A Novel Framework for Indeterminate Numbers Derived from Division by Zero

The expression $\frac{0}{0}$ has long been considered undefined due to its inherent ambiguity, posing a significant challenge in mathematics. In this paper, we present a novel framework that redefines $\frac{0}{0}$ by introducing the concept of an \emph{indeterminate number} $U$. We rigorously define $U$ within the traditional mathematical fields $\mathbb{F}$ (such as the real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$), without extending the number system.Our framework develops arithmetic operations involving $U$ based on set and interval operations, including addition, subtraction, multiplication, and division. These operations are meticulously defined to ensure consistency and avoid contradictions within the existing mathematical framework.We introduce the mechanism of \emph{collapse}, allowing indeterminate numbers to transition into determinate numbers, and explore various types and their implications within the framework. By distinguishing $U$ from unknown variables, we offer a new perspective on indeterminate forms. We demonstrate that operations involving $U$ can be consistently defined without leading to contradictions, adhering to established mathematical principles.Furthermore, we discuss potential applications of this framework across multiple disciplines, including calculus, algebra, quantum mechanics, and computer science. By providing a consistent and practical approach to handling indeterminate expressions, this work has the potential to significantly advance mathematical theory and stimulate further research in mathematics and related fields.