It was another busy night at Hotel Infinity. I think it was about 8pm although it could have been a little later. The time wasn't really all that important. What was important was that word had just been received that the Queen of England would arrive shortly and expected a room for the night.

Unfortunately, every single one of the Hotel's infinite rooms was full.

Fortunately, the concierge, who had a Phd in Mathematics but had gotten tired of driving a taxi and had decided to get a real more discrete job, suggested that the duty manager move each of the guests from their current room to the next higher numbered room as this would make room 1 available without requiring that any guest be evicted. The duty manager replied that this would almost certainly result in a lot of complaints although the concierge was quick to point out that with an infinite number of guests, there were going to be an infinite number of complaints anyways.

That was good enough for the duty manager and the plan was put into operation. When the Queen arrived, she was presented with the keys to room 1 (the Royal Suite, of course) and everyone except the concierge was happy (the concierge had realized too late that although there would probably be a big tip from the Queen in the morning, there wasn't going to be the usual infinite number of tips from satisfied guests).

#### What's the point of this story?

This story is intended to illustrate a few of the more subtle aspects of the concept of infinity:
• that adding one (i.e. the Queen) to a set of infinite size (i.e. the number of guests already in the Hotel when the Queen arrives) doesn't change the size of the set

• that you can map (i.e. match up) the elements of one infinite set (i.e. the rooms in Hotel Infinity) with the elements of another infinite set (i.e. the guests staying at Hotel Infinity) and that adding a finite number of elements to one set (i.e. adding the Queen to the set of guests) doesn't affect our ability to map the elements (i.e. put guests into rooms without leaving any guests without a room).

There's a variation of this story which involves adding an infinite number of guests to the Hotel after it's full - this works also as doubling the size of an infinite set doesn't change the size of the set (i.e. infinity plus infinity is equal to infinity)

• that if there is a non-zero probability that each element of a set of infinite size will have a certain property (i.e. the non-zero probability that each guest will complain about something) then the number of elements of the set with the property (i.e. the guests that complain about something) is infinite

• that increasing the probability that an element of the set has the property (i.e. doing something that DRAMATICALLY increases the probability that a guest will complain) doesn't change the size of the set of elements with the property (i.e. size of the set of guests which complains)

• that a very small amount times infinity (i.e. the average size of a tip left by each guest times the infinite number of guests) is infinity which is (infinitely) larger than the largest possible finite value (i.e. if none of the guests except the Queen leaves a tip then the size of the tip will be finite which is much MUCH worse than the infinite tip that the concierge usually gets when the Hotel is full)
There's also a stupid joke surrounding the term "real" - Hotel Infinity has an infinite number of "discrete" rooms (i.e. like the infinite set of natural numbers and unlike the "larger" infinite set of "real" numbers - see the infinity node for an explanation of why there are more real numbers than integers).

On the other hand, I did (until now) resist the temptation to say that the concierge used to have a continuous job but now has a discrete job (sorry - no explanation of this even stupider "inside joke").

#### Source

A version of this story was told in class by my second year calculus professor. It was a key step along the way to my comprehending the notion of "infinity".

This story is often told with the hotel called the Hilbert Hotel. I prefer the name "Hotel Infinity" as the story flows better with that name and calling it the "Hilbert Hotel" would make it necessary to explain why it is called the "Hilbert Hotel" whereas the name "Hotel Infinity" is, I believe, pretty obvious.