What is Gödel's theorem and why does it matter?

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Please enlighten me.

― a mobius strip, Monday, 22 March 2004 13:07 (twenty-two years ago)

I dunno, but you could try Google's theorem for a start.

― Liz :x (Liz :x), Monday, 22 March 2004 13:12 (twenty-two years ago)

I could tell you but I have to go back to a training course. Its important cos it shows that mathematicians are not (and cannot be) the knowitall boffins they once thought they were.

(Okay, it proves that in any consistent mathematical system, that there will exist at least one statement which is true, but cannot be proved in that mathematical language.)

― Pete (Pete), Monday, 22 March 2004 13:43 (twenty-two years ago)

which means that maths cannot be a self-consistent system that proves all its truths from its own principles. People have tried to work out what this means - for instance, that the human mind must transcend the algorithmic. Roger Penrose uses this as part of his argument for the theorised role in quantum collapse, which I recently talked about on Freaky Trigger, as it happens.

― Martin Skidmore (Martin Skidmore), Monday, 22 March 2004 17:59 (twenty-two years ago)

(Okay, it proves that in any consistent mathematical system, that there will exist at least one statement which is true , but cannot be proved in that mathematical language.)

Is true or might be true?

― Bob Six (bobbysix), Monday, 22 March 2004 18:24 (twenty-two years ago)

Is true. If the statement were false, it would create a paradox. But there would be no way to prove this statement.

It's been a while since I read Gödel, Escher, Bach, so I can't recall if the theorem actually tells you what that statement is, or it if just proves that such a statement exists.

Also it needs to be a "sufficiently complex" system, as I recall.

― Casuistry (Chris P), Monday, 22 March 2004 19:48 (twenty-two years ago)

Chris beat me too it. There is a sufficidently complex clause in there.

― Mr Noodles (Mr Noodles), Monday, 22 March 2004 19:53 (twenty-two years ago)

I'm a mathematician and I still think I'm a knowitall boffin.

― Nick H (Nick H), Monday, 22 March 2004 20:36 (twenty-two years ago)

Martin, are you trying to get Ptee/Alang started?

― Ricardo (RickyT), Monday, 22 March 2004 23:01 (twenty-two years ago)

Don't try it.

― Gödel (nickdastoor), Monday, 22 March 2004 23:09 (twenty-two years ago)

Do they have something against Penrose's bonkers theories? I enjoy them myself.

― Martin Skidmore (Martin Skidmore), Monday, 22 March 2004 23:31 (twenty-two years ago)

His use of Godel's theorem as some sort of justification for his missing inbetweeny quantum/classical physics is really really stupid. (The idea that a computer could not work out Godel's theorem missunderstands massively the software harware divide where his "missing" physics resides). You can even do it in Excel with the CONCATINATE command.

As part of my finals I had to pretty much reproduce Godel's theorem from scratch. I wish I had just memorised it.

― Pete (Pete), Tuesday, 23 March 2004 11:17 (twenty-two years ago)

http://en.wikipedia.org/wiki/Godel
http://en.wikipedia.org/wiki/G%f6del%27s_incompleteness_theorem

― Jaunty Alan (Alan), Tuesday, 23 March 2004 11:23 (twenty-two years ago)

It's a while since I've read about any of this stuff, but I think the statements that are true but can't be proved to be so are self-referential ones, and there's a tie-in with the problem in logic Bertrand Russell identified with self-referntial sets.

If you present a hypothetically all-knowing truth-telling machine with the statement: "The truth-telling machine cannot prove that this is true", then an outside observer will be able to prove that such a statement is true, but the truth-telling machine itself logically cannot. There will always be truths that can only be proven outside the system, which poses a logical problem for any notion of an all-encompassing system of truths.

― Jonathan Z. (Joanthan Z.), Tuesday, 23 March 2004 11:50 (twenty-two years ago)

Indeed, the statement is composed via a Metalangwidge which is why the system cannot read it despite it being true. But yes, see also Godel Escher Bach, but ignore the Escher Bach bits.

― Pete (Pete), Tuesday, 23 March 2004 12:15 (twenty-two years ago)

The question, I suppose, is whether the metalanguage problem really does have profound metaphysical implications or not. I think the jury is still out on that one.

― Jonathan Z. (Joanthan Z.), Tuesday, 23 March 2004 12:17 (twenty-two years ago)

I don't think it does, but that is by no means the majority opinion (question really comes when thinking about ordinary language are we alreay using a meta language, does the very act of using language imply an implicit understanding and even use of a meta language, and how that ties into a degree of self awareness).

― Pete (Pete), Tuesday, 23 March 2004 15:01 (twenty-two years ago)

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