To see this, suppose that we are handed any numbered list (even an
infinite list!), or sequence of elements from our set. We have to
show that such a list cannot account for every element from our
set, and so we want to find an element from our set which does
not appear on the list.
To this end, consider a "typical" list:
You will notice that I have highlighted certain digits:
The first digit of 
, the second
digit of 
, the third digit of
, and so on. We will use these
to build a sequence of 0's and 1's that is guaranteed not to
occur anywhere on the list.
Specifically, set
= 0.10110... (base 2),
where the 
th digit of
is chosen to be the opposite
(or "2's complement") of the th
digit of 
. That is, the first
digit of is taken to be 1 because
the first digit of is 0, the
second digit of is taken to be 0
because the second digit of is 1,
and so on.
By design, can't possibly
occur anywhere on our list. Why? Well,
can't equal because they differ in
the first decimal place. And
can't equal because they differ in
the second decimal place. And ... so on. Cool!
This shows that the collection of all infinite sequences of 0's
and 1's cannot be counted. Since we've managed to identify this set with
the interval [0,1], this means that [0,1] can't be counted either.
In other words, these are examples of uncountable sets.
In conclusion, we've seen that uncountable sets are "more infinite" than
countably infinite sets. You might find it entertaining to mull
over the following characterizations:
Another, even more curious, example of an uncountable set
is provided by the Cantor set,
the next installment in our saga.