elementary set theory - Does Cantor's Theorem and the Continuum Hypothesis imply discrete levels of infinity?

Cantor's Theorem shows that there are an infinite number of distinct infinite
set cardinalities, as there is at least one infinite set, and it provides a method for producing a set with a larger cardinality from another set that works even for infinite sets.

The Continuum Hypothesis (which has been shown to be neither provable nor disprovable, but for the purposes of this question let us assume that it is true) claims that there are no sets with a cardinality between that of integers and the real numbers.

Furthermore there is proof that the cardinality of the integers is the smallest of the infinite cardinalities (Infinite sets with cardinality less than the natural numbers).

And the increment provided by Cantors Theorem (the powerset) happens to take the integers and create a set with the same cardinality as the reals.

Does all this imply an infinite sequence of discrete cardinalities exist? Has this already been studied? Does this give us the potential to have "infinity numbers" With infinity 1 denoting the cardinality of the integers and infinity 2 denoting the cardinality of its powerset, and so on rather than just countable and uncountable infinities?