It’s Wednesday now, and I’m writing this post on my lapcom. There’s no particular reason I’m doing so, though as always, a cause or causes surely exist(s). I just felt the impulse to take the lapcom back to the house with me yesterday afternoon, and I saw no reason not to indulge that impulse. That’s about it.
It’s the 2nd day of September, which should come as no surprise, since yesterday was the 1st day of September. Numbers don’t necessarily have to be put in any given order—and indeed, one cannot properly order the Real numbers, because there is an uncountable infinity of them between any two points on the number line, however arbitrarily close you might want to choose them—but the ordinal numbers have an inherent order. Thus the name. Duh.
I often marvel at the cool facts about uncountable versus countable infinities. Think about it. The “natural” numbers represent a “countable” infinity because you can make progress counting them and, if you had infinite time, you could in principle count them all, though it can be a bit mind-bending to try to imagine what that could really mean. But it is true, mathematically speaking.
It’s also true that this infinity, the so-called countable infinity (ℵ₀, or “aleph nought”), is matchable one-to-one with all other countable infinities. You can assign a natural number to every integer, and to every even or odd number, and to every rational number (i.e. every fraction) and so on. Therefore, they are the same “size” or “cardinality”.
In case anyone wonders why that point might matter, all one has to do is to try to list the Real numbers (designated with ℝ). These are all numbers including the integers and the rational numbers, but also all irrational numbers, i.e., numbers that can’t be expressed as a ratio of any whole numbers or integers. The latter set includes pi (π) and e (e) and phi (φ, the so-called Golden Ratio) and so on, but these are simply a taste of what’s available. All of these numbers have a potentially infinite number of digits* after the decimal point, and those digits do not have any repeating pattern, for if they were consistently repeating, they would be rational numbers.
Cantor’s “diagonal proof” demonstrates that there are more Real numbers than there are integers (or any other aleph-nought level infinities of numbers). I won’t get into it here—it’s not that difficult, and it’s very clever and rather wonderful, but it would take a bit of time, so I just put a link to it for anyone who is curious.
What’s more, once we realize that fact and think about it a little, we realize not only that the Real numbers are uncountably infinite, vastly more infinite than the integers of the natural numbers, but something even more mind-boggling: Between any two numbers you can choose, however arbitrarily close you want to make them, there is an uncountable infinity of Real numbers.
You could pick two numbers that have eighty-thousand zeroes after the decimal place before you start getting non-zero digits—such numbers would be far closer together than any separation we could possibly physically measure using any sensible unit of measurement—and the number of Real numbers between them is much, much larger than the number of integers, and is indeed equal in size to the Real numbers.
You can kind of understand why some of the people who worked with notions such as these, like Cantor and Gödel and so on, got pretty loopy at times**, though it is not possible to say definitively whether the mental atypia were caused by too much contemplation of infinities or were the reasons why such people got into studying them, or were pure coincidence.
Of course, the above raises for me the question whether the infinity of the Complex numbers is larger than that of the Real numbers. I don’t believe it is***; I’m not looking it up to confirm right now, but I am just going to think it through out loud, as it were.
A complex number is made of two “parts”: the so-called Real part and the so-called Imaginary part. But the Imaginary part is just i, the square root of negative one, multiplied by some Real number. So, each complex number is made up of two Real numbers, but two real numbers can be expressed as one Real number in various ways. So, the Complex numbers have a higher dimensionality than the Real numbers (2 versus 1), but they have the same cardinality, as the terms go.
There are higher versions of countable infinities used occasionally in mathematics. I find them a bit more contrived than the preceding, but perhaps that’s just because I’ve never had the call to use them nor to play around with them. These are your ω and ω +1 and so on. The idea behind it is that you can arbitrarily say that, once you’ve counted all the natural numbers (for instance), whatever the hell that might mean****, you can go on and count some more, calling the next number after infinity ω (that’s a small-letter omega, the counterpart to Ω, which is kind of a cooler-looking symbol).
Anyway, that’s enough of me nattering on about matters that don’t really ever seem to interest anyone around me other than me—I guess I’m not really around me, so I didn’t really need to add to caveat—but about which people sometimes smile indulgently for a moment or two, since I sometimes get rather excited when mentioning them.
This doesn’t happen often, though. By which I refer to occasions on which I get excited.
This is all just distraction, anyway, and that’s getting more difficult for me over time. I don’t really feel that I’m going anywhere or getting anywhere. I’m just killing time while waiting for something to kill me. I hate the world (by which I mean the universe, and maybe even the Universe), I hate my life, and I hate myself. This is not a mood (though it entails a mood at times); this is a judgment, this is my (provisional) conclusion, this is my strongly held opinion. I need to stop running from it; I suck, and so does everything and nearly everyone else.
Present company is excluded, of course (not counting me). Indeed, I wish all of my readers the very best possible of every possible thing for which things can be the best.
*Though it is interesting to contemplate whether one can take any arbitrarily large run of digits of, say, π and always be able to find the matching sequence in any other irrational number. I don’t know the answer to that question, so if anyone does, please let me know.
**Poor Gödel became convinced—or at least afraid—that someone was trying to poison him, or might try to poison him, and so he would only ever eat food prepared by his wife. There is something rather charming, almost romantic, to the notion that he distrusted other people, in principle all other people, and yet he believed in his wife completely. I wonder if she was flattered. It was tragic, though, because when she became ill, he stopped eating, and he eventually more or less starved to death.
***It isn’t. I eventually looked it up just to confirm (or refute) my judgement.
****You can say this, in a way, because (for instance) you can count the integers by first counting the positive integers, to positive infinity, and only then starting to count the negative integers toward negative infinity. This means you can continue counting after you’ve counted an infinite number, at least in a sense.
