The Infinite Hotel Paradox

This paradox was proposed by the German mathematician David Hilbert to explain the concept of infinity.

Who contributed to this paradox?

David Hilbert was born on January 23, 1862, in Königsberg (East Prussia). He is globally recognized as one of the most influential mathematicians of the 19th and early 20th centuries. He made contributions to the mathematical framework of quantum mechanics and general relativity.

Georg Cantor was born on March 3, 1845, in St. Petersburg. He created the foundations of modern set theory, using many concepts that were previously only used intuitively. Unfortunately, his ideas were not understood in his time, and this misunderstanding turned into ridicule from the scientific community. It wasn’t until the final years of Cantor’s life that this was rectified.

Shortly after Cantor’s death (in 1920), Hilbert proposed the infinite hotel paradox. It is important to note that this is not a paradox in the strict sense of the word, thanks to Cantor’s work which eliminates any contradictions.

There are different versions of this paradox.

What is the Infinite Hotel Paradox?

This paradox helps explain that some infinities are larger than others.

Let’s consider an example. As the receptionists of Hilbert’s hotel, we are obliged to accommodate all guests who request a room in this hotel. But during peak season, there are more people staying than we can count. We could say there is an infinite number of guests in our infinite hotel.

One More Guest

On a summer night, with an infinite number of guests in the hotel, one more arrives and asks if they can sleep here. At first, we might think it’s impossible to accommodate another person in a hotel with infinite rooms and guests. But, of course, all the receptionists in this hotel are great at math, so after thinking for a moment, we tell all guests to move to the next room. Thus, the guest in room 1 moves to room 2, the one in room 2 moves to room 3, and so on. In summary, they move from room n to room n+1.

Infinitely Many Guests?

Of course, this is the smallest example of problems that may arise. Another case is when infinite guests arrive on a single night. We might believe it’s impossible to accommodate one infinity with another infinity, but if we don’t manage to fit them in, they will leave bad reviews, and the hotel might close. Since we don’t want that, we put our math skills to the test. Since the numbers are infinite, are the even and odd numbers also infinite? The answer is YES.

With this in mind, we can tell the guests to multiply their room number by two. So, the guest in room 1 moves to room 2, the one in room 2 moves to room 4, and so on. In summary, they move from room n to room 2·n. After doing this, we tell the new arrivals to occupy the odd-numbered rooms. Thus, we have managed to accommodate two infinities in one.

Infinites of Infinites?

Of course, this is by no means the greatest problem we might encounter. For example, imagine that infinite buses arrive, each with an infinite number of passengers. It’s an incredible mess.

Of course, Hilbert’s hotel can accommodate all the passengers. But this problem is more complex and has more steps than the previous one. Moreover, there are at least two ways to solve it.

1. To achieve the first solution, we write the natural numbers in the form of a pyramid. In the first row, we write just the number 1. In the second row, we write 2 and 3. In the third, we write 4, 5, and 6, and so on. We can see that if we start from the top and move along the diagonal to the right, we form a sequence of numbers: 1, 3, 6, 10, 15… We assign a room number from this list to each original guest in the hotel, starting from the first. The person in room 1 stays where they are, the person in room 2 moves to room 3, the person in room 3 moves to room 6, and so on. If we follow the same path starting with number 2, we get the sequence 2, 5, 9, 14… In this row, we place the people who arrived on the first bus, starting with the first one. In the next sequence (4, 8, 13…) we place the second bus, and so on.

2. The current guests multiply their room number by 2, just like in the previous problem. Secondly, each bus is assigned a prime number (which is also infinite), excluding two. Third, each seat on each bus is assigned an integer number. Each person then has to multiply their bus number as many times as indicated by their seat number. In other words, raise the bus number to the power of their seat number. So, the passenger on bus 3 with seat number 4 will know their room number by performing the operation 3⁴, which gives them room number 81. In summary, the new arrivals will know their room number by performing bus number^{seat number}. This can also be applied to even more infinities of infinities. For example, if infinite ferries arrive with infinite buses, each carrying infinite passengers, you would have to do (ferry number × bus number) seat number. And so on infinitely, because there are infinite combinations that can happen, or is there?

Can Hilbert’s Infinite Hotel accommodate all types of numbers?

In this case, the answer is NO. At least in this paradox, not all numbers fit. For example, we haven’t included negative numbers, rationals, or real numbers. If we did, the hotel would be chaotic, and even more so for the employees.

Conclusions

• There are infinites that are larger than others.
• If you go to this hotel, make sure to bring a calculator.
• What a headache it is for the workers.