Data Storage in Non-Measurable Mathematical Spaces: A Banach–Tarski Inspired Approach

The conventional approach to data storage is fundamentally constrained by physical limits: more information requires more space. This paper proposes a radical departure from this paradigm by exploring the possibility of storing information within non-measurable mathematical spaces. The concept of non-measurability, rooted in the Banach–Tarski paradox, allows for the counterintuitive notion that a finite object can be decomposed and reassembled into larger copies of itself. Translating this idea to information theory suggests that vast amounts of data could be encoded within a mathematical structure that occupies no measurable volume in the usual sense. Imagine storing the entire content of the internet on a USB key of 34 MB or less. More strikingly, under this framework, as the quantity of information increases, the required storage space may decrease. This is not a compression algorithm in the traditional sense, but rather a fundamental rethinking of how information can be represented geometrically. While this idea may face vigorous mathematical and philosophical scrutiny, its potential implications for information technology are too significant to ignore. This work formalizes this concept by constructing an explicit embedding of finite data into a non-measurable subset of the unit sphere, providing a mathematical foundation for what could become a new era of information storage.