Geodesic
$\Sigma$-Definability of countable structures over real numbers, complex numbers, and quaternions
Algebra i logika, Tome 47 (2008) no. 3, pp. 335-363 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{AL_2008_47_3_a3,
     author = {A. S. Morozov and M. V. Korovina},
     title = {$\Sigma${-Definability} of countable structures over real numbers, complex numbers, and quaternions},
     journal = {Algebra i logika},
     pages = {335--363},
     year = {2008},
     volume = {47},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2008_47_3_a3/}
}
TY  - JOUR
AU  - A. S. Morozov
AU  - M. V. Korovina
TI  - $\Sigma$-Definability of countable structures over real numbers, complex numbers, and quaternions
JO  - Algebra i logika
PY  - 2008
SP  - 335
EP  - 363
VL  - 47
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/AL_2008_47_3_a3/
LA  - ru
ID  - AL_2008_47_3_a3
ER  - 
%0 Journal Article
%A A. S. Morozov
%A M. V. Korovina
%T $\Sigma$-Definability of countable structures over real numbers, complex numbers, and quaternions
%J Algebra i logika
%D 2008
%P 335-363
%V 47
%N 3
%U http://geodesic.mathdoc.fr/item/AL_2008_47_3_a3/
%G ru
%F AL_2008_47_3_a3
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/

[1] Yu. L. Ershov, Opredelimost i vychislimost, Sibirskaya shkola algebry i logiki, Nauchnaya kniga (NII MIOO NGU), Novosibirsk, 1996 | MR | Zbl

[2] J. Barwise, Admissible sets and structures, Springer-Velag, Berlin, 1975 | MR | Zbl

[3] C. C. Goncharov, Yu. L. Ershov, Konstruktivnye modeli, Sibirskaya shkola algebry i logiki, Nauchnaya kniga (NII MIOO NGU), Novosibirsk, 1996

[4] Yu. L. Ershov, “$\Sigma$-definability of algebraic systems”, Handbook of recursive mathematics. Vol. 1: Recursive model theory, Stud. Logic Found. Math., 138, eds. Yu. L. Ershov et al., Elsevier, Amsterdam, 1998, 235–260 | MR | Zbl

[5] Kh. Rodzhers, Teoriya rekursivnykh funktsii i effektivnaya vychislimost, Mir, M., 1972 | MR

[6] R. Rogers, Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967 | MR | Zbl

[7] G. E. Sacks, Higher recursion theory, Perspect. Math. Log., Springer-Verlag, Berlin etc., 1990 | MR