On the Nature of Fractal Numbers and the Classical Continuum Hypothesis (CH)

2025-04-06

AI Summary

This paper addresses foundational issues surrounding the classical Continuum Hypothesis (CH) by challenging the standard cardinality-based framework for characterizing the continuum. Method: It reconceptualizes real numbers as definable objects within a hierarchy of formal languages $F_n$, introducing “fractal numbers”—entities whose existence and properties depend on definitional depth across constructive systems, rather than on absolute set-theoretic ontology. Under this approach, irrationality becomes a relative property, and the countable/uncountable dichotomy is replaced by a continuum of definability density. Contributions: (i) A proof that between $aleph_0$ and $2^{aleph_0}$ lies a countable yet irreducible sequence of definitional stages; (ii) construction of a fractal metric quantifying definability density; and (iii) demonstration that CH loses independent meaning in this framework, as cardinal jumps dissolve into evolutionary formal structures governed by definability constraints.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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.

Problem

• Reinterpreting continuum via stratified definability, not cardinality
• Introducing fractal numbers as layered, constructively defined entities
• Resolving Continuum Hypothesis via gradation, not binary cardinality

Innovation

• Reinterprets continuum via stratified definability hierarchy
• Introduces fractal numbers through layered expressibility
• Replaces CH with gradated spectrum of definability