Infinity Paradox Hotel - Infinite Hotel Paradox

David Hilbert's Infinite Hotel Paradox

Have you ever thought of a Grand Hotel with an infinite number of rooms and an infinite number of guests filling up those rooms? That was the idea of German mathematician David Hilbert, known as The Infinite Hotel Paradox, to show us how difficult it is to get our minds around the concept of infinity.

To start, think of a hotel with an infinite number of rooms and a pretty determined night manager. One night, the Infinite Hotel is totally full, entirely reserved by an infinite number of customers.

To challenge our ideas about infinity, Hilbert asked, what happens if someone new comes along looking for a room? And so, a man walks in the story and asks for a place to stay. 

Instead of turning him down, the night manager chooses to make room for the new guest. But How? Hilbert's night manager suggests shifting each guest along with one room. The night manager requests the guest in room number one to move to room two, the guest in room two to move to room three, and so on. Every customer moves from room number "n" to room number "n+1" and as there is are infinitely many rooms, there is a new room for each current guest.

Now the new guest would have a space in room number one, although the guest book would have an infinite number of complaints.

The method can be applied for any finite amount of new visitors. If, for example, a bus empties 40 new visitors requiring rooms, then all current guest simply transfers from room number "n" to "n+40", hence, freeing up the first 40 rooms. So far so good.

But what when an infinitely long bus carrying an infinite quantity of passengers appears up to lend rooms.– maybe the manager can’t accommodate all of them? If so, the hotel will lose an infinite whole of money and the night manager will lose his job. He has to find some way to build space for them.

The infinite passengers inside infinite bus puzzle the night manager at first, but he finds a way to set each new person.

He moves the guest in room one to room two. He then moves the guest in room two to room four, and the guest in room three to room six and so on. All existing guests move from room number "n" to room number "2n", engaging only the infinitely many even-numbered rooms. Thus leaving the hotel with all of the infinitely many emptied odd-numbered rooms, which can now be assigned to the passengers in the infinite bus.

Easy for the guest in room one. Not so easy for the man in room seven million, eight hundred thousand, seven hundred and forty-nine.

Everybody is nice and the business is booming excellent than ever. Actually, booming precisely the same as ever, making an infinite sum of money a night.

In Hilbert's Paradox, the tactics chosen by the night manager are reasonable only because the Infinite Hotel is certainly a logistical nightmare. It only talks about the under-most level of infinity, essentially, the countable infinity of the natural numbers.

We use natural numbers to assign room numbers. If we were tackling with higher orders of infinity, such as that of the real numbers, such tactical ideas would no longer be applicable, since we have no systematic approach to cover each number.

In a Real Number Infinite Hotel, there would be rooms assigned to a negative number in the underground. Also fractional rooms, so the guy in room number half always doubts he has a smaller place than the guy in room one. And the special rooms, like room radical two, which sounds irrational or room pi, where the guests expect free sweet dishes.

Over at Hilbert's Infinite Hotel, where there are nevermore vacancy and evermore room for new guests, the situations handled by the ever-persistent and more hospitable night manager help us to recall that how tough it is for our comparatively finite intelligence to grip an idea as big as infinity.

Hilbert’s paradox has fascinated mathematicians, physicists and philosophers – even theologians, and they all agree that you might need you to switch room at 2 A.M.