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

TL;DRAbstract

Well-ordering of the Reals presents a major challenge in Set theory. Under the standard Zermelo Fraenkel Set theory  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  i.e. under  theory itself using the Axiom of the Power Set as the guiding principle.

Chat with Paper

AI Agents for this Paper