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 , intuitively, is completely determined by the rationals strictly smaller than and those strictly larger than . 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 are really just excess baggage, we prefer to sway somewhat from Dedekind’s original definition. Instead, we adopt the following definition.
A Dedekind cut is a subset of the rational numbers that satisfies these properties:
is not empty.
is not empty.
contains no greatest element
For , if and , then as well.
Dedekind cuts are particularly appealing for two reasons. First, they make it very easy to prove the completeness, or continuity of the real line. Also, they make it quite plain to distinguish the rationals from the irrationals on the real line, and put the latter on a firm logical foundation. In the construction of the real numbers from Dedekind cuts, we make the following definition:
A real number is a Dedekind cut. We denote the set of all real numbers by and we order them by set-theoretic inclusion, that is to say, for any ,
where the inclusion is strict. We further define as real numbers if and are equal as sets. As usual, we write if or . Moreover, a real number is said to be irrational if contains no least element.
The Dedekind completeness property of real numbers, expressed as the supremum property, now becomes straightforward to prove. In what follows, we will reserve Greek variables for real numbers, and Roman variables for rationals.
Every nonempty subset of real numbers that is bounded above has a least upper bound.
Let be a nonempty set of real numbers, such that for every we have that for some real number . Now define the set
We must show that this set is a real number. This amounts to checking the four conditions of a Dedekind cut.
is clearly not empty, for it is the nonempty union of nonempty sets.
Because is a real number, there is some rational that is not in . Since every is a subset of , is not in any , so either. Thus, is nonempty.
If had a greatest element , then for some . Then would be a greatest element of , but is a real number, so by contrapositive, has no greatest element.
Lastly, if , then for some , so given any because is a real number , whence .
Thus, is a real number. Trivially, is an upper bound of , for every . It now suffices to prove that , because was an arbitrary upper bound. But this is easy, because every is an element of for some , so because , . Thus, is the least upper bound of . We call this real number the supremum of A. ∎
To finish the construction of the real numbers, we must endow them with algebraic operations, define the additive and multiplicative identity elements, prove that these definitions give a field, and prove further results about the order of the reals (such as the totality of this order) – in short, build a complete ordered field. This task is somewhat laborious, but we include here the appropriate definitions. Verifying their correctness can be an instructive, albeit tiresome, exercise. We use the same symbols for the operations on the reals as for the rational numbers; this should cause no confusion in context.
Given two real numbers and , we define
All that remains (!) is to check that the above definitions do indeed define a complete ordered field, and that all the sets implied to be real numbers are indeed so. The properties of as an ordered field follow from these definitions and the properties of as an ordered field. It is important to point out that in two steps, in showing that inverses and opposites are properly defined, we require an extra property of , not merely in its capacity as an ordered field. This requirement is the Archimedean property.
- 1 Courant, Richard and Robbins, Herbert. What is Mathematics? pp. 68-72 Oxford University Press, Oxford, 1969
- 2 Dedekind, Richard. Essays on the Theory of Numbers Dover Publications Inc, New York 1963
- 3 Rudin, Walter Principles of Mathematical Analysis pp. 17-21 McGraw-Hill Inc, New York, 1976
- 4 Spivak, Michael. Calculus pp. 569-596 Publish or Perish, Inc. Houston, 1994
|Date of creation||2013-03-22 12:38:34|
|Last modified on||2013-03-22 12:38:34|
|Last modified by||rmilson (146)|