Originally posted by doublez
UM... "there are even more individual numbers than there are squares".
Sorry doublez, that isn't the way it works. A common error people make when people compare these kinds of sets goes something like this:
"There are more individual numbers than there are squares. If we count from 1 to 100, we've got 100 numbers and only 10 squares. And if we count up to 1,000,000 we've got 1,000,000 numbers and only 1000 squares. Clearly, there are more individual numbers than squares."
For any finite set, that is true. But in an infinite set, we're not bounded by some upper counting value. All we have to count out more squares. So the counting would go something like this:
"If we count from 1 to 100, we have 100 numbers. To find an equal that number of squares, we look just look at the first 100 squares (up to the number 10,000)."
"But wait a minute!" you might say. "If we count up to 10,000 then we would have 10,000 numbers, and that's clearly more than the number of squares!" But all we'd have to do there is count the first 10,000 squares (up to the number 100,000,000).
"But wait a minute!" you might say again. "That's not fair! You're counting much higher to get the same number of squares!" which is true. However, you must remember that these are all finite slices of two infinite sets. When you're dealing with infinity, there is no ceiling to hit, no finish line to cross. This is a progressive embodiment of what Galileo called the "potential" of infinity in the link you provided. So it's not cheating to count ahead with one set (squares), because the other set (numbers) will catch up. There is no "head start" we can give to either set that the other set can't surmount.
That's why there are the "same" number of squares and individual numbers (meaning the sets have the same degree of infinity). Of course, real numbers (1, -2, 1/3, 0.5832298...) are a different kettle of fish... =)
I hope that helps.