A rigorous procedure for generating a well-ordered Set of Reals without use of Axiom of Choice / Well-Ordering Theorem

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. Introduction: In this paper, the author attempts a well-ordering of the Reals. Specifically the wellordering in achieved in the closed interval   1 , 0 . This does not in any way loose generality as the Set of Reals in the open interval   1 , 0 is equinumerous with the Set of Reals in      , via the tangent function   2 tan    x . As stated, well-ordering of the Reals presents a major challenge in Set theory. The most popular version of Axiomatic Set theory is the Zermelo Fraenkel Set theory (ZF) with the Axiom of Choice (ZFC). Under this theory a well-ordering of the Reals is indeed possible. However, a new Axiom, the Axiom of Choice (AC) had to be appended to the original axioms of ZF theory for this purpose. The Axiom of Choice is shown to be equivalent to Zorn’s lemma and the well-ordering theorem [1]. The wellordering theorem simply states that every Set can be well-ordered. The Axiom of Choice, though largely accepted by most mathematicians still retains a few detractors as the AC can establish the existence of certain Sets without actually specifying any way of constructing them. Even for those whose unquestionably accept the AC, any proof given using just ZF theory is considered in some sense ‘superior’ to the same proof given using ZFC. Coming back to the problem at hand, it is to be realized that although a well-ordering of the Reals is possible in ZFC, this remains very much an in-