The Inconsistency of Mathematical Axioms: Premise Conflict Revealed by Zeno's Paradox and Aristotle's Wheel Paradox, and the Missing Projection Layer of "0"

Mathematics is widely regarded as a self-consistent, precise, and universally applicable symbolic system—a reliable language for describing the physical world. However, this paper demonstrates, through a critical analysis of the standard resolutions to Achilles and the Tortoise (Zeno’s paradox) and Aristotle’s wheel paradox, a long-concealed fact:

The same mathematical axiomatic system—real number axioms combined with calculus—relies on two mutually exclusive premises when addressing these two classical problems. In resolving Zeno’s paradox, it implicitly assumes that time possesses a minimal indivisible unit (discreteness). Conversely, in resolving Aristotle’s wheel paradox, it forcibly depends on the assumption that space is infinitely divisible (continuity).

Employing contradictory premises within a single axiomatic framework is not self-consistency—it is premise-level self-contradiction.

More fundamentally, this paper argues that the mathematical symbol “0” has never declared its projection layer in any application. The statement “a quantity is absent in a certain projection layer” has been illicitly equated with “absolute nothingness,” causing every physical expression involving “0” to smuggle unexamined metaphysical assumptions into its symbolic foundation.

We therefore propose: Any mathematical conclusion applied to the physical world must explicitly declare its premises and projection layer. Mathematics without declared premises is not precision—it is intellectual soliloquy disguised as rigor.