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

[joeshipman@aol.com]

  Harvey makes some good points, and in some other cases misses my 
points (which is my fault for not stating them clearly enough.)

 Friedman:
  2. RVM is neither an intuitively justifiable axiom nor is it a 
convention.

 Shipman:
  It may not be intuitive to you, but it certainly represents an 
intuition shared by many mathematicians prior to Banach-Tarski. 
Otherwise, why would anyone have been SURPRISED by Banach-Tarski?

 Friedman:
  It also has the following major drawback: it asserts the existence of 
a
 function from sets of reals to reals without indicating an example.

 Shipman:
  That is a drawback 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?

 *****

 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

 Friedman:
  This is incorrect as indicated above in 3. (3. RVM has only very few 
interesting consequences for the projective
  hierarchy, and almost no interesting consequences for arbitrary sets 
of
 reals.)

 Shipman:
  It is going too far to say that statement of mine is "incorrect". If 
you are going to measure "more" in the crudest possible way, both RVM 
and V=L settle aleph_zero new questions in the language of set theory, 
so only in that sense is my statement 2) incorrect. 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. As far as statements about projective sets, 
etc., are concerned, I won't argue with you about whether the 
aleph_zero new questions about projective sets that RVM settles are 
"less interesting" than the aleph_zero new questions about projective 
sets that V=L settles.

 Friedman:
 >
 > 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".

 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?

 ********

 Friedman:
 >
 > 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.

 Shipman:

  I have said nothing against ZFC. I am sure that for exactly the 
reasons you state, ZFC would have been developed and found of immense 
value in any alternate history; but the existence of a real-valued 
measure on [0,1] would have been considered an important "open 
question" as CH was, and the eventual proof that RVM implied ZFC was 
consistent would be taken as an argument IN FAVOR of RVM.

 *********

 Friedman:

  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.

 Shipman:

  That is your own personal physical intuition, which is well-informed 
by contemporary physics, and which is rightly shared by many of TODAY'S 
physicists. But you are missing my point. 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. That is why the 
Banach-Tarski result was SURPRISING. My point is that if physics and 
mathematics had developed certain results in different sequences than 
they were actually developed, then at the time RVM was formulated 
precisely it would STILL have had the backing of intuition and 
therefore been taken much more seriously as a mathematical axiom, and 
if it had been shown within a few years to establish Con(ZFC) than that 
would have been regarded as even more reason to take it seriously.

 *********

 Shipman, earlier:
 >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).

 Friedman:

  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.

 Shipman:

  I have the highest regard for Solovay's work, but I wrote imprecisely 
and did not mean to imply that Solovay had shown RVM was independent. 
In the alternate history I am proposing, his work would have 
established all the arithmetical consequences of MC, which would 
previously have been open questions, as fully proven theorems.

 **********

 Shipman, earlier:

  > 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.

 Friedman:

  RVM is not widely useful in mathematics or physics. What wide 
applications
 do you have in mind?

 Shipman:

  In addition to Con(ZFC), how about the "Strong Fubini Theorems" I deal 
with in my thesis (see the October 1990 Transactions of the AMS)? I 
show that RVM imples that Fubini's theorem applies to all 
(non-negative) mutlivariate real functions, not just the measurable 
ones, in the sense that iterated integrals may not always exist but 
WHENEVER they exist they are equal. Furthermore, this 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.

  Of course, I am not sure RVM would have become "widely useful", but 
I'll bet it would have become useful enough to withstand challenges 
 from people who say "that really shouldn't be an axiom, it shouldn't be 
regarded as true".

 ********

 Friedman:

 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.

 Shipman:

  I agree that nonmeasurable sets are probably not the problem, but set 
theory is part of the problem. Specifically, it seems to be remarkably 
difficult for physicists to formulate the current crop of fundamental 
theories in a way that is compatible with the current foundation of 
mathematics in Set Theory, but I am not yet prepared to conclude that 
this is entirely the fault of the physicists (though I'm open to 
arguments that it is).

 *********

 Friedman:

 > 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.

 Shipman:

  Because the fundamental theories are still officially formulated in 
terms of real or complex manifolds.

 Friedman:

 >and the
  > full apparatus of analysis is necessary to mathematize our 
fundamental
 > physical theories,

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

 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).

 Friedman:

  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.

 Shipman:

  General Relativity has certainly been adequately mathematized. Quantum 
Field Theory has been pseudo-mathematized in the sense that certain 
simpler subtheories (for example Quantum Electrodynamics) can be 
formulated in a logically impeccable way EXCEPT that there is no proof 
that the procedure for generating experimental predictions, as an 
algorithm, is convergent.

 Friedman:

 >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.

 Shipman:

  Earlier you doubted that space is infinitely divisible, but this is 
exactly the issue here. 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.

 -- JS