Computable Embeddings and Strongly Minimal Theories

Journal of Symbolic Logic 72 (3):1031 - 1040 (2007)
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Abstract

Here we prove that if T and T′ are strongly minimal theories, where T′ satisfies a certain property related to triviality and T does not, and T′ is model complete, then there is no computable embedding of Mod(T) into Mod(T′). Using this, we answer a question from [4], showing that there is no computable embedding of VS into ZS, where VS is the class of infinite vector spaces over Q, and ZS is the class of models of Th(Z, S). Similarly, we show that there is no computable embedding of ACF into ZS, where ACF is the class of algebraically closed fields of characteristic 0

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