v Models of set theory I, Summer 2013

## Models of set theory (V4A4)

### Time and place

• Lecture: Monday 14.15-16.00 (1.008) and Wednesday 12.30-14.00 (Kleiner Hörsaal).

### Contents

Zermelo-Fraenkel set theory (ZF) is effectively axiomatized in the language of first-order predicate logic. It is strong enough to serve as a foundation for all of mathematics. This allows the formulation of self-reflective statements and diagonalizations which imply that the Zermelo-Fraenkel axioms are incomplete. The incompleteness phenomenon is not restricted to peculiar statements of the type "this statement is not provable" but it affects concrete set theoretic principles like the axiom of choice (AC) and the continuum hypothesis (CH). The independence of the continuum hypothesis from the Zermelo-Fraenkel axioms is shown by constructing models of the theories ZF + AC + CH and ZF + AC + non-CH.

Set theory/set theory problems In the course we construct transitive models of set theory by the forcing method: starting from a given model, new models are obtained by a set-theoretic adjunction of "generic" objects. This process is vaguely reminiscent of the the adjunction of transcendental numbers to fields. We shall study various forcing construction mainly directed towards independency in cardinal arithmetic. We shall then introduce symmetry arguments and obtain submodels of the generic extensions in which the axiom of choice is false. References will be given. Lecture notes will be made available.

Lecture notes, 07/09/2013.

### 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 Lücke. The problem sheets will be uploaded weekly. You may solve the problems and write the solution (in English or German) together with one other person.
You should sign up for the tutorial in the first lecture on Monday, April 08.

### Exam

The exam will take place on Wednesday, 31.07.2013, 10 a.m., in Kleiner Hörsaal, Wegelerstrasse. Please be there 5 minutes early.

Last changed: 09 July 2013