General Mathematics

The Countable Real Numbers

Authors: Kesan Yi
Comments: 10 Pages. (Note by ai.viXra.org Admin: For the last time, please cite listed scientific references!)

If it is declared that real numbers are countable, the Cantor's diagonal process must be discussed. We try to find the problem of Cantor's proof by using the same method and logic to prove that a countable set is uncountable. That means real numbers may be a countable set but it was proved uncountable by Cantor's diagonal process. By critically analyzing the ontological foundations of the "Abstraction of Actual Infinity", we expose an illicit methodological shift in classical set theory: the unrestricted extension of actual infinity from the internal generation of infinite sets to external mappings between distinct infinite totalities. By redefining countable infinity purely through an intrinsic single-valued successor operations derived from the Peano framework, we bypass the mapping dilemmas of Cantor's definition of countable set with abstraction of actual infinity. Utilizing this new framework, we construct a "Right-ward Mirroring Successor Operator" capable of sequentially traversing all decimal expansions of real numbers within the interval [0, 1), thereby establishing that real numbers are countable under both potential and actual perspectives of infinity. Finally, we expose a fatal logical rupture within the Dedekind cut framework regarding the synchronization of measure theory derived from uncountable reals and total order set, validating our countable thesis as a necessary resolution to the internal contradictions of classical continuum mechanics.
Category: General Mathematics