Cantor: The Uncountable Continuum, part 1

The problem of comparing different domains of possibility — different infinites — is a difficult one. We'll make use of a powerful tool developed by Georg Cantor, for the demonstration of different sizes (“cardinalities”) of infinites. This video covers the content of this section, but it also gives away some answers. It's up to you whether you'd rather start with the video or the webpage.

To get into Cantor's concept, we'll start with some experiments you can participate in. To start, go through the following images in order. For each one, figure out whether there are more uppercase or lowercase letters:

Slide 1
Slide 2
Now with color!
Slide 3
Slide 4
Slide 5
This one's easy!
Example Frame

I’m sure you noticed that the examples got easier and easier, first through the introduction of color, then by coupling, and then by arranging them neatly. In the last example, it is very easy to see that there are more lowercase than uppercase letters, because the g is all by itself, without a companion uppercase letter. Try looking through the examples again. In the first two examples, you probably counted up all the uppercase letters and lowercase letters separately, and compared your totals. In the third and fourth examples, since the letters are coupled, you just have to look for any solo letters, which makes the extra q easy to spot. And in the fifth case, there's no trouble at all: the lonely g stands out.

Now, we’re going to expand the experiment just a little bit, as we move from finite numbers of objects (in this case, letters), to infinite numbers of objects, as we look at numbers themselves.

Counting infinites

Let’s start with this set of numbers. As you can see, there isn’t enough space to write them all, but there are enough to give you the idea. Now, how many numbers are there in the idea of the series illustrated here?

That’s right, an infinite number. Now, let’s compare that to another unending group of numbers, the even numbers:

Again, the number of even numbers is... infinite! Would it be possible to ask whether it’s the same infinite? It seems reasonable to think that the number of all numbers is larger, twice as big, as the number of even numbers. After all, there are all those odd numbers that aren’t found in the second set! Here’s where a difference between finites and infinites comes in — the way we count things can change how many we have. What if we align the numbers like this:

Now, every whole number has its even number partner, just like the uppercase and lowercase letters in the fifth letter example above. But unlike that example with letters, here there is no solitary g, no extra whole number without a partner. That is, we can't point to any whole number without its even double, or any even number without its half. Since all numbers are in such partnerships, there can’t be more whole numbers than even numbers! Pretty shocking, isn’t it? Of course, we are dealing with infinites, so perhaps the rules have to change.


Next: Cantor: The Uncountable Continuum, part 2