Single-Valued, Multivalued, and No-Valued Numbers

Mathematical operations involving division by zero, square roots, and self-referential paradoxes often lead to indeterminate or undefined results. This paper introduces a novel classification of numbers into single-valued numbers (S), multivalued numbers (M), and no-valued numbers (N). The concept of multivalued numbers provides a structured way to represent and manipulate mathematical uncertainty. We propose that 0/0 is a multivalued number whose range spans the entire number field, while division of zero by any multivalued number results in a no-valued number. This framework unifies numbers and sets by representing sets as numbers based on their cardinality and resolves logical paradoxes such as the Barber paradox and Russell's paradox through a dynamic transition between multivalued and single-valued forms. Additionally, we formalize the collapse of multivalued numbers into single-valued numbers using explicit collapse rules, drawing a connection between this process and quantum measurement.