Bonn Mathematical Logic Group

Mengenlehre / Set Theory (V3A4, F4A1)


Lecturer

Time and place

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 2ℵ0 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

Problem sheets

The problem sheets will be uploaded each Wednesday. You may solve the problems and write the solution together with one other participant. Please hand in your solution before the lecture on the following Wednesday (Briefkästen 6 & 7, Endenicher Allee 60). You need to have at least 50% of the total number of points on the problem sheets to be admitted to the final exam.

Tutorials

Exams

Last changed: 17. Oktober 2018