Abstract: In this talk, I will present a recursion-theoretic view of Levi's theorem: an abelian group is torsion-

free if and only if it orderable. Downey and Kurtz first considered the effective version of Levi's theorem, and

proved the existence of a computable torsion-free abelian group which cannot be effectively orderable.

Simpson and his student later proved that in terms of reverse mathematics, Levi's theorem has strength the

same as WKL0 (Weak Konig Lemma). Kach, Lange and Solomon (APAL 2013) consider the degree-version

of Levi's theorem and showed the existence of a c.e. set C and computable torsion-free abelian group G with

infinite rank, admitting exactly two computable orderings such that every C-computable order on G is computable.

In this paper, Kach, et al. pointed out that there are infinitely many such Cs, and these Cs can have low degree.

Martin shows in his PhD thesis (U. Conn) that C can be of high degree. Our main result provides a close relation between such Cs and PA degrees, and shows that such Cs can be any incomplete c.e. set.

It is a joint work with Frank Stephan (NUS) and Huishan Wu (BLCU).

Bio: Dr. Guohua Wu got his PhD (Math) at Victoria University of Wellington, New Zealand in 2002. He is

now an associate professor at the School of Physical and Mathematical Sciences, Nanyang Technological

University, Singapore. His research interests include computability and complexity, formal languages, mathematical logic and set theory.