2025/04/06 by Станіслав Семенов, Semenov, Stanislav
Economics, Econometrics and Finance · Physics and Astronomy · #03F03 #03F60 #26E40 #Advanced Mathematical Theories and Applications #Complex Systems and Time Series Analysis #F.4.1 #FOS: Mathematics #General Mathematics (math.GM)
paper · pdf · doi:10.48550/arxiv.2504.04637
openalex publication_date 2025/04/06 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28
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 aleph0 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.