[FOM] Intermediate Turing Degrees

[Merlin Carl]

Many thanks for the comments to my question and the links. I will take 
some time to study these papers and read though the earlier discussion.

One approach I thought of was: Can there be a subsystem S of PA such 
that the problem of deciding whether a formula belongs, say, to the 
Sigma(1)-theory of S has a degree between 0 and 0`? An idea could be to 
make the system just strong enough so that a priority method can be 
carried out in it - it just seems that this already gives the strength 
of the halting problem.
Kind regards
Merlin Carl