Fractal Countability as a Constructive Alternative to the Power Set of N: A Meta-Formal Approach to Stratified Definability
Stanislav Semenov
TL;DR
The paper argues that the classical continuum, tied to the power set $\mathcal{P}(\mathbb{N})$ and the uncountable $|\mathbb{R}|=2^{\aleph_0}$, cannot be fully captured within any countable, constructive formalism. It introduces fractal countability as a stratified, process-relative closure generated by a sequence of conservative extensions $\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \dots$, with $S^\infty=\bigcup_n S_n$ and each $S_n$ definable in $\mathcal{F}_n$. The framework connects to the hyperarithmetical hierarchy by showing that, with $S_n$ computable in $\emptyset^{(n)}$, $S^\infty$ coincides with $HYP$, while allowing extensions beyond $HYP$ through stronger definitional principles; it also emphasizes the dependence on the chosen base system $\mathcal{F}_0$. Philosophically, fractal countability reframes the continuum as a constructive horizon rather than a completed totality, enabling a pluralistic foundation where formal systems can progressively approximate the real line via internally coherent, layered definability. Practical implications include a bridge to reverse mathematics and proof theory, providing concrete stratifications (e.g., from computable to arithmetical sets or via bar recursion) that yield principled, constructively justifiable approximations to the continuum. Overall, the approach preserves definability while avoiding non-effective principles, offering a meta-formal lens for understanding the growth of mathematical content within system-relative horizons.
Abstract
Classical set theory constructs the continuum via the power set P(N), thereby postulating an uncountable totality. However, constructive and computability-based approaches reveal that no formal system with countable syntax can generate all subsets of N, nor can it capture the real line in full. In this paper, we propose fractal countability as a constructive alternative to the power set. Rather than treating countability as an absolute cardinal notion, we redefine it as a stratified, process-relative closure over definable subsets, generated by a sequence of conservative extensions to a base formal system. This yields a structured, internally growing hierarchy of constructive definability that remains within the countable realm but approximates the expressive richness of the continuum. We compare fractally countable sets to classical countability and the hyperarithmetical hierarchy, and interpret the continuum not as a completed object, but as a layered definitional horizon. This framework provides a constructive reinterpretation of power set-like operations without invoking non-effective principles.