In second grade our teacher Ms. Bowers introduced us to Cantor’s diagonal proofs. You are already familiar with this, of course. The rational numbers, the fractions, are listed with 1/1, ½, 1/3… on the top row, 1/1, 2/1, 3/1… in the first column and the diagonals always equal to 1: 1/1, 2/2, 3/3… and all the fractions in between. Then Cantor counts them by going up and down diagonally, zig-zagging between them. All the second graders accepted that. Then she showed us that the real numbers, say all the real numbers between .0000… and .9999…., were uncountable because no matter which way we listed them, she could generate a new one by going down the diagonal going on to infinity and generating a new one. Cantor liked diagonals.
I may have lost some of you. I think the reason we got this as second graders is because we knew that if Ms. Bowers was explaining this to us, it couldn’t be that complicated. But adults believe that this stuff should be difficult. So if you don’t get this, don’t feel bad, it just means you are old.
She then told us that it is impossible to prove that there is not an order of infinity between the countable and uncountable. It turns out that almost anytime you can’t prove anything in mathematics it is equivalent to this continuum hypothesis. So after all this work, we are left with yet another metaphor for life. Which seems like a lot of work to get there and it isn’t like there is a shortage of metaphors for life.
Back in second grade a kid came up to me and said:
-If you take all the real numbers of one decimal place, .0 through .9, there aren’t any others. You can’t insert any. Then if you go up by number of decimal places, .00 through .99 and so on, you will get to infinity which makes the real numbers countable.
-Those aren’t numbers, they are parts of numbers.
I told him.
-Then I will pad them with zeroes.
-Then I will insert a new one.
-Not if you follow my rules. It is just a matter of definition.
If you want to make a mathematician uncomfortable, tell them about this assertion made back in second grade. Part of the reason that they will be uncomfortable is that the foundations are shaky. I believe that this assertion is not equivalent to the continuum hypothesis, because it obviates the hypothesis. I suppose it would be boring if the real numbers were countable.
If one day this assertion is proven, proven would mean that most mathematicians agreed with it, which might happen if countable real numbers solved some other problem. If that happened it doesn’t mean that something else isn’t uncountable. But another reason that they would be uncomfortable is that it suggests that most of the high level mathematics done in the last century was a huge waste of time. My girlfriend Christine could have told you that in the first place.