This course is designed as an introduction to basic concepts of Set Theory. We learn how to use set theory as a powerful tool in algebra, analysis, combinatorics, and even in geometry.

Course outline |
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Introduction. 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. |

Ordinals, ordinals arithmetic and its applications. |

Cardinalities revisited. Cofinalities. |

Infinite combinatorics. Continuum hypothesis. |

Axiomatic set theory. |

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,
- understand the need for formalisation of set theory,
- use the basic methods of set theory in other fields of mathematics, in particular, in algebra and in analysis.

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