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Model theory of continuous structures
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Model theory of continuous structures
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Academic year 2023/2024
- Teacher
- Domenico Zambella
- Teaching period
- Sept-Dec
- Type
- Basic
- Credits/Recognition
- 2
- Course disciplinary sector (SSD)
- MAT/01 - mathematical logic
- Delivery
- Formal authority
- Language
- English
- Attendance
- Obligatory
- Type of examination
- Practice test
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Sommario del corso
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Program
Hours: 10
Let L be a first-order 2-sorted language. Let X be some fixed structure. A standard structure is an L-structure of the form ⟨M,X⟩. When X is a compact topological space (and L meets a few additional requirements) it is possible to adapt a significant part of model theory to the class of standard structures. In the last 20 years the most popular approach uses real-valued logic (Ben Yaacov, Berenstein, Henson, Usvyatsov). In this course we present a different, more general, approach which only uses classical logic. This is based on three facts:
- Every standard structure has a positive elementary extension that is standard and realizes all positive types that are finitely consistent.
- In a sufficiently saturated structure, the negation of a positive formula is equivalent to an infinite disjunction of positive formulas.
- There is a pure model theoretic notion that corresponds to Cauchy completeness.
To exemplify how this setting applies to model theory we discuss ω-categoricity and (local) stability. We will revisit the classical theory and compare it with the continuous case.
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Course delivery
First term
Suggested readings and bibliography
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- Other
- Title:
- Standard analysis
- URL:
- Required:
- No
- Enroll
- Open
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