A new spectrum of computable models (Q1943912)

The paper contains a survey of results concerning computable models of the so called Ehrenfeucht theories, i.e., complete theories having a finite number (at least two) of countable models up to isomorphism. A new result is the following Theorem 9: Let \(n\) an \(m\) be natural numbers such that \(1\leq m\leq n\). Then, there exists an Ehrenfeucht theory \(T\) such that {\parindent=6.5mm \begin{itemize}\item[(1)] \(T\) has exactly \(n+2\) nonisomorphic countable models; \item[(2)] \(T\) has \(n\) nonisomorphic almost prime models \({\mathfrak A}_1,\dots,{\mathfrak A}_n\) which are mutually elementary embeddable; and \item[(3)] the models \({\mathfrak A}_1,\dots,{\mathfrak A}_m\) are computable and the models \({\mathfrak A}_{m+1},\dots,{\mathfrak A}_n\) are not computable. \end{itemize}}