A Rigorous Procedure for Generating a Well-ordered Set of Reals without use of Axiom of Choice/Well-ordering Theorem

Abstract

Well-ordering of the Reals@@ presents a major challenge in Set theory. Under the standard Zermelo Fraenkel Set theory (ZF) with the Axiom of Choice (ZFC), a well-ordering of the Reals is indeed possible. However the Axiom of Choice (AC) had to be introduced to the original ZF theory which is then shown equivalent to the well-ordering theorem. Despite the result however, no way has still been found of actually constructing a well-ordered Set of Reals. In this paper the author attempts to generate a well ordered Set of Reals without using the AC i.e. under ZF theory itself using the Axiom of the Power Set as the guiding principle.