NMAK13004U Axiomatic Set Theory and Forcing

Volume 2013/2014
Education
MSc Programme in Mathematics
Content
The language of set theory and formulas. The axioms of ZFC and the hierarchy of hereditary sets. Ordinals and cardinals, Cantor's continuum hypothesis. Classes and recursion. Relative consistency. Posets, filters, names, countable transitive models, generics and generic extensions. The forcing language and the forcing relation. Consistency of the negation of the Continnum Hypothesis.
Learning Outcome
Knowledge: The student is expected to gain a basic understanding of: The goals of axiomatizing set theory; the role of set theory as a foundational theory for mathematics; the role of relative consistency proofs and why relative consistency is the best we can hope for; the notion of independence. Additionally, the student should gain an understanding of what kind of problems in mathematics may be shown to be independent of the axioms of ZFC. 

Skills: At the end of the course, the student must be able to account for the axioms of ZFC, must be able to explain the notions of cardinals and ordinals, and must be able to state and explain the Continuum Hypothesis (CH). The student should be able to define the notions of a poset, a dense set, a filter, a generic filter, a countable transtitive model, a name, and the forcing relation. The student should be able to use forcing extensions to prove the consistency of the negation of CH.

Competences: The primary competence added by this course is that the student will learn to use the method of forcing to prove independence results, and be able to account for the basic strategy behind independence proofs using forcing.
Introduction to Mathematical Logic or a similar logic course.
4 hour lecture + 2 hours of tutorials per week.
  • Category
  • Hours
  • Colloquia
  • 27
  • Guidance
  • 5
  • Lectures
  • 36
  • Preparation
  • 90
  • Project work
  • 30
  • Theory exercises
  • 18
  • Total
  • 206
Credit
7,5 ECTS
Type of assessment
Written assignment
Continuing evaluation based on three problem sets graded on the 7-point scale.
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner.
Re-exam
30 minutes oral exam with preparation time. All aids allowed during the preparation time, no aids allowed during the examination.
Criteria for exam assesment

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