Fractal Analysis on the Real Interval: A Constructive Approach via Fractal Countability
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918025493479424 |
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| author | Semenov, Stanislav |
| author_facet | Semenov, Stanislav |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_13450 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractal Analysis on the Real Interval: A Constructive Approach via Fractal Countability Semenov, Stanislav Logic in Computer Science Functional Analysis 03F60, 26E40, 03F03 F.4.1 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. |
| title | Fractal Analysis on the Real Interval: A Constructive Approach via Fractal Countability |
| topic | Logic in Computer Science Functional Analysis 03F60, 26E40, 03F03 F.4.1 |
| url | https://arxiv.org/abs/2505.13450 |