I was thinking about the fact that some infinities are 'bigger' than others. For example, the cardinality of the reals is literally bigger than the cardinality of the rationals. One cannot match them one-to-one. Georg Cantor proved this several times.
But if I look at the infinities and the items they contain, I noticed a pattern. Aleph-0 is the infinity of the rational numbers. Any rational number is either terminating or repeating. To store any given rational finite number, one would need a finite amount of information. No matter how precise, the precision would be finite, and thus could be representable in a finite amount of information (bits, digits, etc).
But then if I look at the power set of Aleph-0 which is has the cardinality of the reals, to specify any given element from any other requires infinite information. An infinite number of bits is required to name any given irrational number. However, the amount of information needed to name any one element of an uncountable set is infinite, but countable. Each digit of pi corresponds to a natural number.
So if I extend this, to, say, the power set of the set of the real numbers, would that mean, to specify any given element, I would need an uncountable amount of information? There would be no digits in order that could possibly name any given element in the set, even given an infinite number of them?
In summary, is my assumption correct that for any one of the infinity infinite sets of differing sizes, to single out any one element, would require an amount of information equal to the previous smaller infinity?