In reading another question (Explaining Infinite Sets and The Fault in Our Stars) it got me thinking about the way that you can prove that the number of numbers between 0 and 1 and between 0 and 2 are the same. (apologies if my terminology is a bit woolly and imprecise, hopefully you catch my drift though).
The way it is proved is that you can show that there is a projection of all the numbers on [0,1] to [0,2] and vice versa. I'm good with this.
However I then got to thinking that you can also create a projection that takes all the numbers from [0,1] and maps them to two numbers from [0,2] by saying for a number x it can go to x or x+1. This is reversible to so you can say that you can find a pair of numbers in [0,2] such that they differ by one and the lowest is a member of [0,1].
Why is it that this doesn't prove that there are twice as many numbers in [0,2] than in [0,1]. It seems to me that this is the crux of why it runs counter to intuition but I can't work out the flaw.
Or is it just in the nature of infinity that infinity*2 is still the same infinity and thus its just that infinite is "weird"?