Axiomatic Set Theory

6 EC

Semester 2, period 4, 5

5122AXVE6Y

Owner Bachelor Wiskunde
Coordinator N. Bezhanishvili
Part of Minor Logic and Computation, year 1Exchange Programme Faculty of Science, specialisation BSc Mathematics, year 1Bachelor Wiskunde, year 3

Course manual 2018/2019

Course content

Set Theory as a Foundations of Mathematics

Axioms of Set Theory 

Ordinal Numbers

Cardinal Numbers

Cardinal and ordinal arithmetic

Axiom of Choice

Axiom of Foundation

Basics of some additional topics such as large cardinals, constructible universe and the consistency of Continuum Hypothesis, absoluteness, non-wellfounded sets and the Anti-Foundation Axiom.

Study materials

Literature

  • Keith Devlin, 'The Joy of Sets', Springer Verlag, 1993, second edition.

     

  • R. M. Smullyan and M. Fitting, Set Theory and the Continuum Problem, Dover Publications, Inc., 2010.

Objectives

At the end of the course the student should be able to:

  • derive the existence of some simple operations on sets from the axioms of set theory.

  • show properties of ordinals using the method of transfinite induction.

  • make computations with ordinal and cardinal numbers using (infinite) sums, multiplication, and exponentiation.

  • conduct proofs about ordinal and cardinal numbers using the equivalent statements of the Axiom of Choice, such as Zorn's lemma, Koenig's lemma, and Zermelo's theorem.

  • solve basic problems of cardinal arithmetic involving singular and regular cardinals.

  • work with the cumulative hierarchy of sets using the Axiom of Foundation.

     

Teaching methods

  • Hoorcollege
  • Werkcollege
  • Lecture
  • Seminar
  • Self-study

The course is taught in English.

Learning activities

Activiteit

Aantal uur

Hoorcollege

30

Tentamen

3

Werkcollege

26

Zelfstudie

109

Attendance

Programme's requirements concerning attendance (OER-B):

  • Each student is expected to actively participate in the course for which he/she is registered.
  • If a student can not be present due to personal circumstances with a compulsory part of the programme, he / she must report this as quickly as possible in writing to the relevant lecturer and study advisor.
  • It is not allowed to miss obligatory parts of the programme's component if there is no case of circumstances beyond one's control.
  • In case of participating qualitatively or quantitatively insufficiently, the examiner can expel a student from further participation in the programme's component or a part of that component. Conditions for sufficient participation are stated in advance in the course manual and on Canvas.
  • In the first and second year, a student should be present in at least 80% of the seminars and tutor groups. Moreover, participation to midterm tests and obligatory homework is required. If the student does not comply with these obligations, the student is expelled from the resit of this course. In case of personal circumstances, as described in OER-A Article 6.4, an other arrangement will be proposed in consultation with the study advisor.

Assessment

Item and weight Details

Final grade

30%

Tussentoets

50%

Tentamen

20%

Homework

The deadlines for homeworks are strict, no delays are allowed.

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

Weeknummer Onderwerpen Studiestof
1 History of Set Theory  
2 More History. Naive Set Theory. Paradoxes  
3 Naive Set Theory Continued. Axioms of ZFC.  
4 More ZFC. Classes. Ordinals.  
5 Recursion on ordinals. Axiom of Choice (AC) and Well-Ordering Theorem  
6 Ordinal Arithmetic.  
7 Continuous Functions, Fixed Point Theorem, Normal Forms. Applications.  
8 Cardinal Arithmetic. Schroder-Bernstein Theorem.  
9 Cardinal Exponentiation. Generalized Continuum Hypothesis (GCH).  
10 Large Cardinals: Inaccessible cardinals and Models of Set Theory.  
11 Review Other Topics: Boolean algebras, topologies, measure algebras.  
12 Trees, Konig Tree Lemma. Applications to Large Cardinals.  
13 Godel's Constructible Universe. Absoluteness. Montague-Levy Reflection Theorem.  
14 First Order Universes, and the (relative) consistency of AC.  
15 Tarski-Vaught Theorem, MSTV Theorem and the (relative) consistency of GCH.  
16 The Ideas behind the Independence Proofs for AC and GCH. Review.  

Timetable

The schedule for this course is published on DataNose.

Honours information

There is no honors extension for this course.

Additional information

Recommended prior knowledge: Mathematical maturity, decent understanding of first-order logic.

Processed course evaluations

Below you will find the adjustments in the course design in response to the course evaluations.

Contact information

Coordinator

  • N. Bezhanishvili