If you do not like Cantor's Diagnol Argument, possibly because the same real number might not be unique, or the fact that rationals have repeating decimals. Then consider, a another variation of the Diagnol Argument.
Define . Thus this set is the set of all irrationals. Next, by the theory of continued fractions, every irrational number can be expressed as a unique infinite continued fraction. By that you assume that the irrationals are countable thus you can list them for example
Now employ the diagnol argument and you demonstrated that is uncountable.
Switch from number theory to set theory:
we have that,