TITLE: How big is infinity?
QUESTION [39 upvotes]: This might be more philosophy than math, but it’s been bothering me for a while.

Question: If there’s an infinite amount of real numbers between $ 0 $ and $ 1 $, shouldn’t there be twice the amount of real numbers between $ 0 $ and $ 2 $? Wouldn’t that be $ 2 \times \infty $? What about the amount of real numbers between $ 0 $ and $ \infty $? Wouldn’t that be $ \infty \times \infty $?

I guess the whole concept of infinity just kind of evades me. Any help would be appreciated!

REPLY [1 votes]: This is a great anecdotal way to learn about different sizes of infinity.
Wikipedia Article - Hilbert's Hotel
So there are different sizes of infinity.  Suppose you have two sets A and B and both of those sets are full of infinite things, that is, each of those two sets has infinite items.  Not to say that they are they same size, one of their infinities may be bigger than the other.
If for every single item in A I can map it to exactly one item in B and I have all items in B covered, then the two sets are the same size of infinity. 
If in all 1-1 mapping from A to B in which all items in A are used there are always items in B leftover, then the size of B is a bigger infinity than that of A.
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