[FOM] priority arguments in differential geometry?

[Steve Simpson]

[MODERATOR'S NOTE: Almost three years ago,  when Steve Simpson was FOM 
moderator, he proposed the message below. As FOM "ombudsman" (my role at 
the time) I asked him not to post it for fear that in the atmosphere 
prevailing at that time, it would generate more heat than light, and he did 
not post it. Since he has now sent it again, I feel that I have no choice 
but to post it. -Martin]

I thank Soare for his informative message (posted on FOM by Martin
Davis, 11 July 2000) concerning recursion-theoretic results and
methods related to recent work of Nabutovsky and Weinberger in
differential geometry.

In his message, Soare mentioned three recursion-theoretic results:
Theorem 1, Theorem 2, and Theorem 2'. I would appreciate answers to
the following technical/methodological questions concerning these
theorems and their application in differential geometry.

1. Does Theorem 2 require a priority argument for its proof?
Can Theorem 2 be proved more easily than Theorem 2'?
[ Soare has already noted that Theorem 2' requires a priority argument
for its proof. My question here concerns Theorem 2, not Theorem 2'. ]
2. What methods were used in Soare's original proof of Theorem 2?
[ Theorem 2 was announced by Soare in his talk at the Computability
Theorey and Applications meeting, Boulder, June 13-17, 1999. The
proof has not yet appeared. ]
3. Is Theorem 2 one that Nabutovsky/Weinberger requested and found
most useful for their geometrical work?
4. Is Theorem 2' one that Nabutovsky/Weinberger requested?
5. In the Nabutovsky/Weinberger setup, does Theorem 2 have some
interesting geometrical consequences? What are they?
6. In the Nabutovsky/Weinberger setup, does Theorem 2' have some
interesting geometrical consequences which Theorem 2 does not have?
If so, what are they?
7. Does Theorem 1 require a priority argument for its proof?
[ Soare has already noted that Theorem 1 follows easily from the
existence of an infinite descending sequence of r.e. Turing degrees.
And the latter result is standardly proved by means of a finite-injury
priority construction, similar to the famous Friedberg/Muchnik
solution of Post's Problem. And many experts (e.g., my former thesis
advisor, Gerald Sacks), have said that results of Friedberg/Muchnik
type "require a priority argument". My question here concerns Theorem
1 itself, not the result on r.e. Turing degrees which implies it. ]

[ Of course one should also pay attention to Kucera's interesting work
giving a non-priority solution of Post's Problem. ]

-- Steve