[FOM] Choice of new axioms 1 (reply to Friedman)

[Harvey Friedman]

On 2/12/06 8:49 PM, "joeshipman at aol.com" <joeshipman at aol.com> wrote:

> Shipman:
> It [RVM] ... certainly represents an
> intuition shared by many mathematicians prior to Banach-Tarski.
> Otherwise, why would anyone have been SURPRISED by Banach-Tarski?

I don't see that you have made the connection. Again, I doubt any intuition
behind RVM for people who are not confused, once one realizes that there
cannot be any translation invariant countably additive probability measure
on [0,1]. This was known very early on.

So a countably additive probability measure on all subsets of [0,1] can't be
translation invariant, and hence the idea that there is any intuition that
there is a countably additive measure on all subsets of [0,1] is very far
fetched. 
 
> Shipman:
> That is a drawback [not indicating an example] relative to ZF, but not
>relative to ZFC. Is V=L
> "fixing the problem with the Axiom of Choice" a good enough reason to
> adopt it?

It is a clear advantage of V = L over other proposals.

Remember: I do not talk of adopting V = L in the same sense as, say, power
set. It is simply a reflection of a convention to consider only sets
generated by well defined and well accepted processes.

> Shipman, earlier:
>> 
>> 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

It is clear what sense of incorrectness I was using in responding to your
statement. If you take the sum total of published single mathematical
statements about sets of reals and lower, that are known to be independent
of ZFC, then this is a finite number. The number settled by ZFC + V = L
approaches perhaps 99.9%. The number settled by ZFC + RVM approaches perhaps
10% (just an educated guess). Maybe much lower. It depends on whether or not
you count multiplicities.

>But RVM certainly 
> settles infinitely many new ARITHMETICAL questions, questions of a type
> V=L has nothing to say about, including practically all your recent
> independent statements.

My posting concerned what mathematicians would do given the present actual
situation, if they were to expand the accepted axioms. There are at present
no such questions of the kind you are talking about, of any interest to the
mathematical community.

In "Choice of new axioms 2" I will discuss what may happen given various new
developments. 

I don't think that you took into account the severe limitations of RVM for
settling well known statements in the projective hierarchy. If I recall, its
practically nothing at level 3 or higher.

> Shipman:
> 
> Can you be more specific about why the way V=L settles projective
> statements is considered "wrong" by set theorists, and would you
> venture an opinion on whether this attitude of set theorists should be
> considered authoritative?

This appears in the writings of set theorists. Since I do not subscribe to
realism or Platonism in set theory, nor do I subscribe to anti-realism or
anti-Platonism in set theory, I am in no position to offer up a useful
answer to your question of the kind you are seeking.

To be more precise, because of my non positions, I could not conceivably
regard anything like these attitudes of set theorists as "authoritative".
That doesn't mean that I think that they are definitely wrong. Just that
they are not definitely right.
> 
> Friedman:
> 
> I don't see any physical intuition that tells me how to measure an
> arbitrary
> set of real numbers. ...
> 
> Shipman:
> 
>I am not claiming that RVM
> has any justification in physical intuition NOW. I am claiming that it
> USED to have an intuitive justification back around 1900, when both
> mathematical and physical intuition were more naive.

Perhaps too naïve. This is before, e.g., the refutation of translation
invariance. 

I do not know of any evidence concerning the number of mathematicians back
then that were even considering the idea of expansion of the axioms. You
seem to think that this number was significant enough to speculate about
alternate historical paths.
> 
> Shipman:
> 
> In addition to Con(ZFC), how about the "Strong Fubini Theorems" I deal
> with in my thesis ... that iterated integrals may not always exist but
> WHENEVER they exist they are equal.

This is not "wide".

>Furthermore, this [RVM] has applications
> to physics because it allows one to prove a "no hidden variables"
> theorem that rules out the hidden-variables theories that had been
> proposed by the physicist Itamar Pitowsky and the mathematician Stanley
> Gudder in a series of papers.

Do you mean "applications to the theory of some proposed abstract theories
of certain kinds of physics"? This is not the same as "applications to
physics". 

> Shipman:
> 
> I agree that nonmeasurable sets are probably not the problem, [with
foundations of physics[ but set
> theory is part of the problem.

I am sure that there is no current evidence that set theory is the problem.
Any connection between substantial set theory and physics will require an
unforseen breakthrough of a gigantic order.

If set theory will aid in the difficulties, it will most likely be rather
modest doses of it well below, say, ZFC without the power set axiom. It
would already be fantastically surprising if impredicative arguments come up
in a serious way.
> 
> Shipman:
> 
>> although MATTER is not infinitely divisible, SPACE still is,
> 
> Because the fundamental theories are still officially formulated in
> terms of real or complex manifolds.

At this time. But I don't consider personally the idea of the infinite
divisibility of space to be compelling, and the fact that it broke down with
matter makes me feel unsurprised if it were to break down for space.
> 
> Shipman:
> 
> I mean that no reformulation of fundamental physics has been plausibly
> proposed that avoids using the real numbers and only talks about finite
> objects (or even countable objects; all the fundamental theories
> involve ontologies going way beyond second-order arithmetic).

This is incorrect. All of the so called fundamental theories I know anything
about are conveniently formulated in very very weak systems. Z_2 discusses
real numbers and complete separable metric spaces in the obvious way. One
can use very weak fragments of ZF\P if one wishes to avoid some of the
standard coding involved. If one wishes to avoid coding entirely, then one
can use convenient conservative extensions.
> 
> Shipman:
> 
> [For example] General Relativity has certainly been adequately mathematized.

I am referring to the position taken that general relativity is not a
fundamental theory because advanced quantum theory must be taken into
account in any truly fundamental theory.

Of course, at a different level, even these restricted arguably non
fundamental theories are still foundationally incoherent - certainly as
compared to f.o.m.
> 
> Shipman:
> 
> If space is NOT infinitely divisible, and there
> is not an infinite amount of information contained in a finite region
> of spacetime, it's not clear how any "real number", in its entirety,
> can faithfully represent something physical.
> 
Certainly under the idea that space is not infinitely divisible, real
numbers then lose their physical significance. But I was just referring
earlier to "naïve physics", which does have a strong intuitive appeal, and
is inescapably tied up with real numbers.

Harvey Friedman