Computability and Logic

20 DIAGONALIZATION
single infinite list, in our sense of the term. In our sort of list, each entry must come
some finite number of places after the first.
As we use the term ‘list’, Zeus has not produced a list by writing infinitely many
infinite lists one after another. But he could perfectly well produce a genuine list
which exhausts the entries in all the lists, by using some such device as we used
in the preceeding chapter to enumerate the positive rational numbers. Nevertheless,
Cantor’s diagonal argument shows that neither this nor any more ingenious device
is available, even to a god, for arranging all the sets of positive integers into a single
infinite list. Such a list would be as much an impossibility as a round square:
the impossibility of enumerating all the sets of positive integers is as absolute as the
impossibility of drawing a round square, even for Zeus.
Once we have one example of a nonenumerable set, we get others.
2.2 Corollary. The set of real numbers is not enumerable.
Proof: If ξ is a real number and 0