Interesting approach, but it has been proven by mathematicians (I'm not one) that the cardinality of the set of the real numbers between 0 and 1 is the same as the set of all the real numbers. I would suggest discussing your approach and your doubts with a mathematician, as when talking about infinities there is a lot counterintuitive stuff going on. Another commenter already pointed out a possible problem of your "mirror" method: you're relying in being able to read a number backwards. To do that, you need to know all the digits. But it doesn't work well with numbers with infinite decimal digits like pi or its inverse.