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Computable classes of constructivizations for models of finite constructivizability type. (English. Russian original) Zbl 0860.03032
Sib. Math. J. 34, No. 5, 812-824 (1993); translation from Sib. Mat. Zh. 34, No. 5, 23-37 (1993).
Let $$\nu$$ be an enumeration of a countable structure $$M$$. Expanding $$M$$ by the names of all elements $$\nu(n)$$ gives a structure $$M^*$$. The theory of $$M^*$$ is denoted by Th$$(M,\nu)$$. The set of all formulas from Th$$(M,\nu)$$ with at most $$n$$ blocks of quantifiers is denoted by Th$$_n(M,\nu)$$. The structure $$(M,\nu)$$ is called $$n$$-constructivizable if Th$$_n (M, \nu)$$ is decidable. It is called $$n$$-complete if for every $$\varphi ({\mathbf a})\in\text{Th}_n(M,\nu)$$ there exists an $$\exists$$-formula $$\psi ({\mathbf x})$$ realized by $${\mathbf a}$$ in $$M$$ such that $$\psi ({\mathbf x})$$ implies $$\varphi ({\mathbf x})$$ in $$M$$.
The paper is devoted to proving the following theorem. Let $$(M,\nu)$$ be $$(n+1)$$-constructivizable and let $$M$$ be $$n$$-complete but not $$(n+1)$$-complete in any finite expansion by constants. Then for every effective class $$S$$ of constructivizations of $$M$$ one can effectively find a constructivization $$\mu$$ which is not an $$(n+1)$$-constructivization and is not autoequivalent to any element of $$S$$.

##### MSC:
 03C57 Computable structure theory, computable model theory
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##### References:
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