[FOM] Intermediate Turing Degrees

[Merlin Carl]

Dear list members,
it is well known that there's a rich structure of Turing
degrees between 0 and 0`. However, while 0` has many
interpretations in terms of provability, definability etc.
and a halting problem solver would probably be a very
helpful tool for a software developper (for example as part
of a debugger), I have yet neither seen intermediate degrees
occuring in other disciplines of mathematics nor
corresponding to an application.
Once the incompleteness of formal arithmetic was
demonstrated, there was a strive to make the independent
statements more similar to "natural mathematics", i.e.
stronger related to classical areas and theorems. The
Paris-Harrington or the Goodstein Theorem are examples of
such statements.
So my question is: Is there something like Paris-Harrington
for intermediate Turing degrees? Or is there a reason why
there's not?
With kind regards
Merlin Carl