Unfortunately this doesn't get at the heart of GIT. I'm pretty math-ignorant so correct me if I'm getting this wrong, but I'd do it like this:
Hey, $youngster, imagine that we have a Big Book of Mathematics. This book describes addition, multiplication, and some logic. There are pages like this:
Fact: 1 + 4 = 5
Fact: 2 + 3 = 5
Rule: Things that are equal to the same thing are equal to each other.
So now we know: 1 + 4 = 2 + 3
On another page of the book you see this:
Rule 1. The next sentence is false.
Rule 2. The previous sentence is true.
So now we know:... nothing??
It's kind of funny right? Both of those things can't be true within the rules they set up. The whole thing is inconsistent. We don't learn anything, we don't know anything. In fact this one page ruins the whole book. We should just rip it out.
But then, there's another book -- Gödel's book of Mathematics. On yet another page of the book there's this:
Rule: You can't prove the rule on this page!
So now we know: _________
That's just as funny, like the other one right? But here's the really weird part: is it wrong? (Let them get to No.) That's right, you can't prove the rule on this page. But it's still true. Fill in the blank: now we know "it's true!"
That means that there are at least a few weird things that you can't prove are true, but which are true. That's what Gödel thought was funny!
Next steps: try to get them to notice what's weird about both cases, in that they refer to themselves. Walk them through the idea that maybe you could just outlaw referring to a rule on the page. I can't think of an easy way to explain Gödel numbers, so I'd just have to say that he showed that if you allowed a book to have numbers and addition and whatnot, you could still sneak in weird things like that, by using a secret code.
Hey, $youngster, imagine that we have a Big Book of Mathematics. This book describes addition, multiplication, and some logic. There are pages like this:
On another page of the book you see this: It's kind of funny right? Both of those things can't be true within the rules they set up. The whole thing is inconsistent. We don't learn anything, we don't know anything. In fact this one page ruins the whole book. We should just rip it out.But then, there's another book -- Gödel's book of Mathematics. On yet another page of the book there's this:
That's just as funny, like the other one right? But here's the really weird part: is it wrong? (Let them get to No.) That's right, you can't prove the rule on this page. But it's still true. Fill in the blank: now we know "it's true!"That means that there are at least a few weird things that you can't prove are true, but which are true. That's what Gödel thought was funny!
Next steps: try to get them to notice what's weird about both cases, in that they refer to themselves. Walk them through the idea that maybe you could just outlaw referring to a rule on the page. I can't think of an easy way to explain Gödel numbers, so I'd just have to say that he showed that if you allowed a book to have numbers and addition and whatnot, you could still sneak in weird things like that, by using a secret code.