Prof. Dr. Moritz Weber

Set theory and forcing

(Summer term 2019)


For the oral exams, please write an email to Moritz Weber and fix a date.
See below for some information on the material covered in the exams.
The following dates are proposed (but individual dates are also possible):

Time and place of the lecture

Tuesday, 14:00-15:30, HS II (building E2 5)

Part I: Introduction to logical calculus and Gödel's incompleteness theorems Part II: Introduction to set theory and forcing

Exercise sessions

Tuesday, 15:45-17:15, SR6 (room 217, building E2 4)


Set theory is the attempt to axiomatize mathematics and to put it on a solid basis; this system of axioms is usually called ZFC (Zermelo-Fraenkel plus the aciom of choice). It is a central theory in mathematical logic. The main issue is to define what a set actually is, or in other words, to answer the question: Which sets are allowed to be sets?

Forcing is a very powerful tool from set theory with which we may "force" certain sets to be sets. With the help of forcing, we can create a universe of sets in which ZFC and certain additional axioms or statements hold true. A famous example of the use of forcing is the proof that the Continuum Hypothesis (CH) is independent from ZFC (recall that the Continuum Hypothesis roughly states that the cardinality of the power set of the natural numbers is the next biggest cardinality after the natural numbers): There is a universe of sets satisfying ZFC and CH; but there is also another universe of sets satisfying ZFC and the negation of CH. We conclude: We may not deduce CH from ZFC; hence any attempt to "prove" CH (or to disprove it) must necessarily fail.

In the meantime, many other very concrete statements from nowadays mathematics have been proven to be independent from ZFC (like the existence of outer automorphism of the Calkin algebra). So, before trying to prove a very hard problem in mathematics, we might better check whether it is independent from ZFC.

The lecture is held in English. This lecture basically does not require any mathematical background and is somewhat independent from the usual curriculum. Note however, that some of the proofs will be only sketched if not omitted and parts of the lecture might require some background reading by the participants. This style of a lecture is probably quite ambitious for students from the first year.

Oral exams

Here are some questions and topics for the oral exams (20-30 min) in set theory:


Semesterapparat (library)
Link to the lecture Set Theory and Logic by Gunter Fuchs including some lecture notes

Last update: 13 Sept 2019   Moritz Weber Impressum