Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees

Abstract

We show that, under certain assumptions of recursiveness in A \mathfrak {A} , the recursive structure A \mathfrak {A} is Δ α 0 \Delta _\alpha ^0 -stable for α > ω 1 C K \alpha > \omega _1^{CK} if and only if there is an enumeration of A \mathfrak {A} using a Σ α 0 \Sigma _\alpha ^0 set of recursive Σ α {\Sigma _\alpha } infinitary formulae and finitely many parameters from A \mathfrak {A} . This extends the results of [1]. To do this, we first obtain results concerning Δ α 0 \Delta _\alpha ^0 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 Δ α 0 \Delta _\alpha ^0 function from the previous node and its label.