Complexity of index sets of calculable classes with a finite number of constructive systems (Q1096621)

This paper continues the author's investigations into index sets associated with recursive collections of recursive models which he began in Algebra Logika 22, No.4, 372-381 (1983; Zbl 0537.03022). To read the present paper it is necessary to have also read this earlier work. The main results of each of these papers concern recursive collections of Abelian groups \(C=\{G_ 1,G_ 2,...\}\) such that each \(G_ i\) is recursively isomorphic to one of two (or more) groups. In the present paper the groups are \(A_ k\) where \(A_ k=\sum^{\infty}_{n=1}{\mathbb{Z}}(P^ n)\oplus \sum_{k}{\mathbb{Z}}(P^{\infty})\) where \({\mathbb{Z}}(P^ n)\) denotes the cyclic group of order n and \({\mathbb{Z}}(P^{\infty})\) the quasicyclic group). For example, it is shown that there exists a recursive collection C - where each \(G_ i\) is recursively isomorphic to \(A_ k\) for some \(j\leq k\leq n\)- and where the complexity of \(\{G_ i:\) \(G_ i\simeq A_ k\}\) for k fixed is precisely \(\Delta^ 0_ 3\). The techniques are algebraic coding arguments.