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From: slfink@netcom.com (Steven Finkelman)
Subject: Re: rereRe: The end of god
Message-ID: <slfinkCxAKxq.Ls0@netcom.com>
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References: <36vt2m$g6m@scapa.cs.ualberta.ca> <371epj$8gn@engnews2.Eng.Sun.COM> <371lar$qsd@scapa.cs.ualberta.ca>
Date: Fri, 7 Oct 1994 07:32:13 GMT
Lines: 69

To bring fuzzy logic into the fray, to some extent you are all right. 

Kevin Wiebe (kevin@swanlake.cs.ualberta.ca) wrote:
: Tom Wetzel (tlw@Eng.Sun.COM) wrote:
: : ---
: : 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.
: : >

: : 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
: : (Boolean logic) is provably complete, and first-order predicate logic
: : without relations is provably complete.
: : Tom Wetzel

: Hi Tom.  Yes, I know the above.  Thank you for pointing it out.

: I was just making a general statement about the logic this other person was
: using and just got carried away.  I didn't think it was important to flesh
: out the details, but I guess I should have.  (I mentioned later in my post
: that I was refering to STRONG logics only.  I didn't think the original
: poster was interested in weak logics.)

: My statement should have been that all formal systems (logic) that
: are 1) finitely describable, 2) consistent, 3) strong enough to prove
: the basic facts about whole number arithmetic are INCOMPLETE.

: In the context of the previous post, it seemed clear that these types
: of logics were the ones being discussed.  His ignorance of Godel's
: theorems indicated to me that a full explanation and detailed
: discussion would not be possible.  I figured that if I made a statement
: like "all logics stronger than Peano logic", it would not be of help.
: I apologize.  I should have thought about the other people reading 
: this group, and the affect such imprecise statements would have.

: By the way, Euclidean geometry, as it is commonly used, is not
: provably complete, technically.  Tarski, however, proved that it
: could easily be extended to a formal system concerning points, lines,
: and circles that is provably complete.

: Boolean logic and first-order predicate logic (without, say, the 
: existential operator), are provably complete, as you say, but were
: too weak to be relevant to the topic under discussion (set theory,
: etc.)  They can't prove the basic facts about whole number arithmetic.
: They are just as weak as taking the "+" operator out of Peano logic.

: Sorry for the lack of clarity in my previous post - but my original
: points still stand.  One cannot base ones beliefs upon logic
: (first-order predicate logic) in this way:  "If Logic doesn't
: prove it, then it isn't true (doesn't exist)."

: -Kevin-
: --
: |          ]{evin  \^/iebe         /\       Lottery:       |
: |       kevin@cs.ualberta.ca      (  ) A tax on people who |           
: | http://web.cs.ualberta.ca/~kevin )(    are bad at math.  |
: |     Refuse novocaine ; transcend dental medication.      |
-- 
Steven Finkelman
DATA/Massage                                             
slfink@netcom.com
