Transcending Rational Principles: Mathematical Infinity as a Foundation for New Principles – Four Absolute Binary Theorems as a Basis for Absolute Infinite Systems

Abstract

This paper explores the concept of mathematical infinity from a new perspective, redefining infinity as a family of absolute theorems, numbers, principles, and computational systems, rather than a merely quantitative notion. The study criticizes traditional logical and mathematical foundations-such as the principle of identity, the principle of non‑contradiction, quantitative equivalence and inequality, and the classical understanding of the four arithmetic operations as repeated aggregation or subtraction-and argues that these are only limited special cases that obstruct a deeper grasp of infinity and number. The main objectives are to: (1) define infinity in a new mathematical way, (2) define numbers as a special case of infinity, (3) identify primary mental principles as absolute mathematical systems, (4) present four absolute binary theorems as new foundational principles, and (5) establish a framework that transcends set theory and the geometric–quantitative version of Pythagoras’ theorem. Methodologically, the paper constructs and proves four absolute binary theorems, based on the principles of binary rank equivalence, reduction, and absolute binary ordering, and uses them to generate infinite computational systems. The results show that infinity can be understood as absolute infinite hypotheses, principles, theorems, and systems, and that numbers and primary mental principles are forms of infinity defined under absolute principles. Furthermore, the study demonstrates that the four arithmetic operations are, in their true nature, absolute infinite principles, and that the classical principles of identity and non‑contradiction are special finite cases of absolute non‑identity and contradiction. The paper concludes by calling for a comprehensive reformulation of mathematics, logic, and scientific knowledge on the basis of these new absolute principles.