Set Theory and Logic Throughout Mathematics - ALGV19013

• Title: Set Theory and Logic Throughout Mathematics
• Guaranteed by: Department of Logic (21-KLOG)
• Faculty: Faculty of Arts
• Actual: from 2023
• Semester: both
• Points: 4
• E-Credits: 4
• Examination process: oral
• Hours per week, examination: 2/0, Ex [HT]
• Capacity: winter:unknown / unknown (unknown); summer:unknown / unknown (unknown)
• Min. number of students: unlimited
• 4EU+: no
• Virtual mobility / capacity: no
• State of the course: taught
• Language: English
• Teaching methods: full-time
• Teaching methods: full-time
• Note: course can be enrolled in outside the study plan; enabled for web enrollment; you can enroll for the course in winter and in summer semester
• Guarantor: Chris Lambie-Hanson
• Teacher(s): Chris Lambie-Hanson

Annotation

Last update: Chris Lambie-Hanson (23.01.2024)

Set theory and mathematical logic are often studied through the lens of their role in providing a rigorous underpinning for mathematics and metamathematics. In this course, we will consider a slightly less well-known role played by set theory and logic in modern mathematics, surveying a number of instances in which tools and techniques from set theory and logic are used to actually prove new theorems in various fields of pure mathematics, including algebra, analysis, combinatorics, geometry, and voting theory.

Course completion requirements

Last update: Chris Lambie-Hanson (19.02.2024)

To receive credit for the course, all students must complete a small independent research project on an application of set theory or logic not covered in the course. This will then either be presented to the class in a short presentation (5-10 minutes) or written in a short essay.

There will be various homework exercises throughout the course. These are not required, but students are welcome to turn in solutions to be corrected.

The course will have an oral examination. This will include all topics covered in the course, possibly including some of the homework exercises.

Literature

Last update: Chris Lambie-Hanson (19.02.2024)

None of these texts are required for the course, but they provide some additional context for many of the topics that will be covered.

Lecture notes are available at https://github.com/clambiehanson/teaching under the file name lecture_notes.pdf. They will be updated as the semester progresses.

1. T. Jech, The Axiom of Choice. Stud. Logic. Found. Math., vol. 75, North Holland Publishing Company, 1973.

2. I. Goldbring, Ultrafilters Throughout Mathematics. Grad. Stud. Math. vol. 220, American Mathematical Society, 2022.

3. L. Kirby and J. Paris, Accessible independence results for Peano arithmetic, Bull. London Math. Soc., 14(4): 185--293, 1982,

4. P. Komjáth, Three clouds may cover the plane, Ann. Pure Appl. Logic, 109(1-2): 71--75, 2001.

5. P. Eklof, The affinity of set theory and abelian group theory. Rocky Mountain J. Math., 32(4): 1119--1134, 2002.

6. A. Kirman and D. Sondermann, Arrow's theorem, many agents, and invisible dictators. J. Econom. Theory, 5(2): 267--277, 1972.

Syllabus

Last update: Chris Lambie-Hanson (16.02.2024)

Syllabus is provisional and may change based on the background and interests of the students.

1. Applications of transfinite induction and recursion, including constructions of geometrically interesting subsets of Euclidean space

2. Applications of the Axiom of Choice and its relatives: de Bruijn-Erdős theorem, Nielsen-Schreier theorem, existence of non-measurable sets, existence of algebraic closures

3. Applications of ultrafilters and ultraproducts: Arrow's Impossibility Theorem, Ax-Grothendieck theorem

4. Further applications to abelian group theory: Constructions of almost free nonfree groups, slender groups

5. Infinite games

Entry requirements

Last update: Chris Lambie-Hanson (23.01.2024)

Students are expected to have basic knowledge of set theory and logic, equivalent to a one-semester introductory course in each. No other mathematical background knowledge will be assumed.