Hilbert's Hotel is not showing a contradiction. It *is* a paradox, in the sense that it shows how infinite sets differ from finite ones. But there is no specific contradiction you can point out.
And it is even OK to talk about infinite cardinal numbers. They just don't obey the same rules of arithmetic that finite numbers do (although they do obey some of them).
Saying that the infinity of all positive numbers is different than the infinity of even numbers is problematic. It's like saying the three for the set {1,2,3} is different than the three for the set {3,4,5}. Yes, sure, but so what? The sizes are the same in the Cantorian sense. In other senses, the sizes are different. And that is one thing that needs to be addressed when discussing infinite sets: there are many different notions of 'size' and they may disagree here. Proper subsets may be the 'same size' as the 'larger' set. In fact, that is one *definition* of what it means to be an infinite set!
And it is even OK to talk about infinite cardinal numbers. They just don't obey the same rules of arithmetic that finite numbers do (although they do obey some of them).
Saying that the infinity of all positive numbers is different than the infinity of even numbers is problematic. It's like saying the three for the set {1,2,3} is different than the three for the set {3,4,5}. Yes, sure, but so what? The sizes are the same in the Cantorian sense. In other senses, the sizes are different. And that is one thing that needs to be addressed when discussing infinite sets: there are many different notions of 'size' and they may disagree here. Proper subsets may be the 'same size' as the 'larger' set. In fact, that is one *definition* of what it means to be an infinite set!
