To Infinity… But Whatʼs Beyond?

During a conversation with one of my friends, I was told of the ‘Infinite monkey theoremʼ, which states that a monkey hitting keys at random on a typewriter for an infinite amount of time will almost surely type any given text, including the complete works of Shakespeare. Naturally, I found this absurd. How could a monkey, though theyʼve shown themselves to be smarter than we assume, not only accidentally type a coherent sentence but write the complete works of Shakespeare? Weʼre talking about Shakespeare here. Even if a child with literary talent were brought up and trained to write objectively well, youʼd still doubt they could recreate Shakespeareʼs works. But with a monkey, a typewriter, and, well, infinity, you might be able to achieve that (and you wonʼt even have to teach the monkey what alliteration is).

All this to say, infinity is quite vast and provides a framework wherein you can do pretty much anything. Any absurd situation you can think of, as long as it has a non-zero probability of happening, will almost certainly occur given an infinite amount of time or a universe of infinite size. But what exactly is infinity? Well, I can tell you what it isnʼt. Infinity is not a number. It does not fit within the usual arithmetic operations or numerical framework. It is rather a concept or an idea that is boundless— an endless quantity that has neither magnitude nor dimension.

According to mathematician Georg Cantor, also known as ‘The Father of Infinityʼ, there is no largest infinity, and beyond infinity lies another infinity. In fact, there is an infinite variety of different infinities, and some are larger than others. The “smallest infinityˮ is the aleph zero or aleph null (ℵ0). It represents the cardinality of the set of natural numbers and is known as ‘countable infinityʼ. You might be thinking, “Isnʼt infinity supposed to be endless and uncountable? Wouldnʼt the existence of a countable infinity defeat the whole existence of infinity? ˮ. Well, the word “countableˮ here differs slightly from its general use. In mathematical terms, "countable" refers to a type of infinity where the elements of a set can be matched one-to-one with the natural numbers (1, 2, 3, ...). Aleph-null (ℵ0) represents the smallest form of infinity. It describes the size of any set that can be listed in an ordered sequence, even if that sequence never ends. Examples of countably infinite sets include natural numbers, integers, and rational numbers. The key idea is that even though these sets are infinite, you can still list their elements one by one. For instance, if you start counting the natural numbers (1, 2, 3, ...), you'll never run out of numbers, but each number has a specific place in the sequence. This is why it's called "countable."Now, you might be wondering if there are infinities that are uncountable. The answer is yes! Beyond (ℵ0), there are larger infinities that cannot be listed in this way. The most famous example is the set of real numbers (which includes all the decimals and irrational numbers). Unlike natural numbers, you canʼt match each real number with a natural number because between any two real numbers, no matter how close they are, thereʼs always another real number. This makes the set of real numbers uncountable — itʼs a different, larger type of infinity.

This uncountable infinity is represented by the symbol c, also known as the cardinality of the continuum. Cantor showed that $c$ is strictly larger than (ℵ0). In fact, the set of real numbers is just one example of uncountable infinity; there are even larger infinities beyond it! Despite this, it should be noted that infinity isnʼt “becoming biggerˮ. itʼs already complete. While we often think of infinity as something that "goes on and on," suggesting a process of growing or expanding, infinity is not about getting larger over time. Instead, it simply exists in its entirety—boundless and unchanging.

The idea of “beyond infinityˮ or phrases like “to infinity and beyondˮ donʼt imply that infinity can grow or change; rather, they are expressions that highlight the limitless possibilities within the concept of infinity. This ties back to the idea of different sizes of infinity, like (ℵ0). and c. Even though (ℵ0) represents the smallest form of infinity and c is a larger infinity, both are already complete. They donʼt become bigger or progress—they just are. The distinction between them reflects different types of boundlessness, but neither is in a state of becoming. Infinity, in all its forms, represents a totality that defies the limitations of finite thinking. However little we know of infinity, we could say that whether it's a monkey eventually typing Shakespeare or the existence of different infinities, we're dealing with concepts that are not about reaching a goal or surpassing a limit. Infinity contains all possibilities, already fully realized within its endless nature. The expressions “beyond infinityˮ or “to infinity and beyond" represent limitless possibilities. The idea that infinity can come in different forms and magnitudes opens up a world of thought and exploration. Just as infinity invites us to explore beyond our limits and challenge our understanding, letʼs continue to push the boundaries of our knowledge and creativity.

So, channel your inner infinite monkey and aim to create something extraordinary—just like Shakespeare!