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From: ops@houp14.im.Hou.Compaq.com (IM Ops)
Subject: Re: rereRe: The end of god
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Date: Sat, 15 Oct 1994 10:28:19 GMT
References: <36vt2m$g6m@scapa.cs.ualberta.ca> <371epj$8gn@engnews2.Eng.Sun.COM> <1994Oct12.133530.10573@cc.ic.ac.uk> <visser.781973314@galaxy.ph.tn.tudelft.nl>
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In article <visser.781973314@galaxy.ph.tn.tudelft.nl>, visser@ph.tn.tudelft.nl (Boudewijn W. Ch. Visser) writes:
|> ppww@ic.ac.uk (Mr P.P.W. Williams) writes:
|> 
|> > Normally I only read this group, but I feel compelled to add something:
|> 
|> >In article <371epj$8gn@engnews2.Eng.Sun.COM> tlw@Eng.Sun.COM writes:
|> >>
|> >>
|> >>---
|> >>Kevin Wiebe writes:
|> >>>Godel is not a book, he is a famous mathematician.  He used logic to
|> >>>prove that ALL formal systems (ALL logic) is INCOMPLETE.  He also
|> >>>proved that all logic cannot prove that it is consistant.
|> >>>So, since you are so fond of logic, you will have to believe
|> >>>this.  If all you want to depend on is logic, you will have to
|> >>>depend on something that is incomplete.

This preposterous. Godel used logic to prove logic is incomplete?
Then his proof is incomplete. The nature of axioms is such that they
must be true even to show they are false.

|> >>Sorry, Kevin, but this is not correct. What Goedel proved is that
|> >>first order arithmetic is "incomplete" where "incomplete" has a very
|> >>technical definition: A formal system is said to be "incomplete" if
|> >>all of its principles (true statements of that theory) are not
|> >>derivable from a finite set of initial assumptions (axioms and/or rules
|> >>of inference).
|> >>However, there are *many* formal systems that *are* provably complete,
|> >>for example, Euclidean geometry is provably complete, propositional logic
|> >		^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
|> 
|> >Are you sure? Is not his 4th axiom that paralell lines never converge (or
|> >something), which (in real space) is not true and (in theoretical space)
|> >cannot be proven?
|> >flame me if I'm wrong...
|> >-- 
|> >Piers the Hamster, p.p.williams@ch.ic.ac.uk
|> 
|> >"...and I've got this terrible pain in all the diodes down my side..."
|> 
|> About the paralell axiom,it is an axiom because it can't be proved !
|> (if it could be proved from the other (not from an equivalent) axioms,it
|> wouldn't be necessary.
|> It is true that the 'real' world is not euclidean.However,the parallel
|> axiom makes euclidean geometry to be euclidean.If you use some other
|> axiom about parallel lines ,you end up with non-euclidean geometry.
|> Consider geometry on a sphere,for instance.It is definitly non euclidean.
|> No need to flame,or maybe only about 'axiom ... cannot be proven ?'

I must disagree with you here regarding the status of axioms. Axioms are
not arbitrarily chosen nor can they be altered for "use". They represent
fundamental attributes which are self-evident and irrevocable.

Euclidean geometry merely notices that given certain conditions, a planar
field, a line and a point not on the line, there will be only one line
parrallel to the line intersecting the point. It does not specify the
result in a "saddle" field or a spherical field.

We need this sort of information, the field, about space in a local
area to determine the answer to 52nd theorum. This makes in not an
axiom but a synthetic concept based on the spatial field - which is
axiomatic (it's existence, that is).

-- 
John: The thing is, Bob, there's some strange things going on here.
Bob : Yes, John, and that's the strangest thing about it.

                            - Plan 9 from Outer Space
