FOM: c.e. degrees and Turing degrees -- reply to Simpson

Joe Shipman shipman at
Fri Oct 15 09:19:27 EDT 1999

Thanks for clearing that up, Steve.  I think that the semilattice of
c.e. degrees may be getting too much attention in this discussion
compared to the semilattice of Turing degrees.  If a degree is
first-order definable in the larger semilattice, I would regard that as
a  "natural" degree.  So the question is, do Cooper's automorphisms
extend to automorphisms of the entire structure of Turing degrees?  We
have just seen that the set L(1) of c.e. degrees whose jump is 0', while
not definable in the smaller semilattice, is definable in the larger
one; could this be true for an individual c.e. degree?

Barry Cooper is a subscriber to the F.O.M. list; it would be nice for
him to tell us what he knows about the answers to these questions and
the other ones we have been discussing, since as you point out his
proofs may give more information than is implied by his announced

-- Joe Shipman

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