[FOM] natural intermediate r.e. degrees

[Harvey Friedman]

On 2/9/05 8:21 AM, "Stephen G Simpson" <simpson at math.psu.edu> wrote:

> Long-time FOM subscribers may recall that I have a strong interest in
> the topic of natural intermediate r.e. degrees, or the lack thereof.
> See for example my old FOM postings of August 13 and August 16, 1999,
> available in the FOM archives at
> 
>  http://www.cs.nyu.edu/pipermail/fom/1999-August/003376.html
> 
> and
> 
>  http://www.cs.nyu.edu/pipermail/fom/1999-August/003380.html
> 
> My two recent papers relevant to this topic are "Mass Problems and
> Randomness" (BSL 11, 2005, 1-27) and "An Extension of the Recursively
> Enumerable Turing Degrees" (15 pages, submitted for publication), both
> available on my web page http://www.math.psu.edu/simpson/papers/.
> 
> Regrettably, the FOM list as presently constituted is not an
> appropriate venue for a thorough discussion of this topic.  Why am I
> saying this?  Please e-mail me privately.
> 

The FOM subscription list currently has 813 names, with probably at least an
equal number of nonsubscribers who look at the Archives from time to time.
The FOM has been and continues to be by far the most appropriate forum,
worldwide, for the productive discussion of any substantial issues
surrounding the foundations of mathematics.

For a productive discussion of "natural r.e. degrees", there should be some
attempt to propose some definite criteria for "natural" appropriate to this
context. Or at least something that substantially goes deeper than merely
observing that "there are no natural intermediate r.e. degrees".

Note that I provided just that in my posting Complexity of
Notions/Intermediate Degrees, of 2/1/05, 12:16PM.

I stand ready to discuss, here and now on the FOM, my proposals for definite
criteria made in my 2/1/05 posting. I expect that my proposals can be
sharpened up and improved on in various ways.

In your "An Extension of the Recursively Enumerable Turing Degrees", you
consider another kind of degree, and give rather simple examples of various
degrees intermediate between lowest and highest in this other degree
structure. This does not bear on the issue of the meaning/existence of
"natural intermediate r.e. degrees". However, it is a fruitful contribution
to the structure of these alternative degrees, and is entirely appropriate
for discussion on the FOM.

"Mass Problems and Randomness" also concerns alternative degree structures.
So my remarks above apply here as well.

Fortunately, the FOM list as presently constituted is an appropriate venue
for a thorough discussion of this topic. Why am I saying this? Please e-mail
me privately.

Harvey Friedman