I was reading about this topic of my own leisure, and I came across something that I couldn't quite understand. The solution of Galileo's Paradox is that the set of natural numbers and the set of perfect squares are both infinite sets of the same cardinality (namely aleph 0). This I can understand. There can be established a 1:1 correspondence between each element of the two sets. But then why is the set of real numbers larger than the set of natural numbers? Since the latter set is infinite, can't I use the same logic as above to show a 1:1 correspondence? Can anyone explain this to me intuitively?