Infinity within infinity
It was the great 16th-century Italian polymath Galileo Galilei(1564–1642)who wasfirst to alert us to the fact that the nature of infinite collections is fundamentally different fromfinite ones. As alluded to on thefirst page of this book, the size of afinite set is smaller than that of a second set if the members of thefirst can be paired off with those of just a portion of the second. However,infinite sets by contrast can be made to correspond in this way to subsets of themselves(where by the term subset I mean a set within the set itself). We need go no further than the sequence of natural counting numbers 1,2,3,4,···in order to see this. It is easy to describe any number of subsets of this collection that themselves form an infinite list, and so are in a one-to-one correspondence with the full set(see Figure 8):the odd numbers,1,3,5,7,···, the square numbers,1,4,9,16,···and, less obviously, the prime numbers,2,3,5,7,···, and in each of these cases the respective complementary sets of the even numbers, the non-squares, and the composite numbers are also infinite.
The Hilbert Hotel
This rather extraordinary hotel, which is always associated with David Hilbert(1862–1943), the leading German mathematician of his day, serves to bring to life the strange nature of the infinite. Its chief feature is that it has infinitely many rooms, numbered1,2,3,···, and boasts that there is always room at Hilbert’s Hotel.
8. The evensand thesquarespaired with the natural numbers
One night, however, it is in fact full, which is to say each and every room is occupied by a guest and much to the dismay of the desk clerk, one more customer fronts up demanding a room. An ugly scene is avoided when the manager intervenes and takes the clerk aside to explain how to deal with the situation:tell the occupant of Room 1 to move to Room 2 says he, that of Room 2 to move into Room 3, and so on. That is to say, we issue a global request that the customer in Room nshould shift into Room n+1, and this will leave Room 1 empty for this gentleman!