Recursive Labelling Systems and Stability of Recursive Structures in Hyperarithmetical Degrees

Abstract

ABSTRACT. We show that, under certain assumptions of recursiveness in 21, the recursive structure 21 is A^-stable for a < wfK if and only if there is an enumeration of 21 using a E^ set of recursive EQ infinitary formulae and finitely many parameters from 21. This extends the results of [1].To do this, we first obtain results concerning A^ paths in recursive labelling systems, also extending results of [1]. We show, more generally, that a path and a labelling can simultaneously be defined, when each node of the path is to be obtained by a A^ function from the previous node and its label. Introduction.We say that a recursive structure 21 is A\\-stable if, for every recursive structure 03 = 21, every isomorphism from 03 to 21 is A° in Kleene's hyperarithmetical hierarchy. We shall show that, under certain assumptions, 21 is A°-stable iff there exists a "formally A° enumeration of 21."The basic outline of our argument is as in [1], where such a result was obtained for finite a. However, certain technicalities prevent the generalization to infinite a from being as straightforward as might be expected. The argument seems most easily described in terms of recursive labelling systems, which were introduced in [1] for a similar purpose.The two basic results for a-systems are obtained in §1 and applied in the succeeding sections to the question of A°-stability.These basic results are also used in the related topic of A°-categoricity [2]. In §1 we define the notion of a recursively a-guided recursive labelling system, or a-system, and in Proposition 1 state the desired result similar to that of [lj involving the existence of r.e. points of 2N and of labellings of A^ paths in an

Computable completely decomposable groups

“…It could be the case that every ∆ 0 4 -categorical completely decomposable group was already ∆ 0 3 -categorical, similarly to well-orderings [3]. We show that it does not happen in the class of completely decomposable groups: Theorem 4.1.…”

“…McCoy [34] characterized ∆ 0 2 -categorical linear orders and Boolean algebras under some extra effectiveness conditions. Recently, Harris [21] has announced a characterization of ∆ 0 n -categorical Boolean algebras for every finite n. It is also known that in general ∆ 0 n+1 -categoricity does not imply ∆ 0 ncategoricity in the classes of linear orders [3], abelian p-groups [5], and ordered abelian groups [36].…”

On the complexity of categoricity in computable structures

“…Ash [1] has shown that the computable ordinals are all hyperarithmetically stable. So we need only show that, whenever A n ∼ = ω CK 1 (1 + η), A n is not ∆ 0 α -categorical for any computable ordinal α.…”

“…Again from Ash [1] we know that there is some computable ordinal δ that is not ∆ 0 α -stable. Any ordinal is rigid; that is, it has no nontrivial automorphisms.…”

“…The case where Γ is the collection of hyperarithmetic sets has been studied by Ash [1], while the case where Γ is the collection of all computable sets was originally studied by Goncharov [5]. Further work on computable categoricity includes that of Kudinov [12], Cholak, Goncharov, Khoussainov, and Shore [3], Hirschfeldt, Khoussainov, Slinko, and Shore [9], and Hirschfeldt [8].…”