How often did you hear this conversation as a little kid?
Kid 1: Yuh huh!
Kid 2: Nuh uh!
Kid 1: Yuh huh times infinity!
Kid 2: Nuh uh times infinity plus one!
And if you're a normal little kid, you'd think, "Wow, Kid 2 totally won!"
If you're me as a little kid, you'd remind Kid 1 and Kid 2 that you can't count to infinity, so infinity plus one doesn't make sense.

What a buzz kill.

Well, the fact that we cant count to infinity is something we have a lot of trouble wrapping our brains around.  If something is infinite, that means it goes on forever. That's not just a really long time--it's forever!  Okay, see, it's hard to imagine.

But the crazy thing about infinity is that there are different orders of infinity.  It turns out that some things are more infinite than others.  How can that be possible?  Well, I'll show you.

Let's start by talking about the counting numbers: 1, 2, 3, 4, 5, 6, 7...  They go on forever, right?  If you give me the highest number, I'll add one to it and give you a higher one, thereby proving you wrong.  This set of numbers is clearly infinite.

Now, I'm going to show you that there are more numbers between zero and one than there are counting numbers.  It sounds preposterous, I know!

Well, I want you to make a list.  All the numbers between zero and one can be expressed as decimals that go on forever.  1/2 is 0.50000000... where the zeros go on forever.  1/3 is 0.3333333333... where the threes go on forever.  Some numbers are irrational, so they just have a decimal form that goes on forever and never repeats: 0.28894503857220900...  That's not any special irrational number--I just created it by mashing on number keys.

Okay, so make your list, and make sure you include all the numbers between zero and one.  Got it?  Of course not--your list would have to be infinitely long.  But, for the sake of argument, let's say you do have a list of all the numbers between one and zero:
To keep track of how many numbers between zero and one we've thought of, I've numbered the list (using our counting numbers).  This is only the beginning of the list--it goes on forever!

Okay, now, despite the fact that your list goes on forever, I'm going to find you a number that isn't on it.  I'm going to take the number that uses the first digit of your first number, the second digit of your second number, the third of your third and so on.  It will look like this: 0.5349918860...

But, you say, that number is on my list too!  It's way down here at spot 1,875,943!

Oh, but I'm not finished yet.

Now, I'm going to increase every digit of my number by one (and if my digit is nine, I'll make it zero).  Now my number looks like this: 0.6450019971...

This number isn't on your list.  How do I know?  This number has a different first digit from your first number, a different second digit from your second number, a different third from your third, and so on...forever.  So even if it looks like all the digits match up with your 974th number, the 974th digit won't match.

So you concede that maybe your list didn't have all the numbers on it.  You add my number to your list.  "Now it's complete," you say.  Well, I'll just do my trick again, and I'll get another number that's not on your list.  No matter how many numbers you put on your list, I'll keep being able to find a number that's not on it.

And don't forget, you have infinitely many numbers on your list--it goes on forever.  But it can never have all the numbers between zero and 1 since we can always use this technique create a number not on the list.

So when we pair the counting numbers up with the numbers between zero and one, we'll always have more numbers between zero and one than we'll have counting numbers.  The upshot of all this is that we've just seen that there are more numbers between 0 and 1 than there are counting numbers.  The numbers between 0 and 1 are more infinite than the counting numbers!

And you thought infinity was hard to wrap your brain around before...

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