Zero logic

Discussion of the nature of Ultimate Reality and the path to Enlightenment.
User avatar
DHodges
Posts: 1531
Joined: Tue Jan 22, 2002 8:20 pm
Location: Philadelphia, PA
Contact:

Re: Godel's Incompleteness Theorem

Post by DHodges »

The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.
There are several possible responses to Godel.

(1) You can allow statements to not be true or false (the theorem relies on the standard practice of allowing only true or false as truth values). Certainly, in English it's possible to come up with statements that are not completely true or false.

(1 b) Equivalently, you could say that such statements are not "well formed". In English, consider the sentence "It's quicker to go to New York than by train." It's not well-formed enough to say if it is true or false.

(2) You can limit your axiomatic system such that such statements are not valid. Godel applies to a sufficiently powerful system of axioms - i.e., one that allows you to form statements that are essentially of the form "This statement is false."

(3) You can have a system that allows statements and meta-statements about statements in a strict heirarchy, such that only a meta-statement can refer to a statement. With such a heirarchy, statements can not refer to themselves; only a meta-statement can refer to a statement.

(4) You can not worry about it, since the statements that are undecidable are generally self referential; the undecidable statements don't talk about anything other than themselves, and can (generally?) be safely ignored.
Gödel's Theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths ...
This assumes that a computer is representable as a formal system, while a human is not, which is not at all obvious to me. In fact, it looks like begging the question.
User avatar
David Quinn
Posts: 5708
Joined: Sun Sep 09, 2001 6:56 am
Location: Australia
Contact:

Post by David Quinn »

Burt wrote:
He demonstrates it David...

If Godel has demonstrated that this theory is true, then it automatically undemines the idea that truth can't be known or proven.

Godel was underappreciated,I think.
I think he's over-rated. People often trumpet him as some sort of anti-truth saviour, but his theories don't really apply to ordinary human reasoning and its ability to discern truth - as he himself demonstrated with his theorizing.
He also proved that we could not prove anything because we could not place ourselves outside the universe,and observe it.
And yet, for some strange reason, he strove to prove his own theories. He must have suffered from acute memory loss, or was an idiot of the highest order. He did have a mental illness, didn't he?


-
bert
Posts: 648
Joined: Fri Aug 19, 2005 6:08 am
Location: Antwerp

Post by bert »

If Godel has demonstrated that this theory is true, then it automatically undemines the idea that truth can't be known or proven.
one can know that one can not know.Which at least implies that you know that you can not know.
This is the great potential!
Again I say:"All is illusion."

Let me remember you of socrates:
"all I know is I know nothing"
And yet, for some strange reason, he strove to prove his own theories. He must have suffered from acute memory loss, or was an idiot of the highest order. He did have a mental illness, didn't he?
If you don't have affinity with the right knowledge to comprehend this,you are fooled.
Yes,I heard that he died unhappy,or maybe crazy?
lots of great ones did.

pilosophers of this time who are seeking for "the new philosphy" also looking once more to these great thoughtsand see how they can help them offering to the understanding of politics and science ,etc.

In the past to much has been thrown away.Flat-ignored.
hyperqube
Posts: 40
Joined: Thu Jul 28, 2005 7:55 pm

The date of Natalee's return has been found!

Post by hyperqube »

Natalee's Holloway's Miracle

Whew! I have found the day I have been waiting for. Natalee Holloway's miracle is right around the corner.

and this is a school crossing zone...

I pray the you reserve what you may say in haste but to those who disregard the school crossing zone.


here are seven feet of rope that you may hang yourself upon
Locked