Meaning of No-Being

Abstract

In this article, we reinvestigate Gödel’s 1931 proof, mathematically and philosophically, with hope of shedding new light on the study of the completeness of a formal system. First, we develop mathematical proofs to reveal the logic issue in the proof of Theorem VI in Gödel’s seminal works and bring forth an alternative proposition in regard to the provability of the existence of a (true but) undecidable proposition for every ω-consistent recursive class κ of formulas. Next, we undertake a philosophical investigation on the sensibility of our finding. We start with revisiting two enduring paradoxes, Zeno’s Achilles and Tortoise and Russell’s Paradox, to expose the incompleteness of the existing resolutions to them. We proceed to resolve each through novel approaches and introduce new concepts and notions to analytic philosophy. With those developments, we uncover the intrinsic linkage among all three penetrating discoveries, which furnishes our understanding of the completeness of formal systems so as to revive the hope of fulfilling the ambition of Hilbert’s program.

Keywords: Zeno's Achilles and Tortoise, Russell’s Paradox, Godel's Incompleteness Theorem