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

Andrey S. Morozov, M. V. Korovina

• 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}.