v Set Theory, Winter 2012-13

## Set theory (V3A4 Bachelor studies; F4A1 Master studies)

### Time and place

• Lecture: Monday 14.15-16.00 and Wednesday 13.15-15.00, both in kleiner Hörsaal, Wegelerstraße 10.

### Contents

Sets are ubiquitous in present-day mathematics. Basic structures are introduced as sets of objects with certain properties. Fundamental notions like numbers, relations, functions and sequences can be defined from sets. Set theory, together with formal logic, is thus able to provide a universally accepted foundation for mathematics.

Set theory also comprizes a theory of the (mathematical) infinite through the study of infinite sets and their combinatorics. Generalizing the finitary arithmetical operations leads to an infinitary arithmetic of cardinal numbers which has surprising properties. For the smallest infinite cardinal ℵ0 which is the cardinality of the set of natural numbers we have: ℵ0+ℵ0 = ℵ0, ℵ0xℵ0 = ℵ0, whereas the value of 20 is (provably!) undetermined by the common principles of set theory.

The lecture course Set Theory will cover the following basic material: The Zermelo-Fraenkel axioms of set theory; relations, functions, structures; ordinal numbers, induction, recursion, ordinal arithmetic; number systems: natural, integer, rational, real numbers; the axiom of choice and equivalent principles; cardinal numbers and cardinal arithmetic; sets of real numbers, Borel sets, projective sets, regularity properties. Further topics are: the 100 year old Hausdorff paradox which gives an impressive example of a set which is not Lebesgue-measurable; infinitary combinatorics.

The initial development of Zermelo-Fraenkel set theory is rather canonical and is portrayed in similar ways in many books on set theory; references will be given. Lecture notes will be made available.

Lecture notes, 24.01.2013

### Tutorials

You need to have at least 50% of the total number of points on the problem sheets to participate in the exam.

Problem sheets written by Philipp Schlicht (office hour Monday 10-11). The problem sheets will be uploaded each Monday, beginning October 08, and should be handed in before the lecture on the following Monday. You may solve the problems and write the solution (in English or German) together with one other person.

Tutorials by Ronja Reese (Thursday 10.15-12.00, seminar room N0.003, Mathematik-Zentrum, Endenicher Allee 60) and Julian Schlöder (Thursday 16.15-18.00, seminar room 1.007, Mathematik-Zentrum, Endenicher Allee 60). The tutorials begin in the first week of the semester. You should sign up for one of the tutorials in the first lecture on Monday, October 08.

Supplementary material provided by Julian Schlöder and Ronja Reese (disclaimer: no guarantee for correctness): Ordinal arithmetic, Problem 37, Problem 44, Easy training exercises for the exam.

### Exam

The exam will take place on Thursday, 21.02.2013, 9.00-11.00, in Kleiner Hörsaal, Wegelerstrasse (same as for the lectures). Please be there 5 minutes early.

The grades for the exam are available on basis. You have passed the exam if you have at least 40 out of 80 points. Klausureinsicht (possibility of looking at your exam) is on Wednesday, March 6, 14.00-15.00, seminar room 0.006 and you can also come to Philipp Schlicht's office 4.003 on Thursday, March 7, 14.00-14.30 (otherwise please make an appointment with Philipp Schlicht for Friday, March 8 by email: schlicht@math.uni-bonn.de).

The second exam will take place on Tuesday, 19.03.2013, 9.00-11.00, in the seminar room 0.011, Mathematisches Institut. Please be there 5 minutes early. The grades will be posted on basis on the day of the exam. Klausureinsicht for the second exam (possibility of looking at your graded exam) is also on Tuesday, 19.03.2013, 15.00-15.30, in the seminar room 0.011.

Last changed: 05 March 2013