Vaught’s conjecture for quite o-minimal theories. (English) Zbl 1454.03046
Summary: We study Vaught’s problem for quite o-minimal theories. Quite o-minimal theories form a subclass of the class of weakly o-minimal theories preserving a series of properties of o-minimal theories. We investigate quite o-minimal theories having fewer than \(2^\omega\) countable models and prove that the Exchange Principle for algebraic closure holds in any model of such a theory and also we prove binarity of these theories. The main result of the paper is that any quite o-minimal theory has either \(2^\omega\) countable models or \(6^a 3^b\) countable models, where \(a\) and \(b\) are natural numbers.
MSC:
| 03C64 | Model theory of ordered structures; o-minimality |
| 03C15 | Model theory of denumerable and separable structures |
| 03C07 | Basic properties of first-order languages and structures |
| 03C50 | Models with special properties (saturated, rigid, etc.) |
Keywords:
weak o-minimality; quite o-minimal theory; Vaught’s conjecture; countable model; binary theoryReferences:
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