Why does the Dedekind Cut work well enough to define the Reals?

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.

Solution 1:

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.

Solution 2:

Let's use the historically incorrect but nowadays popular definition of a cut as a subset $C$ of $\mathbb{Q}$ such that the following conditions hold:

$\emptyset\neq C\neq \mathbb{Q}$. So a cut is a proper subset.
If $q\in C$ and $p\in\mathbb{Q}$ and $p<q$ then $p\in C$.
$C$ has no maximum, there is no element in $C$ larger than all other elements in $C$.

Now we let all cuts be real numbers. It is a messy task to define (!) all algebraic operations for cuts, but the order is straightforward: $C_1\leq C_2$ if and only if $C_1\subseteq C_2$. The idea is that we can identify every real number with the set of all rational numbers below them. We have used cuts to define real numbers, so how can we formulate the intuition that all real numbers can be represented by cuts? What we can do, is showing how we can construct cuts from decimal numbers. So let $0.d_1 d_2 d_3 d_4\ldots$ be a decimal number. It is certainly at least as large as $0.d_1$ and $0.d_1 d_2$ and so on. And every number larger than each of these should be at least as large as the corresponding number. So we can let the real number (and hence cut) we are looking for be the smallest number at least as large as each of $0.d_1 d_2$ and so on. Now $0.d_1$ is a rational number so we can identify it with the cut $C_1=\{q\in\mathbb{Q}:q<0.d_1\}$ and similarly, we can construct all cuts $C_2$, $C_3$ and so on. Now we want to find a real number, a cut $C$, corresponding to the decimal number. Since it is at least as large as $0.d_1$ we have $C_1\subseteq C$. Since it is at least as large as $0.d_1d_2$ we have $C_2\subseteq C$. Continuing this way, we say that we must have $\bigcup_{n=1}^\infty C_n\subseteq C$. We also want the cut to be not larger than necessary. So we can actually take $C=\bigcup_{n=1}^\infty$ (of course, one must verify that $\bigcup_{n=1}^\infty C_n$ is a cut). In this sense, the set of all cuts contains "every real number".

Now, why is the representation unique? Let $C_1$ and $C_2$ be two different cuts and let $C_1\subset C_2$. Then there is a rational number $q$ in $C_2$ that is not in $C_1$. Also, since the cut $C_2$ has no maximum, there exists a rational number $q'$ in $C_2$ that is larger than $q$ and a even larger number $q''$. Let $C_q$ and $C_{q'}$ be the corresponding cuts. Then $C_1\subseteq C_q\subset C_{q'}\subset C_{q''}\subseteq C_2$, so there lies essentially a rational number $q'$ between $C_1$ and $C_2$, which shows that they are different real numbers.

So, intuitively, we can represent every real number by a cut. And in mathematics, we are bold enough to actually use the cuts as our definition. There are other ways to define the real numbers, but they are in some sense the same, and this "the same" needs lengthy explanation in itself.

Solution 3:

One clue that might lead you to Dedekind cuts is as follows.

Suppose you are oblivious to the real numbers, and only know about the rationals. One day in physics class you're studying the path of a projectile under constant acceleration. In an example you come up with the equation $y = 2 - x^2$ where and $y$ is height and $x$ is time. Solving for height equal to zero you get $x^2 = 2$ or more specifically $2{q^2} = p^2$ if you express $x$ as $\frac{p}{q}$ (where $p$ and $q$ are positive integers).

But then comes the problem. That last equation has no solution. The proof isn't automatic, but essentially $2{p^2}$ has an odd number of $2$'s in it's factorization, and $q$ must have an even number. This is because of what happens if you square the factorization of an integer... the powers afterwards must all be even, right? So it's established that this equation (and many like it) have no solution.

Now if you dig a pit and repeat the experiment so the projectile can pass below height zero you find a very odd situation. A physical object is above a certain plane and then after a period of time is below a certain plane. In the course of it's travel, did it just skip through that plane? Physical intuition suggests that it could not... even in a strictly geometric sense it's hard to imagine a point just phasing through a plane like that. So perhaps the rationals need to be extended to these new 'real' quantities just like the integers needed to be extended to the rationals.

With this in mind, you have a dilemma: how does one approach these new 'real' quantities if they haven't yet been defined. One idea is to use elimination. Whatever this new quantity is (soon $\sqrt{2}$) in comparison to the rationals it is greater than any rational $r$ with $r^2 < 2$ and less than any rational $s$ with $s^2 > 2$. In the experiment, these are the (rational) times before and after the point passed through the plane. This gives you two intervals of rational numbers... an interval being defined by the property that if it contains two numbers, it contains any number in between those two.

Now, to repeat, we can't solve for the time of intersection using the rationals but we can pick two rational times just before and just after our 'missing' intersection. With some work, we can even pick these two as close apart as we want. This is a good clue that as far as approximation goes, the two intervals are doing a good job. We might also notice that we can just keep track of one of the intervals, say the first one, and we'll remember the second for free: it's just the set of rationals greater than all the rationals in the first set.

After more examples like this, we would see more and more examples of intervals of rational numbers that are unbounded below and bounded above. Some are new, some we have already seen before all this speculation, like $(-\infty,r)$. This is convenient, as it provides a way to represent the rationals using these new intervals.

The last step is to somehow define the various mathematical operations for these placeholder intervals so that we can work with the new quantities by using their known properties, and not by how we represent them using Dedekind cuts. This is fairly obvious for Dedekind cuts that represent positive quantities if you remember to first remove the negative, then do the operation on the sets involved, and then add the negatives back in.

[Note: My explanation is all very loose and rough. Also, it turns out that adding roots of polynomials whenever they cross the $x$-axis is not enough. The real numbers add these, and more, so that there is also always a solution to $f(x) = 0$ whenever you have $f(-1) > 0$ and $f(1) < 0$ for a decreasing function $f$ defined on $[-1,1]$, not just polynomials, etc...]

Solution 4:

When we are constructing the reals, we are really, in essence trying to reverse engineer something we already know the properties of from first principals (at least this is how it happened historically). I think mathematicians had a good idea of the properties of real numbers before they had an actual construction for them. Thus "constructing the reals" really means creating a consistent mathematical object that has all the properties that the real numbers should have.

There is more than one construction of $\mathbb{R}$ from $\mathbb{Q}$ (equivalence classes of Cauchy sequences of rationals for instance), but the various methods are equivalent in the sense that the final product has all the same important properties that we care about. In this way we really do not care about uniqueness. You'll come across this same idea again and again in the context of isomorphisms and homeomorphisms.

Think about what truly distinquishes $\mathbb{R}$ from $\mathbb{Q}$...completeness. That is, every subset of $\mathbb{R}$ that is bounded above has a least upper bound. $\mathbb{Q}$ certainly doesn't have this as Dan showed. Dendekind cuts fill in the gaps so to speak by using the flexibility allowed in constructing sets of rational numbers so that the real number they represent is the one that would cause the cut to have a least upper bound if added to the cut.

This always made the most sense to me the first time around when it was put in the context of operations. Let $\alpha, \beta$ be cuts representing the real numbers $1, \pi$ respectively. What does it mean to add $\alpha$ and $\beta$, equivalently $1+\pi$? $$\alpha + \beta =1+\pi = \{x \in \mathbb{Q}|x=r+s \; \text{ such that } \; r \in \alpha, \;\; s \in \beta\}$$ $$ = \{x \in \mathbb{Q}| x <1+\pi\}$$

In essence, after constructing the real numbers and knowing there properties we can manipulate them without thinking about them actually being sets or equivalence classes.

Solution 5:

I haven't looked at the construction in some time, but I seem to recall that a "cut" corresponds to a division of the rational numbers into two distinct parts. Where the "cut" occurs may or may not be a rational number, it is simply a division of the rational numbers into sets $A,B$ such that if $x \in A$ and $y \in B$, then $x < y$.

For example, take the following "cut":

$\{x \in \mathbb{Q}|x^{2} < 2$ or $x < 0\}$
$\{x \in \mathbb{Q}|x^{2} > 2\}$

This set clearly partitions the rationals, but the "cut" itself, the actual point of partition, does not correspond to any rational number. By taking the set of all possible partitions, we are able to not only include the rationals (for we certainly "cut" the rational numbers at rational numbers), but we can also include irrational numbers such as $\sqrt{2}$, as in the example above. Thus we arrive at the set of real numbers, the set of all partitions, the set of all "cuts", in the rational number line.