Rudiments of Axiomatic Set Theory

6 EC

Semester 2, period 4, 5

5314RAST6Y

Owner Master Logic
Coordinator dr. Y.D. Khomskii
Part of Master Logic, Master Logic, track Logic and Mathematics, Master Logic, track Logic and Computation, Master Logic, track Logic and Philosophy, Master Logic, track Logic and Language, Master Logic, track Logic Year, year 1Master Logic, track Exchange, year 1

Course manual 2024/2025

Course content

This course provides an introduction to Axiomatic Set Theory comparable with standard courses of this type. Topics covered are:

  • Naïve set theory and Russel's Paradox
  • The ZFC axioms
  • Construction of basic mathematical objects in ZFC
  • Construction of the natural numbers and other number systems
  • Ordinals and well-orders
  • Ordinal arithmetic
  • Transfinite induction and recursion
  • The  von Neumann Hierarchy and ranks of sets
  • Cardinality
  • The Axiom of Choice
  • Cardinals and cardinal arithmetic
  • Cofinality, regular and singular cardinals
  • Inacessible cardinals and a brief introduction to "models of set theory"

Study materials

Literature

    • Azriel Levý, “Basic Set Theory”, Dover Books on Mathematics, Revised Edition 2002
    • Keith Devlin, “The Joy of Sets”, Fundamentals of Contemporary Set Theory 1993.
      (In the first lecture you can find out more about obtaining these textbooks).

Other

  • Slides created during lectures are posted on Canvas for study and revision

Objectives

  • Understanding the main issues in the foundation of mathematics, the problems with naive set theory, and the solution to these problems provided by the Zermelo-Fraenkel Axioms. Understanding the important of formalizing mathematics in first order logic and the distinction between sets and proper classes
  • Applying the ZFC axioms to formally construct the objects used in mathematics and to prove foundational theorems on which the rest of mathematics is built
  • Understanding more advanced concepts of abstract set theory, such as ordinals, cardinals and the Axiom of Choice.

Teaching methods

  • Lecture
  • Seminar
  • Self-study

Weekly lectures are used to present the material, while weekly exercise sessions are used for students to practice and receive individual feedback. Six longer homework assignments are submitted in pairs, with feedback provided to the students in online comments, and solutions are also made public.

Learning activities

Activity

Hours

Hoorcollege

28

Tentamen digitaal

2

Werkcollege

14

Self study

124

Total

168

(6 EC x 28 uur)

Attendance

This programme does not have requirements concerning attendance (TER-B).

Assessment

Item and weight Details

Final grade

1 (100%)

Deeltoets

The mid-term exam is counted for 30% and final exam for 50% of the final grade, with homework submissions counting for 20%. For exams students are allowed to take hand-written sheets of notes with them. Exams are written on paper and graded manually. Missed assignments are graded as 0%, unless there are exceptional circumstances. Solutions for the homework assignments and exams are posted on Canvas.

Inspection of assessed work

The homework feedback is visible to the students digitally on Canvas. For the mid-term and final exam, students have the opportunity to view their work and compare it to posted solutions.

Assignments

The assessed assignments are submitted in pairs.

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

WeeknummerOnderwerpenStudiestof
1
2
3
4
5
6
7
8

Contact information

Coordinator

  • dr. Y.D. Khomskii