Explaining Gödel's Incompleteness Theorem to a Twelve Year Old

i'm not a mathematician, but his final point about the continuum hypothesis seemed odd to me. the CH is "outside" ZF, but that just means (afaik) that it contains some extra "information" that is not in ZF. adding it to ZF doesn't force any kind of contradiction, in the way that adding "this system is consistent" would. so CH is (just) an example of an independent axiom - it doesn't illuminate what is so weird about the completeness theorem.

(mathematicians: is that right?)

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

#5

Post by Drbble » Sat, Jan 21, 2012, 11:14 PM UTC

andrewcooke wrote: post #4

i'm not a mathematician, but his final point about the continuum hypothesis seemed odd to me. the CH is "outside" ZF, but that just means (afaik) that it contains some extra "information" that is not in ZF. adding it to ZF doesn't force any kind of contradiction, in the way that adding "this system is consistent" would. so CH is (just) an example of an independent axiom - it doesn't illuminate what is so weird about…

Yeah that was a tangent. The author got a bit overexcitrd and overshot.

Also not the OP's reply (paraphrased): "Thanks, but I wasn't interested in the understanding logic or math. I am thinking more along the lines of [metaphysical gobbledygook]. What can you say about that? "

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

#6

Post by Dn_Ab » Sat, Jan 21, 2012, 11:20 PM UTC

This is a great explanation. There are a few things that more emphasis should have been placed on. In going over what an axiom is he should have noted that Godel's work only applies to theories which have a formal axiomatic basis and that the axioms are strong enough to encode the natural numbers enough to do certain arithmetic ops e.g. statements in the theory can be proven using induction. And maybe a bit more emphasis should have been placed on the fact that it is possible for a theory to be proven complete and consistent outside itself.

So for example, in line with the first theorem you can algorithmically verify/decide all statements in a subset of Euclidean Geometry (the subset which does not deal well with circles). And in the second part you can have theories which can verify themselves. Or that it is possible to prove a theory complete and consistent as long as you can find a suitably powerful model outside of it.

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

#7

Post by tyrick » Sat, Jan 21, 2012, 11:22 PM UTC

andrewcooke wrote: post #4

i'm not a mathematician, but his final point about the continuum hypothesis seemed odd to me. the CH is "outside" ZF, but that just means (afaik) that it contains some extra "information" that is not in ZF. adding it to ZF doesn't force any kind of contradiction, in the way that adding "this system is consistent" would. so CH is (just) an example of an independent axiom - it doesn't illuminate what is so weird about…

I think the CH was used just to provide an example of an undecidable statement, not actually demonstrate something weird about the incompleteness theorem. Since the CH cannot be proven to be true or false within ZF (or ZFC), then you are correct, appending it would just provide an addition axiom.

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

#8

Post by Drbble » Sat, Jan 21, 2012, 11:24 PM UTC

yfyf wrote: post #3

Now someone please explain Skolem's paradox to a 12 year old.

It depends on whether the meaning of "exists" is, I think. By analogy to the finite universe (always dangerous in logic), it is like saying "we live in a sphere of diameter 4. We have proven (due some hypothetical variation of relativity) that all matter in the universe will never be able to move more than distance 1 from the original center of our existence. An object of length 3 can be mathematically described, but is theoretically impossible to construct." Or something.

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

#9

Post by Drbble » Sat, Jan 21, 2012, 11:27 PM UTC

tyrick wrote: post #7

andrewcooke wrote: post #4

i'm not a mathematician, but his final point about the continuum hypothesis seemed odd to me. the CH is "outside" ZF, but that just means (afaik) that it contains some extra "information" that is not in ZF. adding it to ZF doesn't force any kind of contradiction, in the way that adding "this system is consistent" would. so CH is (just) an example of an independent axiom - it doesn't illuminate what is so weird about…

I think the CH was used just to provide an example of an undecidable statement, not actually demonstrate something weird about the incompleteness theorem. Since the CH cannot be proven to be true or false within ZF (or ZFC), then you are correct, appending it would just provide an addition axiom.

So it is a weird example, since it NOT an example of a statement that is undecidable in every model.

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

#10

Post by tyrick » Sat, Jan 21, 2012, 11:50 PM UTC

Drbble wrote: post #9

tyrick wrote: post #7

Earlier quoted context omitted.

I think the CH was used just to provide an example of an undecidable statement, not actually demonstrate something weird about the incompleteness theorem. Since the CH cannot be proven to be true or false within ZF (or ZFC), then you are correct, appending it would just provide an addition axiom.

So it is a weird example, since it NOT an example of a statement that is undecidable in every model.

Well, the CH IS an example of a statement that is undecidable in ZF. Once you include is as an axiom, you essentially bypass the need to use ZF to justify it (or prove it). So, just because you are able to include the CH as an axiom, doesn't mean it is not an example of a undecidable statement within a given set of axioms. Does that make sense?