Set Theory and Logic

Relative Set Theory: Definitions of Finite and Infinite Sets

Authors: Wenjie Wu
Comments: 18 Pages. In Chinese (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

In response to Russell's Paradox, which arises from the a priori assumption of actual infinity and the unrestricted comprehension principle in classical set theory, this paper proposes a novel foundational framework: Relative Set Theory. Taking the concept of potential infinity as its ontological core, this theory strictly defines finite sets from the bottom up based on the dynamic generation logic of determined elements, and abolishes the absolute universal set to eradicate the logical root of the paradox. On this basis, the paper innovatively and uniformly defines an infinite set as the "limit of a strictly increasing sequence of finite sets," pointing out that static infinity divorced from a specific evolutionary process lacks logical validity. By introducing an algebra of relative magnitudes benchmarked against the standard sequence of natural numbers, this theory establishes the "principle of relativity" for infinite metrics, revealing that the size of an infinite set is essentially equivalent to the asymptotic growth rate of its generating sequence. This framework not only resolves the counterintuitive dilemmas of classical equipotence theory at the algebraic level but also provides a rigorous and self-consistent relativistic new paradigm for fundamentally circumventing the third foundational crisis of mathematics and re-examining the concept of infinity in mathematical philosophy.
Category: Set Theory and Logic