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Stability of recursive structures in arithmetical degrees. (English) Zbl 0631.03016

A recursive structure \({\mathfrak A}\) is \(\Delta^ 0_ n\)-stable if every isomorphism of \({\mathfrak A}\) with every other recursive structure is \(\Delta^ 0_ n\) in Kleene’s arithmetical hierarchy. The notion of a formally \(\Delta^ 0_ n\)-enumeration of a recursive structure \({\mathfrak A}\) is defined in the paper. It is easy to prove that if a recursive structure \({\mathfrak A}\) has a formally \(\Delta^ 0_ n\)-enumeration, then it is \(\Delta^ 0_ n\)-stable. The converse of this result is proved under the assumption that the existential diagram of \({\mathfrak A}\) and the relations, pointed out in the paper, are recursive. For \(\Delta^ 0_ 1\)-stability the proof uses a finite injury priority argument and it was given by S. S. Goncharov [Algebra Logika 14, 647-680 (1975; Zbl 0367.02023)]. For \(\Delta^ 0_ 2\)-stability the proof uses an infinite injury argument and for \(\Delta^ 0_ 3\)-stability a ‘monstrous’ injury argument. The author shows how this process can be continued for all n.
Reviewer: A.N.Ryaskin

MSC:

03C57 Computable structure theory, computable model theory
03D45 Theory of numerations, effectively presented structures

Citations:

Zbl 0367.02023
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References:

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