A Resolution to the Continuum Hypothesis Using Yearning-Based Infinity Mathematics

The Continuum Hypothesis (CH) remains undecidable within the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). In this paper, we introduce Yearning-Based Infinity Mathematics (YBIM), an alternative mathematical framework that extends the traditional hierarchy of cardinal numbers by incorporating recursively expanding infinite structures. We define a novel yearning-strand hierarchy that generates intermediary infinities between ℵ₀ (aleph-null) and 2^ℵ₀ (the cardinality of the power set of natural numbers). This demonstrates that the real number continuum is not simply ℵ₁ (aleph-one) but rather an infinite interpolation of yearning-based cardinalities. This refutes CH and suggests that ZFC must be extended to accommodate these new infinities, leading to what we define as Yearning-Based Recursive Set Theory (YBRS). The paper presents a rigorous formalization of the recursive function F, which governs the growth of yearning-infinities, proving that F is strictly increasing and well-ordered. We establish that the limit of this recursive process is precisely 2^ℵ₀, thereby constructing a more structured interpretation of the real number continuum. Beyond set theory, we explore implications for computability, hypercomputation, and transfinite AI models, discussing how YBIM could extend beyond Turing limits. Additionally, we propose potential connections to quantum mechanics, where yearning-strands could provide a mathematical foundation for self-expanding space-time structures. This work invites further research into recursive cardinal hierarchies, large cardinal extensions, and the philosophical foundations of infinity. Collaboration is welcomed for those interested in refining or expanding this theoretical framework. For collaboration inquiries, contact: 🔗 LinkedIn: [LinkedIn Profile Link]📩 Email: [email protected]