The material from now on is starred.
Recall that two sets and have the same cardinality (
) if, and only if, there is a bijection between and .
One can show
(the Schröder-Bernstein theorem) that if there is an injection from
to and an injection from to then there is a bijection
between and .
For any set , there is an injection from to
, simply
. However there is never a surjection
, so
, and so

for some sensible meaning of .

Theorem 7.4.1
There is no surjection
.

Proof.
Let
be a surjection and consider
defined by
. Now
such that . If then
but -- a contradiction. But if
then
and -- giving a contradiction either
way.