(in Polish) Practice and Philosophy of Model Theory 3800-PPMT26-M
Model theory can be briefly described as a study of relationships between classes of structures defined axiomatically in a formal language and mathematical properties of the whole classes and particular structures. In two recently published books the authors (Baldwin, and Button and Walsh), describe not only recent progress, but also what John Baldwin calls the paradigm shift: model theory became a useful tool in formulating and studying important classification problems that might have otherwise be hidden to mathematicians. The course is an introduction to model theory with the aim of providing enough content to enable a discussion about major achievements and their relevance in foundations of mathematics and in its philosophy. Selected topics will be presented at a level that does not require advanced background in set theory. Complete proofs of a few fundamental results will be given for slightly simplified versions. Instead of full generality, the focus will be on applications illustrated by examples. A slogan in model theory is that to know a structure is to understand and classify all functions and relations that are definable in it. My aim is to fully explain what this means and to introduce model theoretic tools used to study definability in mathematical structures. After this introduction, the rest of the course will be devoted to a discussion of some philosophical aspects that are brought up in the books of Baldwin and Button and Walsh. All material will be presented in a historical perspective.
The topics will include:
- Tarski’s definition of satisfaction, truth, and the notion of semantic (model-theoretic) inference.
- The compactness theorem and its applications.
- First-order theories, elementary classes, elementary equivalence, isomorphisms and automorphisms of relational structures.
- Basic facts about definability in the standard model of arithmetic and in its reducts and expansions.
- Tarski’s theorem on undefinability of truth.
- Counting the number of models of complete first-order theories. Categoricity of theories and structures.
- Infinitary extensions of first-order logic.
- (If time permits) Abstract elementary classes.
Course coordinators
Type of course
Learning outcomes
Upon completing the course, students will become familiar with basic concepts, terminology, and fundamental results in modern model theory. They will also learn about the methodological framework and the history of the discipline.
Students will learn how to express properties of mathematical structures in formal first-order languages and how to prove that certain properties of a structure are not first-order. They will learn how to use model-theoretic tools to recognize whether two given structures are isomorphic, and, in the case when they are not, how to measure their similarity with respect to their first-order properties. They will also learn how basic model-theoretic tools are used for counting the number of isomorphism types of models of first-order theories.
The student:
- Is able to select and organize information obtained during communication process
- Is able to follow the train of thoughts of others.
- Is able to listen attentively to others.
Assessment criteria
The final assessment of studnents’ achievment will be based on their activity during classes and the quality of written assignments submitted on time
Number of absences 2
Bibliography
- Tim Button and Sean Walsh, “Philosophy and Model Theory”, Oxford University Press, 2018
- Roman Kossak, “Mathematical logic—on numbers, sets, structures, and symmetry”. Second edition Springer Graduate Texts in Philosophy, Springer, 2024.
- John T. Baldwin, “Model Theory and the Philosophy of Mathematical Practice: Formalization without Foundationalism”: Cambridge University Press, 2018 (supplementary reading).
- Ian Hacking, “Why is There Philosophy of Mathematics At All?”, Cambridge University Press, 2014. (supplementary reading).
-Paolo Mancosu, “Abstraction and Infinity”, Oxford University Press, 2016. (supplementary reading).
Notes
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Term 2026Z:
None |