What is Hilbert's Hotel Paradox?

Alright, no problem! Imagine we're sitting at a coffee shop, and I'll tell you about this fascinating thought experiment.

Hilbert's Grand Hotel Paradox: The Magical Hotel That's Always Full, Yet Always Has Room for More

Hey, friend! Have you ever heard of Hilbert's Grand Hotel? It's not an ordinary hotel; it's a thought experiment conceived by the famous mathematician David Hilbert to help us understand just how magical and counter-intuitive the concept of "infinity" truly is.

Imagine there's a hotel with an infinite number of rooms, numbered 1, 2, 3, 4... stretching on forever, with no end.

Even more coincidentally, tonight the hotel is completely full, meaning every single room has a guest in it.

Sounds normal, right? But the paradox (or rather, the counter-intuitive part) is about to begin.

Scenario One: A Single New Guest Arrives

Just then, a weary traveler (let's call him "John") arrives at the front desk, hoping to check in.

The front desk manager, a clever mathematician, smiles and says, "No problem at all. We're full, but we always have a room for you."

How did he do it?

The manager makes an announcement over the intercom to all the current guests:

"To all guests, please move to the room number that is your current room number + 1."

So:

• The guest in Room 1 moves to Room 2
• The guest in Room 2 moves to Room 3
• The guest in Room 3 moves to Room 4
• ...
• The guest in Room n moves to Room n+1

Since there are an infinite number of rooms, no matter how large your room number is, there's always a "+1" room waiting for you.

Once this operation is complete, guess what? Room 1 is now empty! The manager happily hands the key to Room 1 to John.

The first counter-intuitive point: A "full" hotel can magically create an empty room.

Scenario Two: A Finite Number of Guests Arrive (e.g., 40 people)

The situation escalates! A bus carrying 40 passengers pulls up to the hotel entrance.

The manager remains unfazed. He picks up the intercom:

"To all guests, please move to the room number that is your current room number + 40."

So:

• The guest in Room 1 moves to Room 41
• The guest in Room 2 moves to Room 42
• ...
• The guest in Room n moves to Room n+40

After this, what happens? Rooms 1 through 40 are all empty, perfectly accommodating the 40 new guests.

The second counter-intuitive point: No matter how many "finite" guests arrive, this full hotel can always accommodate them.

Scenario Three: An Infinite Number of New Guests Arrive!

This is where things truly get mind-bending. A super-bus with infinite seats, carrying an infinite number of passengers, arrives at the hotel entrance.

Now the manager's brow starts to furrow. Tell guests to move to "n + infinity" rooms? That doesn't make sense. "Infinity" isn't a definite number.

But he quickly comes up with an ingenious solution. He picks up the intercom once again:

"Step One: To all current guests, please move to the room number that is your current room number x 2."

So:

• The guest in Room 1 moves to Room 2
• The guest in Room 2 moves to Room 4
• The guest in Room 3 moves to Room 6
• ...
• The guest in Room n moves to Room 2n

After this step, what happened? All the odd-numbered rooms (1, 3, 5, 7...) are now empty!

And how many odd-numbered rooms are there? That's right, an infinite number!

"Step Two: To all new guests from the infinite bus, please move sequentially into all the odd-numbered rooms."

So:

• The 1st new guest from the bus moves into Room 1
• The 2nd new guest moves into Room 3
• The 3rd new guest moves into Room 5
• ...

You see, all the old guests are in even-numbered rooms, and all the new guests are in odd-numbered rooms. Not a single person missed out, perfectly solved!

So, What Does This All Mean?

This "paradox" is actually not a logical contradiction, but rather a "counter-intuitive" example. It shows us that:

"Infinity" behaves completely differently from the "finite" numbers we're familiar with.

In our everyday experience, a box that's full cannot fit anything else. But in the world of infinity, a "full" set can still accommodate new, even infinitely many, elements.

This illustrates a core concept in set theory: An infinite set can establish a one-to-one correspondence with one of its proper subsets.

• For example, the set of integers (all old guests) and the set of even numbers (the rooms old guests moved into) might seem like the former is twice as large as the latter. However, in the infinite world, they are "the same size" because a one-to-one correspondence can be established (n -> 2n).
• This is why we could free up an infinite number of odd-numbered rooms for the new guests.

Simple Summary

Hilbert's Grand Hotel is like a mind game. It uses a vivid story to help us break free from the confines of finite thinking and glimpse the peculiar and profound nature of infinity. It tells us that we cannot use the simple arithmetic intuition for 1, 2, and 3 to understand the concept of infinite vastness.

Next time someone tells you "it's full, there's no more space," you can tell them about this amazing hotel!