## 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.

### MSC:

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

### Keywords:

recursive structure; arithmetical hierarchy; stability

Zbl 0367.02023
Full Text:

### References:

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