On $\Sigma$--presentable structures over R, C, and H

A.S.Morozov (joint with M.V.Korovina)

At most countable structures $\Sigma$--presentable without parameters
in hereditarily finite superstructures over the ordered field of real
numbers, over  the field of complex numbers, and over quaternions are
characterized. All these structures proven to be hyperarithmetical and
under some natural conditions they appear to be computable. Generally,
the hyperarithmetical upper bound cannot be improved. It is also shown that
in any case these structures are somehow very close to computable ones. In
the proof, a new recursion-theoretic basis theorem was obtained. We
also discuss some methodological consequences of these results.