The last major topic of the course is Dedekind cuts. One can reasonably question the wisdom of including such an abstract topic in the course, and admittedly, part of the original reason for their inclusion in the course is to maintain a rigorous definition of the real numbers in the math ed curriculum, and justify the capstone course as a senior level mathematics course.
My intent in this snapshot is to discuss how and why I teach Dedekind cuts, and to support that they provide an entry to important student outcomes. For those unfamiliar with Dedekind cuts, I will provide a brief background in this section. In the 1800s, Richard Dedekind attempted to teach a rigorous foundational calculus class. While preparing the class, he discovered that he was unable to prove the Intermediate Value Theorem, upon which most of calculus rests. The reason for his difficulty was that mathematicians lacked a proper careful definition of the real numbers. Dedekind undertook to solve this problem. (It should be noted that other mathematicians undertook other routes to solve the same problem, and today we have three distinct methods of constructing the real numbers from that time.)
Dedekind's attack rested on the assumptions that we “know” the rational numbers, and as such can use them to define the real numbers. Dedekind's insight was that the fundamental building block of the calculus was an understanding of the number line. Thus, he endeavored to use the ideas embedded in the number line to rigorously define the real numbers. Dedekind then defined a “cut ” to be a pair of sets of non-empty sets (A,B) of rational numbers such that
The intuition behind this definition was that we can think of a real number as a place to “break” the number line. Consequently, each real number corresponds to the pair of sets that it breaks the number line into. There is, however, one hang-up with this definition, namely that every rational number naturally corresponds to two cuts depending on whether you put the number in question into the left hand set or the right hand set. Modern mathematicians get around this problem (see Rudin for example) by simply defining a cut by the left hand set, since it determines the right hand set. Then they add the additional condition that the left hand set contains no largest element.
The other two standard methods for constructing the real numbers. The first is to define a real number to be an equivalence class of Cauchy sequences and the second is to define a real number as an equivalence class of the set of nested intervals. Since the main purpose of this section is to discuss the use of Dedekind cuts for this constructions, I will not give the more careful definitions here. The reason I chose to present Dedekind cuts in this class over the other two definitions is twofold. First, there is a wonderful article written by Dedekind about his motivation for creating Dedekind cuts, the definitions, and several irrationality proofs that I was hoping to have students read. I have never actually found time for this assignment, however. The other reason was that Dedekind cuts more naturally fit with the theme of filling in the number line. The definition rests on an intuition of the number line as a defining feature for the real numbers, which is less true of the other two, or at least it seems so to me. The Cauchy sequence definition really basis its intuition on convergence, a topic of difficulty as a basic intuition for the students in the class, and the nested interval definition can most easily be treated as an intuitive understanding of decimal representation. Students, however, mostly think of real numbers as their decimal representation, but with the difficulty that they don't recognize problems related to the non-uniqueness of decimal representations. This is a natural mistake. In Cauchy's first attempt to prove that the cardinality of the real plain is equal to the cardinality of the real line, he made a mistake that arises from the non-uniqueness of decimal representations. Thus, I saw an advantage to taking an intuition that the students mostly had correctly, and using it to talk about the other intuitions.