Pardon my ignorance, but I wasn't aware that the "strange" results dealing with infinite sets of various cardinality were related to the "strange" results related to accepting the axiom of choice. Is this a limitation of my mathematics education, or are the infinite set "paradoxical" results independent of the axiom-of-choice sphere cutting "paradoxical" results?

To really push my understanding of the terminology, I thought that definitions of equivalent size for infinite sets based on one-to-one and onto correspondence did not require reference to the axiom of choice.

Alternatively, I'm not understanding the implications I'm supposed to get from:

How about this? Take the set of all natural numbers. Divide it into two sets: the set of even naturals, and the set of odd naturals. Now you have two infinite sets, the set {0, 2, 4, 6, 8, ...}, and the set {1, 3, 5, 7, 9, ...}. The size of both of those sets is the ω - which is also the size of the original set you started with. Now take the set of even numbers, and map it so that for any given value i, f(i) = i/2. Now you've got a copy of the set of natural numbers. Take the set of odd naturals, and map them with g(i) = (i-1)/2. Now you've got a second copy of the set of natural numbers. So you've created two identical copies of the set of natural numbers out of the original set of natural numbers.