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As a kid, I was fascinated by the number infinity. It is hard for us humans to wrap our minds around the concept of infinity. But mathematics makes it possible of course. And fun. I found this great article on the "Hilbert Hotel", I hope you enjoy it too.

 

May 9, 2010, 5:00 pm

The Hilbert Hotel

Steven Strogatz

Steven Strogatz on math, from basic to baffling.

In late February I received an e-mail message from a reader named Kim Forbes.  Her six-year-old son Ben had asked her a math question she couldn’t answer, and she was hoping I could help:

Today is the 100th day of school. He was very excited and told me everything he knows about the number 100, including that 100 was an even number. He then told me that 101 was an odd number and 1 million was an even number, etc.  He then paused and asked: “Is infinity even or odd?”

I explained that infinity is neither even nor odd.  It’s not a number in the usual sense, and it doesn’t obey the rules of arithmetic.  All sorts of contradictions would follow if it did.  For instance, “if infinity were odd, 2 times infinity would be even.  But both are infinity!  So the whole idea of odd and even does not make sense for infinity.”

Kim replied:

Thank you.  Ben was satisfied with that answer and kind of likes the idea that infinity is big enough to be both odd and even.

Although something got garbled in translation (infinity is neither odd nor even, not both), Ben’s rendering hints at a larger truth.  Infinity can be mind-boggling.
 
Some of its strangest aspects first came to light in the late 1800s, with Georg Cantor’s groundbreaking work on “set theory.”  Cantor was particularly interested in infinite sets of numbers and points, like the set {1, 2, 3, 4,…} of “natural numbers” and the set of points on a line.  He defined a rigorous way to compare different infinite sets and discovered, shockingly, that some infinities are bigger than others.

At the time, Cantor’s theory provoked not just resistance, but outrage.  Henri Poincaré, one of the leading mathematicians of the day, called it a “disease.”  But another giant of the era, David Hilbert, saw it as a lasting contribution and later proclaimed, “No one shall expel us from the Paradise that Cantor has created.”

My goal here is to give you a glimpse of this paradise.  But rather than working directly with sets of numbers or points, let me follow an approach introduced by Hilbert himself.  He vividly conveyed the strangeness and wonder of Cantor’s theory by telling a parable about a grand hotel, now known as the Hilbert Hotel.

It’s always booked solid, yet there’s always a vacancy.

For the Hilbert Hotel doesn’t merely have hundreds of rooms — it has an infinite number of them.  Whenever a new guest arrives, the manager shifts the occupant of room 1 to room 2, room 2 to room 3, and so on.  That frees up room 1 for the newcomer, and accommodates everyone else as well (though inconveniencing them by the move).

Now suppose infinitely many new guests arrive, sweaty and short-tempered.  No problem.  The unflappable manager moves the occupant of room 1 to room 2, room 2 to room 4, room 3 to room 6, and so on.  This doubling trick opens up all the odd-numbered rooms — infinitely many of them — for the new guests.

Later that night, an endless convoy of buses rumbles up to reception.  There are infinitely many buses, and worse still, each one is loaded with an infinity of crabby people demanding that the hotel live up to its motto, “There’s always room at the Hilbert Hotel.”

The manager has faced this challenge before and takes it in stride.

 

Read the rest here.

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Hilbert was a sadist, his hotel's guests are masochists, and the manager is both.

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