Better-quasi-ordering

Okay kiddo, imagine you have a bunch of toys and you want to put them in order from your favorite to your least favorite. But sometimes it's just really hard to decide which one is better than another one. For example, you might like your toy car with the red color better than your toy truck with the green color, but you also really like your toy train with the blue color, and it's hard to say which one is better than the other.

Better-quasi-ordering is like a special way of putting things in order, even when it's hard to decide which one is better than another one. It's a bit like making up new rules for how to decide which toy is better than another one, instead of just going with what you feel.

Now, imagine that instead of toys, we're talking about numbers. Sometimes it can be hard to decide which number is bigger than another one, especially if they're very close in value. For example, is 1.34 bigger than 1.33? It's hard to say, right? But with better-quasi-ordering, we can come up with a new set of rules to decide which number is bigger than another one.

These rules can help us put numbers in order in a way that's consistent and makes sense. So if we have the numbers 1.34, 1.33, and 1.35, we can use better-quasi-ordering to say that 1.35 is definitely the biggest, and 1.33 is definitely the smallest, and 1.34 is in the middle. We don't have to worry about whether 1.34 is actually bigger than 1.33, because the rules of better-quasi-ordering give us a way of deciding.

So, that's what better-quasi-ordering is. It's a way of putting things in order, even when it's hard to decide which one is better than another one, by making up new rules to help us decide.