My friend wrote:I've been thinking.

Say you have an X-Y graph and you plot x=y starting at x=0. the area under that is infinite, right?

Then if you have x=2y starting at x=0. The area under that is also infinite.

The mathematical argument goes that both areas are equal because infinity is infinity and you can't have a greater or lesser infinity. I think that argument is bunk.

I responded:

I wrote:The even integers are also the same countable infinity as all the integers. It seems hard to believe but it can be proved. The proof is basically that if you can map every number from one set onto a unique number from another set, than the size of the sets are the same. So to map even integers onto the set of countable numbers, just divide by two (2->1, 4->2, 6->3, etc). It may be hard to wrap your head around, but that's just how infinity works.

He was unconvinced:

My friend wrote:That's the traditional argument used for number sets, but the fact is also that while you can match an even number with a non-even number, you can also fit all the odd numbers in between for the full set while you can't with just the even number set.

The real difference shows when you look at areas like in the x=y example. And yes, each point of infinity can be matched with another point of infinity, but the fact of the matter is that at any given value, the area under x=y is more than twice as great as the area under x=2y. ... Greater and lesser infinities.

I know his reasoning is faulty, but I'm not sure what the counterargument is. Anyone know?