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 at 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.

## No comments:

## Post a Comment