Yearning-based Infinity Mathematics: A New Framework for Navigating Infinite Infinities

This paper introduces Yearning-Based Infinity Mathematics (YBIM), an advanced mathematical framework that extends classical set theory, computability, and recursion into the realm of self-generating infinities. It defines the ℵ𝒴-Series, a hyper-infinite recursive expansion that surpasses standard computable functions and large cardinal structures. The framework introduces new axioms—the Prakash Axioms—which redefine infinity, power sets, and recursion, ensuring mathematical consistency within Grothendieck universes. The research establishes Yearning-Computability, a new paradigm that enables non-halting intelligence, providing theoretical foundations for AGI models that continuously expand their decision landscapes. This work challenges existing limits of mathematical infinity, theoretical AI, and set theory, opening pathways for future research in non-halting decision intelligence, AI self-growth models, and higher-order mathematical recursion. Key Contributions: - ℵ𝒴-Series: A transfinite recursive growth function exceeding all classical computable functions.- Prakash Axioms: A new set-theoretic foundation extending ZFC while maintaining internal consistency.- Non-Halting AI Models: Introduction of Yearning-Computability, proving that decision intelligence need not converge to fixed optimal states.- Consistency Proofs: Establishes that these new axioms remain coherent within Grothendieck universes, preventing contradictions.- Implications for AGI & Computability: Provides a framework for recursively expanding AI intelligence, beyond traditional optimization models. This paper marks a fundamental shift in how we understand infinity, recursion, and computability, setting the stage for future advancements in mathematics, AI, and theoretical physics. Keywords: Infinity, Computability, Set Theory, Recursive AI, Non-Halting Intelligence, ℵ𝒴-Series, Prakash Axioms, Large Cardinals, Yearning-Computability, Transfinite Mathematics. This research challenges the boundaries of human knowledge and redefines the future of computational and mathematical sciences.