Cantor’s Theorem and the Problem of Infinity
Jennifer Lee Lawson, MA
Infinity. Try for a moment to grasp it. Numbers with no end. Is it that it exists and we cannot
grasp it, or is it that it simply does not exist? In this paper, I will argue that infinite cardinal
numbers may not exist. I will analyze Cantor’s Theorem to show this. First, I will explain
Cantor’s Theorem. Then, I will argue why I believe, when tried for infinite Sets, it is truly possible
that he is incorrect. I will argue without using many mathematical or logical symbols, for ease of
reading.
Cantor’s Theorem is well known in mathematics, logic and philosophy of mathematics. It is a
theorem in Set Theory. In the end, Cantor’s Theorem aims to show there is no largest cardinal
number. In other words, there is no largest infinity.
The concept of infinity is taught, in the United States, in public schools in our Math classes. On
our finite chalkboards, we use < — > to symbolize numbers that go on forever in both directions.
Infinity, the concept, however, is rarely ever fully explained. Cantor helps us actually visualize it
with the following picture:
The bottom circle in the picture is called an Empty Set, which is a Set with nothing in it. Just
above that, there are more circles, and the letters inside symbolize cardinal numbers. Those are
called Sets. The top circle appears to show infinite cardinal numbers. An apparent paradox
occurs when one tries to put a circle around the entire picture, otherwise known as The Set of
All Sets. The Set of All Sets contains every number. In other words, if The Set of All Sets is
possible, cardinal numbers may not go on forever. There may be a vast amount of them; too
many for a human being to ever count, possibly; but they may, in fact, end at some point.
Let us not use a chalkboard, then, and simply visualize with our minds. Take a look at the
picture above, Cantor’s Theorem. Just by considering it a graphic, we can see that, clearly, it is
possible to draw a circle around all of the Sets. Is it impossible? No. It is not. We could simply
draw. There is not one thing stopping us from doing that. When we draw a circle around it, is
that paradoxical? No. It is not. It’s simply a bunch of circles and letters with a bigger circle
encompassing them. When we do this, it is The Set of All Sets, which has, for a very long time,
been considered impossible or paradoxical.
Can we conceive of The Set of All Sets now? Maybe you can, especially if you come at this
problem with no prior assumption about the concept of infinity and simply use your hand,
supposing you have one, to just draw a circle.
