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Trivial, strongly minimal theories are model complete after naming constants. (English) Zbl 1035.03013
Summary: We prove that if $$\mathcal{M}$$ is any model of a trivial, strongly minimal theory, then the elementary diagram $$\operatorname{Th}(\mathcal{M}_M)$$ is a model complete $$\mathcal{L}_M$$-theory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are $$\mathbf{0}''$$-decidable, and that the spectrum of computable models of any trivial, strongly minimal theory is $$\Sigma^0_5$$.

##### MSC:
 03C10 Quantifier elimination, model completeness, and related topics 03C35 Categoricity and completeness of theories 03C57 Computable structure theory, computable model theory
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