Uncertain Set Theory

This paper establishes Uncertain Set Theory, a novel foundational mathematical framework built from First Principles to model uncertainty without relying on probabilistic distributions, fuzzy membership degrees, or static interval bounds. Traditional approaches such as Interval Arithmetic frequently suffer from correlation loss and catastrophic numerical noise explosion, while probabilistic models require prior knowledge of probability densities that are often unavailable in real-world high-noise environments. We address these fundamental limitations by introducing the concepts of \textbf{Uncertain Elements} and \textbf{Uncertain Sets}, defined as multi-state superposition entities characterized by possibility domains $\mathrm{Val}(x)$ and configuration spaces $\mathrm{Conf}(S)$. Through this scenario-based structural formulation, we demonstrate that set cardinality itself unifies naturally as an uncertain number $|S| \in \mathbf{U}(\mathbb{N})$, resolving cardinality degeneration and state-collapse seamlessly. Furthermore, algebraic operations on uncertain sets inherit classical binary set relations over their configuration spaces, preserving absolute logical correlation and enabling canonical complexity reduction down to $\mathcal{O}(1)$ via Lazy Evaluation. This framework serves as the core foundation for $U(\mathbb{K})$ arithmetic, scenario-space optimization, and high-dimensional uncertain topology.Keywords: Uncertain Set Theory, Uncertain Elements, Configuration Space, State Superposition, Uncertain Cardinality, Scenario-Based Arithmetic, Correlation Loss, $U(\mathbb{K})$ Spaces, First Principles. About the Author: Tran Manh Cuong Alumnus of Faculty of Mathematics and Informatics (K11) | Thai Nguyen University of Sciences – TNU Research Field: Mathematics of Uncertainty Email: [email protected] Phone: (+84) 353237140 Address: DJ7 Street, Thoi Hoa Ward, Ho Chi Minh City, Vietnam