It’s been a couple months since I’ve written a blog post. Lots has happened since then for me in my life, but I’d like to get back to blogging, no matter my audience size.

Today’s topic? Infinity. Sideways eight, otherwise recognized as $\infty$.

The easiest way to comprehend infinity is that it’s bigger than any ``number’’. 5? Bigger than that. 100? Even bigger. A million and nine? Still bigger. The amount of money Citron and Melvin will lose? Bigger!

It’s big. Although, it really isn’t a number. It’s just a way of describing something that is endless, like the number of counting numbers, or the number of primes.

It comes as shock, personally, how comfortable we are with the notion of infinity.

Zeno’s Paradox

Greek philosopher Zeno once listed a set of paradoxes (no longer paradoxical in modern mathematics), that are pretty famous.

The one I’m specifically referring to is the dichotomy paradox. Let’s say you want to walk $1$ meter away from where you currently stand/sit. But before you get there, you need to walk half the distance, aka $\frac{1}{2}$ meters. (Side note, English is weird with fractions and how they get plural status.) Even in that stretch, you must also walk half of that, which is $\frac{1}{4}$ meters. You can continue this argument, and realize that it must take an infinite number of tasks to complete before you can reach the destination, so how can one travel this distance?

In a modern-day calculus class, you’d actually learn that indeed: \(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots = 1\) In your head, this might make sense, but fundamentally, why are we allowed to do this? Zeno struggled with this idea, and it’s quite interesting that many of us today do not.

In this manner, you’d expect there to be other situations where infinite addition to also work out. But sometimes it does, and sometimes it doesn’t. Why? What’s going on??

Hilbert’s Hotel

Another mindboggling thought experiment is Hilbert’s Hotel. It supposes the existence of an infinite hotel, but specifically, for each counting number, there’s a room for it. So, room one exists, but so does room one million and nine, and as does room six hundred and sixty six.

Situation one

Suppose it’s full one day, meaning every room has someone in it. A tired and weary guest comes by and asks for a room.

Surprisingly, you can still fit them in! Make everyone move one room over, and you can fit them in room one.

Situation two

What if you needed to accomodate the guests from your neighbour hotel that is also infinite?

Solution: For each one of your guests, if they are in room $n$, move them to room $2n$. For the new guests, they all had old room numbers. If they were in room $m$, give them room $2k - 1$.

Convince yourself that this indeed works, and every guest still has a room.

Finishing thoughts

These mind puzzles are absolutely mind-boggling, and there’s really no reason as to why they work, at least on the surface level. Although, it speaks a lot to our modern schooling and media that we are able to conceive and understand these ideas, when even philosophers, long ago, were not able to do so.