Math Mutation 59: Fun With Infinities
'Infinity' is kind of a weird concept. Usually we think of it as
some nebulous way of saying something goes on forever, without
anything more precise than that in mind. Yet you may recall that back
in episode 5, we discussed a proof that the infinite number of real
numbers is actually larger than the infinite number of integers, so
there must be more to this stuff. In fact, there are many different
kinds of infinities, which can be described using what is known as
"transfinite numbers".
For example, draw a number line. You can easily mark numbers on
it like 1, 2, 3, 4, etc. As we commonly know with number lines, you
can move 10 units to the right and get to the number 10, 100 units to
the right and get to the number 100, and so on. Now, let's just
assume we move all the way to the right.
"What?", you might ask. "How do we do that? Aren't number lines
infinite?" Well, yes, that's the point! I'm not asking you to do it
right now on paper with your pencil; I'm pretty sure you don't have
enough paper in your house to draw an infinite line. But let's just
use our imagination, and suppose that it were possible to move all the
way to the right on the number line. This hypothetical number we
would reach, a number which would be greater than any conceivable
integer, is known as Omega. This name probably originates from the
Biblical statement that God is the "Alpha and the Omega", the
beginning and the infinite end.
This Omega number has some very weird properties. That's what you
would expect, I guess, for a number defined this way. For example,
normal integers obey the commutative law of addition: a plus b equals
b plus a, for all a and b. But let's look at 1 + Omega and Omega +
1. 1 + Omega means start at 1 on the number line, then move an Omega
distance to the right. But no matter what integer you start at,
moving "all the way" to the right means the same thing-- so 1 + Omega
is just Omega. But now let's look at Omega plus 1. Here, we are
assuming that we have *already* moved all the way to the right, and
are sitting at the Omega point on the number line, which is larger
than any integer. But now we are moving one unit further. So Omega +
1 is a new transfinite number, which is slightly larger than our
original number Omega!
We can use similar reasoning to see that 2 times Omega and Omega
times 2 are different numbers. 2 times Omega means that we start at
2, move 2 units to the right, and then repeat this Omega times. This
process can't get us any further than Omega, since we are just adding
integers, so 2 times Omega is just Omega. But Omega times 2 is
different: we assume we are already at the Omega point, and move
another Omega to the right-- so Omega times 2 is another number,
infinitely larger than Omega.
Dealing with these numbers can get a little confusing. Rudy
Rucker, in his book 'Infinity and the Mind', proposes a clever method
for depicting Omega on a number line you can draw, based on one of
Zeno's paradoxes. You may recall that Zeno talked about how you can
never cross a room because first you have to go halfway, than half of
the remaining distance, then half of that, and so on to infinity. So
define a warped number line such that the distance from 0 to 1 is 1
inch, the distance from 1 to 2 is 1/2 inch, the distance from 2 to 3
is 1/4 inch, and so on. You may recall that the infinite sum of 1 +
1/2 + 1/4 + 1/8 ... is 2-- so that means that with this definition of
number distances, the Omega point can be clearly drawn 2 inches to the
right of 0.
Now, an inch to the right of Omega, you have Omega times 2,
assuming we are continuing to follow the rule that distances halve as
you try to move further along. Another half inch gets you to Omega
times 3, and by the time you have gone another 2 inches, you are
already at Omega squared! With this warped number line scheme, these
transfinite numbers can almost start to make sense. Until you start
trying to figure out what happens when you travel Omega inches along
this warped number line, and realize that gets you to Omega to the
Omegath power, and start to wonder what that means in practical
terms. After all this, you're still not an an infinity equal to the
number of real numbers.
As usual, I'm just scratching the surface of the concept of
transfinite numbers here-- for a much more elaborate treatment,
I would encourage you to take a look at Rucker's book, linked in the
show notes.
And this has been your math mutation for today.
References:
Online excerpt of
'Infinity and the Mind'
Rudy Rucker at
Wikipedia
Infinity and
the Mind