Infinity Hotel: All Sets Welcome
Vacancy

Here is an excerpt:

The new marketing manager at Infinity Hotel was thrilled that his new marketing campaign was working. Although the previous manager had had an infinite number of vacant rooms, the new manager, George O. Set-Theory, was sure if he directed his advertising to sets, he could fill all the rooms.

Mr. Set-Theory was certainly ambitious and began his advertising in Cantor Village

He expanded his advertising in infinite directions from its heart at Cantor.

No matter how large nor how small, every set was invited to infinity hotel. Lately he’s received a lot of attention from infinite sets. Already, the set of positive rational numbers was lining up at the hotel desk, ready for check-in. Fortunately, they had books they were reading, had knitting or other needlework, or palm-held electronic games. Really, they had an infinite number of ways to keep busy while they waited in line.

Mr. Set-Theory knew he’d better quickly review. He also had to think if there were more members in one set than in another. Counting the rooms and the sets wouldn’t work. It would take too long. Even with an infinite number of things to do, the set of positive rational numbers might soon grow weary.

Mr. Set-Theory thought he could use one-to-one correspondence, so he got his recreation managers to devise a game whereby all the sets of the hotel could pair up to see if each member of the first set corresponded or matched up to a number of the next set with no leftovers. Well, this might work, but, Mr. Set-Theory thought he’d hold off on that game for a while. He really did have to get the rooms assigned. That pairing game could go on ad infinitum. He’d better wait on that!

“Infinity plus 1 is infinity!” exclaimed Mr. Set-Theory. “yes,yes!” he cried. And, infinity times 2 is infinity. What a joy working at Infinity Hotel.

He called on his accountant, Ms. Transfinite A. Rithmetic, to help him. Ms. Transfinite Rithmetic had a sign in her office that was her motto, “Transfinite Rules.” She looked at her sign and quickly hurried to help Mr. Set-Theory. She suggested, “What about infinity times infinity?”

“Yes, yes! That’s the idea,” said Mr. Set-Theory. “Now, what about the set of positive rational numbers, the positive numbers we can express as fractions, {1/2, 1/3, ¾,5/6,…}. Oh, goodness! There are an infinite number of sets within this set. For every number in the set of counting numbers can match a set of rational numbers with that number as a numerator--- Goodness! Fortunately, Infinity Hotel can keep up. We do keep getting more floors, don’t we?”

Mr. Set-Theory had the recreation directors keep the sets busy so he could arrange the rooms.

Ms. Rithmetic started thinking about the sizes of infinity. She remembered reading a book in Cantor’s town library once that talked about hierarchies of ever-larger infinities. The largest one, she recalled, was called, “The Continuum.” “Oh, Mr. Set-Theory,” she stated matter-of-factly, “Your advertising campaign is working so well we may one day become, ‘The Continuum Hotel,’ ---- Imagine!

©2004 J. S. Shipman
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