Computable shuffle sums of ordinals

The main result is that for sets , the following are equivalent:

(1)	The shuffle sum σ(S) is computable.
(2)	The set S is a limit infimum set, i.e., there is a total computable function g(x, t) such that enumerates S.
(3)	The set S is a limitwise monotonic set relative to 0′, i.e., there is a total 0′-computable function satisfying such that enumerates S.

Other results discuss the relationship between these sets and the sets. The author’s research was partially supported by a VIGRE grant fellowship. The author thanks Denis Hirschfeldt and Steffen Lempp for an insightful conversation about LIMINF sets; Christopher Alfeld and Robert Owen for numerous comments and suggestions; and his thesis advisor Steffen Lempp for his guidance.