Fractal Analysis on the Real Interval: A Constructive Approach via Fractal Countability

TL;DR

This paper introduces a fractal-based reinterpretation of the real interval, modeling it as a layered structure of definable points to reformulate classical analysis concepts in terms of stratified definability levels.

Contribution

It presents a constructive, hierarchical approach to real analysis, grounding classical notions in formal systems and stratified definability, enabling new perspectives on computability and approximation.

Findings

Reformulation of continuity, measure, differentiation, and integration in stratified levels

Framework for fractal analysis grounded in syntactic accessibility

Insights into approximation and formal verification through layered structures

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.