Set Theory - M1 - 8EC

Aim of the course:

The course is a combination of an introductory and an advanced course in axiomatic set theory. The  introductory section follows a typical structure and covers the following topics: 
• Naive set theory, the ZFC axioms, sets and classes
• Reconstruction of mathematics in set theory
• Buildup of natural numbers, induction and recursion
• Well-founded relations, ordinals and ordinal arithmetic
• The Axiom of Choice and equivalent principles
• Cardinals and cardinal arithmetic.


The advanced section is a selection of different topics each year. This time, we will cover model-theoretic aspects of set theory, Large Cardinals, and infinitary combinatorics. 
The aim of the course is to provide students with a solid knowledge of axiomatic set theory and to prepare them for research in set theory, as well as using set theory as a tool in mathematical areas such as general topology, algebra and functional analysis.

Prerequisites
We will assume familiarity with basic notions and results from mathematical logic, specifically the syntax and semantics of predicate logic, soundness, completeness and compactness. Additional knowledge of model theory, such as Löwenheim-Skolem theorems, elementary submodels and Łoš’s Theorem, will be beneficial but is not required.
Other than that, very little concrete knowledge is needed, other than the naive use of sets that is very common in mathematics, and a general mathematical maturity. It is possible for students from non-mathematical backgrounds (e.g., philosophy, computer science) to take this course, provided that they have some experience in writing proofs and are able to adapt to the mathematical style and pace.

Homework

There will be six hand-in homework assignments, with weight 5% each, i.e., for a total of 30%, to be submitted in pairs (online via Canvas). Homework will be checked and feedback provided, and solutions will be posted after the  deadline.

Exam
The final exam, as well as the resit, is a 3-hour long paper exam, with a total weight of 70%. The resit replaces the final exam but not the grade for the hand-in homework (this follows standard practice at the Master of Logic programme). Students can take one A4 sheet written on both sides (with hand-writing) to the exam, but other sources are not allowed.
A minimum grade of 5.0 on the final exam/resit is required to pass the course (as well as a minimum 5.5 overall grade).


Literature
For the introductory part of the course, we will mostly use • Hrbacek & Jech, Introduction to set theory (3rd rev. ed.). For the advanced part, we will follow the two books:
• Kanamori, The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Springer-Verlag.
• Jech, Set Theory (The Millennium Edition). Springer, 2003.
Additional supplementary material will be provided. The lectures will be written using a digital tablet and the notes created during lectures will be uploaded, serving as additional study and revision material.