Fractal Origin of the Continuum: A Hypothesis on Process-Relative Definability
Stanislav Semenov
TL;DR
The paper reframes the real continuum as a fractal, process-relative construct arising from layered syntactic growth rather than a preexisting uncountable totality. By formalizing constructive systems $\\mathcal{F}$ with countable syntax and defining countable definable layers $S_n$, it introduces the fractal boundary $S^{\\mathcal{F}}$ and shows it is not globally enumerable within any single system. Extending to all stratified hierarchies $\\mathbb{F}_\\omega$, it defines universal objects $S^{\\mathbb{F}_\\omega}$ and $\\mathbb{R}^{\\mathbb{F}_\\omega}$, each of cardinality $\\mathfrak{c}$ but not capturable by a single constructive procedure, thus modeling the continuum as a union of definability trajectories. The framework avoids AC and Power Sets, offering a constructive foundation where uncountability emerges from meta-structural definability rather than ontological infinity, and it situates the fractal continuum as a constructive core of the classical continuum with precise differences in definability and enumeration. This approach opens pathways to analyze limits, measure, and topology within a fractal-definability context and invites further connections to category-theoretic and type-theoretic perspectives.
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.