From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.physics,sci.math
Subject: Applied inconsistency
Date: 13 Sep 92 20:26:24 GMT
Reply-To: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Followup-To: sci.physics,sci.math
Organization: The Wistar Institute of Anatomy and Biology
In-reply-to: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)

In article <1992Sep13.174721.23818@CSD-NewsHost.Stanford.EDU>, pratt@Sunburn (Vaughan R. Pratt) writes:
>>[#="ZFC is inconsistent"]

>Well, I thought I'd answered just that, but maybe I was jumping the
>gun.  The meaning of ZFC+# as a foundation for mathematics is no
>different from say ZFC+V=L (all sets constructible), or any other
>addition to ZFC.  It just means the same mathematics you've always
>done, plus anything new you can get from this additional postulate.

I'd say # is somewhat different, since it is omega-inconsistent.

>You might object that # has no mathematical content.  That remains to
>be seen.  The test for this is not the form of # but whether
>interesting new mathematics results.  [...]

Or physical.  To give a more concrete analogy, consider the Riemann
Hypothesis.  We all "know" that RH is true.  Physicists will use it
without blinking.  One mathematician got into trouble about ten years
ago when he noticed `amazing' results of mathematical physicists, and
worked out that they implied RH, only to notice later that the whole
set up was circular.  So what happens, if 10 years from now, the
physicists have worked out and established their dream TOE, assuming
RH, yet someone then discovers, by an extremely subtle w^w^w^w^w^w
induction, that there is indeed a distant counterexample?  Do the
physicists abandon ship?  Of course not.  They let the mathematicians
figure it out, and will continue to assume RH.  In essence their TOE
is modeled inside Large_Fragment(PA)+RH, and the only question is how
large is large--it would have to be big enough to do any calculation
physicists run into, but small enough to avoid w^w^w^w^w^w induction.
Physicists have no particular obligation to assume PA or ZFC; they do
have an obligation to work with *some* mathematical model that makes
predictions.

My "of course not" was of course too strong.  Physicists may come up
with an alternate physical model inside PA that correctly takes into
account ~RH, which presumably leads to new experimental predictions.  If
they are verified, the PA model will be adopted, Nobel prizes will be
awarded, the unreasonable reasonableness of mathematics celebrated yet
again, and so on.  If they are not verified, the fragment version with
RH will be kept, leaving the mathematicians bugged to no end.

Note that J Archibald Wheeler, in his dream of dreams of using set
theory and logic as a model for pregeometry, would have used something
like ZFC+# as a model for a black hole.  The paradox of `physics proves
that physics fails' would be tamed as an omega-inconsistency.

All in all, Vaughan's posting regarding ZFC+# and what it could mean for
mathematician-kind is similar, but there the situation is one where
Platonic intuition would be the deciding factor instead of reality.
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
Newsgroups: sci.physics