On $\Sigma$-representability of countable structures over real numbers, complex numbers and quaternions
Andrey S. Morozov, Margarita Korovina · 2008
We study Σ-definability of countable models over hereditarily finite 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 Σ-definable over HF(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 Σ-definable over HF(ℂ), possibly with parameters, has a computable isomorphic copy and that being Σ-definable over HF(H) is equivalent to being Σ-definable over HF(ℝ).