### General

**Set Theory and Forcing, Fall 2014, 7.5 hp**

The course treats modern set theory and independence results. Many statements of mathematics are neither provable nor disprovable from the basic axioms of set theory, they are independent of the axioms. The most famous one is Cantor's Continuum Hypothesis. The course gives an introduction to methods for proving independence results, in particular Cohen's method of forcing.

**Contents**

Classical set theory: Axioms of Zermelo-Fraenkel (ZF) set theory. Ordinal numbers, well-orderings and cardinal number theory in ZF. The Continuum Hypotheses. Independence results for ZF: permutation models and independence of the axiom of choice. Forcing and the independence of the Continuum. Boolean-valued models. Consequences of independence results for mathematics. A selection of the following subjects: Infinitary combinatorics, Gödel's constructible sets and the V=L axiom. Alternative axioms: Projective Determinacy and Martin's axiom. Large cardinals. Constructive set theory: CZF and IZF.

**Course literature**

John L. Bell. Boolean-Valued Models and Independence Proofs. Third Edition. Oxford 2005. (E-book)

Peter Aczel and Michael Rathjen. Notes on Constructive Set Theory. Institut Mittag-Leffler 2001. www.mittag-leffler.se/preprints/files/IML-0001-40.pdf

Yiannis N. Moschovakis. Notes on Set Theory. 2nd ed. Springer 2006. (E-book)

Thomas Jech, Set Theory, Third edition. Springer 2000. (E-book)

**Reference literature**

Kenneth Kunen, Set Theory – an Introduction to Independence Proofs. North-Holland 1980.

**Note:** The "E-books" above can accessed through the Stockholm University Library (www.sub.su.se) provided you have library card (which all students registered at SU can get).

**Teacher**

Palmgren, Erik

**Language:** The course will be given in **English** if there is interest in this.

**Examination:** Home work problems through out the course (75% of total score) and a short written exam (25% of total score) at the end of the course. See grading criteria below. The home work problems will be posted below, marked **Home Work.**

**Written exam:** Tuesday 16 December, 15:15 - 17:15 , Room 22, Building 5, Kräftriket (Department of Mathematics).

The written exam is marked. The last set of home work problems will be marked before January 7.

Schedule: