Yeah, but if natural numbers are infinite, than there will always be a natural number to map to a real one, therefore they're the same cardinality.

Strangely enough, this isn't actually true! Ok, so it is true that if you were to go through one by one and label some real numbers with natural number, you would never run out. But the interesting thing comes when you start with the assumption that you have labelled every real number with a natural number. Turns out there's always some left over...

There is a famous proof about this by Georg Cantor: http://en.wikipedia.org/wiki/Cantors_diagonal_argument

It's only a proof if you take a leap of faith and accept certain things. You could argue that that's how math works, but to me the less axioms we need, the better.

I really love this quote from Wittgenstein:

"Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one".

http://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_the...

Which assumptions do you find troubling?

> That although both are infinite, there are actually more of one than the other.

That's not the assumption, that's the result.

You're right. The assumption would be that there's a countability difference between Naturals and Reals.

That Naturals are infinite but countable and Reals infinite and uncountable.