# Topics in Mathematical Logic

### Course content

Topics that may be covered are: Axiomatic set theory, ordinals, cardinals. Basic structure of the set theoretic universe V. Gödel's constructible universe L and equiconsistency. Infinitary combinatorics. Descriptive set theory, including analysis of Borel sets, analytic sets, and if time allows, descriptive set theory in L. Model theory. Recursion (computability) theory.

Education

MSc Programme in Mathematics

MSc Programme in Mathematics w. a minor subject

Learning outcome

Knowledge: The student should, by the end of the course, know the axioms of set theory, ordinals, cardinals, and the struture of the set theoretic universe V. The student should know the construction of the model L, as well as important combinatorial principles that are true in L, such as the Continuum Hypothesis. The student should know what Borel and analytic sets are, and what properties these sets have, and should know how to prove basic theorems about these types of sets.

Skills: The student should be able to apply set theoretic concepts and result mentioned in the previous paragraph to account for the structure of the universe V, the structure of the constructible universe L, the special combinatorial principles that hold in L, and to account for the structure of Borel and analytic sets.

Competences:  The student should be able to formulate the main results of the course, check whether they are applicable in a concrete problem and use them to solve it.

4 hours of lectures/week + 2 hours of exercises per week for the first 5 weeks. Then 3 weeks of project work.

Examples of literature:

Lecture notes will be provided for some topics.

For other topics, we might use parts of the following examples of course literature:

A. Kechris: Classical Descriptive Set Theory (Springer. Note that this book is available as a pdf for free from the Springer website.)

K. Kunen: Set Theory (North Holland)

D. Marker: Model Theory (Springer)

S. Soare: Recursively enumerable sets and degrees.

Introductory mathematical logic. Some basic knowledge of general topology and measure theory may be required for some topics.

Academic qualifications equivalent to a BSc degree is recommended.

ECTS
7,5 ECTS
Type of assessment
Continuous assessment
Continuing evaluation based on 1 problem set, and 1 written project.
Aid
All aids allowed
Marking scale
Censorship form
No external censorship
One internal examiner
##### Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

Single subject courses (day)

• Category
• Hours
• Lectures
• 20
• Theory exercises
• 12
• Preparation
• 100
• Project work
• 54
• Exam
• 20
• English
• 206