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Originally Posted by BBKing
Ooh - I remember this from a very old maths book I picked up in a second-hand bookshop in Cambridge - it seems that infinity * any real number = infinity, infinity+ any real number = infinity, infinity + infinity = infinity, but infinity ^ infinity is a bigger infinity than the one you started with. At which point my brain explodes.
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Imagine you have a long list of every positive integer, i.e. the length of the list is infinite.
Then to the right of each integer, write down the number you get when the original is doubled. This list (of all even numbers) is also infinite, and the same length of the first.
Now to the right rewrite the second list, but inserting all odd numbers. This list is also infinite, but twice as long as the other two, even though it contains the same numbers as the first.
