Alexey (KAMAZ) Petrov

The Infinium △₁ₓ₁ as a Single Source of the Discrete and the Continuous: A Motivic Bridge Between Arithmetic and the Continuum (Resolution of the Ancient Dichotomy within Δ‑Ontology)

The paper examines a fundamental problem in the foundations of mathematics—the gap between the discrete and the continuous, which can be traced from Zeno’s ancient paradoxes to the modern Millennium Prize problems. It is shown that traditional approaches either postulate the irre...

The Unity of the Discrete and the Continuous as a Structural Principle: The Infinium △₁ₓ₁ and the Theory of Motives

This article is devoted to one of the oldest and deepest problems in mathematics—the gap between the discrete and the continuous. We show that this gap is not an accidental difficulty but a direct consequence of choosing the structureless point as the initial primitive. Within Δ‑...

The Infinium as a Bridge Between the Discrete and the Continuous: The RIT and Its Motive Structure (Why a Single Triangle Resolves the Oldest Paradox of Mathematics)

The problem of the relationship between the discrete and the continuous is one of the deepest and most painful in the foundations of mathematics. Classical approaches either postulate the existence of the continuum without explaining its origin, or reduce continuity to infinite c...

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

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...

Essay: Ontological Remodeling: Demolishing the Walls of Reality

The essay "Ontological Remodeling: Demolishing the Walls of Reality" introduces and develops the concept of ontological remodeling as a diagnostic tool for understanding fundamental shifts in the foundations of human existence. The author argues that contemporary technological an...

The Geometric Quantum as a Universal Generator of Hierarchies: Δ-Ontology, the Principle of Energy Economy, and the Dissolution of Fundamental Paradoxes

We show that the right isosceles triangle with legs 1 and hypotenuse √2 — the infinium 𝕴 = △₁ₓ₁ — is the minimal structural quantum generating all of mathematics through a single mechanism: the diagonal hierarchical transition governed by the principle of energy economy. Gödel's...