**Hotel
is full, some guests arrive**

**Hotel
is full, infinite number of guests arrive**

**Hotel
is full, some guests arrive**

Hilbert’s
Hotel is a hotel with an infinite number
of rooms and infinite number of guests.
Every room is occupied. A new visitor
arrives. Can he be accommodated?

Yes, **arithmetic
with infinite quantities**
allows to do it.

At
first it seems that he cannot, but then
the hotel clerk moves the guest in Room
1 to Room 2, and the guest in Room 2
to Room 3, and so on. Every guest is
moved to the next room along. This leaves
Room 1 vacant for the new visitor. Although
the hotel is full, the new guest can
always be accommodated in Room 1.

If another visitor arrives, the hotel
clerk moves the guests again, and the
new guest can be accommodated in Room
1.

**Hotel
is full, infinite number of guests arrive**

What
will be if hotel is full and **infinite**
number of guests arrive? Can they be
accommodated?

Yes,
they will be accommodated thanks to
**property
**.

For
example, the manager asks each guest
to take note of their current room number
and move to the room whose number is
twice that of his own. After everyone
has done this, only the even-numbered
rooms will be occupied and the odd-numbered
rooms will be empty.

So
Hilbert’s Hotel is full, and yet it
has an infinite number of vacancies.

by
Tetyana Butler