The important feature of Hilbert’s example is not the hotel, or the guests, but the strange consequence of Cantor’s method for ‘counting’ an infinite collection of objects. William Dunham’s Journey through genius does an excellent job with Cantor in his final two chapters. I’m paraphrasing him.

Suppose you want to know if you have more than 5 raspberries. You could count them, or you could stick them on the ends of your fingers. If you have berries left over, there were more than five. If all the berries fit all the fingers, you have put your fingers in a 1-to-1 correspondence with the berries, and concluded that the numbers of each are the same. That’s how small children count, and Cantor’s genius was to extend this idea to sets of infinite objects.

If you can find a 1-to-1 correspondence between the counting numbers (1,2,3,…) and some other set of objects, then the size of the two sets must be equal (in the business, size of sets is called the ‘cardinality’). It is traditional to denote the counting numbers by ‘n’… Hilberts ‘paradox’ arises from Cantors claim that there are as many counting numbers as even numbers. The correspondence is n->2n. For every counting number, I can double it to find every even number. That’s weird.

You can just as easily show that there are as many odd numbers as counting numbers. The scheme is more complicated, but you can also find a 1-to-1 correspondence between the counting numbers and all the fractions between 0 and 1. It seems like the counting numbers can count everything! But that’s not the whole story. The big shock is that there are more numbers between 0 and 1 than counting numbers. That is Cantors ‘nondenumerability of the continuum’. I highly recommend Dunhams book. The proof of the nondeumerabilty theorem is so beautiful and accessible (like accessible to a ten year old).