This is another digression from the Galois theory arc (and Joe, I don’t know if you read the Not About Apples stuff, but you are definitely to blame for the fact that I call it an arc) inspired by a recent conversation with one of my students in my geometry for teachers class.
A major topic of the class is an exploration of some famous attempts in history to prove Euclid’s Fifth Postulate and what we can learn from how they went wrong. Among the proofs considered is one due to the French mathematician Gergonne (a name probably best known for the “Gergonne point” in triangle geometry). In a nutshell, he argued that since there’s no single largest acute angle that can appear in a right triangle, every acute angle can appear in a right triangle. That’s all you really have to know, since geometry really isn’t what I came here to talk about, so much as a resulting conversation from office hours yesterday.
It turns out the flaw in his reasoning is interesting, and it’s something you’ve probably never given a lot of thought to. And it has nothing to do with geometry. Imagine that, in some context, numbers that are small enough are “good” and numbers that are big enough are “bad” (that is, if a number is good, all smaller numbers are also good, and if a number is bad, all bigger numbers are also bad). The idea underlying Gergonne’s mistake is that if there are good numbers and bad numbers, then there must be a single largest good number “at the borderline”. But this is not so. Is there a largest negative number? No. Zero is the best upper bound to the negative numbers (the so-called supremum), but zero isn’t a negative number. Whatever negative number you can think of, there is another negative number closer to zero. Likewise there is no smallest positive number.
What is true is that, if there are good numbers and bad numbers, then there is either a single largest good number or a single smallest bad number, Never neither, never both. In particular, just because a set of numbers doesn’t actually have a maximum doesn’t mean it isn’t bounded; it could just be that the boundary point isn’t the set.
So why is this counterintuitive? Because most people so often think about integers, and integers are not like that. Integers are spaced differently on a number line than real numbers; the technical term is discrete. If you have good integers and bad integers, then it much be that there is a largest good number and a smallest bad number.
This relates to the fact that the counting numbers are well-ordered. In other words, in any nonempty set of counting numbers, there is always a single smallest number. This is the foundational idea for a powerful technique for proving mathematical theorems and grappling with the infinite, so called mathematical induction. But that’s another story. For another time.
Now you might think we’ve said all there is to say. We’ve described the behavior of real numbers (a continuous set) and the behavior (a discrete set), and you might not think any other behaviors would be possible. But rational numbers behave in an even more unexpected way. If you divide a range of fractions into small/good ones and large/bad ones, then there are three possible behaviors.
- there is a largest good number, but no smallest bad number
- there is a smallest bad number, but no largest good number
- there is neither
The last case may be hard to picture. Here is an example. Let’s look at just the positive fractions, and call a number good if and bad if . Then there is no largest good number and no smallest bad! The “borderline” in this case appears to be , but that’s not a rational number! There’s a hole in the number line where the boundary should be!