[FOM] Choice of new axioms 1

[joeshipman@aol.com]

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 
status.

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