2010.5: On $\Sigma$-representability of countable structures over real numbers, complex numbers and quaternions

2010.5: Andrei Morozov and Margarita Korovina (2008) On $\Sigma$-representability of countable structures over real numbers, complex numbers and quaternions. Algebra and Logic, 47 (3). pp. 335-363. ISSN 0002-5232

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DOI: 10.1007/s10469-008-9009-x

Abstract

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).

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