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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 \(\Sigma^0_5\).

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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