Infinite discrete chains and the maximal number of countable models
The paper is aimed at studying the countable spectrum of small linearly ordered theories. The objectives of the research are to study the structural properties of countable linearly ordered theories, as well as to promote the solution to the well-known open problem of model theory, Vaught's con...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Al-Farabi Kazakh National University
2021-12-01
|
| Series: | Вестник КазНУ. Серия математика, механика, информатика |
| Subjects: | |
| Online Access: | https://bm.kaznu.kz/index.php/kaznu/article/view/952/630 |
| _version_ | 1811176006769180672 |
|---|---|
| author | B. Baizhanov T. Zambarnaya |
| author_facet | B. Baizhanov T. Zambarnaya |
| author_sort | B. Baizhanov |
| collection | DOAJ |
| description | The paper is aimed at studying the countable spectrum of small linearly ordered theories. The objectives of the research are to study the structural properties of countable linearly ordered theories, as well as to promote the solution to the well-known open problem of model theory, Vaught's conjecture, which assumes that the number of countable models of a countable complete first-order theory cannot be equal to ℵ1. An important step in solving Vaught's conjecture is the search for conditions under which the theory has the maximal number of countable pairwise non-isomorphic models. By limiting ourselves to linearly ordered theories we do not get special advantages from the viewpoint of studying their countable spectrum. Therefore, in the article, a restriction on 1-types and 1-formulas of the theory is introduced. It is proved that a small countable linearly ordered theory that satisfies the restriction and has an infinite discrete chain has the maximal number of countable non-isomorphic models. To build models, the authors use the method of constructing countable models over countable sets, based on the Tarski-Vaught criterion. It is shown that it is possible to carry out the construction in such a way that the types of unnecessary elements in the resulting model are omitted, what guarantees non-isomorphism of the models and their maximal number. |
| first_indexed | 2024-04-10T19:46:04Z |
| format | Article |
| id | doaj.art-2d5582151c2c41bb8c8fc237933fd089 |
| institution | Directory Open Access Journal |
| issn | 1563-0277 2617-4871 |
| language | English |
| last_indexed | 2024-04-10T19:46:04Z |
| publishDate | 2021-12-01 |
| publisher | Al-Farabi Kazakh National University |
| record_format | Article |
| series | Вестник КазНУ. Серия математика, механика, информатика |
| spelling | doaj.art-2d5582151c2c41bb8c8fc237933fd0892023-01-29T07:22:02ZengAl-Farabi Kazakh National UniversityВестник КазНУ. Серия математика, механика, информатика1563-02772617-48712021-12-0111244656https://doi.org/10.26577/JMMCS.2021.v112.i4.04Infinite discrete chains and the maximal number of countable modelsB. Baizhanov0https://orcid.org/0000-0002-3743-7404T. Zambarnaya1https://orcid.org/0000-0001-7203-1701Institute of Mathematics and Mathematical Modeling, Kazakhstan, AlmatyInstitute of Mathematics and Mathematical Modeling, Kazakhstan, AlmatyThe paper is aimed at studying the countable spectrum of small linearly ordered theories. The objectives of the research are to study the structural properties of countable linearly ordered theories, as well as to promote the solution to the well-known open problem of model theory, Vaught's conjecture, which assumes that the number of countable models of a countable complete first-order theory cannot be equal to ℵ1. An important step in solving Vaught's conjecture is the search for conditions under which the theory has the maximal number of countable pairwise non-isomorphic models. By limiting ourselves to linearly ordered theories we do not get special advantages from the viewpoint of studying their countable spectrum. Therefore, in the article, a restriction on 1-types and 1-formulas of the theory is introduced. It is proved that a small countable linearly ordered theory that satisfies the restriction and has an infinite discrete chain has the maximal number of countable non-isomorphic models. To build models, the authors use the method of constructing countable models over countable sets, based on the Tarski-Vaught criterion. It is shown that it is possible to carry out the construction in such a way that the types of unnecessary elements in the resulting model are omitted, what guarantees non-isomorphism of the models and their maximal number.https://bm.kaznu.kz/index.php/kaznu/article/view/952/630small theorylinear ordercountable modelnumber of countable modelsdiscrete chainomitting types |
| spellingShingle | B. Baizhanov T. Zambarnaya Infinite discrete chains and the maximal number of countable models Вестник КазНУ. Серия математика, механика, информатика small theory linear order countable model number of countable models discrete chain omitting types |
| title | Infinite discrete chains and the maximal number of countable models |
| title_full | Infinite discrete chains and the maximal number of countable models |
| title_fullStr | Infinite discrete chains and the maximal number of countable models |
| title_full_unstemmed | Infinite discrete chains and the maximal number of countable models |
| title_short | Infinite discrete chains and the maximal number of countable models |
| title_sort | infinite discrete chains and the maximal number of countable models |
| topic | small theory linear order countable model number of countable models discrete chain omitting types |
| url | https://bm.kaznu.kz/index.php/kaznu/article/view/952/630 |
| work_keys_str_mv | AT bbaizhanov infinitediscretechainsandthemaximalnumberofcountablemodels AT tzambarnaya infinitediscretechainsandthemaximalnumberofcountablemodels |