The Puzzling World of Supertasks

Once upon a time, in the midst of the holiday season, one of my favorite treats was Gabriel's cake. This remarkable confection was inspired by Gabriel's Horn, and you could whip it up right in the comfort of your own home—well, assuming your home was infinitely large. It might sound puzzling, but stay with me.

The adventure began with baking a cake, a delightful task I had prepared in advance. It was a beautiful cake, slightly oversized, but I was confident I could devour the entire thing in a day if I truly tried. Can't we all? Now, the next step was to cut the cake in half.

As I sliced the cake in half, you probably noticed something intriguing. No new cake magically appeared, but the surface area of the cake increased. It was once entirely covered, and now there were two uncovered regions inside. The journey continued by cutting one of these halves in half again.

The volume of cake on the table remained the same as when we began, but the surface area kept growing. You might guess the next step: cutting one of the quarters in half. This process continued—halving, and halving, and halving again.

As you progressed, the cake grew thinner and thinner. Yet, you kept going, slicing halves into halves, infinitely. And when you finished, you almost had it—a cake that stretched into the heavens. This cake had a fascinating mathematical quirk: its volume, the amount of cake within, remained the same, but its surface area was infinite. It was a cake you could eat but never fully frost. You'd need an infinite amount of frosting to cover it uniformly.

An object with finite volume but infinite surface area, however, didn't have to be endlessly tall. There were bounded super solids, like a cube with an infinite number of smaller and smaller circular holes. But building these in the real world presented obvious challenges, particularly because it required an infinite number of steps, and infinity isn't something you can truly reach—it's unending.

Or so it seemed, until the concept of a Supertask came into play. Instead of taking the same amount of time for each step, imagine accelerating, doing each step in half the time as the last. A two-minute Gabriel's cake? Piece of cake. First, cut the original cake in half and then wait one minute before making the second cut. Continue, halving the time for each successive cut. This allowed you to complete infinite actions within a finite timeframe. A perplexing concept, indeed.

Zeno's famous dichotomy paradox also comes to mind. In this paradox, Achilles runs a race that he should clearly be able to finish. However, as he progresses, he needs to cover half the remaining distance again and again, which implies an infinite number of steps. Yet, Achilles manages to finish, somehow reaching the end despite the infinite steps remaining. It's like trying to figure out which flag Achilles holds up when he crosses the finish line.

Now, let's address the elephant in the room—supertasks are products of our imagination. In the real world, there's a smallest meaningful distance and time. There's the Planck length and Planck time, beyond which interactions make no sense in our current understanding of physics. This seemingly resolves the problem, but it also sidesteps the deeper philosophical questions.

Supertasks are important because they challenge our understanding of the universe and our cognitive limits. They exist in that uneasy handshake between our minds and the cosmos. Thomson's lamp is a classic example of a supertask that diverges. By turning a lamp on and off in a Zeno-like manner, you end up with a paradox—you can't determine if the lamp is on or off because there's no final step.

Similar riddles arise with the Ross-Littlewood paradox, where you move an infinite number of balls with numbers written on them to an urn, then remove balls as you add new ones. Depending on how you handle the task, you end up with conflicting results, either an empty urn or one filled with infinitely numbered balls.

These paradoxes and conundrums teach us that the universe's mysteries are far from being entirely unraveled. As humans, we have an innate curiosity and a penchant for venturing into the unknown, which makes us unique. Just like Homo sapiens who ventured into the vast, uncharted territories of the world, we also explore the abstract domains of mathematics and philosophy, craving to solve problems and uncover the infinite possibilities of the universe. Supertasks may seem like mere intellectual riddles, but in our quest for understanding, we don't just solve the existing problems; we create new ones and chase the impossible, fostering a deep love and obsession with the enigmatic.