Mr. Cantor’s Motel: An Infinite Number of Rooms

Mr. Cantor's MotelONCE UPON A TIME there lived a very ambitious man, Mr. Cantor.

Combining energy with philanthropic interests, his aim was to acquire money so he could be of service to mankind. He decided to open up his own motel… the Cantor Motel. Mr. Cantor’s motel looked something like this:

Mr. Cantor's Motel Things were going well, but people liked Mr. Cantor’s Motel so well that they started telling their friends. Mr. Cantor had to turn people away. This made him unhappy. So he had an idea: “let’s expand”. He added more rooms, but, again, there were more potential customers than rooms. So, guided by his mathematical friend, he built The Ultimate Mr. Cantor’s Motel. Cantor called it UCM for short. This motel had an infinite number of rooms and looked something like this:

Mr. Cantor's Motel The “…” means that the rooms continue on indefinitely. How did the new motel solve Mr. Cantor’s problem? Well, there are many rooms, and if the new motel had empty rooms, new guests could be accommodated. But what if the motel were full? That was certainly no problem for Mr. Cantor’s mathematician. If a new guest came he was simply put in room 1 and

Guest from room 1 was put in room 2, Guest from room 2 was put in room 3, Guest from room 3 was put in room 4, and so on.¹

  Mr. Cantor became rich beyond belief, but he became more and more eager to increase his wealth.  So he built UCM 2, and, again, it experienced total occupancy. But fate was against him. Just as things were going well, UCM 2 burned to the ground. Fortunately, all the guests escaped, but unfortunately they were left un-housed. “But there is no problem”, said the motel owner’s mathematical friend. On his advice Mr. Cantor put

Guest in room 1 of UCM 1 in room 2 of UCM 1, Guest in room 2 of UCM 1 in room 4 of UCM 1, Guest in room 3 of UCM 1 in room 6 of UCM 1, and so on, – and – Guest from room 1 of UCM 2 in room 1 of UCM 1, Guest from room 2 of UCM 2 in room 3 of UCM 1, Guest from room 3 of UCM 2 in room 5 of UCM 1 and so on.²

Was Mr. Cantor happy with the solution? Ambitious people never are. He built a whole chain of UCMs – in fact, he built an infinite number of them. In addition to UCM 1 and UCM 2, there was UCM3, UCM 4 … There stood the UCMs built all around the world, monuments to one man’s ambition. But again fate showed its nasty hand. All but the original UCM burned. But not only that! All of the doomed UCMs had been fully occupied and all the survivors of the fire rushed by plane and helicopter to the surviving UCM 1.³ The harried Mr. Cantor was given the problem of housing all of these former and current UCM occupants.

But what are mathematicians for? “The solution to this problem is simple”, said the mathematician friend. He created the table shown, in part, below:

Mr. Cantor's Motel

An enlightened Mr. Cantor instructed the residents of UCM 1 to vacate the UCM 1 motel. Then he read the table from top to bottom and from left to right. Reading:

11 sent occupant 1 of UCM 1 to room 1 of UCM 1, 12 sent occupant 2 of UCM 1 to room 2 of UCM 1, 21 sent occupant 1 of UCM 2 to room 3 of UCM 1, 13 sent occupant 3 of UCM 1 to room 4 of UCM 1,

22 sent occupant 2 of UCM 2 to room 5 of UCM 1, 31 sent occupant 1 of UCM 3 to room 6 of UCM 1, 41 sent occupant 1 of UCM 4 to room 7 of UCM 1, and so on.⁴

In this way, all the guests were happy and all the rooms were occupied. Mr. Cantor,⁵ after recovering the trauma of all the fires, gave most of his fortune to aspiring students of mathematics. He died a contented man.

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¹ This shows infinity + 1 = infinity.

² This shows infinity + infinity = infinity. It’s here that the reader might wonder why the second UCM was built. Since infinity + infinity = infinity, two UCMs would yield no more money than having one.

³ Why would so many people who had experienced fire in a motel chain desire to go back to a motel in the same chain? Maybe they had a free happy hour.

This shows infinity x infinity = infinity.

Georg Cantor, who inspired this fable, was a German mathematician. He invented set theory, which included infinite sets. The rooms in UCM illustrate the smallest infinity he created. There is no largest infinity. Set theory was condemned by the mathematics community when first introduced. David Hilbert defended set theory by declaring, “No one shall expel us from the Paradise that Cantor has created”. Later set theory achieved wide acceptance.

The candidate for the next largest infinity is the set of real numbers, the numbers we use in everyday life: 1; 3.7; -10; 3,000,000; … This infinity is called C. There is a $1,000,000 reward if you can prove that C is the second largest infinite set. Good luck.

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