Duality of Continuity and Discreteness of the Universe at the Ultimate Scale——A Unified Analysis Based on Mathematical Infinity, Physical Limits and Cognitive Boundaries

Abstract The debate between continuity and discreteness runs through the history of mathematics, physics and philosophy of science. From the perspectives of the constructibility of mathematical infinity, the Planck limit of physical observation and Gödel's incompleteness of human cognition, this paper systematically demonstrates that under the premise that existing theories and experiments cannot reach the ultimate scale of spacetime, the two hypotheses of "the ultimate continuity of the universe" and "the ultimate discreteness of the universe" are strictly symmetric, logically equivalent and credibly equivalent in epistemology. The universe perceived by humans within observable and computable ranges inevitably presents the characteristic of "brokenness", namely discreteness, quantization, non-integrity and non-infinite divisibility. Such brokenness is not an ultimate attribute of the cosmic ontology, but an inevitable result of the boundaries of human cognition. This paper further discusses the significance of this conclusion for quantum gravity research, the unified field theory and the philosophy of science, and proposes a consistent and unfalsifiable epistemological framework. 摘要 连续与离散之争贯穿数学、物理学与科学哲学史。本文从数学无限的可构造性、物理观测的普朗克极限以及人类认知的哥德尔不完备性三个维度出发,系统论证:在现有理论与实验无法抵达时空终极尺度的前提下,“宇宙最终连续”与“宇宙最终离散”两种假设在认识论上严格对称、逻辑等价、可信度等价。人类在可观测与可计算范围内所感知的宇宙,必然呈现“破缺性”——即表现为离散、量子化、非完整、不可无限细分的特征。这种破缺并非宇宙本体的终极属性,而是人类认知边界的必然结果。本文进一步讨论该结论对量子引力研究、统一场论及科学哲学的意义,提出一种自洽且不可证伪的认识论框架。