The Omega Number System: Toward a Transfinite Extension of Complex Analysis

We present the Omega Number System, an extension of the complex number system that incorporates both infinitary and infinitesimal scales into a unified, hierarchical framework. Our construction is anchored by a fundamental scaling element, \(\Omega\), which is rigorously defined as the hyperreal number corresponding to the equivalence class of the standard sequence of natural numbers obtained via the ultrapower construction. This canonical infinitary element organizes transitions across transfinite hierarchies of magnitude and serves as the basis for our extended arithmetic. From \(\Omega\) we derive key foundational objects, including the absolute zero \(\underline{0}\), the almost zero \(\overline{0}\) (representing the transfinite continuity of infinitesimal numbers), including its fundamental member, the canonical zero \(0^*\), defined as the multiplicative inverse of \(\Omega\); such that together with the identity element \(1\), these objects integrate consistently with classical arithmetic. We illustrate our approach through two foundational models—a linear model that extends familiar arithmetic in a straightforward manner and a non-linear model incorporating hyper-exponential growth that captures phenomena well beyond classical constructs at each index level. These models yield unique hierarchical expansions and demonstrate how infinitesimals and infinite scales can coexist systematically. We also discuss potential applications, such as the reinterpretation of classical singularities and the regularization of divergent behaviors. Although some aspects—such as multivalued or probabilistic interpretations of certain functions—are presently exploratory, the Omega system proposes a flexible foundation for further analytical developments. Future work will pursue more complete axiomatic foundations, abstract algebraic generalizations, and connections to advanced problems in pure mathematics and theoretical physics, thereby laying the groundwork for Omega Analysis—an extension of the methods of classical complex analysis into the transfinite realm.