Theory of Systems: A First-Principles Foundation for Mathematics

We formalize the Theory of Systems—a foundational framework derived from a single first principle: something exists (A = exists). Beginning with the Laws of Logic (L1-L5) as structural properties of distinction, we derive a complete theory of mathematical objects. Main contributions: - E/R/R Framework: Every determinate system exhibits Elements (what exists), Roles (why significant), and Rules (how structured). - Four Principles (P1-P4): Hierarchy, Criterion Precedence, Intensional Identity, and Finite Actuality—each derived from the Laws of Logic. - Coq Formalization: 385 proven theorems including uncountability of [0, 1], Extreme Value Theorem, and Intermediate Value Theorem—without the Axiom of Infinity.