[FOM] RE: FOM Continuum Hypothesis
Matt Insall
montez at fidnet.com
Sun May 18 15:09:41 EDT 2003
I previously wrote:
``The following statement is true about finite
sets:
For n>0, there is a cardinal k>n such that the cardinality of the power set
of n is greater than k.''
Thanks to Bill Tait for reminding me that I should have put ``n>1'',
instead of ``n>0''. He also points out that my statement about this axiom
allowing for ``many more'' cardinals needs to be made more precise, since
inside any model of set theory - including those which fail to satisfy CH -
one may find a constructible universe, and it will have the same cardinals
as
the original model. I am not sure how to make it more precise, but
properties
of cardinals like the one above (with ``n>1'' instead of ``n>0'') are
satisfied
in models I prefer, whereas they are not for models of, for example, V=L.
(Just
as in models of V=L, all sets are constructible, while I personally think
there
are ``many'' sets that are not constructible. In a sense, I think that even
a
``local'' violation of V=L holds, so that the universe is ``almost
everywhere''
not constructible, and in some sense, ``most'' sets are not constructible.)
Also, I wrote:
``I think an approach that could be tried, but has not
received a lot of attention, is to try to connect CH or its negation to some
physical theory, and then perform physical experiments aimed at providing
support for the adoption of one or the other.''
Thanks to Joe Shipman for pointing out one connection to such physical
theories.
He mentioned offlist that he has written a paper in which he references work
of
Pitowsky that makes use of CH to prove the existence of spin-1/2 functions
in
deterministic versions of quantum mechanics, and of Gudder in probability
manifolds.
(I have not yet been able to obtain a copy of Gudder's article, but I have
access to
Shipman's article and Pitowsky's paper.) The work of Shipman connects CH -
and other
axioms beyond ZFC - to various strengthenings of Fubini's theorem on the
order of
integration for iterated integrals of functions other than measurable ones.
He
relates this through the work of Pitowsky and Gudder to measurement of
noncommuting
observables. It is interesting to note that although Pitowsky uses CH to
prove
his theorem 1 (``There exists a spin-1/2 function.''), he comments in a
footnote that
CH itself is not necessary for the proof, but a ``much weaker'' assumption
may be
used instead. The existence of electrons (which are spin-1/2 particles) may
in some
sense constitute physical evidence that we should assume CH, or at least the
weaker
assumptions to which Pitowsky alludes. If then CH can also be used to
predict that
some specific particles do not exist, and we find those particles in nature,
and if
it is shown that the assumptions to which Pitowsky alludes are required to
prove
existence of spin-1/2 functions, then we have a case in which nature
provides us
with a clear demarcation about which axioms to assume for our set theory to
be of
use in the realm of a particular physical theory. In addition, it would be
interesting
to see if assumptions closely related to the negation of CH can be used to
prove the
negation of theorem 1, in which case the physical theory points clearly to
CH in particular
as an axiom that should be accepted.
Thus while I still prefer to accept the negation of CH, perhaps nature will
encourage
me to revise this to an acceptance of either CH or even the much more
``rigid'' V=L.
But let me briefly draw attention to the setting in which Pitowsky was
working: ``deterministic'' models. There has been much discussion of
determinism vs non-determinism
for a basic philosophy of natural processes. Perhaps my adherence to not CH
and
nonconstructibility is actually consistent with non-deterministic models of
nature -
specifically of quantum theories - while CH and V=L are consistent with
deterministic
models. The details may yet need working out, but that is where all the fun
is, right?
References
S. Gudder, ``Probability Manifolds'', in J. Math. Phys. 25 (1984),
2397-2401.
I. Pitowsky, ``Deterministic Models of Spin and Statistics'', in Phys. Rev.
D
27 (1983), 2316-2326.
J. Shipman, ``Cardinal Conditions for Strong Fubini Theorems'', in Trans.
Amer.
Math. Soc. 321, no. 2 (October 1990), 465-481.
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