Limit Computability and Ultrafilters

We study a class of operators on Turing degrees arising naturally from ultrafilters. Suppose U is a nonprincipal ultrafilter on ω. We can then view a sequence of sets A = (Ai)i∈ω as an “approximation” of a set B produced by taking the limits of the Ai via U : we set limU (A) = {x : {i : x ∈ Ai} ∈ U}. This can be extended to the Turing degrees, by defining δU (a) = {limU (A) : A = (Ai)i∈ω ∈ a}. The δU — which we call “ultrafilter jumps” — resemble classical limit computability in certain ways. In particular, δU (a) is always a Turing ideal containing ∆2(a). However, they are also closely tied to Scott sets: δU (a) is always a Scott set containing a′. Our main result is that the converse also holds: if S is a countable Scott set containing a′, then there is some ultrafilter U with δU (a) = S. We then turn to the problem of controlling the action of an ultrafilter jump δU on two degrees simultaneously, and for example show that there are nontrivial degrees which are is “low” for some ultrafilter jump. Finally, we study the structure on the set of ultrafilters arising from the construction U 7→ δU ; in particular, we introduce a natural preordering on this set and show that it is connected with the classical Rudin-Keisler ordering of ultrafilters. We end by presenting two directions for further research.