Real Analysis – Why the Dedekind Cut Defines Real Numbers Effectively

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I am a seventeen year old high school student and I was studying some Real Analysis on my own. In the process, I encountered the Dedekind Cut being used to construct the Reals.
I just can't get the hang of the definition that $\mathbb R =\{ \alpha \mid \alpha \text{ is a cut} \}$.

Why does this work? Why is this a good definition for the reals? What got me really thinking was the fact that cuts are subsets of $\mathbb Q$. Why are they used to construct the reals?

Also, to define the reals, we are to consider cuts at exactly the "real" points on the number line, right? If we don't know what they are (the Reals) how can we find a cut corresponding to a real (at the "rational" points there seems to be no problem) and how can I Prove that they are unique.

Maybe my naive thoughts are hindering my progress but I just can't seem to understand the cuts being used here. So could you please elaborate your answers a bit more than necessary so that I can get the concept right.

Any help is much appreciated!

Thanks in advance.

Best Answer

Devlin K.: The Joy of Sets (Springer, Undergraduate Texts in Mathematics)

In naive set theory we assume the existence of some given domain of 'objects', of which we may build sets. Just what these objects are is of no interest to us. Our only concern is the behavior of the 'set' concept. This is, of course, a very common situation of mathematics. For example, in algebra, when we discuss a group, we are (usually) not interested in what the elements of the group are, but rather in the way the group operation acts upon those elements.

The above quote is mentioned in connection with "definition" of sets, but it shows that this situation is quite common in mathematics.

It is not important how the real numbers are represented, the important thing are their properties.

In the case of Dedekind cuts the starting point is that we suppose we already have defined the rational numbers $\mathbb Q$, and we what somehow get a new set $\mathbb R$, which will have nicer properties. This means that we want somehow define a set $\mathbb R$ together with operations $+$ and $\cdot$ and relation $\le$, such that

  • they have "all familiar properties"; i.e. $(\mathbb R,+,\cdot)$ is an ordered field;
  • they "contain" rational numbers; which formally means that there is an injective map $e:\mathbb Q\to\mathbb R$, which preserves addition, multiplication and inequality;
  • they "improve" the set of rational numbers in the sense that it contains all "missing" numbers; every non-empty subset of $\mathbb R$ which is bounded from above has a supremum, see wikipedia: Least-upper-bound-property.

Note that rational numbers do not have least-upper-bound-property, the set $\{x\in\mathbb Q; x^2<2\}$ does not have a supremum in $\mathbb Q$.

We can give many different definitions which will fulfill the above properties; theoretically they are all equally good; for practical purposes some of them might be easier to work with.

The construction of reals using Cauchy sequences has a similar spirit, in this case the property which we want to add is completeness as a metric space. (Rational numbers do not have this property.)

Let me mention two books, which deal with this topic:

  • Artmann B: The concept of number (Ellis Horwood, 1988). This books mentions several constructions of reals (Dedekind cuts, Cauchy sequences, decimal representation, continued fractions). Advantages and disadvantages of various approaches are mentioned in this book. (Although all construction lead to "the same" - isomorphic - set of reals, some properties of $\mathbb R$ are easy to prove and some might be more difficult, depending on the chosen approach.)

  • Ethan D. Bloch: The Real Numbers and Real Analysis, Springer, 2001. This book is intended as a textbook for a course in real analysis, but it discusses the two most usual definitions of real numbers in detail in the first two chapters.

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