S-Definability of countable structures over real numbers, complex numbers, and quaternions

• Autores: Andrey S. Morozov, M. V. Korovina
• Localización: Algebra and logic, ISSN 0002-5232, Vol. 47, Nº. 3, 2008, págs. 193-209
• Idioma: inglés
• DOI: 10.1007/s10469-008-9009-x
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• Resumen

  • We study ?-definability of countable models over hereditarily finite {ie193-01} superstructures over the field R of reals, the field C of complex numbers, and over the skew field H of quaternions. In particular, it is shown that each at most countable structure of a finite signature, which is ?-definable over {ie193-02} 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 ?-definable over {ie193-03}, possibly with parameters, has a computable isomorphic copy and that being ?-definable over {ie193-04} is equivalent to being ?-definable over {ie193-05}.