The prediction of singularities within black holes presents a major paradox in modern theoretical physics. General relativity suggests that curvature diverges at the center of black holes, leading to incompleteness of spacetime. However, quantum corrections and alternative gravity models indicate that singularities may not physically exist. In this paper, we present a rigorous mathematical analysis proving the non-existence of singularities, incorporating geodesic completness, modified Einstein equations, and quantum gravity effects.