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

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.

Defining Infinity: Georg Cantor

Until the beginning of the 19th century, mathematicians believed that infinity was a single, straightforward concept. But as it turns out, there’s more to infinity than meets the eye.

Georg Cantor, the father of modern mathematics, made groundbreaking discoveries about infinity in the early 20th century. He found that not all infinities are created equal — some are bigger than others, some are countable while others are too vast to count, and there are even infinitely many different versions of infinity. Amidst all these fascinating findings, Cantor was labeled as a madman by many of his peers. However, a few brave mathematicians, such as David Hilbert, supported Cantor’s unconventional ideas about infinity.

Back in 1924, mathematics lost one of its greatest minds, Georg Cantor. In the wake of Cantor’s death, David Hilbert, another brilliant mathematician, gave a lecture entitled “Über das Unendliche,” or “On the Infinite.” It was during this lecture that Hilbert posed his now-famous hotel question.

What would happen if an infinite hotel were completely full, and a new guest showed up looking for a room? The question became known as the Infinite Hotel Paradox or Hilbert’s Grand Hotel. George Gamow later popularized the problem in his book, One Two Three … Infinity.

The paradox is quite simple yet challenging to understand. Surprisingly, the answer lies in Cantor’s ideas on infinity. Though it may seem like a mind-boggling concept, the Infinite Hotel Paradox is a fascinating topic that can lead to insights into the nature of infinity.

Hilbert’s Infinite Hotel

David Hilbert’s fascinating scenario starts by imagining a hotel with an infinite number of rooms, each numbered by positive integers: 1, 2, 3, and so on forever. This hotel is unique and defies our usual understanding of numbers because even though every room is always occupied — there isn’t a single empty room — new guests can still be accommodated, regardless of the number.

One such day, when the hotel was fully booked, a new customer showed up at the front desk without a reservation. The receptionist, Georg Cantor, was unfazed by this seeming impossibility. Despite every room being occupied, Cantor confidently assured the guest that a room would be made available. This might seem impossible under normal circumstances, but at Hilbert’s Infinite Hotel, the impossible becomes the norm.

As soon as the new guest arrived, Cantor quickly formulated a plan. He grabbed the microphone and made an announcement that echoed through the hallways of the infinite hotel. His voice was steady and confident as he said,

“Dear customers, due to the arrival of a new guest, we ask for your cooperation in a small change. Please, each guest move from your current room to the next one. The person in room number 1, please move to room number 2. The guest in room number 2, please move to room number 3. The guest in room number n, please move to room number n+1, and so on. We thank you for your understanding and cooperation.”

As Cantor’s words resonated through the hotel, there was a buzz of activity as guests started moving their belongings to their new rooms. The plan was ingenious in its simplicity. By moving each guest just one room down, they freed up room number 1 for the new guest, while still accommodating everyone else. It was a perfect demonstration of the infinite potential of Hilbert’s Hotel.

By the way, it’s important to note that one should not ask, “Where will the last customer go?” in the context of Hilbert’s Infinite Hotel. The very nature of infinity dictates that there can be no ‘last’ customer. This is akin to the impossibility of having a ‘last’ room in an infinite hotel. Infinity, by definition, is unbounded and limitless, implying that there is always room for one more. Just as there are infinite numbers — there’s always a next number, no matter how high you count — there will always be a next customer at Hilbert’s Infinite Hotel. Thus, the concept of the ‘last’ customer is essentially a paradox within this paradoxical scenario.

It is imperative to bear in mind the stark difference between traditional hotels and Hilbert’s Infinite Hotel. In any hotel with a finite number of rooms, no matter how grand or expansive, the method of making room for additional guests by shifting occupants to the next room will invariably hit a roadblock. This is simply because in a finite hotel, there is no extra room available for the guest in the highest numbered room to transition to. The finite nature of real world hotels inevitably leads to a limitation on the number of guests it can accommodate at any given time. This distinct contrast underscores the intriguing and counterintuitive properties of infinity that are so elegantly encapsulated by Hilbert’s Infinite Hotel.

Infinite Number of New Customers

Just as Cantor was savoring his coffee, triumphantly celebrating his solution to the previous conundrum, a rumbling diverted his attention to the hotel entrance. He watched, eyes widening, as a humongous bus pulled up, disgorging an endless stream of new customers. Cantor sighed, put down his coffee, and rose to the challenge. This time around, he had to accommodate not one, but an infinite number of new guests.

Cantor, however, remained undeterred. He picked up the microphone and announced confidently,

“Dear customers, we have a busload of infinitely new guests arriving. We know this might sound surprising, but we have a plan in place to accommodate everyone. Each guest already in the hotel, please move from your current room to the one numbered twice your current room number. The guest in room number 1, please move to room number 2. The guest in room number 2, please move to room number 4. The guest in room number n, please move to room number 2n, and so on. This way, we will free up all the odd-numbered rooms for our new set of guests.”

The hotel guests were once again flabbergasted but complied with the instructions. As they moved, the infinite number of odd-numbered rooms became available for the new customers, once again proving the limitless possibilities in Hilbert’s Infinite Hotel.

An Infinite Bus Full of Infinite New Customers

Just as Cantor was about to take a sip of his long-awaited coffee, he caught sight of an unbelievable spectacle unfolding outside the hotel. One by one, an infinite number of buses pulled up to the entrance of Hilbert’s Infinite Hotel. To his amazement, each bus was packed with an infinite number of eager new customers.

The hotel’s fame had spread far and wide and people from everywhere were rushing to experience the wonders of the Infinite Hotel. The magnitude of the task ahead was enormous, even by the hotel’s standards. Cantor, for a moment, felt a pang of nervousness. However, he quickly regained his composure, realizing that this was yet another opportunity to showcase the remarkable potential of infinity. He knew he had to formulate a plan that would accommodate this mind-bending influx of guests — a plan that would once again make the impossible, possible. Cantor picked up the microphone once again and addressed the guests,

“Dear valued customers. We have a new request for you. Please everyone move to a room number equal to twice the number of the room they are staying in. This will free up our odd-numbered rooms and make room for our new guests.”

Upon hearing Cantor’s announcement, the guests quickly started moving, filling up the even-numbered rooms and vacating the odd-numbered ones.

Once the odd-numbered rooms were vacant, Cantor turned his attention to the new guests. He asked the customers of the first bus to move to rooms that were multiples of 3, such as 3, 9, 27, 81, 243. 

The customers of the second bus were directed to rooms that were multiples of 5, like 5, 25, 125, 625. 

The guests from the third bus were accommodated in rooms that were multiples of 7.

As each subsequent bus arrived, Cantor assigned its passengers to rooms that were multiples of the next prime number. 

The nth bus would have its customers accommodated in rooms that were multiples of the (n+1)th prime number. 

Given that there are infinitely many primes and all primes after 2 are odd, this strategy fit perfectly with the vacancy of the odd-numbered rooms.

As Cantor meticulously placed each customer, there were still infinitely many empty rooms available. Rooms 6, 10, 12, 14, 15, 18 and so forth remained unoccupied. So, every new customer who arrives at the last minute can also stay in this hotel.

The complex puzzle of accommodating an infinite number of buses each carrying an infinite number of guests had been solved. Once again, Cantor had proven the mind-bending properties of infinity and the inherent potential of Hilbert’s Infinite Hotel.

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