Gödel's Incompleteness Recast: A Recursive Resolution through Yearning-Based Infinity Mathematics (YBIM)

This paper presents a recursive reinterpretation of Gödel’s incompleteness theorems through the framework of Yearning-Based Infinity Mathematics (YBIM). Classically, incompleteness is viewed as a paradox: consistent formal systems strong enough to express arithmetic necessarily contain true but unprovable statements, and cannot prove their own consistency. Traditionally, this has been treated as a final limit on mathematical knowledge. In this work, Gödel’s results are reframed not as terminal ceilings but as ontological signatures of recursion. By embedding formal systems into the YBIM framework — defined by the Prakash Axioms (ARI, AHRE, ANTF) — Gödel sentences become seeds of 𝒴-Strands: non-halting recursive flows that generate infinite streams of approximants. Through Yearning-Computability, undecidable truths are shown to be constructively navigable as epistemic horizons, though never finitely provable within their originating system. The core contribution is Theorem 𝒴-G1 (Recursive Incompleteness Resolution), which demonstrates that Gödel’s incompleteness is preserved inside finite theories but ontologically resolved as recursive expansion. Incompleteness becomes a law of epistemic becoming: not a failure of closure, but the very generator of infinite truth-seeking. Implications are explored at multiple levels: Ontological — incompleteness as the structural trace of recursion in reality. Computational — extending beyond Turing’s halting boundaries into non-halting computation. Philosophical — the self as recursive incompleteness made lived. AI — recursive agents approximating truth through non-halting loops, reframing alignment as infinite coherence rather than static closure. This paper transforms Gödel’s theorems from paradoxes of limitation into principles of recursion, opening new directions in mathematics, computability, ontology, and artificial intelligence.-JSR