Omniscient Mathematics: An Axiomatic Theory of the Limits of Superintelligence

This paper introduces the concept of Omniscient Mathematics as a formal framework for analyzing the epistemological and computational limits of superintelligent systems operating in spaces of total information. Inspired by Jorge Luis Borges’ Library of Babel, the proposed framework investigates axiomatically environments in which all possible symbolic combinations already exist, including all true statements, false statements, mathematical proofs, scientific theories, and random structures. I demonstrate that the existence of total information does not imply the existence of accessible knowledge. In such environments, exhaustive search becomes computationally and physically infeasible due to combinatorial explosion, semantic entropy, and the impossibility of brute-force verification. Consequently, intelligence emerges not as a mechanism for generating information, but as a necessary process of semantic selection, compression, interpretation, and truth extraction. The paper formally distinguishes structural omniscience from cognitive omniscience, showing that possessing all possible information is fundamentally different from understanding or identifying meaningful knowledge within it. Based on this distinction, we establish the Theorem of the Necessary Emergence of Intelligence, demonstrating that intelligent agents necessarily arise whenever informational spaces become sufficiently large, dense, or combinatorially complete. The proposed theory establishes deep connections between information theory, computability, artificial intelligence, semantic search, and superintelligence. Furthermore, it suggests that future superintelligent systems will not primarily depend on infinite storage or exhaustive computation, but rather on increasingly efficient mechanisms for semantic navigation in spaces of extreme informational complexity. The results provide a new theoretical perspective on the nature of intelligence, knowledge discovery, and the fundamental limits of artificial superintelligence.