Recently I read an article on infinity which struck me quite a lot. Here is the summery of the article:

Some of the strangest aspects of infinity 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.

First he does the doubling trick. That reassigns the current guests to the even-numbered rooms and clears out all the odd-numbered ones — a good start, because he now has an infinite number of rooms available.

But is that enough? Are there really enough odd-numbered rooms to accommodate the teeming horde of new guests? It seems unlikely, since there are something like “infinity squared” people clamoring for these rooms.

(Why infinity squared? Because there were an infinite number of people on each of an infinite number of buses, and that amounts to infinity times infinity, whatever that means.)

This is where the logic of infinity gets very weird.

To understand how the manager is going to solve his latest problem, it helps to visualize all the people he has to serve.

Of course, we can’t show literally all of them here, since the diagram would need to be infinite in both directions.

But a finite version of the picture is adequate. The point is that any specific bus passenger (your Aunt Inez, say, on vacation from Louisville) is sure to appear on the diagram somewhere, as long as we include enough rows and columns. In that sense, everybody on every bus is accounted for. You name the passenger, and he or she is certain to be depicted at some finite number of steps east and south of the diagram’s corner.

The manager’s challenge is to find a way to work through this picture systematically. He needs to devise a scheme for assigning rooms so that everybody gets one eventually, after only a finite number of other people have been served.

Sadly, the previous manager didn’t understand this and mayhem ensued. When a similar convoy showed up on his watch, he became so flustered trying to process all the people on bus 1 that he never got around to any other bus, leaving all those neglected passengers screaming and furious. Visualized on the diagram below, this myopic strategy would correspond to a path marching eastward along row 1, never to return.

The new manager, however, has everything under control. Instead of tending to just one bus, he zigs and zags through the diagram, fanning out from the corner as shown below.

He starts with passenger 1 on bus 1 and gives her the first empty room. The second and third empty rooms go to passenger 2 on bus 1, followed by passenger 1 on bus 2, both of whom are depicted on the second diagonal from the corner of the diagram. After serving them, the manager proceeds to the third diagonal and hands out a set of room keys to passenger 1 on bus 3, passenger 2 on bus 2, and passenger 3 on bus 1.

I hope the manager’s procedure — progressing from one diagonal to another — is clear from the picture above, and that you’re convinced that any particular person will be reached in a finite number of steps.

So, as advertised, there’s always room at the Hilbert Hotel.

The argument I’ve just presented is a famous one in the theory of infinite sets. Cantor used it to prove that there are exactly as many positive fractions (ratios p/q of positive whole numbers p and q) as there are natural numbers (1, 2, 3, 4, …). That’s a much stronger statement than saying both sets are infinite. It says they are infinite to precisely the same extent, in the sense that a “one-to-one correspondence” can be established between them.

You could think of this correspondence as a buddy system in which each natural number is paired with some positive fraction, and vice versa. The existence of such a buddy system seems utterly at odds with common sense — it’s the sort of sophistry that made Poincaré recoil. For it implies we could make an exhaustive list of all positive fractions, even though there’s no smallest one!

And yet there is such a list. We’ve already found it. The fraction p/q corresponds to passenger p on bus q, and the argument above shows that each of these fractions can be paired off with a certain natural number 1, 2, 3,…, given by the passenger’s room number at the Hilbert Hotel.

The coup de grace is Cantor’s proof that some infinite sets are bigger than this. Specifically, the set of real numbers between 0 and 1 is “uncountable” — it can’t be put in one-to-one correspondence with the natural numbers. For the hospitality industry, this means that if all these real numbers show up at the reception desk and bang on the bell, there won’t be enough rooms for all of them, even at the Hilbert Hotel.

The proof is by contradiction. Suppose each real number could be given its own room. Then the roster of

occupants, identified by their decimal expansions and listed by room number, would look something like this:

Room 1: .6708112345…

Room 2: .1918676053…

Room 3: .4372854675…

Room 4: .2845635480…

Remember, this is supposed to be a complete list. Every real number between 0 and 1 is supposed to appear somewhere, at some finite place on the roster.

Cantor showed that a lot of numbers are missing from any such list; that’s the contradiction. For instance, to construct one that appears nowhere on the list shown above, go down the diagonal and build a new number from the boldface digits:

Room 1: .6708112345…

Room 2: .1918676053…

Room 3: .4372854675…

Room 4: .2845635480…

The decimal so generated is .6975…

But we’re not done yet. The next step is to take this decimal and change all its digits, replacing each of them with any other digit between 1 and 8. For example, we could change the 6 to a 3, the 9 to a 2, the 7 to a 5, and so on.

This new decimal .325… is the killer. It’s certainly not in Room 1, since it has a different first digit from the

number there. It’s also not in Room 2, since its second digit disagrees. In general, it differs from the nth number in the nth decimal place. So it doesn’t appear anywhere on the list!

The conclusion is that the Hilbert Hotel can’t accommodate all the real numbers. There are simply too many of them, an infinity beyond infinity.