Instructor: Dr. Lajos SOUKUP
: The course is based on printed handouts distributed in class.
: Some familiarity with "higher" mathematics.
No specific knowledge is expected.
This course is designed as an introduction to Axiomatic Set Theory.
|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
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.
Herbert B. Enderton, Elements of Set Theory
Homepage of the course: http://lsoukup.kedves-soukup.net/bsm