From: kevin@vaxc.cc.monash.edu.au
Newsgroups: sci.philosophy.tech
Subject: God's and Zorn's existence.
Date: 12 Sep 92 19:08:21 +1000
Organization: Computer Centre, Monash University, Australia

Well, a few interesting things have been said about the proof of God's
existence. Let's in particular look at Meyer's paper.
First, it has been commented that, regardless of how firmly Meyer's 
tongue was placed in his cheek when he came up with his proof, it is
still significant, or non-trivial.
Well, I am afraid I disagree. 
The fact that causality has certain properties and Zorn's lemma is true
implies there is a first cause , is a trivial statement, be it mathematical,
philosophical or anything else. 
As someone who has tutored set theory courses - I can make the following 
observation. People often find Zorn's lemma a bit complicated. It's
slightly technical, and often applied inapropriately. Thus, many students
find it insuperably difficult to prove theorems from Zorn's lemma in a basic
way. Moreover, they find results derived from Zorns's lemma surprising, and
have a feeling that something underhand is being done when Z.L. is applied.

However, once the student really understands Z.L. (which perhaps occurs
after the student sees Z.L. is equivalent to A.C.), the student often realizes
the consequences are easy. While the student may have not been able to begin
a proof that in a poset, every antichain is contained in a maximal antichain,
the student now will instantly be able to see this result. Moreover, the 
student will recognize Z.L. as an intuitively acceptable, and  in fact, 
simple statement (assuming they accepted A.C.)

THe problem is, because of the technical nature of Z.L., it is often viewed
as mysterious, and proofs using it (to the uninitiated) seem bizarre. But,
once one really understands Z.L, (which any basically intelligent person
can with a bit of work), such applications will seem completely transparent.

My claim: Meyer's result is completely transparent. Moreover, it will be
completely transparent to anyone who understands Z.L. That's a fact.
I'm sure even Meyer agrees with this. Meyer had philosophical reasons
for wanting to expose this result, but still,  mathematically, it is trivial.
Its triviality cannot be denied.

Well, the more interesting claim was the following.
There are no non-trivial a-priori statements that can be made about the 
real world.

Here, a-priori and trivial are a bit vague, but I'll take a-priori as
intuitively understood by a philospher, and trivial as understood by a 
mathematician.

Unfortunately, I thing the above statement is true. Reason ? Not on
logical grounds (so I don't leave myself open to the same attack on
positivism) Reason? Empirical. What is the cardinality of the set
of a-priori, non-trivial statements that have been made about the real
world? Zero. Zilch. The empty set. Not a bean. The big O. None.
The number of people in an empty room. 2 minus 2. The greatest integer
less than 1. lim as x approaches zero of ((sin x)/x)-1. 24 minus the
sum of the divisors of 24. The cardinality of phi, the only urelement
of the standard models of ZF(C). 

It seems (though I may be wrong) that a few people disagree. Surely,
in order to maintain credibility, such a person must produce an example.

But please don't give rubbish examples of 10 line tautologies. Reasons?
   Just because a statement is linguistically technical doesn't
make it non-trivial. Eg? Most statements of Z.L. A statement's triviality
doesn't rest on the linguistic inadequacy of notatio , etc.

Note that this is not an anti philosophical push. It is just saying
that philosphers can't do science, in some broad sense.

Yours,
Kevin Davey.

                                                                                                                                                                                                      non-trivial statement about the real world should