FOM: structural questions

[Harvey Friedman]

I wish to thank Shavrukov 12:34PM 8/28/98 for the various references and
historical points. He closes with the following question:

>I would greatly appreciate it if you could comment on the following
>proposition:
>
>Questions like
>>1. Are PA[pi-0-n] and PA[pi-0-m] isomorphic?
>>2. Is there a nontrivial automorphism of PA[pi-0-n]?
>>3. Is the first order theory of PA[pi-0-n] decidable?
>>4. Is PA[pi-0-n] isomorphic to some {x: x >= y}?
>only have any intellectual signficance in as much as they provide
>background to the problem
>'Why is everything important linearly ordered?'

First of all, let me point out that there are some analogous questions for
recursion theorists.

"Questions about the structure of the degrees of unsolvability somewhat
analogous to the questions 1-4 above only have any intellectual
significance in as much as they provide background to the problem *Why are
all important degrees of unsolvability linearly ordered?*"

"Questions about the structure of the r.e. degrees somewhat analogous to
the questions 1-4 above only have any intellectual significance in as much
as they provide background to the problem *Why are all important r.e.
degrees either 0 or 0'?*"

My own view is that we have the following standard situation.

1. There is a class of objects, only some of which are really natural.
2. There is a striking phenomonon that appears in the natural ones.
3. This phenomenon is demonstrably false for the class of objects.
4. Basic information about the structure of the class of objects, which
includes mostly the uninteresting objects, is sought.
5. But this information about the entire class of objects appears to be
completely different from the information about the interesting objects.
6. Nobody has a good candidate for formalizing an appropriate subclass of
objects so that the phenomonon can be explained or derived.
7. Then: how is research in 4 to be evaluated?

My own view is that a certain amount of 4 is very worthwhile, and that one
reaches fairly early a point of diminishing returns. And also, even small
advances towards explaining or deriving 2 is much more important. However,
it may be very hard to make progress. But it must be emphasized and
discussed as being of trandscendent interest by the professionals who do 4
for a living.

Specifically, with regard to proof theoretic degrees, the subject is not
all that well explored, and 4 is in the worthwhile stage. These are
judgment calls. But, after all, we make judgement calls all the time.