Fried's NAMS 54:9 letter

In the current issue of the Notices of the AMS (Vol.54, #9), Mike Fried
published a letter to the editor in which he writes (among other things):
"My first Annals paper solved a problem posed by Ax and Kochen. Prior
to my result, someone well-connected to the area 'proved' there could
be no such theorem as mine."

I would like to know:
(1) Who was the "well-connected" individual Fried is referring to?
(2) What is a precise statement of the result that this "well-connected"
individual claimed to have proved?
(3) Was this claim made in a printed form, such as a preprint or a thesis?

Henceforth in this posting, I will refer to the "well-connected individual"
as WCI.

I believe the "first Annals paper" of Fried is the one in which he
presented a decision procedure and elimination of quantifiers for the
elementary theory of finite fields. That a decision procedure existed
was never in doubt, in the wake of Ax's fundamental paper, "The
Elementary Theory of Finite Fields". The problem was to give it
explicitly and the expected method for doing so was to provide an
explicit elimination of quantifiers.

The existence of an elimination of quantifiers for a first order
theory is somewhat delicate, since it depends on the language in
which the theory is presented and not only on the class of models
of the theory. It was certainly known, before Fried's Annals paper
(with Sacerdote) was published, that there is no elimination of
quantifiers for certain formulations of the language and that there
does exist elimination of quantifiers for certain others. For example,
it lacks elimination of quantifiers if formulated in the language
of fields (as Fried and Sacerdote themselves note in their Annals
paper) and in the language of fields supplemented by a sequence of
predicates R_n(b) which say that b has an n-th root, and some
Artin-Schreyer analogues (as Ax had originally hoped), but it
does have elimination of quantifiers if instead one supplements
the language of fields with predicates \phi_n(a_0,...,a_n) that
say, roughly, that the polynomial in x of degree n with coefficients
a_0,...,a_n has a root. The reason for the qualification "roughly"
is that it is easier to work with the theory of pseudofinite fields,
where the statement holds without qualification, but when one wants
to talk about the theory of finite fields, one has to worry about
particular finite fields that arise as special cases.

The elimination of quantifiers given by Fried and Sacerdote is with
respect to yet another, at that time entirely unconventional formulation
of the language. If someone else had written even an incorrect paper
containing this formulation, and if Fried had been aware of it at
the time he and Sacerdote wrote their Annals paper, it would have
been appropriate for them to have cited it. For, the invention of
that language itself would have been a genuine innovation in its
own right, even if the author had incorrectly claimed to prove that
one could not eliminate quantifiers using that language. I think it
is more likely that the "well-connected" individual (WCI) that Fried
has in mind was not referring to the language that Fried and Sacerdote
used. Therefore it is hard to take at face value Fried's assertion that
WCI had claimed to prove that there could be no such result as Fried
and Sacerdote had proved.

So, as I have asked above, I would like to know what Mike Fried is
talking about.

Since I have not seen it remarked in print before, I would also like
to point out an error at the beginning of their article (pp.204-205)
where they write: "The method demonstrates anew Ax's striking result
that a sentence of L_K is true for all residue fields of {\cal O}_K
if and only if it is true for all finite extensions of residue fields
of {\cal O}_K." The sentence "(2=0)->\forall x(x^2=x)" is a counterexample
with K the field of rational numbers and no such silly "result" occurs in
the writings of Ax. This error somehow got past both authors, the five
referees and the editor(s).

Incidentally, I learned recently that James Ax died last year after a
6 month battle with colon cancer. I certainly hope that Fried is not,
in his letter to the current Notices of the AMS, referring to Ax, who
can no longer defend himself.

The results quoted above about the nonexistence of elimination of quantifiers
for the theory of pseudofinite fields with respect to the language based
on the relations R_n described above was proved by myself and Ax, in a
conversation in his office, thereby shooting down his original idea. At
that time, I was Ax's thesis student and my thesis problem was to give an
explicit elimination of quantifiers for the theory of finite fields. The
result that the theory of pseudofinite fields does have elimination of
quantifiers with respect to the language based on the relations \phi_n
described above is my own discovery. Largely because I didn't want to get
caught in the middle of the Ax's disputes with Fried, who claimed to have a
solution of my thesis problem, I abandoned this thesis problem and wrote my
thesis with Michio Kuga instead. My thesis topic was taken over by Catarina
Kiefe, who went much further than I did, and the results I quoted can probably
be found in her thesis; I haven't read it in a very long time, and never very
carefully, so I can't cite it more explicitly.

How it is that it was impossible for Fried to share his solution with Ax
and his thesis students is an interesting study in its own right and could
probably fill a book, a musical or a sitcom. But I did learn a valuable
lesson from the experience: even if it is clear that a person doesn't know
the most elementary and necessary facts and definitions about a subject in
which he claims to have solved an important problem and even if he is
incoherent and rambling in his attempts to explain his solution, it is
still possible that he does in fact have a solution; that, moreover, this
can only be determined by a careful reading of the solution when he finally
manages to write it down; and that one must not be misled in that
determination by idiotic statements such as the one I mentioned from
pages 204-5, no matter how vague the rest of the paper seems and no
matter how hard it is to read it.
--
Allan Adler <ara@xxxxxxxxxxxxxxxxxxxx>
* Disclaimer: I am a guest and *not* a member of the MIT CSAIL. My actions and
* comments do not reflect in any way on MIT. Also, I am nowhere near Boston.


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