v Set Theory, Winter 2016-17

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

### Time and place

• Lecture: Monday 14.15-16.00 and Friday 14.15-16.00, 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; infinitary combinatorics and large cardinals.

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.

### 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. The problem sheets will be uploaded each Friday, beginning October 21, and should be handed in before the lecture on the following Friday. You may solve the problems and write the solution (in English or German) together with two other people.

Tutorials by Sebastian Gurke (Thursday 08.15-10.00, Thursday 10.15-12.00) and Andreas Lietz (Tuesday 10.15-12.00), in room N0.008.

### Exam

The minimum percentage of points for participating in the exam is 45%. The exam took place on Friday,03.03.2017, 9.00-11.30, in Grosser Hörsaal, Wegelerstrasse 10.

The Klausureinsicht wil tke place in the Hausdorffraum 1.012 on Tuesday, 07.03., 14.00-15.00. The grades will be available on basis on Monday, 06.03.

The second exam will take place Thursday, 23.03.2017, 9.00-11.30 in Grosser Hörsaal, Wegelerstrasse 10. Please be there at 8.55.

The Klausureinsicht wil tke place in the Hausdorffraum 1.012 on Monday, 26.03., 14.00-14.30. The grades will be available on basis on Friday, 24.03.