Instructor: Dr. Lajos SOUKUP
Text: The course is based on printed handouts
Prerequisite: Some familiarity with "higher" mathematics.
No specific knowledge is expected.
Course description: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 |
Introduction. Set theory as the study of infinity.
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Countable sets and their combinatorics.
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Cardinalities. Cardinal arithmetic.
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Axiom of Choice. Ordered and well-ordered sets. Zorn lemma and its applications.
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Well-ordering Theorem. Transfinite induction and recursion.
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Ordinals, ordinals arithmetic and its applications.
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Cardinalities revisited. Cofinalities.
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Infinite combinatorics. Continuum hypothesis.
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Axiomatic set theory. |
Learning Outcomes
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.
Books:
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