Knowable and Unknowable: Revision of the Four Major Contradictions in Mathematics and Physics -- And Reestablishing the Original Meaning of Godel's Incompleteness Theorems

Up to now, the development of human mathematical and physical systems has been trapped in the game between the exploration of the knowable and the limitations of the unknowable. The four fundamental contradictions in existing theories-the opposition between 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 irreconcilable paradoxes inherent in the universe itself. Instead, they result from the academic community's long-standing equating of the cognitive shortcomings of human civilization at the current stage with the essential laws of the objective universe, the forced implantation of unproven implicit rules, the cobbling together of contradictory systems, and the malicious distortion of Godel's Incompleteness Theorems. Taking objective reality as the core, this paper complements key concepts to achieve an absolute logical closed loop. Based on the original geometric formula of π and the essence of differential calculation, it completely overthrows the century-old dogma that "π is infinite and non-repeating" and fundamentally proves that π is a definite finite real number. It clarifies the unified classification standard for infinite recurring decimals and reconstructs a contradiction-free number system. By introducing the spatial curvature parameter, it realizes the unification of geometry, adheres to the law of excluded middle to eliminate intuitionistic fallacies, and solves the puzzle of quantum apparent randomness with cosmic determinism, thus comprehensively revising the four core contradictions. Strictly based on the original text of Godel's Incompleteness Theorems and the author's realist stance, this paper upholds the core authoritative position: acknowledging the proof boundary of formal systems, rejecting vulgar distortion, adhering to the knowability of objective truth, and clarifying that the revision of the system is fully consistent with the original theorems. Embedding the core viewpoints of top scholars such as Plato, Gauss, Hilbert, Einstein, and Godel, it consolidates the foundation of academic authority, and constructs a unified underlying framework of mathematics and physics that adapts to human current applications, takes into account future exploration, has no internal conflicts, and can be extended for a long time. This framework breaks the conservative barriers of the existing system, returns mathematics and physics to their essence of describing the deterministic truths of the universe, and solves the industry dilemma of "unattained knowability and misinterpreted unknowability".