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Explaining Gödel's Incompleteness Theorem to a Twelve Year Old

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Re: Explaining Gödel's Incompleteness Theorem to a Twelve Year Old

#31

Post by cdog46 »

i have an advanced graduate degree & am not 12 years old. The guy on reddit with his explanation was good reading. Now-fr the interesting part-it's people who can understand the theorem AND explain it to a third party are the really RARE people. Putting complexity into simple terms is what takes the real talent. Amen

Re: Explaining Gödel's Incompleteness Theorem to a Twelve Year Old

#32

Post by cdog46 »

RockofStrength wrote: post #16

This is my layman's understanding of Godel's incompleteness theorems: Any set of axioms powerful enough to decide all its theorems will always have inconsistencies in its theorems (e.g. "if true then false"). To be consistent, the set of axioms must be incomplete (some of its theorems are undecidable). The following sentences have some of the 'flavor' of Godel's theorems: "This sentence is false." →if true, then fals…

Thank god-you are one of the rare individuals who can use the English language to explain a complex idea. The hideous bias I have encountered in the "hi tech" world is because my mind works like yours and I have always found your talent to be rare gem. Why anyone would think the reduction of complexity into simple terms means putting it into "12 year old" terms is beyond me. So I remain an exile.

Re: Explaining Gödel's Incompleteness Theorem to a Twelve Year Old

#33

Post by [deleted] »

jules wrote: post #30
ot wrote: post #25

If the 12yo in question knows a bit of programming, a much simpler thing to explain would be the Halting problem, both the statement and the proof. The bulk of Godel's proof is introducing recursive functions and the "godelization", which means mapping formulas to natural numbers and functions to arithmetic/first order expressions. This all comes for free if you know any programming language; if you don't like natura…

Indeed, the proof of the halting problem is much easier to sketch than the proof of Godel's incompleteness theorem (which the OP doesn't even attempt). So here he goes: a proof sketch of the halting problem for 12 year old programmers. Suppose that somebody wrote us a program that can check whether another program will loop indefinitely or eventually halt when run on a particular input: bool halts(string program, str…

[deleted]

Re: Explaining Gödel's Incompleteness Theorem to a Twelve Year Old

#36

Post by neilk »

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.

Re: Explaining Gödel's Incompleteness Theorem to a Twelve Year Old

#37

Post by VMG »

baddox wrote: post #19
VMG wrote: post #12

To be clear this is an explanation of the consequences of Gödel's Incompleteness Theorem

Is that so? I think you mean that this is an explanation of the theorem, but not of a proof of the theorem.

You are correct - I was expecting an explanation of the proof because that's the part that is more difficult for me to understand, not because it was advertised.
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