This course is designed as an introduction to Axiomatic Set Theory.

Course outline |
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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