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On using transfinite numbers to navigate The Library Of Babel
We can use transfinite numbers to navigate the library, because we can imagine all possible books in it. It is only a matter of how many duplicates there are. For instance, if you take a book off of one shelf and then put it back on that same shelf, that does not count as two different books (or even one). If you took another copy from the same shelf and placed it on top of the first copy, this would be counted as two copies.
To make this clear, imagine that every book in the library is numbered. Then each copy of a given number refers to one and only one book - no more or less than that.
So, the entire library can be conceived as a single book numbered 1. Each distinct copy of this book is really just an additional number added to it (the number 2 refers to another book with all of the same books in it that are also in Book 1). This way, you could navigate through all of the books without having any duplicate copies count.
If you wanted to know how many books there were without having to actually count them all, then you would use the Aleph-One number system. This is a special transfinite number system that has only one book (Book 1) in it, and each copy of Book 1 counts as zero books.
This is because Book 1 contains every possible book in it. So, the number zero multiplied by itself as many times as there are books would be exactly equal to all of the possible books.
Now, we can use the Aleph-One number system to count how many books there are in The Library Of Babel. Each copy of Book 1 counts as zero, because it is just a duplicate of every possible book.