[FOM] Choice of new axioms 1

[Harvey Friedman]

On 2/10/06 7:11 AM, "joeshipman at aol.com" <joeshipman at aol.com> wrote:

> 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

RMV is not nearly as good as V = L for the mathematical community AT LEAST
BEFORE ANY NEW DEVELOPMENTS RELATED TO RECENT WORK SINK IN, for two reasons.

1. V = L is most attractive as a *convention*, as I mentioned in my posting.
In many ways, a convention is more acceptable than an axiom, particularly if
that axiom does not have a good story for its consistency among people who
are thinking of ZFC as their starting point.

2. RVM is neither an intuitively justifiable axiom nor is it a convention.
It also has the following major drawback: it asserts the existence of a
function from sets of reals to reals without indicating an example. Such a
criticism is not available against the axioms of ZF and not against V = L.
In a way, V = L fixes the problem with the axiom of choice.

3. RVM has only very few interesting consequences for the projective
hierarchy, and almost no interesting consequences for arbitrary sets of
reals.

> 1) V=L is practically impossible to STATE to a mathematician who hasn't
> had a lot of logic and set theory

This can be finessed somewhat talking about extreme robustness. Of course,
to pull this kind of exposition off, one needs a lot of careful thought and
experience. Moreover, it is a challenge to completely get around this. I
have no doubt that one can completely get around this, with some hard work.
It will happen if this becomes a critical issue at some point.
> 
> 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

This is incorrect as indicated above in 3.

> 3) There is an intuitive justification for it

See 2 above. What is true is that the statement RVM fits very well into
classical mathematical subjects - i.e., measure theory. So it has a familiar
meaning.
> 
> 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".

Whatever few statements in the projective hierarchy settled by RVM, they are
the "right way" according to set theorists. If my memory serves me right,
"most" of the copious statements in the projective hierarchy settled by V =
L (and not ZFC), are settled the "wrong way" according to set theorists, and
some of them are settled the "right way" according to set theorists. But
this correctness for some statements in the projective hierarchy is,
according to set theorists, in the category of "a stopped clock is correct
twice a day".  
> 
> 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. 

This I do not believe. It is "obvious" that ZFC is some sort of "canonical
extrapolation of finite set theory to infinite set theory", in a sense that
remains to be fully elucidated. I am sure that this has been at least a
decisive factor in the ultimate source of its acceptance as the current gold
standard. So I don't see any chance of a different historical development.
> 
> 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,

I don't see any physical intuition that tells me how to measure an arbitrary
set of real numbers. This is on several levels. I don't see the physical
significance of an arbitrary set of reals, and I don't see the physical
meaning of a mapping of all sets of reals into the reals, and I don't see
the physical meaning of arbitrary infinite sequences of arbitrary sets of
reals. 

There is also the matter of overstating the influence of heavily theoretical
investigations related to physics on physics. E.g., physicists demand
interaction with the functioning of the real world via actual hard nosed
experiments before accepting theoretical developments.

>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).

A serious misreading of mathematical politics, if only for the fact that
Solovay's work appears in the proceedings of a 1967 conference, with a
footnote "the main results of this paper were proved in the spring of 1966",
and Cohen was awarded the Fields medal in summer of 1966. Also, this result
of Solovay occupies only a small fraction of that paper, in which much more
original deep results appear. Exact quote from Solovay at the beginning of
the relevant 3 page section: "Our method is the method of inner models of
Godel [2]. We use in an essential way some recent work of Jack Silver".

These comments have nothing to do with any kind of evaluation of the total
of Solovay's impressive (early, middle, late) work in set theory, and
qualifications for Fields Medals.
 
> 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.

RVM is not widely useful in mathematics or physics. What wide applications
do you have in mind?
> 
> Even today, I have never heard any mathemtician give me a good reason
> why RVM should be regarded as false.

Mathematicians are not sufficiently concerned with RVM these days, to have
an opinion, because of the move to concreteness, and because they don't know
how to use it.
> 
> 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 --

I agree that the foundations of physics is in an extremely problematic
state, but this does not have any apparent connection with nonmeasureable
sets and set theory.

In fact, the only substantial subject I know of for which the foundations is
not in an extremely problematic state is mathematics. And my work can be
viewed as a persistent attempt to throw its foundations also into an
extremely problematic state (there are other interpretations of what I am
doing).

> although MATTER is not infinitely divisible, SPACE still is,

Why do you think that space is? I consider it a coin toss whether physicists
at the end of the century will be adopting physical theories with space
infinitely divisible, or space not infinitely divisible. I would not predict
one way or the other.
 
>and the 
> full apparatus of analysis is necessary to mathematize our fundamental
> physical theories,

I don't know what you mean by "full apparatus".

Also, there seems to be doubt that there are any truly fundamental physical
theories left that have been mathematized. I have heard the view that any
truly fundamental theory must incorporate quantum mechanics (quantum field
theory), which has not been properly mathematized.

But I have never heard that the problem in mathematizing physical theories
has anything to do with set theoretic issues. Of course, it would be
exciting if it did.

>even though the "physical meaning" of the real
> numbers is completely opaque.
> 
I don't know quite what this means. At least at some naïve intuitive level,
the physical meaning of the real numbers is apparent. Arbitrary sets of real
numbers is a different matter.

Harvey Friedman