## Set Theory - SET

Instructor: Dr. Lajos SOUKUP

Text: The course is based on printed handouts distributed in class.

Prerequisite: Some familiarity with "higher" mathematics. No specific knowledge is expected.

Course description:
This course is designed as an introduction to basic set theoretic notions and methods.

Course outline
Introduction. Elementary Set Theory
Set theory as the study of infinity.
Countable sets and their combinatorics.
Cardinalities. Cardinal arithmetic.
Axiom of Choice. Ordered and well-ordered sets. Zorn lemma and its applications.
Well-ordering Theorem. Transfinite induction and recursion.
Applications in algebra, analysis, combinatorics and geometry
Ordinals, ordinals arithmetic and its applications.
Cardinalities revisited. Cofinalities.
Infinite combinatorics. Continuum hypothesis.
Axiomatic Set Theory
Learning Outcomes
After successfully completing the course, the student should be able to:
• understand the various kinds of infinities,
• master cardinal and ordinal arithmetic,
• carry out proofs and constructions by transfinite induction and recursion,
• apply variants of the axiom of choice, in particular, the Zorn lemma,
• use the basic methods of set theory in other fields of mathematics, in particular, in algebra and in analysis.
• understand why axiomatic set theory can be viewed as a "foundation of mathematics''
• understand how one can build a rich theory from simple axioms,

Books:
• A. Shen, and N. K. Vereshchagin, Basic Set Theory, AMS Student Mathematical Library 17,
• P. Halmos: Naive Set Theory
• P. Hamburger, A. Hajnal: Set Theory
• K. Ciesielski: Set Theory for the Working Mathematician

Homepage of the course: http://lsoukup.kedves-soukup.net/bsm