The Aleph-Field: A New Algebraic Framework for Finite Representation of Transcendental Numbers

We introduce a novel algebraic framework called the ℵ-Field, built on the concepts of Schema Numbers and Symbolic Convergent Operators (SCOs). In this system, every element is defined by a finite symbolic description that encodes an infinite convergent process. Schema numbers are finite objects whose values arise from limits, series, or products of rational functions, and SCOs are new primitive operations that encapsulate infinite summation, limits, and products as atomic symbols. The ℵ-Field extends the rational numbers with these schema-based elements under a complete set of algebraic axioms. We formalize the syntax and semantics of schema numbers and SCOs and present axioms governing the ℵ-Field. We sketch key results, including that the ℵ-Field forms a commutative field closed under all SCOs and that classical transcendental constants admit finite schema definitions within it. Crucially, by design every ℵ-field element has a finite symbolic name, solving the traditional finite-representation problem for transcendentals without relying on classical real analysis. We also introduce two new concepts: Generalized Schema Numbers, allowing multi-index and nested convergent constructions, and Meta-Schema Operators, which operate on schema descriptions themselves (for example, a symbolic exponential operator). Finally, we discuss implications and open questions of this foundational theory and outline directions for future work.