Well, you need books of infinite length.
First you start by considering books with an infinite (but countable) pages: perhaps, these can be imagined as displays of some sort that let you enter a number n and return you the contents of the n-th page.
Then you put all these books in your library - the practical details are left as an exercise. These are already enough to have |R
| books, but why stop here?
If you want to go even higher in the transfinite hierarchy, you treat each one of these books as if they were page numbers instead of books: that is, you develop another kind of "book" which is a display that lets you enter one of the first-level "books" and return the corresponding page.
In a certain sense that can be made precise, there are more "second level books" of this sort than there are "first level books".
Then you repeat the procedure again, and build "third level books", "fourth level books", and so on.
You do this for for all integer levels. Then what can you do? Why, of course, you can build "omega-th level books" that take in input any
book of a previous level and return a page. Then you build "omega-plus-one-th" level books, "omega-plus-two-th" ones, and so on, up to infinity. And then you do it again, and again, and again, and do this too infinitely many many times, and then you do this
infinitely many times, and so on.
Again, the practical details are left to the reader - what am I, an engineer?
edited 25th May '11 4:53:07 AM by Carciofus