> Eric Boyd wrote:
>
> > Hi,
> >
> > "Gifford, Nate F" <giffon@SDCPOS3B.DAYTONOH.ncr.com>
> > > ... If you can build a model with a truth value of 1 then either
> > > Godel's Theorem is false or your model is equivalent to the
> > > universe. You pick.
> >
> > False dilemma.
> >
> > Godel's theorem applies only to "formal systems", which means, for
> > instance, that it doesn't apply to geographical maps.
No, it doesn't apply to geographical maps or acts of congress or
farts in the bathtub. It only applies to the "formal systems" of formal logic.
That consists of a set of symbols and some rules for putting them together
in strings called statements, along with some axioms saying that a few of the
statements are true, and some rules for when some statement(s) imply another.
[snip!]
> Call me totally ignorant, but what is Godel's Therom, if I can understand it
> already (I'm about halfway throung Calculus IV)
>
> --
> Nathan Russell
> frussell@frontiernet.net
You're obviously *far* from totally ignorant. A web search
for "godel" turns up lots of good stuff on Godel's Theorem. See
http://users.ox.ac.uk/~jrlucas/index.html
for example. So ERiC can read all about it before he gets back to his dorm room....
Also about 20 years ago Douglas Hofstadter wrote a very popular book
called "Godel, Escher and Bach", which requires no mathematical background.
I would highly recommend this book--my copy is worn out by all the students I
have lent it to.
The basic idea is that any formal system must contain statements which are sensible
(i.e., must be either true or false) but cannot be proved or disproved within the system.
This is kind of surprising because you would think that, given the axioms and the rules,
you could mechanically go about deriving true statements in such a way that any given
true statement would eventually be proved.
Of course that is an oversimplification. This is one of the most profound ideas in all
of mathematics, and it takes some time to digest.
--Bill Haloupek "Everything should be made as simple as possible, but not simpler." -Einstein