A decidable Ehrenfeucht theory with exactly two hyperarithmetic models
T is an Ehrenfeucht theory if it has finitely many countable models up to isomorphism and is not \(\omega\)-categorical. T is persistently Ehrenfeucht if for every n-type \(\Gamma\) (x) of T, the theory \(\Gamma\) (c) in the language of T with a new constant c added is Ehrenfeucht. This lengthy paper is devoted to a proof of the following theorem. For each \(n<\omega\) there is a complete, decidable, persistently Ehrenfeucht theory T having exactly five non-isomorphic countable models: three decidable models and two models decidable exactly in the hyperarithmetic set H(n). By a theorem of Millar this is the best that can be achieved if a decidable Ehrenfeucht theory has a countable model which is not \(0''\)- decidable. The T exhibited is a theory of trees which is inspired by earlier examples of Millar and Peretyat'kin.
scientific article; zbMATH DE number 4212016