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

Course outline |
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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 |

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,

- 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