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.

翻译：经典集合论通过幂集P(N)构造连续统，从而假设了一个不可数的整体。然而，基于构造性和可计算性的方法表明，任何具有可数语法的形式系统都无法生成N的所有子集，也无法完整地捕获实数线。在本文中，我们提出分形可数性作为幂集的一种构造性替代。我们不再将可数性视为绝对的基数概念，而是将其重新定义为一种分层的、过程相关的、在可定义子集上的闭包，该闭包由对基础形式系统的一系列保守扩展生成。这产生了一个结构化的、内部增长的构造性可定义性层级，它保持在可数领域内，但逼近了连续统的表达丰富性。我们将分形可数集与经典可数性及超算术层级进行比较，并将连续统解释为一个分层的定义性视界，而非一个已完成的对象。该框架为幂集类操作提供了一种构造性重释，而无需诉诸非有效原则。