| CODE | MAT3000 | ||||||||
| TITLE | Introductory Set Theory | ||||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 6 | ||||||||
| ECTS CREDITS | 4 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | The notion of set is the fundamental notion of modern mathematics. In the late 19th century Georg Cantor developed a revolutionary concept of transfinite numbers that can be used to compare the ‘sizes’ of possibly infinite sets. A naive approach to set theory leads to paradox and it was left to Zermelo to propose an axiomatic approach that puts set theory on a sound rigorous basis. Axiomatic set theory can be viewed as a foundation of mathematics in the following sense: all mathematical notions can be defined in purely set theoretical terms and their properties can be proved using only the set theoretical axioms. Furthermore the language of set theory has played a central unifying role in modern mathematics. The study-unit will cover: - The Zermelo-Fraenkel Axioms - Relations, Functions, and Orderings - The Natural Numbers - Finite, Countable, and Uncountable Sets - Cardinal Numbers, Alephs, and Cardinal Arithmetic - Ordinal Numbers - The Axiom of Choice - Transfinite Induction - Examples - Filters and Ultrafilters: Closed Unbounded and Stationary Sets - Combinatorial Set Theory Study-Unit Aims: To introduce students to the elements of axiomatic set theory and its role as a foundation for mathematics, in the sense that all mathematical concepts can be characterized in terms of the primitive notions of set and membership. On the other hand set theory is also a branch of mathematics, like algebra or geometry, with its own subject matter, basic results, open problems. The aim of this study-unit is to give a general introduction to both aspects, with an eye for the unifying philosophical issues that lie behind them. Applications to other areas of Mathematics like Topology and Combinatorics are also treated. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - Understand such topics as the theory of partial orderings and well orderings, cardinality, ordinal numbers, and the role of the Axiom of Choice. Also further topics such as stationary sets and combinatorics of infinite sets which are important for later stages of development in the mathematics education of students. - Appreciate the role of set theory as a foundation for mathematics, and of the part that axiomatic set theory has to play. 2. Skills: By the end of the study-unit the student will be able to: - Clarify exactly what is meant by “set”, although we do not know the complete answer to the question: What is a set? - Demonstrate facility with the notions of set theory and show an understanding of how axiomatic set theory can be a foundation for mathematics. - Apply set-theoretic results to other areas of mathematics such as topology, functional analysis and combinatorics. Main Text/s and any supplementary readings: Karel Hrbacek and Thomas Jech: Introduction to Set Theory, Third Edition, Revised and Expanded, Marcel Dekker, 1999. Paul Halmos: Naive Set Theory, Van Nostrand Reinhold, 1960. Keith J. Devlin: Fundamentals of Contemporary Set Theory, Springer, 1979. |
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| ADDITIONAL NOTES | Follows from: MAT1100 | ||||||||
| STUDY-UNIT TYPE | Lecture and Independent Study | ||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | David Buhagiar |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2023/4. It may be subject to change in subsequent years. |
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