Recall that we have set up a correspondence between the points of and the collection of all sequences of L's and R's.

But what's so special about L's and R's? Not a thing. If we can agree to use sequences of 0's and 1's instead, then it's nearly immediate that is a BIG set! Why? Well, a sequence of 0's and 1's

looks just like a **binary decimal**

Now each point in [0,1] has a binary decimal expansion and, conversely, each binary decimal represents some point in [0,1]. Thus, we have a correspondence between the points in the Cantor set and the points in the interval [0,1]. Amazing!!

In truth, there are some details to check here. For example, a given element of [0,1] may have more than one binary decimal expansion; indeed, 1/2 can be written as

(How would you check this?)

The
key observation here is that we can specify a mapping,
or correspondence, from
**onto** [0,1]. Since
is a subset of [0,1],
this means that [0,1] and
must be the same "size" (or have the same **cardinality**) --
but let's save the actual details for another day.

Is this cool or what? A set so small that we couldn't possibly draw it on paper (or, in this case, on a computer screen) is still, somehow, as big as an entire interval.

Here's that
"big versus small" deal again: The Cantor set is apparently
"big", since it has the same cardinality as the interval [0,1]
(an **uncountable** set).
If you stick around to read
**yet more on the Cantor set**,
you would see that the Cantor set is pretty darn small, too.
Of course, if you're tiring of magic and wonder, you could always
wimp out.

Neal Carothers - carother@bgnet.bgsu.edu