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Categoricity of computable infinitary theories. (English) Zbl 1161.03020
Summary: Computable structures of Scott rank \(\omega_1^{CK}\) are an important boundary case for structural complexity. While every countable structure is determined, up to isomorphism, by a sentence of \(\mathcal{L}_{\omega_1 \omega}\), this sentence may not be computable. We give examples, in several familiar classes of structures, of computable structures with Scott rank \(\omega_1^{CK}\) whose computable infinitary theories are each \(\aleph_0\)-categorical. General conditions are given, covering many known methods for constructing computable structures with Scott rank \(\omega_1^{CK}\), which guarantee that the resulting structure is a model of an \(\aleph_0\)-categorical computable infinitary theory.

MSC:
03C57 Computable structure theory, computable model theory
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