Geodesic
$\Sigma$-Definability of countable structures over real numbers, complex numbers, and quaternions
Algebra i logika, Tome 47 (2008) no. 3, pp. 335-363.

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We study $\Sigma$-definability of countable models over hereditarily finite ($\mathbb{HF}$-) superstructures over the field $\mathbb R$ of reals, the field $\mathbb C$ of complex numbers, and over the skew field $\mathbb H$ of quaternions. In particular, it is shown that each at most countable structure of a finite signature, which is $\Sigma$-definable over $\mathbb{HF}(\mathbb R)$ with at most countable equivalence classes and without parameters, has a computable isomorphic copy. Moreover, if we lift the requirement on the cardinalities of the classes in a definition then such a model can have an arbitrary hyperarithmetical complexity, but it will be hyperarithmetical in any case. Also it is proved that any countable structure $\Sigma$-definable over $\mathbb{HF}(\mathbb C)$, possibly with parameters, has a computable isomorphic copy and that being $\Sigma$-definable over $\mathbb{HF}(\mathbb H)$ is equivalent to being $\Sigma$-definable over $\mathbb{HF}(\mathbb R)$.
Keywords: countable model, computable model, $\Sigma$-definability. 
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A. S. Morozov; M. V. Korovina. $\Sigma$-Definability of countable structures over real numbers, complex numbers, and quaternions. Algebra i logika, Tome 47 (2008) no. 3, pp. 335-363. http://geodesic.mathdoc.fr/item/AL_2008_47_3_a3/

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