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 Axiomatic Set Theory.

Course outline
Logic in nutshell
The first axioms. Relations and functions
Natural numbers and ordinal numbers.
Replacement Axiom. Transfinite Induction and Transfinite Recursion
Arithmetic of natural and ordinal numbers.
Axiom of Choice. Zorn Lemma. Well-ordering Theorem.
Cardinalities. Equinumerosity. Cardinal arithmetic. Cofinality
Applications in algebra, analysis, combinatorics and geometry

Learning Outcomes
After successfully completing the course, the student should be able to:

• understand why axiomatic set theory can be viewed as a "foundation of mathematics''
• understand the importance of the axiomatic method
• understand how one can build a rich theory from simple axioms,
• understand the need for formalisation of set theory,
• 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.

Books:
Herbert B. Enderton, Elements of Set Theory

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