29 May 2001 Infinity is NOT a number III: The Search for Lebesgue ------------------------------------------------------ I'm tired of the mundane look at infinity. You see, generally when people do stuff about infinity, the most mind-blowing they will get is to show you that you can put a proper subset of an infinite set in a 1-1 relation with itself. Fooey. That's not the strangest infinity can get. It can really blow your mind, and often messes up one's "intuition" about various things. (By the way, "mathematical intuition" is not something one is born with. One has to sit around for long periods of time, thinking about examples and counterexamples of various properties. It comes from meeting every statement of definition with questions such as: what's an equivalent expression for this? what would happen if this property =didn't= hold? let me try it for x=pi... etc. You gotta work to develop "intuition".) One of the best ways to learn about what one thinks about infinity, limits, set sizes, etc., is to consider paradoxes -- things that just don't seem to resolve themselves. In some cases, the paradoxes are real, in that the inherent contradiction can't go away without changing the problem: "This statement is false". Sometimes it is only an =apparent= paradox, in which a better understanding of the processes involved takes one to a reasonable conclusion, as opposed to an earlier absurd one: like Zeno's paradoxes. Zeno's paradoxes are, in a way, the easiest to deal with, because they have to do with limits. The one most people will remember is this one: "You can't cross the room; for, to get from one side to the other, one must first go halfway across, and then go halfway across the remaining distance, and then halfway across that... One can never reach the other side, for there is always some non-zero distance left to go! Of course, one can simply stride across the room and say you've disproved Zeno's hypothesis, but then Zeno says -- no, no, the motion is an =illusion=. Being able to move just doesn't make sense, so even though you thought you were moving, you weren't. Yeah, convincing argument, eh? Funny how psychics get away with that kind of explanation all the time... Anyway, if we look at the distances Zeno talks about, one can see there's a limit to the sum: 1/2 + 1/4 + 1/8 + 1/16 +.... = 1 -- but that's an equality only when one has the countably infinite number of terms on the left... how can we possibly cross the room in finite time? Well, we couldn't if we kept changing our pace so that we can step up only to the halfway mark. But as we generally have a fixed pace, we get a =converging series= for the time it takes us to cross as well. So each of those smaller and smaller steps take a linearly related smaller and smaller time, so infinity is wrapped up in a nice, small package. By the way, when one adds up a countably infinite number of terms, it's called a =series=. When one gets the next term by multiplying by a constant, it's called a =geometric series= (like 1/3 + 1/9 + 1/27 + 1/81 + ...). How do we prove that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1? Well, we =could= do it the rigorous way, with limits of sequences, with Ns and epsilons, but let's not waste our time, shall we? Let's just assume that the sum =is= some number S. So S = 1/2 + 1/4 + 1/8 + 1/16 + ... let's multiply both sides by 1/2! 1/2*S = 1/4 + 1/8 + 1/16 + .... Now look at the right hand side above. It looks =just= like the original S, except it's missing the initial 1/2. So let's add that to both sides. 1/2 + 1/2*S = 1/2 + 1/4 + 1/8 + ... = S Now we have a simple algebra problem: 1/2 + 1/2*S = S Multiply both sides by 2 : 1 + S = 2*S Subtract S from both sides: 1 = S There, our series = 1! However, one can get into deep doo-doo when one eschews rigor in math. Physicists walk this line to their peril sometimes (though physics students are in the worst danger) -- luckily, things tend to be bounded because though Nature doesn't really abhor a vacuum, it DOES abhor unbounded quantities. Why is this a danger? Let's try another series without thinking: S = 1 + 2 + 4 + 8 + .... Lovely geometric series there, the ratio from one term to the next is 2. So let's do our magic: 2*S = 2 + 4 + 8 + ... 2*S + 1 = 1 + 2 + 4 + 8 + .... = S 2*S + 1 = S 2*S = S - 1 S = -1 Okay, so the limit is -1. But we keep adding positive numbers together! Have we learned a secret about infinity? If one goes high enough, one wraps around through the negative numbers? Does this make any sense? And the answer is: no, this doesn't work. We goofed. We couldn't say: S = 1 + 2 + 4 + 8 + ... because by putting down S, we were assuming the limit was a =number=. And, class, we all know that: (all together) Infinity is NOT a number. We can just look at the series and tell it is unbounded, and has an infinite limit. We get slapped with a wet noodle for doing what we're not supposed to do. So let's try another infinite series - something that caused no end of trouble in the Middle Ages, and made people befuddled because they had no rigorous idea of mathematics. I'm not even sure what most of the Greeks would've made of it (considering the difficulty Zeno had, and it seems noone satisfactorily answered his paradoxes until about Newton's time): look at this: 1 - 1 + 1 - 1 + 1 - 1 + 1, etc. now, this is addition an subtraction -- those are associative operations, so it shouldn't matter how we group them: (1 - 1) + (1 - 1) + (1 - 1) + ... = 0 + 0 + 0 + 0 + 0 .. It doesn't matter how many times we add zero, we'll always have zero, so the sum is obviously zero. But wait, let's try a different grouping: 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ... = 1 + 0 + 0 + 0 + .... So wait, it looks like the answer is 1. Want to really confuse yourself? Notice that what we have is a geometric series, with a ratio of -1, so let's do our S trick again: S = 1 - 1 + 1 - 1 + ... -1*S = - 1 + 1 - 1 + ... 1-1*S = 1 - 1 + 1 - 1 + 1 ... = S 1 = 2*S S = 1/2 1/2? What?! Again, the answer is that one is not being rigorous. Amazingly enough, when one is adding an infinite number of terms, order and grouping =do= matter. In some series they don't matter (like the 1/2 + 1/4 + ... one), but some series which =do= converge can sometimes be made to have a different sum if you shuffle the terms around. That can blow one's mind, but there it is. That's all I want to talk about that for now. So let's think about other infinity paradoxes. How about this one - it's in the class of paradoxes called =supertasks=, and you'll soon see why: I'll do the most obvious one -- you've got a light switch, and a super-precise clock: at time = 1/2, you turn the light on at time = 1/2 + 1/4, you turn the light off at time = 1/2 + 1/4 + 1/8, you turn the light on at time = 1/2 + 1/4 + 1/8 +1/16, you turn the light off . . . The time these flips are taking place are converging to 1. At time 1, is the light on or off? This is equivalent to asking if infinity is even or odd. I'm not quite sure how to answer this question without saying this task simply cannot be done. People have pointed out that at some point a connection will have to be made at faster than the speed of light, for a switch of constant height. So other people said, okay, we'll have the switch drop half its height over the connection on each step. Then other people said, well, you know, eventually the switch will be so close to the connection, they will fuse on an atomic level -- or at least become attached permanently through electromagnetic forces. Then the switch would =have= to be on. This is all ignoring the fact that at a certain point one will reach quantum limits in being able to measure time. Simply, you can't possibly do this in real life. But that never stopped Schroedinger or Einstein when though experiments were on the line. So let's try this in our minds. How can we do this? Again, we really can't resolve this. There are several other supertasks of this nature, all of which are unworkable in real life (as opposed to the "crossing the room" supertask as Zeno describes it, which gives one no real paradox). Here's another one to think about: Let's say we have an infinite number of balls (countably infinite, of course -- how can one have an =uncountable= amount of balls?) and they're all numbered with the counting numbers: 1, 2, 3, etc. And we have an infinite box to throw them into. At time 1/2, we throw in 1-10, and remove 1 At time 3/4, we throw in 11-20, and remove 2 At time 7/8, we throw in 21-30, and remove 3 At time 15/16, we throw in 31-40, and remove 4 . . . . etc. The time, again, is converging to 1 (supertasks are generally like this). The question is: is the box empty or not at the end of this procedure? Well, at first glance it would seem that the box would be empty, because, even though every ball is thrown in at some step, it is removed at a later step. So there can't possibly be any balls remaining. On the other hand, at each step, there is a net gain of 9 balls in the box. If one keeps throwing in 9 balls for an infinite number of steps, one obviously has an infinite number of balls in the box. This, in my mind, is a true paradox, relating to the nature of infinity. I can't really wrap my mind around it other than to say this task can't be done. And that may be the only answer possible. Just because one can state a problem doesn't mean it has a solution or the question even really makes any sense. Think about the hypothetical meeting between the irresistible force and the immovable object. Or the stone that God makes that is so heavy, God can't lift it. There is unboundedness implied in both of the above problems, the limitlessness of the forces, the infinite inertia of the object, and the omnipotence of God. If infinity weren't a part of any of these things, if there were a sort of upper bound, then none of these would be paradoxes. So let me tell you of one final paradox, which does indeed relate to infinity, and, again, caused no end of trouble in the Medieval period: Say you have two concentric circles, one of which has a radius twice the length of the other. The one with the longer radius, we know from simple geometry, has a circumference that is twice that of the smaller circle. However, one can take a line segment from the center of the two circles through the smaller circle to the larger circle. This radius hooks up and two points on the circles uniquely. For every point on the larger circle there is a unique point on the smaller circle - they have the same number of points... So they must have the same circumference? 2 = 1? What is going on here? I shall leave that until my next installment on infinity, in which we actually find Lebesgue, and see he's up to no good, consorting with Cantor and his fractal set.

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