On the Continuum Hypothesis and Diagonal Method

Abstract Cantor used a hypothetical list to prove that real numbers in the range (0,1) are uncountable. The indexes of the numbers in this list are natural numbers, and I agree that Cantor's proof is true for a list where the indexes are natural numbers. However, Cantor proved that the cardinality of real numbers in the in the range (0,1) is greater than ω in this proof. He also defined the number ω1 as greater than all natural numbers. Therefore, I add a real number to this list that I assume is in the range (0,1) and has an index of ω1. Then, I use a similar method to what Cantor did with his hypothetical list; I mean I try to find a real number that is not in the range (0,1) in my new hypothetical list. However, this new method differs from the Cantor diagonal method because in this new list, it is assumed that there is a real number whose index is not a natural number. I call this new method DM2 and using this method, I study the cardinality of digits of real numbers that exist in a defined range. I present some new results regarding the Continuum Hypothesis via this method.