This post is dated April 16th, which is Easter Sunday on both the Gregorian and Julian (Orthodox) Calendars. (The next common Easter is not until April 20th, 2025.) It is the start of spring break. This is the 106th day of the year, and so we read page 106 of Pappas.
This is what Theoni Pappas writes on page 106 of her Magic of Mathematics:
"We are familiar with the natural numbers, the whole numbers, the integers and how and what they are used to count. But what about Georg Cantor's transfinite numbers?"
Yes, what about them -- one person who explains them well is Eugenia Cheng, in her newest book, Beyond Infinity. This Pappas page is titled "Cantor & the Infinite Cardinal Numbers," and cardinal numbers are exactly what Cheng writes about in Chapter 8, "Comparing Infinities." She begins:
"When children learn to climb up one step, they get very excited and climb up another and another. They marvel at how high they can go just by iterating one new thing that they have learned."
In previous chapters, Cheng derives several facts about infinity. But we're starting Chapter 8 in today's post because I'm bound by the order in Pappas (which is less logical than Cheng's order). So let's just summarize the facts right here:
-- The real numbers are more infinite than the natural numbers.
-- If the infinity of natural numbers is written as "omega," then there are 2^omega real numbers.
Cheng tells us that our aim is to define infinity in such a way that these hold. She admits that this is backwards from what we think math is. Normally we define objects (like natural numbers) first and see what properties they hold, rather than start with the desired properties (the "desiderata"). But she explains that even as early as algebra, we are given an equation, say
3x + 4 = 10
And we're saying, I want this to be true. What values of x make this true? And so this is what we're doing with our definition of infinity.
Cheng continues:
"In this previous chapter [Chapter 6, covered later in Pappas and hence on the blog -- dw] we showed that the real numbers are uncountable by showing that any attempted pairing with the natural numbers was doomed to leave out at least one real number. We can use this to compare any two sets of things."
She provides a simple example:
1 --> 4
2 --> 8
3 --> 12
16
This is closely related to the concept of functions that we teach in Common Core math. Indeed, Cheng explains:
"If every attempted pairing between sets A and B is doomed to leave something out on the right, then we say that the set on the right is 'bigger.'"
She does add one caveat, though:
"The technical situation is slightly more complicated than this. The sentence above says that a surjective function from left to right is impossible, but technically we also have to say that an injective function from left to right is possible."
"Injective" essentially means "one-to-one." Cheng writes that this is related to the Axiom of Choice, which she discusses in Chapter 6 (and I discuss in a later post).
Anyway, she tells us that we can tell whether a set has exactly ten elements or not by attempting to find a function with that set as the domain and:
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
as the range. What we are finding is the size or "bigness" of the set -- its cardinality.
Cheng writes:
"The smallest possible set is empty. So the smallest possible cardinality is 0."
We know what zero and all the finite cardinal numbers all, but now we want infinite cardinals. She tells us that the set of all natural numbers has the smallest infinite cardinal. She explains that any subset of the natural numbers has the same cardinality as all the natural numbers, using the Hilbert Hotel analogy. Even if many of the rooms are empty, if there are infinitely many guests in the hotel, then they can fill all the rooms, as follows:
-- Whoever has the smallest room number will count zero people with smaller room numbers. So they move into room number 0 + 1 = 1.
-- Whoever has the next smallest room number will count one person with a smaller room number. So they will move into room number 1 + 1 = 2.
Likewise all the guests can move down to the smallest available room number. And so the infinity of the naturals is the smallest infinite cardinality. Cheng adds:
"Remember when we were trying to see if infinity was a number, we kept running up against the problem of subtracting infinity from both sides of an equation. We kept finding that subtracting infinity from both sides of an equation led to a contradiction."
And now we can see why there was a contradiction -- the naturals are the smallest infinity. Cheng tells us that this smallest infinite cardinal is given the name aleph-0, named after the first letter of the Hebrew alphabet. As Pappas writes:
"Aleph-0 (aleph-null) refers to the number of counting numbers. Any set that can be put in a one-to-one correspondence with the counting numbers is said to have aleph-0 number of elements."
Returning to Cheng:
"We have seen many infinities that turned out to be the same size as the natural numbers. There are ones that seem to be smaller but aren't, such as the even numbers or the odd numbers."
She tells us that there is a set that is larger than the natural numbers -- the real numbers. But is this the next largest cardinality? The idea that the reals are indeed the next cardinality after the naturals is called the Continuum Hypothesis. Cheng writes:
"It has been proved that the Continuum Hypothesis can't be proved using a standard type of logic called Zermelo-Frankel logic. It has also been proved that it can't be disproved!"
Therefore the Continuum Hypothesis (like the Axiom of Choice) is independent of ZF. Similarly, Euclid's Fifth Postulate (on parallels) is independent of the first four. It's possible to find worlds in which CH is true or false, just as with Euclidean and non-Euclidean geometry.
Logically, the next infinity up from aleph-0 should be aleph-1. Therefore, CH is the statement that the cardinality of the real numbers is aleph-1. Pappas lists other sets that would have this cardinality -- the points on a line, sphere, or cube.
But Cheng takes this a step further. She shows us that a short interval, such as [0, 1/100], has the same cardinality the unit interval [0, 1]. In this case, the one-to-one correspondence mapping the shorter interval to the longer interval is to scale up by 100 -- in other words, a dilation. She writes:
"This argument would work for any size interval, from any real number to any other real number. To scale the portion from 0 to 1 up to the whole of the reals, we have to do something more sneaky as we can't just 'multiply everything by infinity.'"
Her trick is to use the function mapping x to 1/x, which really does map a finite interval (0, 1) to an infinite interval.
Earlier, we wrote that if the naturals have cardinality infinity, the reals have cardinality 2^infinity. So we can also write CH as an equation:
aleph-1 = 2^aleph-0.
Cheng writes that there is a general form of the continuum hypothesis, GCH. It states that at every stage, we form the next higher infinity by raising 2 to the power of each infinity:
aleph-0, aleph-1, aleph-2, aleph-3, ...
where in each case aleph-(n + 1) = 2^aleph-n. Cheng doesn't provide us with any examples of sets with cardinality aleph-2, but Pappas does -- the set of all curves. Pappas adds that "no one has yet come up with examples of sets that use aleph-3 or aleph-4 or any higher ones."
Let's return to Cheng and try to figure out why she uses 2^aleph-0 for the cardinality of the reals:
"If aleph-0 is 'the size of the natural numbers,' what could it possibly mean to raise 2 to the power of it? Usually we define 2^n to mean '2 multiplied by itself n times,' but this won't work with infinity because you can't 'multiply 2 by itself an infinite number of times.'"
She is trying to explain the concept of power set. Suppose we have a pair of sneakers and a pair of sandals, then we have these choices:
-- Take no shoes. Go barefoot.
-- Just take the sandals.
-- Just take the sneakers.
-- Take both the sneakers and the sandals.
Then with two pairs of shoes there are 2^2 = 4 possibilities. If there's a third pair of shoes, let's say flip-flops, then there are 2^3 = 8 possibilities -- and Cheng draws a binary tree to show them all.
So if these were all the natural numbers, of which there are aleph-0, then there would in fact be 2^aleph-0 combinations of naturals (that is, subsets of naturals). But as Cheng points out:
"If you remember that 2^aleph-0 is supposed to be the size of the real numbers, you might think that it's a bit random to try and understand the 'number of real numbers' via the 'number of possible subsets of the natural numbers.'"
But Cheng points out that these are related -- a real number can be written in decimal notation, or better yet in binary notation. Then all of the digits are either 0 or 1 -- that is, a subset of the digits have the value 1. So this is the missing link between reals and subsets of naturals.
Then Cantor's diagonal argument (explained in Chapter 6 -- sorry!) is then used to show that the power set has larger cardinality than the set itself, so we conclude, again assuming GCH:
aleph-0
aleph-1 = 2^aleph-0
aleph-2 = 2^aleph-1 = 2^2^aleph-0
aleph-3 = 2^aleph-2 = 2^2^2^aleph-0
is a hierarchy of ever-increasing infinities.
This post is labeled "traditionalists." Last year (in October 2015), I actually wrote about something called "ordinal numbers" and mentioned them in an argument about Common Core. Ordinal numbers are actually the subject of Chapter 9 in Cheng's book.
Unfortunately, Chapter 9 is not one of the chapters I selected to cover on the blog. But still, since "traditionalists" are mentioned in the label and title of this post, I want to cut-and-paste what I wrote last year about ordinals. A fuller explanation of ordinals can be found in Chapter 9 of Cheng (and at the link I gave in my original post):
So now let's look at the first problem. Mehta tries to explain here why 5 * 3 should be written only as 3 + 3 + 3 + 3 + 3 and not as 5 + 5 + 5. But I disagree with Mehta here. For just as we appealed to higher math to explain how to draw the array, I can appeal to higher math to explain why, in fact, we should say that 5 + 5 + 5 is correct and 3 + 3 + 3 + 3 + 3 is wrong.
To find out why, the higher math we seek out is set theory -- the same theory that mathematicians all the way from Georg Cantor to Randall Holmes studied. In particular, we consider something called an "ordinal." Instead of 5 * 3, let's consider omega * 3, where omega -- the last letter of the Greek alphabet -- is a particular ordinal. Just as with matrix multiplication, ordinal multiplication is not commutative -- and in fact, omega * 3 is exactly omega + omega + omega. It is definitely not the same as 3 * omega. So using ordinal arithmetic, 5 * 3 = 5 + 5 + 5 is exactly right.
I've decided that it would detract from this Geometry blog to get into a deep discussion of what exactly ordinals are -- except that they have something to do with order (which helps to explain why the order matters). Here's a link that describes ordinal multiplication in more detail:
http://mathworld.wolfram.com/OrdinalMultiplication.html
I can see why students would be frustrated when they lose points for multiply in the wrong order. As Geometry teachers, we can compare this to the student who complains about losing points for not putting square units for area or cubic units for volume. Of course, there are actual reasons that students must put square or cubic units. But juxtaposing these two situations in which students lose points even after doing all of the calculations correctly, we can see why from the students' perspective they are just losing points for silly reasons in both cases.
And it was a poster with the single letter b who responded with ordinals:
b:
You are not teaching students a useful skill. You are teaching them an arbitrary rule that has no applicability outside passing a test that you wrote. No mathematician would say that pq can only mean p groups of q, never q groups of p. It can be either, or neither.
After thinking for a while, I've come up with two cases where a product has an asymmetrical grouping interpretation in real mathematics.
One is that matrices are more commonly divided into column vectors than row vectors, so if you think of a 5-component vector as a group of 5 things, then a 5×3 matrix is more likely to be 3 groups of 5 than 5 groups of 3.
The other is ordinal multiplication, where ω·3 means ω+ω+ω, while 3·ω means 3+3+3+... (ω times), which is less than ω+ω+ω.
Note that in both cases the order is the opposite of what you teach.
[end b's response]This concludes the post. My next post will be on Tuesday, April 18th, first day after spring break.
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