Hierarchies of torsion-free Abelian groups
Using the arithmetic hierarchy of sets one can define the arithmetic hierarchy of Abelian groups. Let N be the set of all natural numbers, and let X denote one of the symbols \(\Sigma^ 0_ n\), \(\Pi^ 0_ n\), \(\Delta^ 0_ n\), where \(n\geq 1\). A map \(\nu\) of the set N onto the Abelian group A is said to be an enumeration of this group. The pair (A,\(\nu)\) is called an X-group if for any numbers x and y one can effectively find the number of the element \(\nu x+\nu y\) and the set \(\{<x,y>|\nu x=\nu y\}\) is an X-set. The group A is said to be X- representable, if there exists an enumeration \(\nu\) of the group A, such there the pair (A,\(\nu)\) is an X-group. We note that in a different terminology \(\Delta^ 0_ 1\)-representable groups are called constructivizable groups. We call the class of X-representable groups the X-class. The scheme of inclusions between these classes is the following: \(\Delta^ 0_ 1\subseteq \Sigma^ 0_ 1\subseteq \Delta^ 0_ 2\), \(\Delta^ 0_ 1\subseteq \Pi^ 0_ 2\subseteq \Delta^ 0_ 2\) etc. In this paper we investigate this hierarchy for the class of torsion-free Abelian groups.
scientific article; zbMATH DE number 4014696