Baizhanov, B. S.; Tazabekova, N. S.; Yershigeshova, A. D.; Zambarnaya, T. S. Types in small theories. (English) Zbl 1338.03059 Mat. Zh. 15, No. 1, 38-56 (2015). Summary: In this paper quasi-neighborhoods, neighborhoods, their properties and connection with each other are studied. Also we are going to describe how orthogonality of types connected with definability of quasi-neighorhoods.We study the relations of weakly orthogonality and almost orthogonality of types and relation of them with the concept as the weakly and strongly convergence of formula to type. In Theorem 4 there is proved that some applied conditions with orthogonality save the property of non-homogeneity and keep the number of countable, non-isomorphic models. For any model \(\mathfrak{M}\) of a theory \(T\), \(\mathfrak{D}(\mathfrak{M})\) is called to be a finite diagram of \(\mathfrak{M}\), it is the set of all types that realized in \(\mathfrak{M}\). We prove that if there is a counterexample to Vaught’s conjecture then there is a finite diagram with \(\omega_1\) countable non-isomorphic models. MSC: 03C45 Classification theory, stability, and related concepts in model theory 03C15 Model theory of denumerable and separable structures Keywords:neighborhood; orthogonality of types; convergence of formula to a type PDFBibTeX XMLCite \textit{B. S. Baizhanov} et al., Mat. Zh. 15, No. 1, 38--56 (2015; Zbl 1338.03059)