On the Nature of Fractal Numbers and the Classical Continuum Hypothesis (CH)
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912397129678848 |
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| author | Semenov, Stanislav |
| author_facet | Semenov, Stanislav |
| contents | We propose a reinterpretation of the continuum grounded in the stratified structure of definability rather than classical cardinality. In this framework, a real number is not an abstract point on the number line, but an object expressible at some level Fn of a formal hierarchy. We introduce the notion of "fractal numbers" -- entities defined not within a fixed set-theoretic universe, but through layered expressibility across constructive systems. This reconceptualizes irrationality as a relative property, depending on definability depth, and replaces the binary dichotomy between countable and uncountable sets with a gradated spectrum of definability classes. We show that the classical Continuum Hypothesis loses its force in this context: between aleph_0 and c lies not a single cardinal jump, but a stratified sequence of definitional stages, each forming a countable yet irreducible approximation to the continuum. We argue that the real line should not be seen as a completed totality but as an evolving architecture of formal expressibility. We conclude with a discussion of rational invariants, the relativity of irrationality, and the emergence of a fractal metric for definitional density. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04637 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Nature of Fractal Numbers and the Classical Continuum Hypothesis (CH) Semenov, Stanislav General Mathematics 03F60, 26E40, 03F03 F.4.1 We propose a reinterpretation of the continuum grounded in the stratified structure of definability rather than classical cardinality. In this framework, a real number is not an abstract point on the number line, but an object expressible at some level Fn of a formal hierarchy. We introduce the notion of "fractal numbers" -- entities defined not within a fixed set-theoretic universe, but through layered expressibility across constructive systems. This reconceptualizes irrationality as a relative property, depending on definability depth, and replaces the binary dichotomy between countable and uncountable sets with a gradated spectrum of definability classes. We show that the classical Continuum Hypothesis loses its force in this context: between aleph_0 and c lies not a single cardinal jump, but a stratified sequence of definitional stages, each forming a countable yet irreducible approximation to the continuum. We argue that the real line should not be seen as a completed totality but as an evolving architecture of formal expressibility. We conclude with a discussion of rational invariants, the relativity of irrationality, and the emergence of a fractal metric for definitional density. |
| title | On the Nature of Fractal Numbers and the Classical Continuum Hypothesis (CH) |
| topic | General Mathematics 03F60, 26E40, 03F03 F.4.1 |
| url | https://arxiv.org/abs/2504.04637 |