[FOM] Choice of new axioms 1
joeshipman at aol.com
Fri Feb 10 07:11:49 EST 2006
Friedman remarks that V=L has many advantages as an axiom from the
point of view of ordinary mathematicians, but that set theorists don't
like it because it settles many questions in the "wrong way".
I propose that RVM (there exists a countably additive real-valued
measure on all subsets of [0,1]) ought to be even MORE acceptable to
mathematicians than V=L, because
1) V=L is practically impossible to STATE to a mathematician who hasn't
had a lot of logic and set theory
2) RVM settles even more questions than V=L does, in particular it
implies Con(ZFC) and lots of other new arithmetical statements while
V=L proves no new arithmetical statements
3) There is an intuitive justification for it
4) If V=L settles a lot of set-theoretic questions "the wrong way",
then either RVM settles them "the right way" or else there is more than
one "wrong way" and Friedman shouldn't have used the word "the".
This relates to the ongoing discussion of the real numbers as a
foundation for physics. It is my belief that it is something of a
historical accident that the axioms of set theory arose in their
current form. The discovery of the Banach-Tarski paradox was shocking,
but because the notion that matter was infinitely divisible had already
been called into question by the atomic theory, it was not a fatal blow
to physical intuition, and the result was instead that
non-Lebesgue-measurable sets were deprecated as unphysical.
I claim the theory of general relativity could have been discovered
BEFORE the atomic theory was verified (since before the very late
1800's there was no direct evidence of elementary particles and the
only evidence for the atomic theory was the indirect evidence of fixed
mass ratios of chemical compunds).
If that had occurred, then the Banach-Tarski paradox would have induced
physicists and mathematicians to SACRIFICE A DIFFERENT PHYSICAL
INTUITION. That is, the discovery of general relativity would have
called into question that "space is homogeneous and isotropic", so the
"rotation invariance" assumption would have been sacrificed instead of
the "infinite divisibilty" assumption. ("translation invariance" would
not have been threatened by Banach-Tarski).
THAT would have led to non-measurable sets being less deprecated, so
that the axiom RVM would have been considered to STILL HAVE THE BACKING
OF PHYSICAL INTUITION. Therefore, more mathematics would have been
developed using it, and it would eventually have been found to be a
proper extension to ZFC (in this alternate universe Solovay would have
shared the Fields Medal with Cohen, for the epochal verification that
RVM was indeed more powerful).
The eventual discovery that matter was not infinitely divisible would
not have threatened the use of RVM, because once it had become accepted
and found widely useful, it would have attained a valued MATHEMATICAL
Even today, I have never heard any mathemtician give me a good reason
why RVM should be regarded as false.
In the real universe, non-Lebesgue-measurable sets were deprecated and
RVM was neglected. But the choices that were actually made have left
the foundations of physics in an extremely problematic state --
although MATTER is not infinitely divisible, SPACE still is, and the
full apparatus of analysis is necessary to mathematize our fundamental
physical theories, even though the "physical meaning" of the real
numbers is completely opaque.
-- Joe Shipman
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