On Sun, 1 Dec 2002, Sarad AV wrote:
> We can't define completeness.
We can define it, as has been done.
What we can't do is -prove- any set of rules of arrangement that describe
symbol manipulation as -complete- -within the rules of arrangement-.
Complete means that we can take any and all -legal- strings within that
formalism and assign them -one of only two- truth values; True v False.
The fundamental problem is axiomatic. The rules define -all- statements as
being -either true or false-, no other possibility is allowed -by
principle-.
We create two lists 'true' and 'false', we are -required- to put -any-
string (or formula in Godel-speak, or 'sequence' and 'inside or outside'
with regard to Cauchy Completeness) we write in one of these two, and
only these two lists.
However, as Godel shows, we -can- write strings (some of them are quite
simple which is what makes it so shocking) that we can't put in -either-
of these lists.
There is -no- place to write it down. It just hangs there in Limbo. There
is no -I don't know- list.
There is a parallel (but I don't think fully equivalent) situation with
Geometry and Euclid's V Postulate. It turns out not to be so universal
after all.
One approach to dealing with this situation is Para-Consistent Logic. Time
will tell how usefull that is.
Personal Note:
I don't believe that the value of Godel is really the utility of
mathematics as much as demonstrating the imperfect reasoning of -all-
human beings. Mankind, all mankind, is on a hunt for universality in a
quest for transcending the mortal coil. It's the concept of
'transcendence' that keeps getting us in trouble. Intelligence isn't all
it's cracked up to be.
"We arrive at truth not by reason only, but also by the heart."
Blaise Pascal
"Intellectual brilliance is no guarantee against being dead wrong."
David Fasold
"It is not clear that intelligence has any long-term survival value."
Stephen Hawkings
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We don't see things as they are, [EMAIL PROTECTED]
we see them as we are. www.ssz.com
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Anais Nin www.open-forge.org
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