Leader Science and Technology Development Series

A series to educate leaders on various topics I have useful

The Countable Infinity Paradox

Also known as Hilbert’s Paradox of the Grand Hotel, is a captivating thought experiment that reveals the peculiar behavior of infinite sets.

Let’s explore it

The Setup

Imagine a hotel with an infinite number of rooms, each numbered sequentially (1, 2, 3, …).

Initially, every room is occupied by guests.

Infinite Guests Arrive

New guests arrive, expecting their own rooms.

Surprisingly, even though the hotel is fully occupied, we can still accommodate an infinite number of additional guests.

How It Works

To make room for one more guest, we shift every existing guest to the next room (Guest in Room 1 moves to Room 2, Guest in Room 2 to Room 3, and so on).

Now Room 1 is vacant, and the new guest can occupy it.

By repeating this process, we can accommodate any finite number of new guests.

Infinite Guests? No Problem!

But what if infinitely many new guests arrive?

We can cleverly assign each existing guest to a room with a number twice their current room number (Guest in Room 1 moves to Room 2, Guest in Room 2 to Room 4, and so on).

All odd-numbered rooms (which are countably infinite) become available for the newcomers.

Conclusion

Hilbert’s paradox demonstrates that even with infinite occupancy, the hotel can always make room for more guests.

Infinity behaves in fascinating and counterintuitive ways!