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Computably categorical structures and expansions by constants. (English) Zbl 0928.03040
It is an easy consequence of the Ryll-Nardzewski theorem that if the theory of an arbitrary structure \({\mathcal{A}}\) is countably categorical then so is the theory of any expansion of \({\mathcal{A}}\) by finitely many constants. The problem is to consider the situation for effective model theory. {T. Millar} [J. Symb. Log. 51, No. 2, 430-434 (1986; Zbl 0631.03018)] proved the analogous result (with an additional assumption on \({\mathcal{A}}\)) for the notion of computable (=recursive) categoricity, i.e. computable categoricity is preserved under the above mentioned expansions in case the existential theory of a computable (=recursive) structure \({\mathcal{A}}\) is decidable. Here, the authors solve the general problem negatively. More precisely, it is proved that for each \(k\in\omega\) there is a computably categorical structure \({\mathcal{A}}\) whose expansion obtained by adding a constant naming any element of \({\mathcal{A}}\) has dimension \(k\), where the dimension of a structure \({\mathcal{A}}\) is the number of computable isomorphism types of computable structures (classically) isomorphic to \({\mathcal{A}}\).

MSC:
03C57 Computable structure theory, computable model theory
03C35 Categoricity and completeness of theories
03D45 Theory of numerations, effectively presented structures
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