Fractal Origin of the Continuum: A Hypothesis on Process-Relative Definability

2025-04-05

AI Summary

This paper addresses the foundational source of the uncountability of the real continuum, challenging the standard assumption of actual infinity. Method: It introduces the “fractal genesis hypothesis,” modeling the real continuum as a dynamically accumulating hierarchy of countable layers of reals—each layer corresponding to the set of reals definable within a finite syntactic capacity—and constructs a rigorously countable yet non-uniformly-enumerable real universe via constructive syntactic modeling and a countable-by-construction framework. Contribution: The work reveals that apparent uncountability arises not from set-theoretic completed infinity, but from the metatheoretic continuity of definability and the intrinsic fractal structure of syntactic hierarchies. Consequently, it provides a processual, syntax-based, and semantically open reconstruction of the continuum—free from the Axiom of Choice or other non-constructive assumptions—thereby recasting uncountability as an emergent property of hierarchical definability rather than an ontological feature of the real numbers.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Argumentation
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We propose a new constructive model of the real continuum based on the notion of fractal definability. Rather than assuming the continuum as a completed uncountable totality, we view it as the cumulative result of a vast space of stratified formal systems, each defining a countable layer of real numbers via constructive means. The union of all such definable layers across all admissible chains yields a set of continuum cardinality, yet no single system or definability path suffices to capture it in full. This leads to the Fractal Origin Hypothesis: the apparent uncountability of the real line arises not from actual infinity, but from the meta-theoretical continuity of definability itself. Our framework models the continuum as a process-relative totality, grounded in syntax and layered formal growth. We develop this idea through a formal analysis of definability hierarchies and show that the resulting universe of constructible reals is countable-by-construction (that is, each element is definable within some finite syntactic system, but no single procedure enumerates all of them uniformly) yet inaccessible to any uniform enumeration. The continuum, in this view, is not a static set but a stratified semantic horizon.

Problem

• Proposes fractal definability model for real continuum
• Explains continuum uncountability via meta-theoretical definability
• Constructs continuum as process-relative layered formal systems

Innovation

• Fractal definability models real continuum
• Stratified formal systems create countable layers
• Process-relative totality avoids actual infinity