Paradoxes of Naive Set Theory and Their Resolution in Δ‑Ontology: From the Structureless Point to the Structural Quantum

Abstract

This paper examines the classical paradoxes of naive set theory: Russell’s paradox, the barber paradox, the Grelling–Nelson paradox, the Burali‑Forti paradox, as well as the Banach–Tarski paradox and Zeno’s paradox (Achilles and the tortoise). It is shown that their emergence is due to two factors: the use of a structureless primitive (a point or an arbitrary set) and the unrestricted comprehension principle, which allows the formation of sets from any properties. We analyze the standard solution within ZF(C), which introduces external restrictions on set formation but does not eliminate the deeper cause—the absence of constructive structure in the basic objects. The alternative approach of Δ‑ontology, based on the structural quantum—the right isosceles triangle △₁ₓ₁—naturally avoids these paradoxes, since the existence of an object in this system means presenting its constructive structure (a △‑mosaic). It is shown that Russell’s paradox cannot be formulated in Δ‑ontology, the Banach–Tarski paradox loses its force due to the absence of non‑measurable sets, and Zeno’s and Burali‑Forti’s paradoxes are resolved through discreteness and the constructive hierarchy of types. We conclude that the paradoxes are symptoms of a structureless foundation, and their elimination requires replacing the primitive, not merely restricting the axioms. Δ‑ontology works like a microscope: magnifying a point reveals a fractal RIT inside it, not emptiness.