A Universal Foundational Theory

This paper presents a Universal Foundational Theory, a rigorous mathematical framework designed to unify all consistent mathematical systems while resolving foundational challenges, such as self-referential paradoxes and the Continuum Hypothesis (CH). Building on set theory, we develop a structured hierarchy comprising Morphing Theory, Compositional Theory, and Hierarchical Theory, with morphing categories extending category theory to enable dynamic structural unification. Through formal definitions and theorems, we demonstrate the theory’s ability to embed paradoxical structures and address cardinality questions, exemplified by a case study on CH. The framework’s applicability to cosmology and computational complexity underscores its interdisciplinary relevance, offering novel insights into foundational mathematics and its connections to logic, physics, and computer science, in alignment with Axioms’ focus on advancing rigorous theoretical developments.