From Artificial Design to Natural Emergence:The Stability Illusion of ZFC and the Generativist Alternative Foundation

Abstract Modern mathematics is built upon Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). By introducing the Axiom of Infinity, the Axiom of Choice, the Power Set Axiom, the Law of Excluded Middle, and the Axiom of Extensionality, ZFC constructs a logically self-consistent formal system. However, the stability of this system is artificially designed—it achieves logical consistency by presupposing entities that do not physically exist: infinite sets, a global choice function, and the set of all possible subsets. From the perspective of generativist mathematics, this paper systematically reveals five conflicts between ZFC axioms and physical reality, and argues that these artificial presuppositions have led to four profound consequences: a fundamental cleavage between local and global descriptions, an exponential explosion of energy consumption for global coordination with system scale, the absence of self-evolutionary principles and mechanisms in artificial intelligence, and a long-standing lack of a suitable mathematical foundation for the deep study of complex systems. The four axioms of generativism—Distinguishability, Finiteness, Determinacy of the Successor, and Local Coupling—offer an alternative kind of stability. This stability is not logical non-contradiction, but inevitable evolutionary convergence. It is not presupposed, but rigorously derived from the dynamics of local coupling. This paper does not aim to overturn the logical self-consistency of ZFC, but to expose its physical irrelevance and its systematic obstruction to scientific research, and to present an alternative foundation aligned with physical reality. Keywords: ZFC; generativism; stability; mathematical foundation; physical reality; evolutionary convergence; complex systems; artificial intelligence