive mathematics, proof theory, and computable semantics, providing a new lens to study the continuum through syntactic accessibility and formal verifiability.
Abstract
This paper develops a technical and practical reinterpretation of the real interval [a,b] under the paradigm of fractal countability. Instead of assuming the continuum as a completed uncountable totality, we model [a,b] as a layered structure of constructively definable points, indexed by a hierarchy of formal systems. We reformulate classical notions from real analysis -- continuity, measure, differentiation, and integration -- in terms of stratified definability levels S_n, thereby grounding the analytic apparatus in syntactic accessibility rather than ontological postulation. The result is a framework for fractal analysis, in which mathematical operations are relativized to layers of expressibility, enabling new insights into approximation, computability, and formal verification.