From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Report on Philosophies of Physicists
Organization: Computer Science Department,  Stanford University.
Date: Sun, 13 Sep 1992 05:02:06 GMT

In article <TORKEL.92Sep12080037@bast.sics.se> torkel@sics.se (Torkel Franzen) writes:
>
>In article <1992Sep12.022132.4581@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.
>Stanford.EDU (Vaughan R. Pratt) writes:
>...
>  Your remaining comments in response to my question about "yielding a
>consistent foundation for mathematics" miss the point of the question.
>You said that adding either a certain axiom A or its negation to ZFC
>yields a consistent foundation for mathematics, since A is independent of
>the axioms of ZFC. Now, what I pointed out was that adding e.g. "ZFC
>is inconsistent" to the axioms of ZFC also would, according to your argument,
>yield a "consistent foundation for mathematics" (assuming of course the
>consistency of ZFC). I say "according to your argument", since you invoke
>only the undecidability of A. Now my question is, in what sense is
>the theory obtained by adding to ZFC the axiom "ZFC is inconsistent"
>a foundation for mathematics?

You're right, I misinterpreted your question altogether, very sorry.

Let # denote the Goedel coding of "ZFC is inconsistent."  What I
interpret you to be asking is, what reality R could possibly be
described by ZFC+# under the assumption that ZFC is consistent in R?
Right?

Looks bad for me.  ZFC+# purports to describe a reality, but what sort
of a reality is it that lies about reality?  If it gets *that* truth
wrong (saying that ZFC is inconsistent when it isn't), what other
truths might it have wrong as well?  How could an ordinary
mathematician ever expect to be able to work reliably in such a
reality?  You'd have to be a computer, feeling your way blindly along
with only the raw axioms of ZFC+# to guide you, and we know how bad
computers are at doing creative mathematics in that style (or any other
for that matter).

I see the following chink in the armor of this situation.  The Goedel
coding of "ZFC is inconsistent" must have been worked out in some
reality Q different from R, a model of ZFC but evidently not of ZFC+#.

Now while this coding was fine for Q, it fails to reflect some detail
of R, perhaps the behavior of proofs, which in R might behave slightly
differently from Q, i.e. from how we had always understood them up
until now.

Now you might respond "But I always understood exactly how proofs
worked, it's very clear and it will never change."  This is the
response of the typical 19th century physicist to quantum mechanics: we
know exactly how mechanics (Q) works and it does not involve quanta
(R).

But when an irresistible datum meets an immovable physicist we all know
which has gotta give.  It's the same with adding # to ZFC.  This
informs us that # does not assert "ZFC is inconsistent" after all, due
to some mishap in the Goedel coding process, possibly a trivial coding
bug but (since we exercised every precaution, just as do experimental
physicists) more likely a fundamental gap in our understanding of proof
mechanics.  We then have to look for a way to patch up our
understanding that fits all the information we have to date, EVEN IF IT
MEANS UPROOTING OUR DEEPEST PREJUDICES ABOUT PROOF MECHANICS.

Physicists apparently have to play by these rules, and I don't see why
mathematicians should have it any easier.

-- 
======================================================| God found the positive
Vaughan Pratt   pratt@cs.Stanford.EDU   415-494-2545  | integers, zero was
======================================================| there when He arrived.
Newsgroups: sci.physics