Bad Math Books and Cantor Cardinality

A bunch of readers sent me a link to a tweet this morning from Professor Jordan Ellenberg:

The tweet links to the following image:

(And yes, this is real. You can see it in context here.)

This is absolutely infuriating.

This is a photo of a problem assignment in a math textbook published by an imprint of McGraw-Hill. And it’s absolutely, unquestionably, trivially wrong. No one who knew anything about math looked at this before it was published.

The basic concept underneath this is fundamental: it’s the cardinality of sets from Cantor’s set theory. It’s an extremely important concept. And it’s a concept that’s at the root of a huge amount of misunderstandings, confusion, and frustration among math students.

Cardinality, and the notion of cardinality relations between infinite sets, are difficult concepts, and they lead to some very un-intuitive results. Infinity isn’t one thing: there are different sizes of infinities. That’s a rough concept to grasp!

Here on this blog, I’ve spent more time dealing with people who believe that it must be wrong – a subject that I call Cantor crackpottery – than with any other bad math topic. This error teaches students something deeply wrong, and it encourages Cantor crackpottery!

Let’s review.

Cantor said that two collections of things are the same size if it’s possible to create a one-to-one mapping between the two. Imagine you’ve got a set of 3 apples and a set of 3 oranges. They’re the same size. We know that because they both have 3 elements; but we can also show it by setting aside pairs of one apple and one orange – you’ll get three pairs.

The same idea applies when you look at infinitely large sets. The set of positive integers and the set of negative integers are the same size. They’re both infinite – but we can show how you can create a one-to-one relation between them: you can take any positive integer $i$, and map it to exactly one negative integer, $0 - i$.

That leads to some unintuitive results. For example, the set of all natural numbers and the set of all even natural numbers are the same size. That seems crazy, because the set of all even natural numbers is a strict subset of the set of natural numbers: how can they be the same size?

But they are. We can map each natural number $i$ to exactly one even natural number $2i$. That’s a perfect one-to-one map between natural numbers and even natural numbers.

Where it gets uncomfortable for a lot of people is when we start thinking about real numbers. The set of real numbers is infinite. Even the set of real numbers between 0 and 1 is infinite! But it’s also larger than the set of natural numbers, which is also infinite. How can that be?

The answer is that Cantor showed that for any possible one-to-one mapping between the natural numbers and the real numbers between 0 and 1, there’s at least one real number that the mapping omitted. No matter how you do it, all of the natural numbers are mapped to one value in the reals, but there’s at least one real number which is not in the mapping!

In Cantor set theory, that means that the size of the set of real numbers between 0 and 1 is strictly larger than the set of all natural numbers. There’s an infinity bigger than infinity.

I think that this is what the math book in question meant to say: that there’s no possible mapping between the natural numbers and the real numbers. But it’s not what they did say: what they said is that there’s no possible map between the integers and the fractions. And that is not true.

Here’s how you generate the mapping between the integers and the rational numbers (fractions) between 0 and 1, written as a pseudo-Python program:

 i = 0
for denom in Natural:
for num in 1 .. denom:
if num is relatively prime with denom:
print("%d => %d/%d" % (i, num, denom))
i += 1


It produces a mapping (0 => 0, 1 => 1, 2 => 1/2, 3 => 1/3, 4 => 2/3, 5 => 1/4, 6 => 3/4, …). It’ll never finish running – but you can easily show that for any possible fraction, there’ll be exactly one integer that maps to it.

That means that the set of all rational numbers between 0 and 1 is the same size as the set of all natural numbers. There’s a similar way of producing a mapping between the set of all fractions and the set of natural numbers – so the set of all fractions is the same size as the set of natural numbers. But both are smaller than the set of all real numbers, because there are many, many real numbers that cannot be written as fractions. (For example, $\pi$. Or the square root of 2. Or $e$. )

This is terrible on multiple levels.

1. It’s a math textbook written and reviewed by people who don’t understand the basic math that they’re writing about.
2. It’s teaching children something incorrect about something that’s already likely to confuse them.
3. It’s teaching something incorrect about a topic that doesn’t need to be covered at all in the textbook. This is an algebra-2 textbook. You don’t need to cover Cantor’s infinite cardinalities in Algebra-2. It’s not wrong to cover it – but it’s not necessary. If the authors didn’t understand cardinality, they could have just left it out.
4. It’s obviously wrong. Plenty of bright students are going to come up with the the mapping between the fractions and the natural numbers. They’re going to come away believing that they’ve disproved Cantor.

I’m sure some people will argue with that last point. My evidence in support of it? I came up with a proof of that in high school. Fortunately, my math teacher was able to explain why it was wrong. (Thanks Mrs. Stevens!) Since I write this blog, people assume I’m a mathematician. I’m not. I’m just an engineer who really loves math. I was a good math student, but far from a great one. I’d guess that every medium-sized high school has at least one math student every year who’s better than I was.

The proof I came up with is absolutely trivial, and I’d expect tons of bright math-geek kids to come up with something like it. Here goes:

1. The set of fractions is a strict subset of the set of ordered pairs of natural numbers.
2. So: if there’s a one-to-one mapping between the set of ordered pairs and the naturals, then there must be a one-to-one mapping between the fractions and the naturals.
3. On a two-d grid, put the natural numbers across, and then down.
4. Zigzag diagonally through the grid, forming pairs of the horizontal position and the vertical position: (0,0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2), (3, 0), (2, 1), (1, 2), (0, 3).
5. This will produce every possible ordered pair of natural numbers. For each number in the list, produce a mapping between the position in the list, and the pair. So (0, 0) is 0, (2, 0) is 3, etc.

As a proof, it’s sloppy – but it’s correct. And plenty of high school students will come up with something like it. How many of them will walk away believing that they just disproved Cantor?

And indeed, he was right. Phil Plait the Bad Astronomer, of all people, got taken in by a bit of mathematical stupidity, which he credulously swallowed and chose to stupidly expand on.

We’ll consider three infinite series:

S1 = 1 - 1 + 1 - 1 + 1 - 1 + ...
S2 = 1 - 2 + 3 - 4 + 5 - 6 + ...
S3 = 1 + 2 + 3 + 4 + 5 + 6 + ...


S1 is something called Grandi’s series. According to the video, taken to infinity, Grandi’s series alternates between 0 and 1. So to get a value for the full series, you can just take the average – so we’ll say that S1 = 1/2. (Note, I’m not explaining the errors here – just repeating their argument.)

Now, consider S2. We’re going to add S2 to itself. When we write it, we’ll do a bit of offset:

1 - 2 + 3 - 4 + 5 - 6 + ...
1 - 2 + 3 - 4 + 5 + ...
==============================
1 - 1 + 1 - 1 + 1 - 1 + ...


So 2S2 = S1; therefore S2 = S1=2 = 1/4.

Now, let’s look at what happens if we take the S3, and subtract S2 from it:

   1 + 2 + 3 + 4 + 5 + 6 + ...
- [1 - 2 + 3 - 4 + 5 - 6 + ...]
================================
0 + 4 + 0 + 8 + 0 + 12 + ... == 4(1 + 2 + 3 + ...)


So, S3 – S2 = 4S3, and therefore 3S3 = -S2, and S3=-1/12.

So what’s wrong here?

To begin with, S1 does not equal 1/2. S1 is a non-converging series. It doesn’t converge to 1/2; it doesn’t converge to anything. This isn’t up for debate: it doesn’t converge!

In the 19th century, a mathematician named Ernesto Cesaro came up with a way of assigning a value to this series. The assigned value is called the Cesaro summation or Cesaro sum of the series. The sum is defined as follows:

Let $A = {a_1 + a_2 + a_3 + ...}$. In this series, $s_k = Sigma_{n=1}^{k} a_n$. $s_k$ is called the kth partial sum of A.

The series $A$ is Cesaro summable if the average of its partial sums converges towards a value $C(A) = lim_{n rightarrow infty} frac{1}{n}Sigma_{k=1}^{n} s_k$.

So – if you take the first 2 values of $A$, and average them; and then the first three and average them, and the first 4 and average them, and so on – and that series converges towards a specific value, then the series is Cesaro summable.

Look at Grandi’s series. It produces the partial sum averages of 1, 1/2, 2/3, 2/4, 3/5, 3/6, 4/7, 4/8, 5/9, 5/10, … That series clearly converges towards 1/2. So Grandi’s series is Cesaro summable, and its Cesaro sum value is 1/2.

The important thing to note here is that we are not saying that the Cesaro sum is equal to the series. We’re saying that there’s a way of assigning a measure to the series.

And there is the first huge, gaping, glaring problem with the video. They assert that the Cesaro sum of a series is equal to the series, which isn’t true.

From there, they go on to start playing with the infinite series in sloppy algebraic ways, and using the Cesaro summation value in their infinite series algebra. This is, similarly, not a valid thing to do.

Just pull out that definition of the Cesaro summation from before, and look at the series of natural numbers. The partial sums for the natural numbers are 1, 3, 6, 10, 15, 21, … Their averages are 1, 4/2, 10/3, 20/4, 35/5, 56/6, = 1, 2, 3 1/3, 5, 7, 9 1/3, … That’s not a converging series, which means that the series of natural numbers does not have a Cesaro sum.

What does that mean? It means that if we substitute the Cesaro sum for a series using equality, we get inconsistent results: we get one line of reasoning in which a the series of natural numbers has a Cesaro sum; a second line of reasoning in which the series of natural numbers does not have a Cesaro sum. If we assert that the Cesaro sum of a series is equal to the series, we’ve destroyed the consistency of our mathematical system.

Inconsistency is death in mathematics: any time you allow inconsistencies in a mathematical system, you get garbage: any statement becomes mathematically provable. Using the equality of an infinite series with its Cesaro sum, I can prove that 0=1, that the square root of 2 is a natural number, or that the moon is made of green cheese.

What makes this worse is that it’s obvious. There is no mechanism in real numbers by which addition of positive numbers can roll over into negative. It doesn’t matter that infinity is involved: you can’t following a monotonically increasing trend, and wind up with something smaller than your starting point.

Someone as allegedly intelligent and educated as Phil Plait should know that.

Genius Continuum Crackpottery

This post was revised on June 25, 2014. Mr. Wince has been threatening to sue me for libel. I don’t think that that’s right, but one thing that he’s complained about is correct. I called him a high school dropout. In his article, Wince refers to “when he dropped out of high school”, but in the same sentence, he goes on to say that he dropped out to attend community college. Calling him a dropout is a cheap shot, which I shouldn’t have included, and for that, I apologize. I’ve removed the line from the post. I still think that his math is laughably wrong, but I shouldn’t have called him a dropout.

There’s a lot of mathematical crackpottery out there. Most of it is just pointless and dull. People making the same stupid mistakes over and over again, like the endless repetitions of the same-old supposed refutations of Cantor’s diagonalization.

After you eliminate that, you get reams of insanity – stuff which
is simply so incoherent that it doesn’t make any sense. This kind of thing is usually word salad – words strung together in ways that don’t make sense.

After you eliminate that, sometimes, if you’re really lucky, you’ll come accross something truly special. Crackpottery as utter genius. Not genius in a good way, like they’re an outsider genius who discovered something amazing, but genius in the worst possible way, where someone has created something so bizarre, so overwrought, so utterly ridiculous that it’s a masterpiece of insane, delusional foolishness.

Today, we have an example of that: Existics!. This is a body of work by a guy named Gavin Wince with truly immense delusions of grandeur. Pomposity on a truly epic scale!

I’ll walk you through just a tiny sample of Mr. Wince’s genius. You can go look at his site to get more, and develop a true appreciation for this. He doesn’t limit himself to mere mathematics: math, physics, biology, cosmology – you name it, Mr. Wince has mastered it and written about it!

The best of his mathematical crackpottery is something called C3: the Canonized Cardinal Continuum. Mr. Wince has created an algebraic solution to the continuum hypothesis, and along the way, has revolutionized number theory, algebra, calculus, real analysis, and god only knows what else!

Since Mr. Wince believes that he has solved the continuum hypothesis. Let me remind you of what that is:

1. If you use Cantor’s set theory to explore numbers, you get to the uncomfortable result that there are different sizes of infinity.
2. The smallest infinite cardinal number is called ℵ0,
and it’s the size of the set of natural numbers.
3. There are cardinal numbers larger than ℵ0. The first
one larger than ℵ0 is ℵ1.
4. We know that the set of real numbers is the size of the powerset
of the natural numbers – 20 – is larger than the set of the naturals.
5. The question that the continuum hypothesis tries to answer is: is the size
of the set of real numbers equal to ℵ1? That is, is there
a cardinal number between ℵ0 and |20|?

The continuum hypothesis was “solved” in 1963. In 1940, Gödel showed that you couldn’t disprove the continuum hypothesis using ZFC. In 1963,
another mathematician named Paul Cohen, showed that it couldn’t be proven using ZFC. So – a hypothesis which is about set theory can be neither proven nor disproven using set theory. It’s independent of the axioms of set theory. You can choose to take the continuum hypothesis as an axiom, or you can choose to take the negation of the continuum hypothesis as an axiom: either choice is consistent and valid!

It’s not a happy solution. But it’s solved in the sense that we’ve got a solid proof that you can’t prove it’s true, and another solid proof that you can’t prove it’s false. That means that given ZFC set theory as a basis, there is no proof either way that doesn’t set it as an axiom.

But… Mr. Wince knows better.

The set of errors that Wince makes is really astonishing. This is really seriously epic crackpottery.

He makes it through one page without saying anything egregious. But then he makes up for it on page 2, by making multiple errors.

First, he pulls an Escultura:

x1 = 1/21 = 1/2 = 0.5
x2 = 1/21 + 1/22 = 1/2 + 1/4 = 0.75
x3 = 1/21 + 1/22 + 1/23 = 1/2 + 1/4 + 1/8 = 0.875

At the end or limit of the infinite sequence, the final term of the sequence is 1.0

In this example we can see that as the number of finite sums of the sequence approaches the limit infinity, the last term of the sequence equals one.
xn = 1.0
If we are going to assume that the last term of the sequence equals one, it can be deduced that, prior to the last term in the sequence, some finite sum in the series occurs where:
xn-1 = 0.999…
xn-1 = 1/21 + 1/22 + 1/23 + 1/24 + … + 1/2n-1 = 0.999…
Therefore, at the limit, the last term of the series of the last term of the sequence would be the term, which, when added to the sum 0.999… equals 1.0.

There is no such thing as the last term of an infinite sequence. Even if there were, the number 0.999…. is exactly the same as 1. It’s a notational artifact, not a distinct number.

But this is the least of his errors. For example, the first paragraph on the next page:

The set of all countable numbers, or natural numbers, is a subset of the continuum. Since the set of all natural numbers is a subset of the continuum, it is reasonable to assume that the set of all natural numbers is less in degree of infinity than the set containing the continuum.

We didn’t need to go through the difficult of Cantor’s diagonalization! We could have just blindly asserted that it’s obvious!

or actually… The fact that there are multiple degrees of infinity is anything but obvious. I don’t know anyone who wasn’t surprised the first time they saw Cantor’s proof. It’s a really strange idea that there’s something bigger than infinity.

Moving on… the real heart of his stuff is built around some extremely strange notions about infinite and infinitessimal values.

Before we even look at what he says, there’s an important error here
which is worth mentioning. What Mr. Wince is trying to do is talk about the
continuum hypothesis. The continuum hypothesis is a question about the cardinality of the set of real numbers and the set of natural numbers.
Neither infinites nor infinitessimals are part of either set.

Infinite values come into play in Cantor’s work: the cardinality of the natural numbers and the cardinality of the reals are clearly infinite cardinal numbers. But ℵ0, the smallest infinite cardinal, is not a member of either set.

Infinitessimals are fascinating. You can reconstruct differential and integral calculus without using limits by building in terms of infinitessimals. There’s some great stuff in surreal numbers playing with infinitessimals. But infinitessimals are not real numbers. You can’t reason about them as if they were members of the set of real numbers, because they aren’t.

Many of his mistakes are based on this idea.

For example, he’s got a very strange idea that infinites and infinitessimals don’t have fixed values, but that their values cover a range. The way that he gets to that idea is by asserting the existence
of infinity as a specific, numeric value, and then using it in algebraic manipulations, like taking the “infinityth root” of a real number.

For example, on his way to “proving” that infinitessimals have this range property that he calls “perambulation”, he defines a value that he calls κ:

$sqrt[infty]{infty} = 1 + kappa$

In terms of the theory of numbers, this is nonsense. There is no such thing as an infinityth root. You can define an Nth root, where N is a real number, just like you can define an Nth power – exponents and roots are mirror images of the same concept. But roots and exponents aren’t defined for infinity, because infinity isn’t a number. There is no infinityth root.

You could, if you really wanted to, come up with a definition of exponents that that allowed you to define an infinityth root. But it wouldn’t be very interesting. If you followed the usual pattern for these things, it would be a limit: $sqrt[infty]{x} lim_{nrightarrowinfty} sqrt[n]{x}$. That’s clearly 1. Not 1 plus something: just exactly 1.

But Mr. Cringe doesn’t let himself be limited by silly notions of consistency. No, he defines things his own way, and runs with it. As a result, he gets a notion that he calls perambulation. How?

Take the definition of κ:

$sqrt[infty]{infty} = 1 + kappa$

Now, you can, obviously, raise both sides to the power of infinity:

$infty = (1 + kappa)^{infty}$

Now, you can substitute ℵ0 for $infty$. (Why? Don’t ask why. You just can.) Then you can factor it. His factoring makes no rational sense, so I won’t even try to explain it. But he concludes that:

• Factored and simplified one way, you end up with (κ+1) = 1 + x, where x is some infinitessimal number larger than κ. (Why? Why the heck not?)
• Factored and simplified another way, you end up with (κ+1) = ℵ
• If you take the mean of of all of the possible factorings and reductions, you get a third result, that (κ+1) = 2.

He goes on, and on, and on like this. From perambulation to perambulating reciprocals, to subambulation, to ambulation. Then un-ordinals, un-sets… this is really an absolute masterwork of utter insane crackpottery.