Computable isomorphisms, degree spectra of relations, and Scott families

A computably categorical structure whose expansion by a constant has infinite computable dimension

“…This result raises the following question, left open in [1], as well as in [9], where an easier proof of Theorem 1.3 was given: Does there exist a computably categorical structure whose expansion by some set of finitely many constants has computable dimension ω? In this paper we give the following positive answer to this question.…”

“…The proof of this theorem, which will be given in Section 3, uses techniques from [7], which in turn builds on [1,6,9]. The original source for the method common to all these papers is the work of Goncharov [3,4].…”

“…We call x 0 the top of [n] and x n+2 the coding location of [n]. (We keep the terminology "coding location" used in [6,7,9], although in this paper we will be diagonalizing rather than coding. )…”

“…For readers familiar with papers such as [6,7,9], we note that these catchup operations will take the place of the isomorphism recovery procedure used in those papers. This change will be needed in the proof of our main result, because there each special component will have infinitely many images.…”

Effective Model Theory: The Number of Models and Their Complexity

“…(Khoussainov and Shore [1998] ). This convention is important for establishing computability properties of the graphs being constructed.…”