Dedekind cuts∗

The purpose of Dedekind cuts is to provide a sound logical foundation for the real number system. Dedekind’s motivation behind this project is to notice that a real number α, intuitively, is completely determined by the rationals strictly smaller than α and those strictly larger than α. Concerning the completeness or continuity of the real line, Dedekind notes in [?] that If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions. Dedekind defines a point to produce the division of the real line if this point is either the least or greatest element of either one of the classes mentioned above. He further notes that the completeness property, as he just phrased it, is deficient in the rationals, which motivates the definition of reals as cuts of rationals. Because all rationals greater than α are really just excess baggage, we prefer to sway somewhat from Dedekind’s original definition. Instead, we adopt the following definition. Definition. A Dedekind cut is a subset α of the rational numbers Q that satisfies these properties: 1. α is not empty. 2. Q \ α is not empty. 3. α contains no greatest element 4. For x, y ∈ Q, if x ∈ α and y < x, then y ∈ α as well. ∗〈DedekindCuts 〉 created: 〈2013-03-21 〉 by: 〈rmilson 〉 version: 〈32907 〉 Privacy setting