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
(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
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 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
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
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
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:
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
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
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
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
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.