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Effectively infinite classes of weak constructivizations of models. (English. Russian original) Zbl 0824.03013
Algebra Logic 32, No. 6, 342-360 (1993); translation from Algebra Logika 32, No. 6, 631-664 (1993).
Weak constructivizations of strongly constructive models are studied. An enumerated model $$(M, \nu)$$, where $$\nu$$ is an enumeration of $$M$$, is called strongly constructive (constructive) if there exists an algorithm to recognize all (quantifier-free) formulas $$\varphi(\overline x)$$ and tuples $$\overline m$$ of natural numbers for which $$M\models \varphi(\nu\overline m)$$ holds. A model $$M$$ is called $$n$$-complete if, for each formula $$\varphi(x_ 1, \dots, x_ m)$$ which has at most $$n$$ alternations of quantifiers and for each tuple $$a_ 1,\dots, a_ m\in M$$, there exists an $$\exists$$-formula $$\psi(x_ 1,\dots, x_ m)$$ such that $$M\models \psi(x_ 1,\dots, x_ m)$$ and $$M\models \forall\overline x(\varphi\to \psi)$$. A model $$M$$ is called limit-$$\omega$$-complete if, for any $$n$$, it possesses a finite $$n$$-complete enrichment by constants, but it has no finite complete enrichments by constants. Theorem 1. If a model $$M$$ is limit-$$\omega$$-complete and possesses a strong constructivization, then, given any computable class $$S$$ of constructivizations of $$M$$ we can effectively build a non-strong constructivization that is not equivalent to any constructivization from $$S$$. Theorem 2. If $$M$$ is strongly and weakly constructivizable, then, for a given computable class of its constructivizations, we can effectively build a weak constructivization of $$M$$ that is not autoequivalent to any constructivization in this class. It follows that the class of weak constructivizations of a strongly constructivizable model is either empty or effectively infinite, and in this case, it is not computable.

##### MSC:
 03C57 Computable structure theory, computable model theory 03D45 Theory of numerations, effectively presented structures
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##### References:
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