Understanding the Infinity Concept: A Dive into Hilbert’s Infinite Hotel Paradox

Ali
9 min readJan 1, 2024

When I was a child, my parents took me to the beach for the first time. I remember standing on the beach, gazing out into the vast expanse of the sea. It was like nothing else I had ever seen. The horizon seemingly stretched out forever, the deep blue colors merging with the limitless sky. I remember thinking, “This must be what eternity looks like.” I could stay there for hours, transfixed by its beauty and majesty. Even now, years later, its endless beauty calling to me, reminding me. It’s like an old friend — always there, forever changing, reminding me of the unfathomable depth of this world.

After years, I stumbled upon a book with an intriguing title “Hitchhiker’s Guide to the Galaxy”. Leafing through the pages, I found myself captivated by the author’s wit and humor. Little did I know that this book would soon alter my perception of infinity forever. As I read on, the author presented an innovative definition of infinity that caught my attention.

INFINITE: Bigger than the biggest thing ever and then some. Much bigger than that in fact, really amazingly immense, a totally stunning size, real “wow, that’s big,” time. Infinity is just so big that by comparison, bigness itself looks real titchy. Gigantic multiplied by colossal multiplied by staggeringly huge is the sort of concept we’re trying to get across here.”

When I first entered university, I finally got the idea of infinity. Mathematicians all over the world have been captivated by the idea of infinity for centuries. The term is thrown around a lot, and it’s easy to assume that infinity must encompass everything that could ever exist. But, as I would learn, it’s not quite like that. It was mind-boggling to learn that infinite didn’t necessarily mean all-encompassing.

As an example, consider the infinite set of rational numbers between 0 and 1. There are, quite literally, an infinite number of these numbers — but none of them are equal to 2. It’s a small detail perhaps, but it represents the kind of intricate complexities that mathematicians have been grappling with for millennia. And as someone who wasn’t quite sure what to expect from university, I realized that I had a lot to learn about the world — both big and small.

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Ali

Math Teacher. Content Curator. Soccer player. Maradona fan. Mostly write about the lectures I love to learn better. alikayaspor@gmail.com