On Disintegrated Models
Nurlan Markhabatov
Definition 1. [1] Let \(\mathcal{T}\) be a family of theories and \(T\) be a theory such that \(T\notin \mathcal{T}\). The theory \(T\) is said to be \(\mathcal{T}\)-approximated, or approximated by the family \(\mathcal{T}\), or a pseudo-\(\mathcal{T}\)-theory, if for any formula \(\varphi \in T\) there exists \(T'\in \mathcal{T}\) for which \(\varphi \in T'\).
Definition 2. An infinite model \(\mathcal{M}\) of a \(\mathcal{T}\)-approximated theory \(T\) is called disintegrated if it is a disjoint union of models, \(\mathcal{M}=\bigsqcup_{i\in \omega} \mathcal{M}_i\), where \(\mathcal{M}_i\) is finite (possibly either \(\omega\)-categorical or minimal). Otherwise, \(\mathcal{M}\) is called integrated.
Theorem. Any disintegrated infinite model is pseudofinite.
Theorem. There is an integrated model, which is pseudofinite.
This research was funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP19677451, AP22782938).
Bibliography
- S.V. Sudoplatov, Approximations of theories, Siberian Electronic Mathematical Reports, 17, (2020), pp. 715–725. https://doi.org/10.33048/semi.2020.17.049