Mathematical Logic 2
A.Y. 2024/2025
Learning objectives
The course aims to provide students with an understanding of the most important general aspects of Model Theory,
so as to allow a possible continuation of Logic studies in more specific directions.
so as to allow a possible continuation of Logic studies in more specific directions.
Expected learning outcomes
At the end of the course the student should be able to recognize the models of a logical theory, identifying those with further
specific properties. He/she will also need to be able to abstract the logical structure from explicit structures. Learning the general topics of Model Theory, at the end of the course the student will be able to continue with the independent study of more specific topics of Mathematical Logic.
specific properties. He/she will also need to be able to abstract the logical structure from explicit structures. Learning the general topics of Model Theory, at the end of the course the student will be able to continue with the independent study of more specific topics of Mathematical Logic.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
The course will focus on classical topics in model theory. In particular:
- introduction to the concepts of structure, model, elementary embedding, elementary equivalence, isomorphism of structures, definable sets, Tarski-Vaughn criterion;
- Lowenheim-Skolem theorems;
- Back and Forth, Scott isomorphism theorem;
- Ultraproducts and elementary extensions, Los Theorem;
- Elimination of quantifiers and related topics (model completeness, o-minimality, separation between theories) with examples and applications (infinite sets, algebraically closed fields, closed real fields, Nullstellensatz and Artin's Theorem);
- Topological space of types, their realization and omission, saturated models, prime models, atomic models.
- introduction to the concepts of structure, model, elementary embedding, elementary equivalence, isomorphism of structures, definable sets, Tarski-Vaughn criterion;
- Lowenheim-Skolem theorems;
- Back and Forth, Scott isomorphism theorem;
- Ultraproducts and elementary extensions, Los Theorem;
- Elimination of quantifiers and related topics (model completeness, o-minimality, separation between theories) with examples and applications (infinite sets, algebraically closed fields, closed real fields, Nullstellensatz and Artin's Theorem);
- Topological space of types, their realization and omission, saturated models, prime models, atomic models.
Prerequisites for admission
Previous knowledge of the basic relationships between first-order syntax and semantics (standard topics of "Logica Matematica 1") is recommended, although not strictly necessary.
A minimum prior knowledge of ordinal numbers, transfinite induction and cardinality theory is recommended, but not necessary, as is some basic knowledge of algebra notions (groups, rings, fields).
A minimum prior knowledge of ordinal numbers, transfinite induction and cardinality theory is recommended, but not necessary, as is some basic knowledge of algebra notions (groups, rings, fields).
Teaching methods
Frontal lessons through screen projection of what is written on a graphic tablet. Notes from each lesson will be shared on Ariel.
Various exercises will be left to be carried out optionally to test the achieved level of comprehension autonomously.
Various exercises will be left to be carried out optionally to test the achieved level of comprehension autonomously.
Teaching Resources
On the Ariel page of the course, the following will be uploaded:
- complete notes of the course;
- the notes of each lesson;
- a workbook;
- (minimal) notes on logical and set theory prerequisites.
Although it will not be followed exactly, the suggested textbook for the course is "Model Theory: an Introduction" by David Marker.
- complete notes of the course;
- the notes of each lesson;
- a workbook;
- (minimal) notes on logical and set theory prerequisites.
Although it will not be followed exactly, the suggested textbook for the course is "Model Theory: an Introduction" by David Marker.
Assessment methods and Criteria
The exam will consist of an oral in which both theory and exercises will be asked. If the student requests it, part of the oral can be replaced by a seminar on topics related to the course, to be agreed upon in advance with the teacher.
Students will be able to test the level of learning achieved during the course by solving some assigned exercises.
Students will be able to test the level of learning achieved during the course by solving some assigned exercises.
MAT/01 - MATHEMATICAL LOGIC - University credits: 6
Lessons: 42 hours
Professor:
Luperi Baglini Lorenzo
Shifts:
Turno
Professor:
Luperi Baglini LorenzoProfessor(s)