[FOM] Re: Empirically relevant undecidability

Timothy Y. Chow tchow at alum.mit.edu
Fri Feb 20 10:08:21 EST 2004

Neil Tennant wrote:
>Question: does anyone know, for the various likely current theories T
>(say, ZF; or ZFC; or ZF+V=L; or Z_2; or ...) of a mathematical
>sentence S, and any scientific hypothesis H, such that S is an
>H-desirable would-be theorem of T, but S is provably independent of T,
>on the assumption that T is consistent?

This question, or a variant, gets batted around on USENET every now and
then.  Here is one example that Matthew Wiener likes to point out (I quote
from http://www.google.com/groups?selm=46p2on%249v5%40netnews.upenn.edu ):

  And indeed, one can construct a local deterministic interpretation
  of QM *assuming* CH (or even just Martin's Axiom).  This does not
  contradict Bell's inequality, since the interpretation does some
  clever tricks with non-measurable sets.  It's "physics" enough to
  have been published in PRD.

  See Gudder's book on quantum probability, and the references therein.

The book in question is _Quantum_Probability_ by Stanley Gudder,
Academic Press, ISBN 0123053404.  I have never looked at this book,
so I can't comment further.

Although you mention ZFC as one of your acceptable options, many people
find it implausible that AC is used in an essential way in science.  I
gather that the situation in mathematical physics is rather similar to
that in math in general: There are many theorems that are freely used by
mathematical physicists and are usually proved using AC (since their
most general form requires AC); the use of AC can almost certainly be
removed a posteriori in applications, but nobody bothers to do so.  If
this picture is accurate, then it seems likely that the answer to your
question is no, but that demonstrating this would require carrying out
in detail some kind of "reverse mathematics" program for physics.

Also, perhaps the parallel thread in FOM on hypercomputation is relevant
to your question.  Advocates of hypercomputation are interested in the
scientific hypothesis H that one can build a hypercomputer that solves the
halting problem or Hilbert's Tenth Problem.  This would seem to suggest
that every true Pi_1 sentence is an "H-desirable would-be theorem of T,"
and so no consistent recursive T containing Robinson's arithmetic could
prove all such would-be theorems.


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