Hilbert’s Paradox of the Grand Hotel

OldTrout, Going Postal

Mr. RotherhamPoofta visited the site the other day and, as is often the case, he asked me if he should book a room.  He is very sweet and kind and asks this of many people.

I suggested that he should book a room at Hilbert’s Grand Hotel.

This hotel has an infinite number of rooms which are occupied.

OldTrout, Going Postal

The paradox is that a room may always be found due to the nature of infinity.

This can be done in various ways.  We can make use of the counting numbers \(\mathbb{ℕ}\).  These are the natural numbers \(n = 1, 2, 3, . . . n\).  This is an infinite sequence so the room \(n + 1\) is available.

OldTrout, Going Postal

Or we can make use of the odd or even numbers because they are also infinite sequences nested within \(\mathbb{N}\)

OldTrout, Going Postal

Then there are the prime numbers; another infinite sequence which is nested in \(\mathbb{ℕ}\).  New guests may be arriving by an infinite number of buses.

OldTrout, Going Postal

We can make use of the negative numbers which are included in the sequence of all integers \(\mathbb{ℤ}\)  or the rational numbers  \((p/q)\) which are the sequence \(\mathbb{ℚ}\).

Then we have the real numbers which is the sequence \(\mathbb{ℝ}\).

OldTrout, Going Postal

They are nested like this :

OldTrout, Going Postal

From this we can see that a room is always available.  However, another paradox may arise because all the rooms are single rooms.

OldTrout, Going Postal

 

© OldTrout 2016