Um, sorry if I came across as a jerk, I didn't mean to. Angela, I am trying to point out that talking about cardinality is a way of talking about size for infinite sets. But that the whole idea of size isn't even rational without number or distance or some other metric since as soon as that is introduced, it makes infinite division an immediate contradiction. Thus talking about cardinality in terms of size, and then talking about differently sized cardinalities, doesn't make much sense to me. Once you divorce size from the equation, it is gone.
Also, if looking at the real number line divided into one unit segments, couldn't one ignore the irrationals and see this also as an infinity of infinities of rationals (since each one unit is infinitely divided into rationals), and thus argue that the cardinality of the reals is the same as the cardinality of the rationals, which is aleph-naught? Am I way off here?
Sorry, this is just super interesting to me, and yes I know that the generally accepted view is that the cardinality of the reals is greater than the cardinality of the rationals. I would agree with that if the rationals were somehow limited to not divide out infinitely, but flying in the face of that is the fact that between any two rationals, I can actually find an infinity of other rationals. So theoretically the rationals are not limited from infinite division, despite this somehow breaking the definition of a rational number. Its all very weird. Thanks for any insight.