The Infinite Dance of Cantor's Diagonal Argument
An infinite set may have the same cardinality as a proper subset of itself, as the depicted bijection f(x)=2x from the natural to the even numbers demonstrates. Nevertheless, infinite sets of different cardinalities exist, as Cantor’s diagonal argument shows – HiTe, Public domain, via Wikimedia Commons

Cantor’s Diagonal Argument. Imagine for a moment that you are sitting in a cozy library, surrounded by towering shelves filled with books. Each book represents an infinite set, and you have the ambitious task of listing all these infinite sets. You might think that infinity is just one big, unending concept, but Georg Cantor, a brilliant mathematician from the 19th century, shattered this notion with his groundbreaking work on the nature of infinity.

Cantor’s Diagonal Argument is one of those mind-bending ideas that makes you question everything you thought you knew about numbers and infinity. To understand it, let’s start with a simpler example: listing all the real numbers between 0 and 1. Imagine writing them down in an infinite list:

0.123456… 0.987654… 0.111111… 0.222222…

And so on.

Now, Cantor proposed a clever way to show that no matter how comprehensive your list is, there will always be some real numbers missing from it. Here’s how he did it: he constructed a new number by taking the first digit from the first number on the list, the second digit from the second number, the third digit from the third number, and so forth. Then, he altered each digit slightly—say by adding 1 to each digit (if it’s a 9, wrap around to 0). This new number will differ from every number on your list in at least one decimal place.

For example: – If your first number is 0.123456…, take the first digit (1) and change it to 2. – If your second number is 0.987654…, take the second digit (8) and change it to 9. – If your third number is 0.111111…, take the third digit (1) and change it to 2.

Following this method, you’ll end up with a new number like 0.292…, which is guaranteed not to be on your original list because it differs from each listed number at least in one decimal place.

This simple yet profound idea shows that no matter how you try to list all real numbers between 0 and 1, there will always be some numbers left out. Therefore, the set of real numbers is uncountably infinite, meaning its size is larger than that of countable infinity—like the set of natural numbers (1, 2, 3,…).

Cantor didn’t stop there; he went on to show that there are different sizes of infinity by comparing other sets as well. For instance, he demonstrated that while both the set of natural numbers and the set of real numbers are infinite, they are not equal in size—the latter being a larger type of infinity.

This revelation was nothing short of revolutionary because it contradicted centuries-old beliefs about infinity being a singular concept. Before Cantor’s work, most mathematicians thought of infinity as one monolithic entity—an unending stretch without any variation in size or scope.

Cantor’s ideas were initially met with resistance and skepticism from his contemporaries. The notion that infinities could come in different sizes was so counterintuitive that even some prominent mathematicians found it hard to accept. However, over time, Cantor’s theories gained acceptance and became foundational in modern mathematics.

One fascinating implication of Cantor’s work is its impact on our understanding of mathematical structures and functions. For example, his diagonal argument laid the groundwork for later developments in set theory and topology—fields that explore more abstract properties of mathematical spaces.

Moreover, Cantor’s insights have philosophical implications as well. They challenge our intuitive understanding of concepts like size and quantity, pushing us to think more deeply about what it means for something to be infinite.

In essence, Cantor’s Diagonal Argument invites us into an infinite dance—a never-ending exploration where each step reveals new layers of complexity and wonder. It reminds us that even within the boundless realm of infinity, there are still mysteries waiting to be uncovered.

So next time you find yourself pondering over an endless horizon or contemplating the vastness of space and time, remember Georg Cantor’s groundbreaking work. His discoveries remind us that infinity isn’t just one thing; it’s an entire universe filled with countless infinities dancing together in an intricate ballet beyond our wildest imagination.

Don Leith

By Don Leith

Retired from the real world. A love of research left over from my days on the debate team in college long ago led me to work on this website. Granted, not all these stories are "fun" or even "trivial" But they all are either weird, unusual or even extraordinary. Working on this website is "fun" in any case. Hope you enjoy it!