Cardinality of infinite sets Thank you everybody... when I posted the original question, which was really addressed to the moderators, I had no expectation of such a lengthy, interesting, and varied list of replies. I am surprised to learn that some mathematicians are inclined to be suspicious of transfinite theory, and question its value, because all of the philosophical problems of TF can be ultimately traced back, in logical sequence, to the 19th C Peano/Dedekind axioms of arithmetic - and if you are going to question those, then you can no longer assume that 1+1=2, and the entire science of mathematics falls on its butt in a crumbling heap!
As to different levels of infinity - the proofs that there must be at least two levels of infinity are so childishly simple, they can be understood even by high-school students who are not particularly bright in mathematics. The validity of the countable or aleph-null infinity is embedded in the properties of the natural number line. The validity of the aleph-1 or non-countable infinity is adequately demonstrated by Cantor's Diagonal Argument, which is very simple, and readily intellgible to non-mathematicians. Anybody who wishes to deny that higher-level infinities are valid in mathematical philosophy, must begin by pointing out where Cantor went wrong.
Are aleph-1 infinities useful in the sciences? Difficult question. We know that aleph-null infinities are not useful for investigating the origins of the universe, because every variable that we would wish to measure simply approaches infinity (or 0) as we approach closer to the singularity, and this of course is non-informative. Will the solution entail finding a way to apply the concept of the aleph-1 infinity? Some day, we'll know.