Aside #sqrt(2)/2: I promised Gene-Bob that I would discuss the cardinality of the continuum the next day, but failed to follow up, citing a discontinuity in the continuum. Interestingly enough, it was a discontinuity in Fourier series that led Georg Cantor to the discovery that all infinities were not the same.
Aside #aleph_0 + 1: My family's fascination with infinity, or at least my and 2 of 2's fascination with infinity, probably began with Buzz Lightyear, who always departed with the phrase, "To infinity, and beyond!"So, what is infinity? You might be tempted to say that it is the largest number. But couldn't I define a number as infinity plus 1, and wouldn't that be bigger than infinity? Well, yes and no. Mathematicians avoid these kinds of problems by dodging the issue and saying that infinity is more of a concept than a quantity. It's similar to how physicists say that light is sort of a particle and sort of a wave.
Aside #many: A very strange, but very beloved friend of mine at the Looniversity of Tex-Mex at Autism, is fond, after a few beers and a couple of slices of Milto's pizza, of saying that there are really only a few numbers that really matter. For example, 1 and 2. Anything more than a very small number can be represented as "many". I think he used to cite some ancient culture that had such a limited number system. "Many" was their infinity.
Aside #0: I can't recall if my friend had a concept of zero in his numbering system. Oh, by the way, he was the accountant for one of the departments at the University.So back in the 70's (the 1870's), a curious fellow named Georg started looking into the concept of the continuum. An example of a continuum would be the points on a line. How many are there? Is it the same number as the number of cardinal numbers?
Curious Georg decided that there were more points in the continuum than there were counting numbers (i.e. cardinal numbers). In this off-Cantor world, there are orders, or levels, of infinity. The lowest order of infinity is called aleph null.
Aside #128: As they say at the First Brobdignagian Babatist Church of Suburbia when discussing the King James Version of the Bible, "if it was good enough for JEE-sus then it's good enough for me." I think a similar statement can be made for the ASCII character set and we should be immediately suspicious of someone who proposes the use of characters that aren't in ASCII.Aleph is the first letter of the Hebrew alphabet. And not only is it not found in ASCII, it isn't in the ISO 8859-1 character set either. So since I doubt that ℵ happens to produce a funny looking X on your screen, I'll just have to write aleph_0 and you should think "aleph null". Similarly, aleph_1 is "aleph one".
Now that we have notation nailed down, let's get back to infinity. Aleph_0 is defined as the number of counting numbers, or the number of elements in any countably infinite set. Counting numbers are a good example since it is fairly easy to construct the next in the sequence no matter where you are. So we can accept that, given infinite time we could count (list) all of the counting numbers. Don't you just love circular definitions? Okay, let's just say that you can list all of the counting numbers from zero on up, without missing any, for as long as you have the time. For example, you could list every last counting number from 1 to 1,000,000. With any luck, you now understand the concept of a set being countably infinite. The size of that set (its cardinality) is aleph_0.
The next issue is to figure out if a given infinite set is countable or not. We do this by trying to do a one-to-one mapping between the set of interest and the set of cardinals. For example, is the set of even numbers countably infinite? Well, yes it is because we can map the cardinal 1 to the even number 2, the cardinal 2 to the even number 4, and so on, for all possible even numbers. We'll never finish the job, but since we can define a one-to-one mapping that clearly won't miss any of the even numbers, and since we've already accepted that the cardinals are countably infinite, we have proven that the set of even numbers is also countably infinite. So both sets are the same size, and the size is aleph_0.
Stop! Back up. Did I just say that there are as many (positive) even numbers as there are cardinals? Yep. I did. Maybe now you're starting to understand why Curious Georg lost his mind and wound up in the looney bin.
Let's try another set: the real numbers between zero and one. Is it countably infinite? Let's make a list. Each number will be represented by an infinite set of digits, so 0.2 is written in our list as 0.20000000... (with an infinite number of 0's off to the right, so that we can distinguish it from 0.200000000...1). Here's a bit of my list (not in order):
1) 0.1000000000...Okay, I've listed every cotton-picken real number between 0 and 1! It took a while, but I did it! Now lets check to make sure that I didn't miss any. To do this I'm gonna make a new number and see if it's in my list. Like all of my numbers, my new number has an infinite number of digits in it. The first digit (to the right of the decimal) will be different from the first digit in the first entry on my list. The second digit will be different from the second digit of the second entry in the list. You following me? Let's look at my list and highlight the digits I'm looking at:
2) 0.1844765194...
3) 0.1368698456...
4) 0.1684413685...
5) 0.1277569545...
6) 0.9423655486...
7) 0.7526952756...
8) 0.3149712951...
9) 0.8211036501...
10) 0.3741098120...
1) 0.1000000000...The red digits in my list are going to be used to create my new number using this formula: new digit = (old digit + 1) modulo 10. That's a fancy way of saying that I'll change a 0 into a 1, a 1 into a 2, a 2 into a 3, and so on, with a 9 becoming a 0. Here's the transform:
2) 0.1844765194...
3) 0.1368698456...
4) 0.1684413685...
5) 0.1277569545...
6) 0.9423655486...
7) 0.7526952756...
8) 0.3149712951...
9) 0.8211036501...
10) 0.3741098120...
0.1864552900... becomes 0.2975663011...Guess what? It isn't in my list. How do I know? Well, because it is different from the first number in my list in at least the first decimal place. And it is different from the second number in my list in at least the second decimal place. And it is different from the third number in my list in at least the third decimal place. And so on into infinity. So my list wasn't complete after all. I failed in producing that one-to-one mapping from the cardinals to the reals without missing any! And it can't be done!
Therefore, the set of reals (between 0 and 1, and by extension the set of all reals) is uncountably infinite. We define the number of elements in this uncountable set as aleph_1.
Later on, in a proof I haven't seen, Cantor proposes that aleph_1 = 2aleph_0. I'll have to take his word for it.
3 comments:
speaking of conundrums, is ∞/0 still undefined (division by zero) or does it fall into one of the Infinity rules, where ∞/anything is still ∞?
Also, after reading this blog entry, I'm curious if you have any thoughts about the danger of dihydrogen monoxide?
Division by zero is one of those things that mathematicians dodge because they can't handle it. My personal belief is that dividing by zero results in infinity. It is logical, after all.
So, Dividing infinity by infinity should result in infinity, not because dividing infinity by anything results in infinity, but because dividing by zero results in infinity. In fact, I might go so far as to posit that aleph_0 divided by zero is aleph_1!
As for DHMO, my position is that nothing is either completely good or completely bad. Moderation is the key. So use DHMO in moderation and you should be fine.
I theorize that ∞/∞ is equal to 1, not because it's logical but because of the "any number on the top being the same as any number on the bottom must make the result equal to one" theory. it worked for me in first grade, and Everyone Knows™ that Everything Taught In First Grade must be true (witness the recent villification of the music teacher, who dared suggest that Santa was Faux!).
Also, I'm working on a better name for this theory .. it's a bit cumbersome at the moment .. doesn't just Roll Off The Tongue.
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