Suppose I were to tell you that you could show me a long, straight
piece of cake, and I could start eating it in such a way that if you
are just patient enough to wait an infinite number of minutes, I would
hand you back an amount of cake that in some sense was the same as
what you started with, despite having eaten enough that I now had as
many pieces in my tummy as you had on your plate. Would you believe
me? Well, there are some physical limitations here, such as the
slight matter of the inifinite minutes, and the slight explosion we
might get when I start biting into the cake at the atomic level, if my
teeth were capable of such a feat. But, if it's a purely mathematical
cake, you might be surprised to learn that this can actually be done.
The principle I'm talking about here is known as the 'Cantor Set',
named after the famous Georg Cantor, who made many breakthroughts in
our understanding of infinities. That name is actually a bit unfair,
since another mathematician, Henry John Stephen Smith, discovered the
set a bit earlier, but Cantor is credited for introducing it more
widely. Here's the basic idea.
Take a length-1 line segment, and cut out the middle third. Then
cut out the middle third of the remaining two segments. Continue the
process to infinity, cutting out a third each time. The Cantor Set
consists of all the points remaining in the line after all our cuts.
Now let's ask ourselves: what is the total length of the Cantor Set?
You can quickly see that each step multiplies the total length by a
factor of 2/3: if we started at length 1, then our total length after
the first cut is 2/3; the total length after the second cut is
2/3*2/3; the total length after the third is (2/3)^3, and so on. You
can quickly see that this total value approaches zero as our cuts
approach infinity. So the total length, or measure, of the Cantor set
is zero-- and thus the parts that you cut out will eventually have a
total length equal to the full line segment we started with.
So, if we have cut out an amount equivalent to the full line, we
must not be left with that many points then, right? On the contrary,
our Cantor set has just as many points as the line we started with.
To see this, let's try expressing each point on the line as a
(possibly infinite) base-3 decimal: that's a decimal point followed
by numbers which can only be 0, 1, or 2. If you think carefully about
how we constructed the set, you will see that our first step, of
cutting out the middle thirds, precisely removed the points who had
a 1 rather than a 0 or 2 in the first position of their base-3 value.
The second step removed, from among the remaining points, the points
that had a 1 in the second position of their base-3 value. After our
infinite number of steps, the Cantor set contains precisely those
numbers with only 0 or 2 in their base-3 value: .22, .020, .0022200,
and so on.
Now comes the clever part. For each number in our Cantor set, map
it to a base-2 decimal, where all the 2s are replaced with 1s. Because
our set consisted of all possible decimal combinations of 0s and 2s,
this means we have mapped our set to all possible decimal combinations
of 0s and 1s. But in base 2, every number is a decimal combination of
0s and 1s-- so this mapping shows that for every point in the real
number line, a corresponding point exists in the Cantor set. And thus
we have as many points in the Cantor set as we did in the original
line, despite its total length being 0.
The Cantor set has a number of other interesting properties. It
can be seen as an early example of a fractal, and in fact the well
known 'Sierpinski_carpet' and 'Menger Sponge' fractals, which look
like crazy squares or cubes with lots of holes, can be seen as
higher-dimensional analogues of this set. It's also a totally
disconnected set: for any two points, look at the first ternary digit
where they differ, and you can divide the Cantor set into closed
subsets based on that digit, where the two points will be in different
ones. A bunch of other interesting properties, a bit too complex to
detail in the podcast format, are also described on the Wikipedia
page linked in the show notes.
Overall, I think this is yet another great example of the
counterintuitive consequences of dealing with infinity. This idea was
said to be an important step for Cantor on the way to developing his
general theories of infinite sets.
And this has been your math mutation for today.
Cantor Set at Wikipedia