# Hilbert’s Paradox of the Grand Hotel

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.

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.

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

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.

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 $$\mathbb{ℝ}$$.

They are nested like this :

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