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. Integers. Rational numbers. Real numbers
Cardinalities. Equinumerosity
Well orders. Ordinals and ordinal arithmetic
Replacement Axiom. Transfinite Recursion
Axiom of Choice. Zorn Lemma. Well-ordering Theorem.
Cardinals and cardinal arithmetic. Alephs.

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.

Reference book:
Herbert B. Enderton, Elements of Set Theory

Course Homepage: lsoukup.herokuapp.com/bsm