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Degree spectra of prime models. (English) Zbl 1069.03025
Summary: We consider the Turing degrees of prime models of complete decidable theories. In particular we show that every complete decidable atomic theory has a prime model whose elementary diagram is low. We combine the construction used in the proof with other constructions to show that complete decidable atomic theories have low prime models with added properties.
If we have a complete decidable atomic theory with all types of the theory computable, we show that for every degree \(\mathbf d\) with \({\mathbf 0} < {\mathbf d} \leq {\mathbf 0}'\), there is a prime model with elementary diagram of degree \(\mathbf d\). Indeed, this is a corollary of the fact that if \(T\) is a complete decidable theory and \(L\) is a computable set of c.e. partial types of \(T\), then for any \(\Delta^0_2\) degree \({\mathbf d} > 0\), \(T\) has a \(\mathbf d\)-decidable model omitting the nonprincipal types listed by \(L\).

MSC:
03C57 Computable structure theory, computable model theory
03B25 Decidability of theories and sets of sentences
03D28 Other Turing degree structures
03D45 Theory of numerations, effectively presented structures
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