NMAK23010U Logic in Analysis and Topology

Volume 2023/2024
Education

MSc Programme in Mathematics
MSc Programme in Mathematics with a minor subject

Content

This course is an introduction to how basic concepts in logic and set theory connect to general topology (e.g., metric spaces) and analysis. Specifically, we will introduce:

  • The metric space X_L of countable models of a countable first order language L.
  • Logic with infinitary conjunction ("and") and disjunction ("or"), which will give us a new language called L_{\omega_1,\omega}, whose formulas can express more than formulas in ordinary first order logic can.
  • The use of ordinals in logic and analysis.
  • We will introduce Borel sets, which you were already introduced to briefly in measure theory and analysis courses, and we will analyse and describe the Borel sets using ordinals and the formulas of the language L_{\omega_1,\omega}.
  • The "logic action" on the space X_L, and Scott's analysis of isomorphism of countable models of a countable language.
  • The "Baire Category Theorem", and its applications in logic and model theory, such as the "Omitting Types Theorem".
  • The "Baire Property" of sets.
  • Analytic sets and co-analytic sets and their basic theory.
  • If time allows, we may also briefly discuss "Continuous Logic", where truth values between 0 and 1 (true and false) are permitted.
Learning Outcome
  • Knowledge: To display knowledge of the course topics and content.
  • Skills: The aim is that, upon completing the course, the student should have the skills that would allow the student to read a short research paper in this field of mathematics.
  • Competences: The student should be able to apply the theory to solve problems of moderate difficulty within the topics of the course. 

Examples of literature:

Lecture notes will be provided for some topics.

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

  • D. Marker: Model Theory: An Introduction (Springer, GTM 217)
  • A. Kechris: Classical Descriptive Set Theory (Springer, GTM 156).
  • K. Kunen: Set Theory (North Holland, 1980 edition is preferred over the newer, revised version).
It is strongly recommended that you have already taken the course "NMAA13036U Introduction to Mathematical Logic", or a similar course (covering the same material) in first order logic and set theory.

Additionally, it is strongly recommended that you have already taken a course such as "NMAA04016U Analysis 1" covering the basics of metric spaces, and a course such as "Lebesgue Integral and Measure theory" covering the basics of measure theory.


Overall, academic qualifications equivalent to a BSc degree are recommended.
The course will be taught in a "flipped classroom" format for the first 5 weeks of the block. This means that each week, the students will be given 2 screencast video lectures (each video lecture approx. 45 minutes), and these videos will be the basis of an in-person traditional double (2 hour) lecture in a classroom. The students will also be given tasks and questions related to the videos (i.e., the screencast lectures) to prepare for the in-class discussion. There will also be 2 hours of exercises per week for the first 5 weeks of the block. During this phase of the course, there will be 1 mandatory homework assignment worth 45% of the course grade. The expected workload on this homework assignment is 20 hours.

For the remaining 3 weeks of the block, each student will write a short independent project on a topic related to the course material. The choice of topic for the independent project will be made in agreement with the lecturer. The course lecturer may set certain exercises or problems to be a required part of the independent project, if this is deemed relevant by the course lecturer. The lecturer and tutor (exercise instruktor) will be available for consultation for 1 hour/week each during the project phase of the course.
  • Category
  • Hours
  • Lectures
  • 10
  • Preparation
  • 106
  • Theory exercises
  • 10
  • Project work
  • 54
  • Guidance
  • 6
  • Exam
  • 20
  • Total
  • 206
Continuous feedback during the course of the semester
Credit
7,5 ECTS
Type of assessment
Continuous assessment
Type of assessment details
Continuing evaluation based on 1 problem set (assignment), and 1 written project. Towards the final grade, the problem set counts 45%, and the written project counts 55%.
Aid
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner
Re-exam

25 min oral examination, no preparation time. During the exam, the student is allowed to consult a note the student has made, of length no longer than 1 A4 page per exam question/topic, briefly at the beginning after drawing an exam question. No other material may be used of consulted during the exam.

Criteria for exam assesment

The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.