Transfinite cardinals in da house!
Are you an earnest young actor? I ask only because so many of the actors my own age that I see onstage seem to be of the impression that eyes-wide, gulping-down-air, semi-innocence is the only way to go. I know Oscar Wilde was all about the importance of being earnest, but it’s okay to bring a little gravity to your roles. The audience will still get it.
Also, never yell. There’s a difference between projecting and yelling, and it isn’t “heightened drama”.
Now that we’ve got that out of the way, I’m going to talk about math. Because the grand majority of my blog entries are about humanities stuff like theatre and literature and philosophy, and, you know, that’s only half of what I do. Well, let’s go with a figure of about 80%*, but still.
Admittedly, I don’t have the passion for numbers that some of my friends do. I like getting the right answer, and I like understanding how I got it. I like taking things apart logically, too. I like challenges like “Can every positive even integer be expressed as the sum of two primes?”**, or “How do you divide a cake for an unspecified number of recipients so that each recipient feels she got a fair share?”. But my very favourite mathematical topic (well… at the moment) is Cantor’s system of transfinite cardinals.
A cardinal number is a number that expresses an amount of something: 1, 9, and 389798 are all cardinal numbers. Transfinite cardinal numbers are a system of numbers you can use to say how big a certain kind of infinity is. Because, while an infinity of items is, well, endless, it’s still possible to show that, say, the number of real numbers (infinite) is greater than the number of integers (also infinite).
(This may be a good time to mention that Cantor, the guy who first suggested all this, went crazy-paranoid and spent a lot of time having nervous breakdowns.)
Anyway, chapters nine and ten of William Dunham’s Journey Through Genius explain transfinite cardinals pretty well, but I’m going to give it a shot, just for kicks. Whaddayasay?
So, the first question a would-be master of transfinite cardinals has to ask himself may seem to have an obvious answer: how can you tell when you have the same amount of two different items? Quoth the would-be M. of T. C.: “Uh… I count them?” Okay, good start. But what does counting show?
Would-be M. of T. C.: “That the number of one is the same as the number of the other, so I have the same number of both?”
All right; now suppose you’re going through thousands and thousands of votes by hand. Each voter has deposited either a red slip to support candidate A or a blue slip to support candidate B. You could count the red slips first, record the number, and then count the blue slips to see if they match up, but, let’s face it: counting is long and boring and easy to mess up, and all you really want to know is who won. So, instead, you grab two slips at a time, one red and one blue, and toss them aside until you’re either out of slips or left with only one colour. If there’s only red left, there must have been more reds than blues; if there’s only blue left, there must have been more blues; and, if nothing’s left, there must have been exactly the same number of reds as blues.
But you don’t know exactly how many slips there were. You just know there’s the same number of reds as blues in the discard pile because you matched each and every red to a single blue.
The magnitudes of infinite sets work kind of like that. Just like the thousands and thousands of voting slips, you can’t count them directly***. But you can try to match their individual members on a one-to-one basis. If it works, they have the same magnitude. If it doesn’t, one is bigger than the other****.
Cantor started with the natural numbers: 0, 1, 2, 3, 4… He decided to call their infinite magnitude aleph-nought. Then he tried to match other fields of numbers to the natural numbers on a one-to-one basis. (Isomorphically, if you feel like using the fancy-dancy-pants term.) Basically, matching something to the natural numbers just means that you have to be able to put them in some sort of order without leaving out a single one. Because, if you can do that, you can put them in a long column, and, on the other side of the column, you can put “0, 1, 2, 3, 4…”. One-to-one. So the integers (… -2, -1, 0, 1, 2…) have the same magnitude as the natural numbers because you can list the integers in the order: 0, -1, 1, -2, 2, -3, 3… You’ll get every integer without leaving any out.
Cantor found a way to match the rational numbers (all numbers of the form q/r, where q and r are both integers and r is not equal to zero) to the natural numbers, too.
At this point, you may be wondering whether there’s any point to this one-to-one matching stuff if every infinity seems to be of the same magnitude as that of the natural numbers. To this I say – but wait, there’s more!
Cantor then turned his attention to the real numbers: the stuff that includes pi, the square root of two, 6, ½, 0.38473892057657…, etc. Can you put the real numbers in an order without leaving out a single one? (I know, I know, the suspense is terrible.) Well, as it turns out, no, you can’t. And you can prove that. In fact, you can even prove that the infinite number of real numbers between 0 and 1 is “bigger” than the infinite number of natural numbers.
First, you write every real number as an infinite decimal of the form:
0.a1a2a3a4a5a6a7….
where the a#s are all digits, and, if you have the choice between ending in …0000…. instead of ending with ….9999…, you choose the former*****.
Then, you assume that you can somehow order the real numbers between 0 and 1 – that you can match each with a unique natural numbers. So, you have a list that looks like this:
0 – 0.a1a2a3a4a5a6a7….
1 – 0.b1b2b3b4b5b6b7….
2 – 0.c1c2c3c4c5c6c7….
etc.
But wait a minute! There has to be at least one number of the form 0.a1a2a3a4a5a6a7…. that isn’t on your list. How do you know? Well, imagine doing the following:
1. Take digit a1 from the first number on your list, and either add or subtract one to get a new digit, x1. If one of the two operations would make the new digit 0 or 9, choose the other option.
2. Do the same for digit b2 of the second number on your list to get x2.
3. And the same for digit c3 of the third number to get x3.
4. You see where this is going?
5. So, once you’ve gone down your (infinite) list and made a new digit xn for each nth entry, you have a new number: 0.x1x2x3x4x5x6x7…
6. Let’s call that new number George.
7. Now, you know George isn’t on your original list because you’ve gone through very carefully and constructed George so that it’s different from each number by at least one digit. You know its first digit is different from the first digit of the first number, its second digit is different from the second digit of the second number, etc., etc. ad infinitum.
8. So, no matter how many real numbers are on your initial list, you’ve always left at least one out.
9. CONTRADICTION WITH YOUR ORIGINAL ASSUMPTION!!!1!!eleven!!!
10. Therefore, your assumption was incorrect: you can’t order the real numbers between 0 and 1.
11. By similar arguments, you can’t order the real numbers.
12. The infinite number of real numbers is of a different magnitude than the infinite number of natural numbers, integers, or rational numbers.
13. And so’s your mom.
* If you’re gonna do math, you gotta be precise.
** If you can prove this, you’re only six years too late to be in the money!
*** Unless you have an infinity of time. Which you don’t, unless you’re the archangel Gabriel or something. But then you’d probably have more interesting things to do.
**** Which means something sort of different when we’re talking about infinities. But don’t you worry your pretty little head! (Or, alternatively, you can check out Wolfram Mathworld.)
***** This is necessary because 0.39999… = 0.40000…, so if you don’t explicitly make a choice, you’ll wind up repeating some numbers. But you don’t have to take my word for it:
Let x=0.39999…
10x=3.9999…
100x=39.9999…
100x-10x=90x
39.9999… – 3.9999… = 36.0000…
90x=36.0000…
x=0.4000…
QED, mofos!