Using the fundamentals of set theory, explore the mind-bending concept of the “infinity of infinities” — and how it led mathematicians to conclude that math itself contains unanswerable questions.
When I was in fourth grade, my teacher said to us one day: “There are as many even numbers as there are numbers.”
“Really?”, I thought.
Well, yeah, there are infinitely many of both, so I suppose there are the same number of them.
But even numbers are only part of the whole numbers, all the odd numbers are left over, so there’s got to be more whole numbers than even numbers, right?
To see what my teacher was getting at, let’s first think about what it means for two sets to be the same size.
What do I mean when I say I have the same number of fingers on my right hand as I do on left hand? Of course, I have five fingers on each, but it’s actually simpler than that. I don’t have to count, I only need to see that I can match them up, one to one.
In fact, we think that some ancient people who spoke languages that didn’t have words for numbers greater than three used this sort of magic. For instance, if you let your sheep out of a pen to graze, you can keep track of how many went out by setting aside a stone for each one, and putting those stones back one by one when the sheep return, so you know if any are missing without really counting.
As another example of matching being more fundamental than counting, if I’m speaking to a packed auditorium, where every seat is taken and no one is standing, I know that there are the same number of chairs as people in the audience, even though I don’t know how many there are of either.
So, what we really mean when we say that two sets are the same size is that the elements in those sets can be matched up one by one in some way. My fourth grade teacher showed us the whole numbers laid out in a row, and below each we have its double.