In college I learned that there are different sizes of infinity (actually I had heard this in high school, but didn’t understand what it meant, nor had I heard anything about why it was true). I thought that I could explain this fact to my family. So I gave it a shot:

Me: You know the counting numbers 1, 2, 3, 4, 5, etc. How many are there?

My mom: Well…infinitely many.

Me: Right. Now think of all of the numbers between 0 and 1. Not just whole numbers, but fractions and any number that can be written out as a decimal number like 0.421355… etc, but just think about the ones between 0 and 1 for now. How many of those are there?

My mom: I guess I’m not sure.

Me: Well, there is 1/2 and 1/3 right? but also 1/4, 1/5, 1/6, etc. And there are lots of others.

My mom: OK, I guess there are infinitely many.

Me: OK, so you’ve got the number of counting numbers (which is infinity) and you’ve got the number of numbers between 0 and 1 (which is also infinity). But are they the same infinity? Are there the same number of counting numbers as numbers between 0 and 1?

My mom: Well, they’re both infinity, so they must be the same. Right?

Me: That’s the point. People always assumed they were. It seems like they should be, but they’re not! And I’ll tell you why.

My mom: OK.

Me: If they were the same number, then I could pair them off. I mean I could pair off the counting numbers with numbers between 0 and 1. Just like if I had the same number of apples and oranges I could pair off each apple with an orange. So if those two infinities are the same I could make a list:

1 0.034898…

2 0.572543…

3 0.625464…

etc. Right?

My mom: Sure.

Me: But I’m claiming that you can’t make any such list. There are just too many numbers between 0 and 1. Here is why: if you have any such list of numbers between 0 and 1 then I can come up with at least one that isn’t on your list. I’ll just pick one whose first digit is different than the first digit of the first number on your list. I’ll make sure my second digit is different than the second digit of the second number on your list and my third digit will be different than the third digit of the third number on your list. Etc. So, if you think about it, my number is different than any of the numbers on your list.

After thinking about it, I think my mom understood the argument. I told my siblings about it too, and they got it too. Finally, I tried to explain it to my grandma:

Me: You know the counting numbers 1, 2, 3, 4, 5, etc. How many are there?

My grandma: Oh, I guess there are 5.

Me: No, but you can keep going …,6, 7, 8,… I’m talking about all of those. How many are there?

My grandma: Well, I guess there are 8 then.

Me: No. Grandma, I’m talking about all the whole numbers. You know, you could just keep going 9, 10, 11, 12, 13,… Right?

My grandma: OK, I guess there are 13 of them then.

Me: But grandma, you could just keep going 14, 15, 16, 17, 18,…

My grandma: Then I guess there are 18.

Me: Never mind grandma.

I was a little hesitant to tell this story for fear that people would think “his grandma was just not a smart person,” which is not true. My grandma was very smart. I think she did well in high school. I’m pretty sure she didn’t have any formal education after high school, but she was an intelligent woman. She could carry on an informed conversation, take care of her finances and run a household. She read quite a lot. She kept a journal that she wrote in every night. Among other things she recorded in her journal daily high and low temperatures. So please, don’t think that my grandma was not intelligent. In fact, she told me of some math that she had discovered on her own: “Start with any number. Like 17. That’s S-E-V-E-N-T-E-E-N” she held up a finger for each letter she spelled out, counting them. “That’s 9. N-I-N-E. That’s 4. F-O-U-R. That’s 4 again. Start with a different number, like 12. T-W-E-L-V-E. That’s 6. S-I-X. That’s 3. T-H-R-E-E. That’s 5. F-I-V-E. That’s 4. No matter what you start with you always end up at 4!” I can see her in my mind counting out the letters with her fingers. Now there’s nothing deep about this discovery and lots of other people have observed the same thing, but it is math and my grandma did it on her own and she did it for fun, which drives home three points that I have been trying to make in my blog:

1) People can do original math.

2) They like to discover it for themselves.

3) They actually have fun doing it.

But to this day I cannot understand why she didn’t get what I was alluding to when I asked how many counting numbers there are. Maybe she was joking with me (I don’t think so). Maybe she didn’t want to hear what I had to say. Maybe I was just wording it in the wrong way for her to see what I was saying (this is the most likely explanation, I think). But I can’t help but ask myself “did she grasp the concept of infinity?” I think she did. Everybody does. Right? That got me to thinking about just how natural it is to think about infinity. I’ll have more to say about that in a later blog.

I have never really tried explaining any math to my grandparents. I can imagine my grandmother responding the same way your grandmother did because she wouldn’t understand what I was getting at. She is very practical-minded, and I don’t think she understands that the math I do is not just solving complicated equations to reinforce buildings or thwart hackers or figure out optimal tax rates or something. It might just be that most people don’t know what math is, so they don’t understand what you’re doing when you talk about the set of all natural numbers.

The rest of this comment might end up being more appropriate for a later post, but I’m leaving it here anyway. I was really upset when I learned that the rationals are countable. It offended my sensibilities. I was taking a Moore method class basically on naive set theory and how to write proofs, and one of the things we were supposed to work on was whether the set of ordered pairs in NxN was countable. I tried everything I could think of to show that they weren’t. I was OK with 2N, 1/2N, etc. being countable (even though 2N was smaller in some sense), but I felt like an “infinite multiple” of an infinite set could not possibly be countable. Multiplying something by an infinite something just HAD to ratchet up the “degree of infiniteness”. It was probably one of the first times my mathematical intuition was dead wrong.

I still have trouble with uncountableness when I really think about it. Basically, any of the numbers that we do think about are countable. So we’ve got the rationals. Then we can add the algebraic numbers, which are most of the first irrationals we think about. Then we can add pi and all the rational multiples of it and numbers algebraic over Q[pi], and then the same for e. Those are all countable sets. What’s left? Measure theoretically, everything! But you can’t say what it is. (I guess you could go with the infinite sequences of decimal/binary/whatever representations.) It’s almost like the act of naming or recognizing these numbers moves them to the “countable” pile in the big pile of numbers. Recently I’ve gotten a little more comfortable with just thinking about the reals as the completion of the rationals, which are the really meaningful numbers. The reals are the mortar holding the bricks together.

In the comment above, the sentence “Basically, any of the numbers that we do think about are countable” would be better worded as “Basically, the set of numbers that we do think about is countable.”

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Cory remembers when you explained this. He likes the concept.

I like the concept. Thank you for simply explaining a complex concept. As for your Grandma, some people are very literal. It has nothing to do with how smart they are, just how they process the world.

“It has nothing to do with how smart they are, just how they process the world.” I like it. Thanks for sharing this with me!