# There’s always room at the Hilbert Hotel

I stumbled upon this series on mind-blowing math facts from a couple of years ago.  It’s by Cornell mathematician Steven Strogatz and treats things like the weirdness of pi, the quirks of probability, Zeno’s paradox, and some of the fun things you can do with calculus.

(Homeschoolers and other educators, take note:  Recovering mathematics and its different applications is urgently needed today and is the missing piece of a truly classical education.  We are doing things with the language part, the trivium, but we now must bring back the mathematics part, the quadrivium, which is far more than just Saxon math.  What Strogatz does here is show that math is far more than memorizing tables and working out problems, showing that it is wonderful, mysterious, philosophical, and imaginative, something that students need to realize.)

Anyway, here he treats the mathematics of infinity, along with the paradox that some sets are more infinite than others:

Some of its [infinity’s] 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.

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.

Does it ever.   Including a set of guests that there is no room for after all.  Read the whole post and the whole series (which is reportedly coming out as a book).

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• Carl Vehse

BTW, Georg Cantor, in addition to creating set theory was a devout Lutheran. Cantor also made many contributions to number theory.

Among many Lutherans who made important contributions to mathematics:

Gottfried Wilhelm Leibnitz, a Lutheran mathematician who developed differential calculus in the late 1600s.

Johann Carl Friedrich Gauss, a devout Lutheran mathematician and scientist, who first proved the fundamental theorem of algebra, and introduced the method of least squares, the Gaussian gravitational constant, and invented the heliotrope, and made numerous other mathematical and scientific contributions.

Bernhard Riemann, a Lutheran (and PK) mathematician (and Gauss’s student), on whose mathematical works Einstein’s theory of relativity rests, and who postulated in 1859 his famous hypothesis, the greatest unsolved prime number problem in mathematics.

Friedrich Wilhelm Bessel a Lutheran astronomer and mathematician who systematized the Bessel mathematical functions and first measured the distance to a star.

Max Born, a Jew who later converted to Lutheranism, was a physicist and mathematician who developed the now-standard interpretation of the probability density function in the Schrödinger equation of quantum mechanics for which he received the Nobel Prize. A number of QM rules and equations have his name associated with them.

• Carl Vehse

BTW, Georg Cantor, in addition to creating set theory was a devout Lutheran. Cantor also made many contributions to number theory.

Among many Lutherans who made important contributions to mathematics:

Gottfried Wilhelm Leibnitz, a Lutheran mathematician who developed differential calculus in the late 1600s.

Johann Carl Friedrich Gauss, a devout Lutheran mathematician and scientist, who first proved the fundamental theorem of algebra, and introduced the method of least squares, the Gaussian gravitational constant, and invented the heliotrope, and made numerous other mathematical and scientific contributions.

Bernhard Riemann, a Lutheran (and PK) mathematician (and Gauss’s student), on whose mathematical works Einstein’s theory of relativity rests, and who postulated in 1859 his famous hypothesis, the greatest unsolved prime number problem in mathematics.

Friedrich Wilhelm Bessel a Lutheran astronomer and mathematician who systematized the Bessel mathematical functions and first measured the distance to a star.

Max Born, a Jew who later converted to Lutheranism, was a physicist and mathematician who developed the now-standard interpretation of the probability density function in the Schrödinger equation of quantum mechanics for which he received the Nobel Prize. A number of QM rules and equations have his name associated with them.

• Kirk

As a former home schooler, you lost me at “math.”

• Kirk

As a former home schooler, you lost me at “math.”

• SKPeterson

Nice list, Carl. We can give the Presbyterians some credit for Pr. Thomas Bayes, though. And I guess, the Romans get to hold up Laplace in the world of modern mathematics and statistics.

• SKPeterson

Nice list, Carl. We can give the Presbyterians some credit for Pr. Thomas Bayes, though. And I guess, the Romans get to hold up Laplace in the world of modern mathematics and statistics.

• Dr. Luther in the 21st Century

Be careful, Veith, lest you light a fire you didn’t intend. The fights over how to teach math can get pretty vicious. Home schoolers who care about math take no prisoners.

• Dr. Luther in the 21st Century

Be careful, Veith, lest you light a fire you didn’t intend. The fights over how to teach math can get pretty vicious. Home schoolers who care about math take no prisoners.

• Klasie Kraalogies

Set theory was one of my greatest joys in undergraduate maths.

• Klasie Kraalogies

Set theory was one of my greatest joys in undergraduate maths.

• john

Pretty sure the idea that some infinities are bigger than others comes down to Cantor’s ideas on countability (which he either developed or inspired, I do not know). The notion of countability means that there is an algorithm that will put any x in the set of an infinite set in order with the natural numbers. The integers, for example, can be counted by starting at zero then going positive and negative, and eventually any x you choose can be counted to with natural numbers. For instance -5 is the 11th number in this infinity, because I count: 0, 1, -1, 2, -2, 3, … , 5, -5.

Mathematics, whatever else it may seem to be, at this level is often about the power to choose values that meet certain conditions. You have to fully understand the conditions, then it becomes demystified and yet retains it beauty.

Then with other sets it is more tricky, the rational numbers can be counted this way by using a technique that puts the numerators and denominators into a systematic count.

But as far as I know that still leaves plenty of uncountable infinities, such as the real numbers and the complex numbers, which cannot be counted along a 1-to-1 basis with the natural numbers, so that they are said to have a different cardinality, a different kind of infinity.

It is not insignificant in philosophy and theology either. I was just reading Spinoza, and he puts out a notion of “finite after its kind” which is his idea that an object or thought or similar thing can be circumscribed by another object of the same type which is larger. But in our practice, this can mean it is part of infinite set, possibly, just as three is contained by four. At the same time, he calls God a being of infinite attributes and infinite essentiality. So he was trying to remove all concept of limitation from God with these devices. The attributes of God, such as goodness, are conceived of as infinite goodness, and infinitely essential so that nothing can countermand the essentiality of it.

When we talk about God as being infinite, it is useful to be aware of these kinds of things. Also, infinity by definition just means a lack of limit, not a certain countable quantity towards which infinite sets proceed. There is a certain similarity in how the form of certain mathematical expressions is infinite and the essence of God is infinite, but one is merely formal and the other exists as such.

Also, 100% appreciate the point on the need to re-claim mathematics.

• john

Pretty sure the idea that some infinities are bigger than others comes down to Cantor’s ideas on countability (which he either developed or inspired, I do not know). The notion of countability means that there is an algorithm that will put any x in the set of an infinite set in order with the natural numbers. The integers, for example, can be counted by starting at zero then going positive and negative, and eventually any x you choose can be counted to with natural numbers. For instance -5 is the 11th number in this infinity, because I count: 0, 1, -1, 2, -2, 3, … , 5, -5.

Mathematics, whatever else it may seem to be, at this level is often about the power to choose values that meet certain conditions. You have to fully understand the conditions, then it becomes demystified and yet retains it beauty.

Then with other sets it is more tricky, the rational numbers can be counted this way by using a technique that puts the numerators and denominators into a systematic count.

But as far as I know that still leaves plenty of uncountable infinities, such as the real numbers and the complex numbers, which cannot be counted along a 1-to-1 basis with the natural numbers, so that they are said to have a different cardinality, a different kind of infinity.

It is not insignificant in philosophy and theology either. I was just reading Spinoza, and he puts out a notion of “finite after its kind” which is his idea that an object or thought or similar thing can be circumscribed by another object of the same type which is larger. But in our practice, this can mean it is part of infinite set, possibly, just as three is contained by four. At the same time, he calls God a being of infinite attributes and infinite essentiality. So he was trying to remove all concept of limitation from God with these devices. The attributes of God, such as goodness, are conceived of as infinite goodness, and infinitely essential so that nothing can countermand the essentiality of it.

When we talk about God as being infinite, it is useful to be aware of these kinds of things. Also, infinity by definition just means a lack of limit, not a certain countable quantity towards which infinite sets proceed. There is a certain similarity in how the form of certain mathematical expressions is infinite and the essence of God is infinite, but one is merely formal and the other exists as such.

Also, 100% appreciate the point on the need to re-claim mathematics.

• Heck, I’ll potentially add more fuel to the fire and note that I first learned about the Hilbert Hotel in high school. Public high school. We spent some time talking about infinity in one of my math classes.

Anyhow, infinity is fun. I remember being somewhat wowed when one of my applied math classes taught me the distinction between countably infinite sets and uncountably infinite sets. The idea of there being different sizes of infinities is as counterintuitive as everything else about infinity, I guess. But then, in the set of natural numbers, there are only two members between 0 and 1 inclusive (those being, well, 0 and 1). But in the set of real numbers, there’s 0, 0.1, 0.001, 0.0001, 0.00001, … and you’ll never even get to 0.2 that way!)

Another fun fact (others here likely know more about this than me; I’m just sharing what I barely remember) concerns pi, which, being an irrational number, has an infinite number of digits in it. Which means that any finite number you can think of, no matter how long, shows up somewhere in the representation of pi. So, yes, every birthdate, every phone number, and, well, anything else you can think of, occurs somewhere in pi.

• Heck, I’ll potentially add more fuel to the fire and note that I first learned about the Hilbert Hotel in high school. Public high school. We spent some time talking about infinity in one of my math classes.

Anyhow, infinity is fun. I remember being somewhat wowed when one of my applied math classes taught me the distinction between countably infinite sets and uncountably infinite sets. The idea of there being different sizes of infinities is as counterintuitive as everything else about infinity, I guess. But then, in the set of natural numbers, there are only two members between 0 and 1 inclusive (those being, well, 0 and 1). But in the set of real numbers, there’s 0, 0.1, 0.001, 0.0001, 0.00001, … and you’ll never even get to 0.2 that way!)

Another fun fact (others here likely know more about this than me; I’m just sharing what I barely remember) concerns pi, which, being an irrational number, has an infinite number of digits in it. Which means that any finite number you can think of, no matter how long, shows up somewhere in the representation of pi. So, yes, every birthdate, every phone number, and, well, anything else you can think of, occurs somewhere in pi.

• Joe

Don’t forget Lutheranism’s biggest contribution to math:

the finite can contain the infinite; where ever and whenever God wants it too …

• Joe

Don’t forget Lutheranism’s biggest contribution to math:

the finite can contain the infinite; where ever and whenever God wants it too …

• Joe

*to* – not a lot of contributions to grammar going on in my world today …

• Joe

*to* – not a lot of contributions to grammar going on in my world today …

• john pawlitz

Let me just say that I really appreciate your bringing this point up. I strongly believe that mathematics is at the foundation of clearly thinking, both in its power to critique the economy of certain choices and in its power to convey through abstraction the form of ideas. It is absolutely imperative that we not neglect it.

• john pawlitz

Let me just say that I really appreciate your bringing this point up. I strongly believe that mathematics is at the foundation of clearly thinking, both in its power to critique the economy of certain choices and in its power to convey through abstraction the form of ideas. It is absolutely imperative that we not neglect it.

• Fernando

Nice account of some of the basic ideas about countability. Strogatz is very good at this.

Cantor was indeed Lutheran, but in the later part of his life he found that it was Catholic theologians who showed the most interest in his ideas. There’s a recent book about this (in German, alas) with the marvelous title “Cardinals and Cardinality”.

And yes, it would be very cool if home schoolers began to take mathematics more seriously and less mechanically.

• Fernando

Nice account of some of the basic ideas about countability. Strogatz is very good at this.

Cantor was indeed Lutheran, but in the later part of his life he found that it was Catholic theologians who showed the most interest in his ideas. There’s a recent book about this (in German, alas) with the marvelous title “Cardinals and Cardinality”.

And yes, it would be very cool if home schoolers began to take mathematics more seriously and less mechanically.