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        riddles >> general problem-solving / chatting / whatever >> Gödel in a Nutshell
        
(Message started by: THUDandBLUNDER on May 26^th, 2004, 6:07pm)

Title: Gödel in a Nutshell
Post by THUDandBLUNDER on May 26^th, 2004, 6:07pm

From Fermat's Last Theorem (Singh, 1997):

"Gödel reinterpreted the liar's paradox and introduced the concept of proof.
The result was a statement along the following lines:

This statement does not have any proof.

If the statement were false then the statement would be provable, but this would contradict the statement. Therefore the statement must be true in order to avoid the contradiction. However, although the statement is true it cannot be proven, because this statement (which we now know to be true) says so."

And André Weil's reaction to Gödel's theorems:

"God exists since mathematics is consistent, and the Devil exists since we cannot prove it."

Title: Re: Gödel in a Nutshell
Post by Sir Col on May 28^th, 2004, 12:08pm

Or as Gödel would have put it...
([exists]y) ~([exists]x) Dem (x,y) [supset] ~([exists]x) Dem(x,Sub(n,17,n))

Title: Re: Gödel in a Nutshell
Post by grimbal on May 28^th, 2004, 12:14pm

Are you sure that you know that the statement is true?

It could be that the statement is true and that indeed you cannot prove it. But if your set of axioms is inconsistent, every statement is provable. In that case, the statement is false and it can be proven true.

Title: Re: Gödel in a Nutshell
Post by rmsgrey on May 30^th, 2004, 3:49pm

I think that depends upon your definitions of "true" and "provable" and how they interact. Certainly some ways of approaching the problem imply that everything provable is true - in which case, for inconsistent axioms, everything is "true".

Title: Re: Gödel in a Nutshell
Post by Sir Col on Jun 4^th, 2004, 5:18pm

Certainly the Intuitionists (Kronecker, Poincaré, and Brouwer, to name a few) would not accept Gödel's arguments. They reject one of the fundamental tools of the modern mathematician, the law of excluded middle: every statement is either true or false. They argue that because something is not true does not mean that it is necessarily false; it is simply unknowable. They reject existence proofs and defend the need for the assertion to be constructible to be meaningful.

Title: Re: Gödel in a Nutshell
Post by Leonid Broukhis on Jun 8^th, 2004, 6:28pm

on 05/26/04 at 18:07:26, THUDandBLUNDER wrote:

Paradox, shmaradox... Who said that every phrase that starts with "This statement" is indeed a statement?

Title: Re: Gödel in a Nutshell
Post by Icarus on Jun 8^th, 2004, 7:05pm

Webster does, for one. Are you going to be so gauche as to argue with dead man? ;)

Title: Re: Gödel in a Nutshell
Post by Leonid Broukhis on Jun 8^th, 2004, 8:38pm

on 06/08/04 at 19:05:51, Icarus wrote:

Webster does, for one. Are you going to be so gauche as to argue with dead man? ;)

If you mean "an expression in language or signs of something that can be believed, doubted, or denied or is either true or false" via proposition(2) (see http://www.m-w.com ), then your argument is circular. We do not know whether we should believe, deny, or attempt to find out the boolean value of a phrase before we are sufficiently convinced that the phrase is a statement.

Is the phrase "This statement is sleeping furiously" a statement? What about the original one with colorless green ideas?

Title: Re: Gödel in a Nutshell
Post by rmsgrey on Jun 9^th, 2004, 3:03am

Actually, the quibbling about what is and isn't a statement is beside the point - Godel came up with a version which took the form of a mathematical expression, which had equivalent meaning to "Expression G is unprovable" where G is the unique identifier for the expression whose meaning is "Expression G is unprovable". A way of paraphrasing the essential property is the statement "This statement is false", provided you accept that it is a statement. If you reject it as a statement, then you dodge the issue in the case of the paraphrase, but the Godel expression is constructed according to the inductive definition of "expression" as used in the expression, so the equivalent quibble fails in this case.

Title: Re: Gödel in a Nutshell
Post by Leonid Broukhis on Jun 9^th, 2004, 10:37am

rmsgrey, I'm not saying that Gödel is wrong, I'm saying that the "nutshell" is broken. Most if not all self-referential paradoxes arise because of two ways a statement can be constructed out of a phrase: theme & rheme, theme => rheme (not in everyday speech), or we can refuse to construct a statement if the theme is false, but it is beside the point.

Consider A ::= "this statement is false". Its theme is IsStatement(A), its rheme is NOT(A).
As we can see, if we let IsStatement(A) be true, the first two expressions lead to contradiction, therefore IsStatement(A) is false, and the paradox disappears: A = false & NOT(A).

If this is confusing, consider pointing to the sky and saying "This dog has no tail". We consider this statement (if we decide to deem that utterance a statement) false, despite the fact that the sky has no tail because its theme (the object pointed to is a dog) is false, no matter what the rheme says.

Title: Re: Gödel in a Nutshell
Post by Leonid Broukhis on Jun 10^th, 2004, 11:10pm

Going on a tangent here. The following is not related to Gödel or nutshells, but rather to theme and rheme. Earlier today while listening to the radio while driving I heard a talk show host say "The late Ray Charles died today at 73 of liver disease." (For the purists: I could have transposed "of ..." and "at ..."). This is a good example of what happens when someone hesitates about the boundary of theme and rheme after starting a sentence.