If we take the definition of an infinite set as any set for which some proper subset has the same cardinality of its parent set. We can then state that the set of all Natural numbers, as an infinite set, has the same cardinality as the set of all even Natural numbers. Therefore the cardinality of the set of all odd Natural numbers has the same cardinality as the set of all even Natural numbers and the set of all Prime numbers, where prime numbers begin with 2 and progress 3, 5, 7, 11, … are within the set of all Natural numbers. Also the set of half of all Primes is equal in cardinality to half of the Natural numbers which would also be equal to the set of all Natural numbers in cardinality. It would, then, not be unreasonable to state that any geometric division (by geometric division, I mean any division obtained by some mathematical means other then counting out arithmatically a subset of the parent set; i.e. every other element, every fouth element, every power of 2 element and so forth) of an infinite set can itself be geometrically divided into an infinite set of equal cardinality to all its ancestral sets.

We can thus consider the set of all sets and subsets of an infinite set as having a cardinality equal to the cardinality of the base set (in this case, the Natural numbers set N). But, what about that 1-1 correspondence between the set of all Real numbers to the set of all Natural numbers? If we can divide the set of all Natural numbers up into an infinite number of subsets whos cadinality is equal to the parent set then how can we fail to find a corresponding element in the set of Natural numbers to the set of Real numbers? I have shown that I can divide the Natural numbers into an infinite set of infinite sets. Further more, each subset of infinite cardinality can contain completely unique elements from any other division of the parent set. So, before I have even made a dent in exhausting the elements of the Natural numbers, I can divide off a subset that could itself be divided off into a collection of infinite sets just to cover the elements between 0 and 1 in the Real number set. If there aren't enough, I can divide that sub sub sub set again and still have an infinily divisible set to cover the correspondance with. I would never lack for a unique element from the Natural numbers set to match up with an element to the Real numbers set.

I know that the numbers in the Natural number set would become absolutly enormous by the time we covered the interval from 0 to 1 in the Reals but, so what, I still have plenty to move to the next interval. In fact I still have the same number of elements in the Natural numbers that I had before I used up an infinitely small fraction of them to cover all previous intervals. I am not going to run out of Natural numbers by placing them in a 1 to 1 correspondence with any other set!