Room at the Hilbert Infinity Hotel

-Suchi Dagli

Imagine you are packing to stay at your favorite Hotel for a 3 nights 4 days stay, you are super excited and cannot wait to reach there. But to your surprise, you find out that the Hotel has NO VACANCY! That might be very disheartening for you. But when your favorite Hotel is “The Hilbert Infinity Hotel”: there’s vacancy even when there is no vacancy.

In the 1920s, the German mathematician David Hilbert devised a famous thought experiment to show us just how hard it is to wrap our minds around the concept of infinity. Hilbert’s paradox of the Grand Hotel ( Infinite Hotel Paradox or Hilbert’s Hotel) illustrates a property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often.

Imagine a hotel with an infinite number of rooms and a very hardworking night manager. One night, the Infinite Hotel is completely full, totally booked up with an infinite number of guests. 

How will you get a room?

You walk into the hotel and ask for a room. Rather than turn you down, the night manager decides to make room for you. How? Take a look at this.

He will ask the guest in room number 1 to move to room 2, the guest in room 2 to move to room 3, and so on. Every guest moves from room number “n” to room number “n+1”. Since there are an infinite number of rooms, there is a new room for each existing guest. This leaves room 1 open for you. This process can be repeated to accommodate any finite number of new guests like you!

Now what if you tell me that not one, but a tour bus unloads 70 new people looking for rooms, then every existing guest just moves from room number “n” to room number “n+70”, thus, opening up the first 70 rooms. 

Have you noticed where we’re going? 

∞ + N = ∞

Infinite buses with countably infinite people:

But the challenge isn’t over yet. Let’s say now an infinitely large bus with a countably infinite number of passengers pulls up to rent rooms. Now, the infinite bus of infinite passengers confuses and amazes the night manager at first, but he realizes there’s still a way to place each new person. 

This is how he does it:

He asks the guest in room 1 to move to room 2. He then asks the guest in room 2 to move to room 4, the guest in room 3 to move to room 6, and so on. Each current guest moves from room number “n” to room number “2n” — filling up only the infinite even-numbered rooms. 

By doing this, he has now emptied all of the infinitely many odd-numbered rooms, which are then taken by the people filing off the infinite bus. 

However, the business, if one might look at it, is booming the same as ever, banking an infinite number of dollars every night. 

The very popular Hilbert Hotel:

Sooner or later, the popularity of this incredible Hotel increases and people, like you and me, start coming in from far and wide. Now one day the night manager looks outside and sees an infinite line of infinitely large buses, each with a countably infinite number of passengers. What can he do? He cannot lose out on an infinite amount of money!

 Now remember  what Euclid proved that around the year 300 B.C.E., that there is an infinite quantity of prime numbers.

So, to accomplish this task, to make rooms for infinite buses having infinitely large travelers, he assigns every current guest to the first prime number ”2″ raised to the power of their current room number. So, the current occupant of room number 7 goes to room number 2^7, which is room 128. 

Assigning prime numbers to the buses will be the correct way!

The night manager then takes the people on the first of the infinite buses and assigns them to the room number of the next prime, 3, raised to the power of their seat number on the bus. So, the person in seat number 7 on the first bus goes to room number 3^7 or room number 2,187. This continues for all of the first bus. The passengers on the second bus are assigned powers of the next prime, 5. The following bus, powers of 7. Each bus follows: powers of 11, powers of 13, powers of 17, etc. Since each of these numbers only has 1 natural number power of their prime number base as factors, there are no overlapping room numbers. 

All the buses’ passengers fan out into rooms using unique room-assignment schemes based on unique prime numbers. In this way, the night manager can accommodate every passenger on every bus. And the task is accomplished! 

Because we have successfully accommodated the passengers in our hotel having infinite rooms. So, we’ve successfully proved :

∞ * ∞ = ∞

But will there be any vacant rooms?

Although, there will be many rooms that go unfilled, like rooms 6,12, 15, 18 since these numbers are not powers of any prime number. But that doesn’t matter as long as everyone gets a room and everyone is happy! Right?

 So is ∞ * ∞ less than ∞? No, because we’ve already proved that infinity + infinity = infinity, so even if there are infinitely many rooms left it will be filled by infinitely many passengers waiting in the queue.

What about the other numbers on the number line?

The night manager’s strategies are only possible with the lowest level of infinity, mainly, the countable infinity of the natural numbers, 1, 2, 3, 4, and so on.

We use natural numbers for the room numbers as well as the seat numbers on the buses. 

“If we were dealing with higher orders of infinity, such as that of the real numbers, these structured strategies would no longer be possible as we have no way to systematically include every number.”

However it will look something like this

However, The “Real Number” Infinite Hotel will have negative number rooms in the basement. Fractional rooms, so the guy in room 1/2 will always suspect that he has less room than the guy in room 1. Square root rooms like room radical 2, and room pi will be such where the guests will expect free dessert hahah!

Hilbert’s paradox is a veridical paradox: it leads to a counter-intuitive result that is provably true. The statements “there is a guest to every room” and “no more guests can be accommodated” are not equivalent when there are infinitely many rooms. Initially, this state of affairs might seem to be counter-intuitive. The properties of “infinite collections of things” are quite different from those of “finite collections of things”. Thus, while in an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert’s aptly named Grand Hotel, the quantity of odd-numbered rooms is not smaller than the total “number” of rooms

But over at Hilbert’s Infinite Hotel, where there’s never any vacancy and always room for more, the scenarios faced by the extra hospitable night manager trying to accommodate everyone in should remind us of just how hard it is for our relatively finite minds to grasp a concept as large as infinity.

But honestly, I suggest you to not take a lot of baggage as they might need you to change rooms at 2 a.m. when people like us will walk in, desperately trying to know which room we will get!

Sources:

  1. Jeff Dekofsky: The Infinite Hotel Paradox: TED talk
  2. Hilbert’s Hotel Paradox- on wordpress- by soyoungsocurious
  3. Wikipedia- Hilbert’s Paradox of the Grand Hotel
  4. @daily_math_ on Instagram

Published by mscnm

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