[FOM] Absoluteness of Clay prize problems

[Harvey Friedman]

On 8/17/04 8:28 AM, "John T. Baldwin" <jbaldwin at uic.edu> wrote:

> I had lunch with a mathematician the other day who opined that the Clay
> problems were probably all provable
> or refuatable in ZFC.    One way to argue this (without finding proofs)
> is to give a syntactic analysis showing that the varous statements
> are absolute.  For instance,  I have heard that the Riemann Hypothesis
> is pi^0_1.  Can anyone provide references on specific
> statements or even better a summary article on this subject.
> 

Another good source to examine for absoluteness is

Steve Smale, Mathematical problems in the next century, in: Mathematics:
Frontiers and Perspectives, ed: Arnold, Atiyah, Lax, Mazur, International
Mathematical Union, American Mathematical Society, 2000, p. 271-294.

Problem 1. The Riemann Hypothesis.
Problem 2. The Poincare Conjecture.
Problem 3. Does P = NP?
Problem 4. Integer zeros of a polynomial of one variable.
Problem 5. Height b ounds for diophantine curves.
Problem 6. Finiteness of the number of relative equilibria in celestial
mechanics. 
Problem 7. Distribution of points on the 2-sphere.
Problem 8. Introduction of dynamics into economic theory. (Conceptual
problem only).
Problem 9. The linear programming problem.
Problem 10. The Closing Lemma.
Problem 11. Is one-dimensional dynamics generally hyperbolic?
Problem 12. Centralizers of diffeomorphisms.
Problem 13. Hilbert's 16th Problem.
Problem 14. Lorenz attractor.
Problem 15. Navier-Stokes equations.
Problem 16. The Jacobian Conjecture.
Problem 17. Solving polynomial equations.
Problem 18. Limits of intelligence. (Conceptual problem only).

Smale also lists three additional "lesser" problems.

Add.1. Mean value Problem.
Add.2. Is the three-sphere a minimal set?
Add.3. Is an Anosov diffeomorphism of a compact manifold topologically the
same as the Lie group model of John Franks?

This important matter of "absoluteness" raises some subtle but productive
issues. 

Let me start by discussing what we could mean by saying that an open
mathematical conjecture is arithmetical.

One definition is this. Suppose we give a formalization of the statement S
in the language of set theory, as we would using standard definitions of the
relevant mathematical concepts used in S in the language of set theory. Then
there is an arithmetical sentence A such that

ZFC proves "S if and only if A".

The problem with this definition is that for trivial reasons, this is
obviously follows from

ZFC proves or refutes S.

This is clearly not what we have in mind when we say that S is arithmetical.

A more appropriate definition is this. We say that "S is known to be
arithmetical" if and only if an arithmetical sentence A has been presented
together with a proof in ZFC that S is equivalent to A. Obviously this
notion is time dependent.

An important project would be to take Smale's list and/or the Clay list and
appropriately document assertions that "such and such problem is known to be
arithmetical". 

In fact, in my opinion, the essence of mathematics is mostly Pi01 and
predominantly at most Pi02. What do I mean by this?

For almost all centrally fundamental statements S in core mathematics,
either 

1. One can appropriately document that S is Pi02 in the sense above; or
2. Failing this, one can find a statement S' and give a proof that S'
implies S over ZFC, where a large majority of people who believe in S also
believe in S', and where one can appropriately document that S' is Pi02 -
and, most usually, appropriately document that S' is Pi01.

HOWEVER, just because a mathematical conjecture is arithmetical, does that
mean that it can be proved or refuted in ZFC?

Let me give my views on this, and related, matters.

1. Already when I was a student in the mid 1960s, it was evident that there
was a gaping hole in the foundations of mathematics - the independence
results were very far away from treating any mathematically compelling
arithmetical statement.

2. In fact, this gaping hole seemed so serious that its continued presence
threatened to cripple the long range interest and importance of work in the
foundations of mathematics.

3. I thought: if I were to work in f.o.m. for a sustained amount of time, I
had better do something substantial about this gaping hole.

4. I believe that it is very likely that all specific conjectures already
made in core mathematics - especially those that are arithmetical in the
sense above - are provable or refutable in ZFC.

5. HOWEVER, I believe that there is a vast array of compelling conjectures
waiting to be made, as mathematics inexorably advances and widens its scope
and horizons, which are even Pi01 in the sense above, and which are neither
provable nor refutable in ZFC.

6. For instance, BRT is, in a certain sense, far more ambitious than other
mathematics. For instance, a tiny tiny tiny tiny corner of BRT consists of a
compelling list of 2^512 statements - very very very few of which are
themselves compelling in their own right. Partly because 2^512 is so large a
number of mathematical problems, some of them (no one being compelling) are
now known to be neither provable nor refutable, and also at the same time,
Pi02 in the sense above.

7. In postings I expect to make shortly, individual mathematical statements,
neither provable nor refutable in ZFC, which are steadily more and more
compelling than any I have described previously, will be presented. There
are already tangible glimpses of how they will be morphed into statements of
a more familiar combinatorial and analytic flavor.

I close by taking up the matter of a classic success for f.o.m., but which,
over time, has shown its age. See

Donald A. Martin, Hilbert's First Problem: The Continuum Hypothesis, in:
Mathematical developments arising from Hilbert problems, Proceedings of
Symposia in Pure Mathematics, Volume XXVIII, Part 1, 1976, p. 81-92.

It has been 28 years since this article was written. It closes with the
following:

"Throughout the latter part of my discussion, I have been assuming a naïve
and uncritical attitude toward CH. While this *is* in fact my attitude, I by
no means wish to dismiss the opposite viewpoint. Those how argue that the
concept of set is not sufficiently clear to fix the truth-value of CH have a
position which is at present difficult to assail. As long as no new axiom is
found which decides CH, their case will continue to grow stronger, and our
assertion that the meaning of CH is clear will sound more and more empty."

Some of my own views are as follows.

a. As I indicated above, the essence of mathematics is mostly Pi01 and
predominantly at most Pi02.

b. As a consequence of well known results going back to Godel, this
indicates that the continuum hypothesis issue, as well as other related set
theoretic problems, can be safely ignored by the mathematics community.

c. However, as BRT and related developments show, one cannot safely ignore
large cardinals, or "equivalent" forms, if one prefers, such as the
existence of a probability measure on all subsets of the unit interval, etc.

d. In order for the continuum hypothesis and related problems to reappear on
the mathematical landscape, it would seem that, at the very least, a
particularly compelling form of the axiom that Donald Martin speaks of in
the quote above would have to be discovered. I know of no scholar who is now
at all optimistic about finding such a compelling axiom.

Harvey Friedman