An Introduction to Elementary Set Theory

Introduction

Georg Cantor (Source: Wikimedia Commons)

We provide a student project on elementary set theory based on the original historical sources by two key figures in the development of set theory, Georg Cantor (1845–1918) and Richard Dedekind (1831–1916). The project develops the basic properties of sets, and discusses how to define the size of a set and how to compare different sizes of sets. The project concludes by exploring a rather unusual world of infinite sets.

Georg Cantor, the founder of set theory, is considered by many as one of the most original minds in the history of mathematics. Initially, Cantor’s revolutionary ideas were not accepted by the leading mathematicians of his time. Eventually, though, Cantor’s new mathematics became the standard of the twentieth century. David Hilbert (1862–1943), one of the greatest mathematicians of the twentieth century, described Cantor’s work as “the most astonishing product of mathematical thought” [1, p. 359] and claimed that “no one shall ever expel us from the paradise which Cantor has created for us” [1, p. 353].

Richard Dedekind was a friend and an important ally of Cantor. The beginning of Dedekind’s friendship with Cantor dates back to the early 1870s, when they met while on holiday at Interlaken. Their friendship and mutual respect lasted until the end of their lives. Dedekind was one of the first who recognized the importance of Cantor’s ideas, and became an ally of Cantor in promoting set theory.

At the end of the nineteenth century, Cantor and Dedekind wrote two influential treatises that laid the foundations of set theory and mathematics. These were

• the 1888 treatise by Dedekind, “Was sind und was sollen die Zahlen?” (often translated as “The nature and meaning of numbers,” but, more literally, “What are the numbers and what should they be?”), and
• the 1895 treatise by Cantor, “Beiträge zur Begründung der transfiniten Mengenlehre. I” (“Contributions to the founding of the theory of transfinite numbers. I”).

Our project, An Introduction to Elementary Set Theory, uses these two primary sources to introduce students to basics of set theory. The Latex source is also available for instructors who may wish to modify the project for students. The comprehensive “Notes to the Instructor” presented next are also appended to the project itself. Our primary source project module for students is part of a larger collection published in Convergence, and an entire introductory discrete mathematics course can be taught from various combinations selected from these projects. For more projects, see Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science.

Notes to the Instructor

Richard Dedekind (Source: Wikimedia Commons)

This project is a self-contained treatment of the topics from elementary set theory typically covered in a first course in discrete mathematics. It may be used as a text for the set theory unit of a standard one-semester course at the freshman or sophomore level, and should require approximately three/four weeks of class time to complete. A first course in discrete mathematics typically covers logic, set theory, and number theory units. This project may be covered immediately after the logic unit or at the end of the course when both the logic and number theory units have already been covered. No specific prerequisites are assumed other than a basic familiarity with pre-calculus and/or college algebra.

The first part of the project covers standard material about basic properties of sets; membership, subset, and equality relations between sets; and set operations. It ends with an introduction to Russell’s paradox, which typically stimulates interesting in-class discussion. In particular, the purpose of Exercise 2.24.5 is to stimulate such a discussion. A rigorous treatment of Russell’s paradox is, of course, beyond the scope of this project.

The second part of the project covers set equivalence, cardinal numbers, and countable and uncountable sets. Several exercises, especially in the second part of the project, are slightly open-ended. In our opinion, this stimulates independent thinking, as well as provides an opportunity for further in-class discussion. In our experience, such discussions enhance students’ understanding of the material.

Typically, students have little to no difficulty understanding the material presented in the first part of the project. However, the material in the second part of the project requires instructor guidance. It is advisable to have a detailed class discussion on some of the excerpts from Cantor and Dedekind, as well as on set equivalence, the cardinality of a set, and countable and uncountable sets. In particular, the open-ended exercises about the Cantor-Bernstein-Schröder theorem, the Cantor and Dedekind definitions of finite and infinite sets, and the Axiom of Choice provide the instructor with an opportunity to have a more detailed class discussion on these topics.

The project may be covered in its entirety or only parts of it may be selected for class discussions. Alternatively the instructor may wish to assign it as outside class reading. The two parts of the project may be assigned independently, and each requires approximately one/two weeks of class time. The project has a large variety of exercises. The instructor may wish to pick and choose which exercises to assign, depending on what parts of the project will be covered.

Download the project, An Introduction to Elementary Set Theory, as a pdf file ready for classroom use.

Download the modifiable Latex source file for this project.

For more projects, see Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science.

Bibliography

[1]  Hollingdale, S., Makers of Mathematics, Penguin Books, New York, 1994.

Acknowledgment

The development of curricular materials for discrete mathematics has been partially supported by the National Science Foundation's Course, Curriculum and Laboratory Improvement Program under grants DUE-0717752 and DUE-0715392 for which the authors are most appreciative. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.