Cantor’s Diagonal Argument
Recall that. . .
•A set Sis finite iff there is a bijection between Sand {1,2,...,n}for some positive integer n, and
infinite otherwise. (I.e., if it makes sense to count its elements.)
•Two sets have the same cardinality iff there is a bijection between them. (“Bijection”, remember,
means “function that is one-to-one and onto”.)
A set Sis called countably infinite if there is a bijection between Sand N. That is, you can label the
elements of S1, 2, . . . so that each positive integer is used exactly once as a lab el.
Why “countably infinite”? Such a set is countable because you can count it (via the labeling just mentioned).
Unlike a finite set, you never stop counting. But at least the elements can be put in correspondence with N.
On the other hand, not all infinite sets are countably infinite. In fact, there are infinitely many
sizes of infinite sets.
Georg Cantor proved this astonishing fact in 1895 by showing that the the set of real numbers is not
countable. That is, it is impossible to construct a bijection between Nand R. In fact, it’s impossible to
construct a bijection between Nand the interval [0,1] (whose cardinality is the same as that of R).
Here’s Cantor’s proof.
Suppose that f:N→[0,1] is any function. Make a table of values of f, where the 1st row contains the
decimal expansion of f(1), the 2nd row contains the decimal expansion of f(2), . . . the nth row contains the
decimal expansion of f(n), . . . Perhaps f(1) = π/10, f(2) = 37/99, f(3) = 1/7, f(4) = √2/2, f(5) = 3/8,
so that the table starts out like this.
nf(n)
1 0 .3 1 4 1 5 9 2 6 5 3 ...
2 0 .3 7 3 7 3 7 3 7 3 7 ...
30.1 4 2 8 5 7 1 4 2 8 ...
4 0 .7 0 7 1 0 6 7 8 1 1 ...
5 0 .3 7 5 0 0 0 0 0 0 0 ...
.
.
..
.
.
Of course, only part of the table can be shown on a piece of paper — it goes on forever down and to the
right.
Can fpossibly be onto? That is, can every number in [0,1] appear somewhere in the table?
In fact, the answer is no — there are lots and lots of numbers that can’t possibly appear! For example, let’s
highlight the digits in the main diagonal of the table.
1
