On Knowability and Unknowability: Correction of Four Core Contradictions in Mathematical Physics -- with a Rectification of Gödel's Incompleteness Theorems

Abstract

Up to now, human mathematical and physical systems have been trapped in the tension between exploring the knowable and facing the limits of the unknowable. The four fundamental contradictions in existing theories—the opposition of finiteness and actual infinity, the conflict between the law of excluded middle and intuitionistic logic, the artificial division of Euclidean and non-Euclidean geometry, and the incompatibility of classical and quantum mathematics—are not inherent paradoxes of the universe, but stem from the academic community equating human cognitive shortcomings with objective cosmic laws, implanting unproven implicit rules, and maliciously distorting Gödel's Incompleteness Theorems. Taking objective reality as the core, this paper complements key concepts to form an absolute logical closed loop. Based on the original geometric formula of π and the essence of differential calculation, it overthrows the dogma that "π is infinite and non-repeating", proving it a definite finite real number. It clarifies the unified classification of infinite recurring decimals to reconstruct a contradiction-free number system, introduces spatial curvature parameters to unify geometry, adheres to the law of excluded middle to eliminate intuitionistic fallacies, and resolves quantum apparent randomness with cosmic determinism, thus revising the four core contradictions comprehensively. Strictly based on Gödel's original theorems and the realist stance, this paper acknowledges the proof boundary of formal systems, rejects vulgar distortion, and upholds the knowability of objective truth, confirming that the system revision is fully consistent with the original theorems. Embedding core viewpoints of Plato, Gauss, Hilbert, Einstein and Gödel, it constructs a unified underlying framework of mathematics and physics—adaptable to current applications, conflict-free and extendable. This framework breaks conservative barriers, returns mathematics and physics to their essence of describing deterministic cosmic truths, and solves the industry dilemma of "unattained knowability and misinterpreted unknowability".

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