It has been acknowledged that this material presenting a real number system is a Dedekind -complete ordered field was inserted here as a result of google search.
It is true
that the real numbers are 'points on a line,' but that's not the whole truth.
This web page explains that the real number system is a Dedekind-complete
ordered field. The various concepts are illustrated with several other fields
as well. Version of 11 Nov 2009 by Eric Schechter. If you find any errors, or see
anything that isn't explained clearly enough, or have any other comments about
this page, please write to me.
What
are the "real numbers," really?
The short, simple answer used in
calculus courses is that a real number is a point on the number line.
That's not the whole truth, but it is adequate for the needs of freshman
calculus. The freshman calculus course (at most universities nowadays) follows
the 17th century style of Newton and Leibniz, emphasizing computations and omitting
many proofs. The omitted proofs depend on a careful explanation of what the
"real numbers" really are. That explanation and those proofs were not
discovered until the 19th century, after Newton and Leibniz were long dead.
A proper explanation of the real
numbers nowadays is covered, if at all, in a course in "real
analysis" in the junior or senior year of students who are majoring in
mathematics. Surprisingly few students take such a course; perhaps that's
because it is too algebraic for the analysts' taste and too analytic to please
the algebraists.
In this web page, I'll discuss the
mathematical meaning of "real number." Before that, I want to discuss
this more elementary question: where did the name "real" come from?
(It turns out to have little to do with the deeper properties of real numbers.)
To answer that question, I first need to talk about complex numbers.
Treating
points in the plane as numbers
There is a natural way to
"add" or "multiply" two points in the Euclidean plane. By
"natural" I mean that the definitions have turned out to be useful
for many applications, and that the definitions are fairly simple.
Unfortunately, the definitions take their simplest forms if we use different
coordinate systems for the addition and multiplication operations.
If we use
that coordinate system, then the formula for vector addition is very simple:
The first coordinate of V1+V2 is the sum of the first
coordinates of V1 and V2, and the second coordinate of V1+V2
is the sum of the second coordinates of V1 and V2. That
is,
(a,b) + (c,d) = (a+c, b+d)
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- The "multiplication" that we want to use can
also be described in Cartesian coordinates: (a,b) ⋅ (c,d) = (ac−bd, ad+bc). But that's a bit complicated
and nonintuitive; it looks somewhat arbitrary and contrived. We get a much
simpler, more geometrically appealing definition if we switch to polar
coordinates. Let a point be represented by <r,θ> if it has radius r
and angle θ -- i.e., if it is located r units away from the origin, and on
a ray that is θ radians counterclockwise from the ray that points toward
the right. That point has Cartesian coordinates (r cosθ, r sinθ). If you
substitute those values into our Cartesian formula for multiplication, and
then simplify using some trigonometric identities, you'll end up with this
much simpler definition of multiplication:
If P1 has polar
coordinates <r1,θ1> and P2 has polar
coordinates <r2,θ2>, then
the product P1P2 is defined to be the point with polar coordinates <r1r2, θ1+θ2>. |
- In other words, multiply the radii and add the angles.
The effect of multiplying points in the plane by P2 is to rotate
the plane through an angle of θ2 and stretch (or shrink)
the plane by a magnification factor of r2. This concept is very
simple, and it's quite useful in engineering, which is often concerned
with describing rotations (e.g., of engines).
When addition and multiplication are
defined as above, then the points in the plane are called complex numbers,
for reasons that will be discussed a few paragraphs from now.
Since (a,0)+(c,0)=(a+c,0) and
(a,0)×(c,0)=(ac,0), the points along the horizontal axis have an arithmetic
just like "ordinary" numbers; we will write (a,0) more briefly as a.
For instance, (5,0) will be written as 5. The points along the vertical axis
also have a shorter notation: the point (0,b) will be written more briefly as
bi; for instance, (0,5) will be written as 5i. The i stands for "imaginary",
for reasons explained below.
Important exercises. Using either the formula (a,b) × (c,d) = (ac−bd, ad+bc) or
the definition in terms of polar coordinates, the beginner should now verify
that i2 = −1. That will be important in the discussion below.
Here are the answers to those two
exercises: Using the Cartesian coordinate system, we compute i2 =
(0,1) × (0,1) = (0•0−1•1, 0•1+1•0) = (−1,0) = −1. Or, using polar coordinates:
The number i has radius 1 and angle π/2. Hence the number i2 has
radius 1•1=1 and angle (π/2) + (π/2) = π; the complex number with those polar
coordinates is −1.
Probably the simplest way to
understand "complex numbers" is to start with points in the plane, as
I have done in the preceding paragraphs. However, by a historical accident, the
simplest explanation was not the first explanation discovered. Indeed, the
geometric, points-in-the-plane viewpoint wasn't discovered until the 19th
century, long after the algebraic computations had been investigated. As early
as the 16th century, mathematicians were devising new "numbers" as a
way of solving polynomial equations; they were thinking in terms of algebraic
formulas rather than pictures. They were particularly interested in the third
and fourth degree equations at that time, but they even had new insights into
the quadratic equation. The attitude that they took was something like this:
We all know that there isn't really
any "number" p that can satisfy the equation p2 = −1.
Such a "number" can only exist in our imagination. But if it
somehow did exist, what kind of arithmetic rules would it have to
follow?
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You have to admire the genius of the
16th century mathematicians: They correctly worked out the arithmetic rules of
the complex numbers despite their lack of the simple geometric model; they
calculated with "numbers" whose existence they didn't even believe
in!
Their terminology was unfortunate,
however. There is nothing fictitious or dreamlike about rotations of engines,
but the name stuck. The points on the vertical axis are now called imaginary
numbers, despite the fact that they have very tangible applications. The
points on the horizontal axis are (by contrast) called real numbers. All
the points in the plane are called complex numbers, because they are
more complicated -- they have both a real part and an imaginary part.
Thus ends our tale about where the
name "real number" comes from. But we have barely begun investigating
the mathematical properties associated with that name.
Getting
rid of the pictures
The "point on a line"
answer is not a fully satisfactory answer, because it is not axiomatic or
algebraic. It relies on pictures that we don't really understand. For instance,
the set of real numbers and the set of rational numbers have essentially the
same picture, but their algebraic properties differ in ways that are very
important for analysts.
Imagine studying that picture of a
line under a super microscope. If you could magnify the line at a very high
power -- say at a magnification of a googolplex, or better yet a magnification
of infinity -- would it still look the same? Or would you see a row of dots
separated by spaces, like the dots in a picture in a newspaper? (It turns out
that, in some sense, the real numbers would still look like a line under
infinite magnification, but the rational numbers would be dots separated by
spaces. But that is only a vague and intuitive statement, not anything precise
that we can use in proofs.)
The only way to get rigorous answers
to these questions is to set up a very careful system of axioms about geometry
... but that amounts to the same thing as setting up a careful set of axioms
about the algebraic properties of the real numbers. It turns out that the
latter is a little easier, so we may as well concentrate on the algebraic
aspects of the situation. To answer questions like this, ultimately we have to
get away from the pictures; we have to understand the real numbers entirely in
terms of formulas.
As a preview, here is the definition
that we're going to end up with: the real line is a Dedekind-complete
ordered field. That's complicated, so we'll work our way up to it in
stages. We'll discuss:
- What is a field?
- What is an ordered field?
- What is a Dedekind-complete ordered field?
- Why do I say that the real line is a
Dedekind-complete ordered field? How can that be a definition?
Groups
and fields
First of all, a group is a
mathematical object; it is a triple (X,e,*) with these properties:
- X is a nonempty set.
- e is a specially chosen member of the set X. It is
called the identity of the group.
- * is a binary operation on X, which we may call the group
operation. This means that whenever p and q are members of X, then p*q
is also a member of X.
- (p*q)*r = p*(q*r) for all p,q,r in X.
- p*e = e*p = p for every p in X.
- For each p in X, there exists at least one
corresponding q in X that satisfies p*q = q*p = e. (It can be shown that
there is at most one such q, and thus q is uniquely determined
by p; we call q the inverse of p.)
Exercises:
- The identity is uniquely determined --- i.e., if p*e1
= e1*p = p and p*e2 = e2*p = p for all p
in X, then e1 = e2.
- Inverses are uniquely determined --- i.e., if p*q1
= e and p*q2 = e then q1 = q2,
The group is said to be abelian
(or commutative) if it also satisfies this property:
- p*q = q*p for all p,q in X.
Examples:
- (Z,0,+) is an abelian group, where Z is the set of all
integers
- ({even integers}, 0,+) is an abelian group
- ({-1,1}, 1, x) is an abelian group
- (R+,1,x) is an abelian group, where R+ is the set of
all positive real numbers
- (R\{0},1,x) is an abelian group, where R\{0} is the set
of all nonzero real numbers. (Here "\" means the difference of
two sets.)
- (T,1,x) is an abelian group, where T is the set of all
complex numbers that lie along the unit circle centered at 0
Now, a field is a quintuple
(Y,0,+,1,×) with these properties:
- Y is a set, 0 and 1 are two specially chosen members of
Y, and + and × are two binary operations on Y.
- 0≠1.
- The triple (Y,0,+) is an abelian group.
- The triple (Y\{0},1,×) is an abelian group. (Note that
this group has for its set of members, all the members of Y except 0.)
- p×(q+r) = (p×q) + (p×r) for all p,q,r in Y.
(Exercise: A few
mathematicians do not include the requirement that 0≠1. Prove that there is
only one "field" in which 0=1. For that field, the set Y has only one
member.)
Here are a few examples:
- The rational numbers (i.e., numbers like 3/4 and
-171/25) are a field.
- The real numbers (i.e., numbers like 87.324116279...)
are a field.
- The complex numbers are a field. (Exercise:
Verify all the axioms. Also, what is the multiplicative inverse of 3+2i ?)
- The set of all numbers of the form p+q√2, where p and q
are rational numbers, is a field; it is a subset of the reals and a
superset of the rationals. (Exercise: Verify all the axioms. Also,
what is the multiplicative inverse of 3+2√2 ?)
Following is one more example. We
will present a finite field -- that is, a field with only finitely many
members. For the set Y, we'll use Y={0,1,2,3,4}. For its addition and
multiplication operations, we'll use ordinary addition and multiplication,
modified by this rule: If the result of addition or multiplication results in a
number greater than 4, subtract 5 or 10 or 15, to get a number in the set Y
again. In other words, we'll use these tables for addition and multiplication:
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This field is sometimes called arithmetic
modulo 5. (Exercises: Show that a similar field can be given with 5
replaced by any prime number. Show that there is also a field with 4 elements,
and a field with 9 elements, but there is no field with exactly 6 elements.
Much much harder: It can be shown that there is a field with exactly n
elements, for some integer n, if and only if n is of the form pr for
some prime number p.)
Ordered
fields
Next, we need to define an ordered
field. This is a sextuple (Y,0,+,1,×,<) where
- (Y,0,+,1,×) is a field.
- < is a binary relation on the set Y. This means that
for each p and q in the set Y, either p < q is a true statement or p
< q is a false statement. That can also be described this way: We are
given some subset S of the set of all ordered pairs of elements of Y, and
we abbreviate the sentence "(p,q) is a member of S" with the
notation p < q.
- For each p and q in Y, one and only one of these three
statements is true:
p < q, p = q, q < p.
(That's
called the Trichotomy Law, because we are cutting the
possibilities into three cases.)
- For all p,q,r in Y, if p < q, then p+r < q+r.
- For all p,q in Y, if 0 < p and 0 < q, then 0 <
p×q.
The reals and the rationals, with
their usual orderings are two familiar examples of ordered fields. A slightly
less familiar example is given by the set of all numbers of the form p+q√2,
where p and q are rational numbers. (Exercise: Show that that set is an
ordered field.)
It can be shown that every
ordered field contains, as a subset, an isomorphic copy of the rational numbers
-- i.e., a set that is identical to the rational numbers in all its arithmetic
operations; it may differ only in the names of some things, via a change in
labeling. If you relabel things a bit, you can say that the rational numbers
are a subset of every ordered field.
In particular, every ordered field
contains infinitely many members. Therefore, the field of the arithmetic modulo
5 cannot be made into an ordered field by defining < in some clever way.
It can also be proved that, in any
ordered field
- −1 < 0, and
- if p ≠ 0, then p2 > 0.
Since i2 = −1, it follows
that there is no way that we can make the complex numbers into an ordered
field, no matter how we define <.
Infinitesimals
This next part is optional -- i.e.,
you can get through the definition of the real numbers without ever thinking
about infinitesimals. But I think this next part is interesting, and also makes
the definition of the real numbers easier to understand.
About
300 years ago, Newton and Leibniz invented calculus. Well, that's an
oversimplification. Some of the ideas of calculus were already around, but they
cleaned it up and knitted it together with what we now call the Fundamental
Theorem of Calculus. Newton also showed some of the ways calculus can be used
-- he worked out many of the basic laws of physics, and showed how to compute
the orbits of the planets much more simply and accurately than anyone had ever
done before. In doing so, he contributed greatly to the beginning of the Age of
Enlightenment -- an age in which people realized that they can accomplish quite
a lot through reasoning, and that they don't have to just live in fear,
superstition, and confusion. This may have indirectly contributed to things
like the industrial revolution and the birth of democracy.
Anyway,
Newton and Leibniz knew how to do many of the computations that we now teach in
calculus, but they didn't know how to do satisfactory proofs of the
theory behind calculus. They tried to do proofs, but their explanations were a
bit lacking. Many of their explanations were based on infinitesimals --
i.e., numbers that are infinitely small but not zero. For instance, in their
explanations, dy/dx did not represent a limit of changing numbers. It
represented a quotient of unchanging numbers, but those numbers were
infinitesimals.
The
computations of Newton and Leibniz were accepted by other mathematicians, but
the proofs were not. The explanation of infinitesimals didn't entirely make
sense, and mathematicians were uncomfortable with it. In the following
centuries, Cauchy and Weierstrass produced the epsilon-delta proofs that we now
find in calculus textbooks. Those proofs involve numbers that are of
"ordinary" size (not infinitesimal), but the numbers would vary
through many different ordinary sizes; thus we take the limit as epsilon
changes toward zero. In our textbooks, dy/dx represents the limit of a changing
quotient of two ordinary numbers. In the late 19th century, Dedekind finally
gave a clear explanation of the real numbers (which we'll sketch at the end of
this web page), and we can prove that in Dedekind's number system there are
no infinitesimals. Arguments with infinitesimals were no longer needed and
fell out of favor. Ultimately, infinitesimals were discredited and discarded by
mathematicians (though they continued to be mentioned in some physics books
many decades later).
In
the 1960's, mathematician Abraham Robinson finally figured out how to make
sense out of infinitesimals. Thus nonstandard analysis was born. It
involved some nonstandard real numbers, among which we can find
some infinitesimals. In the paragraphs below, I will give an example of an
ordered field that has some infinitesimals. The discussion below is based on
20th-century ideas, not just on those of Newton and Leibniz. I should mention,
however, that the example that I will present is not the approach
preferred by the nonstandard analysts. They prefer an approach that is more
complicated but also more powerful. (It involves making careful logical
analysis of a formal first-order language, but we don't need to discuss that
here.)
Some
of the nonstandard analysts now actually feel that infinitesimals yield a
better understanding of calculus. After all, it gave Newton and Leibniz the
intuition that they needed. We can actually make rigorous mathematics, with
only slight adjustments in the ideas of Newton and Leibniz. (For instance, the
derivative should be the standard part of that quotient of
infinitesimals; this term is explained in a later paragraph below.) But most
mathematicians still prefer the epsilon-delta approach, which they feel is
simpler. (Both methods are correct, and both yield the same results.) At any
rate, some discussion of infinitesimals may be helpful in our explanation of
ordered fields.
Definitions. Suppose that Y is an ordered field. An infinitesimal
member of Y is a member r, other than 0, that satisfies all of these
infinitely many conditions:
−1
< r < 1 , −1/2 < r < 1/2 , −1/3 < r < 1/3 , ...
Two members of Y are said to be infinitely
close if their difference is an infinitesimal.
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Some
ordered fields have infinitesimals, and some don't. The ordered fields that
have no infinitesimals are called Archimedean fields; we'll see later that
the real number system (i.e., Dedekind's number system, also known as the standard
real numbers) is Archimedean. The ordered fields that do have infinitesimals
are called non-Archimedean fields; we'll give an example of such a field
in the next few paragraphs.
The
example will be based partly on rational functions. By a rational function
in the variable t, we will mean a function of the form p(t)/q(t), where
p(t) and q(t) are polynomials with standard real coefficients, and q is not the
constant polynomial 0. Note that each real number can be viewed as a rational
function -- for instance, the number 7 can be viewed as 7/1, where 7 and 1 are
both polynomials of degree 0. Thus the set of real numbers is a subset of the
set of rational functions. (Of course, to make sense of this, we have to assume
that we already have some understanding of the real numbers. But we won't need
a very deep understanding; the "points on a line" conception will
suffice for now.)
We define addition and
multiplication of rational functions in the usual fashion, as in high school
algebra. However, we make this one alteration in the usual treatment of
rational functions: We will consider two rational functions to be "the
same" if they agree except at finitely many values of t. For instance,
these two functions
t−3
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t2−t−6
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and
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1
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t+2
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are
not really the same, because the first one is defined at t = −2 and the second
one is not. But the two functions are identical for all other values of t, so
we will view them as "the same" for purposes of the present discussion.
With that convention, it can be shown that the set of all rational functions
is a field.
Also,
the real numbers are a subset of the rational functions. For instance, the
constant 1 and the constant 7 are polynomials of degree 0, so the constant 7/1
is a rational function. In this fashion we can view every real number as a
rational function.
We
can make the rational functions into an ordered field, if we just define
the right ordering. To do so, we will make use of the following theorem. (We
will omit the proof of the theorem, which is a bit harder, but it just involves
some advanced calculus and some college algebra.)
Theorem. Suppose that q(t) and r(t) are given rational functions
in the variable t. Then there exists some real number t0 (which may
depend on the choice of q and r) such that exactly one of these three cases
holds:
Furthermore, if case 2 holds, then
q(t) = r(t) for all but finitely many values of t.
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We now define an ordering on the
rational functions, by saying that
q < r or
q = r or q > r
if cases 1, 2, or 3 hold,
respectively. In other words, one rational function is less than another if it
is eventually less -- i.e., if it is less when we go far enough to the
right on the graphs of the two functions. How far to the right we have to go
may depend on which two functions we're looking at; but the theorem says that
for each choice of two rational functions, there is some point after which one
function stays below the other (unless they're the "same").
With this definition of ordering, it
turns out that the set of rational functions is an ordered field. But it also
turns out that the functions
1/t, 2/t,
1/t2, etc.,
are infinitesimals. Thus, the field
of rational functions is non-Archimedean, when ordered as we have described.
How does this relate to Newton's
view of numbers? I'm sure that Newton wasn't thinking of his infinitesimals as
rational functions. But we can get some idea of his viewpoint, as follows:
There are no infinitesimals among
the standard real numbers. But we could imagine that, with a sufficiently
powerful microscope, we might discover some additional "nonstandard"
numbers that we had not noticed before. Nestled around each standard real
number r, infinitely close to it, are infinitely many new nonstandard numbers.
(Then r is the standard part of any of those new numbers.) In
particular, nestled around 0 are the infinitesimals. We can also get some other
nonstandard numbers by taking the reciprocals of the infinitesimals; those
numbers are infinitely large. The collection of all the numbers -- both
"standard" and "new", together -- is an ordered field. Its
ordering is the same as the ordering of the set of rational functions.
Least
upper bounds
Suppose that Y is an ordered field,
and S is a nonempty subset of Y, and b is a member of Y. We say that b is an upper
bound for the set S if we have s < b satisfied for every s in S.
If
the set S has an upper bound, then in general it has many upper bounds. Say B
is the set of upper bounds of S, and B is nonempty. Does B have a lowest
member? If it does, that member is called the least upper bound of the
set S.
The
word "complete" has different meanings in different branches of
mathematics. Generally, an object is called "complete" if there are
no "holes" in it -- i.e., if nothing that seemingly "ought
to" be there is missing. This vague description has different meanings for
different kinds of mathematical objects -- a complete ordered field, a complete
measure space, a complete logic, etc. Here, we will only consider the meaning
of completeness for ordered fields.
An
ordered field Y is said to be complete, or Dedekind complete, if
it has this property, also known as the least upper bound property:
Whenever S is a nonempty subset of
Y, and S has at least one upper bound, then S has a least upper bound.
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Dedekind completeness turns out to
be crucial in analysis, because it enables us to take limits.
Some ordered fields are Dedekind
complete, and some aren't. Here are two quick examples of ordered fields that
aren't complete:
- The set of rational numbers is incomplete (i.e., not
complete). To see this, let S be the set of all rational numbers r that
satisfy r2 < 5. Then S has many upper bounds -- for
instance, 3 is an upper bound, and 2.24 is another upper bound, and 2.23607
is another upper bound. We can keep finding more of these numbers --
whatever rational number we propose for an upper bound for S, it is
possible to find another rational number that is still a little lower and
that is also an upper bound for S. You can probably see why already: These
numbers are converging to
√5 = 2.23606797749978969640917366873128...
But
that number is not rational. Any rational upper bound for S would have
to be slightly higher than √5, and between that rational number and √5
we can always find still another rational number. In the field of rational
numbers, the set S does not have a least upper bound.
- If Y is a non-Archimedean field -- i.e., an ordered
field that has infinitesimals -- then Y is incomplete. One way to see this
is to let S be the set of all infinitesimals. Since some of the infinitesimals
are positive, any upper bound for S must be greater than 0. Note that 1 is
an upper bound for S, and 1/2 is another upper bound for S, and 1/3 is
another upper bound for S, and so on. Suppose (for contradiction) that b
were the least upper bound for S. Then b must be positive, and must
be less than or equal to all of the numbers 1, 1/2, 1/3, etc. -- thus b
must be a positive infinitesimal. Then 2b is also an infinitesimal, so 2b
is a member of S. Since b is an upper bound for S, that tells us 2b <
b. But b < 2b since b is positive. This is a contradiction.
(Note that, conversely, any complete
ordered field must be Archimedean.)
Complete
fields
We have used the real numbers in
some of our preceding discussions. For instance, the complex numbers are
ordered pairs of real numbers, and our example of infinitesimals involved
rational functions with real coefficients. In effect, we "borrowed"
the real numbers -- we used the reals in examples, even though we hadn't
formally defined them yet; we just relied on the informal and intuitive
understanding that students already have, based on the geometric line. Trust
me, there is no circular reasoning here -- I won't use the "borrowed"
concepts when I finally get around to defining the real numbers. You'll see
that if you actually work through all the details. (I'm not claiming that this
web page is more than an outline.)
The definition of the reals depends
on two more theorems, both of which are difficult to prove.
Theorem 1. There exists a Dedekind-complete ordered field.
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The literature contains many
different proofs of this theorem. I think three are simple enough to deserve
mention here:
- Proof using decimal expansions. Let Y be the set of all infinite decimal expansions --
i.e., expressions such as 3.682951... and −17.311897... . Adopt the
convention that 2.719999... is the "same" as 2.7200000..., etc.
Use the usual operations of addition and multiplication. Then Y is a
complete ordered field, but verifying that fact is extremely tedious. It
generally isn't worked out in full detail. One place that you can find it
in fairly complete detail is in J. F. Ritt, Theory of Functions,
1946. It is also sketched in M. Rosenlicht, Introduction to Analysis,
reprinted by Dover.
- Proof using Dedekind cuts. Let Q be the set of rational numbers; we assume that
we already have a good understanding of those. By a Dedekind cut we
mean a pair (A,B) with these properties:
- A and B are nonempty subsets of Q whose union is Q
- a < b, for every a ∈
A and every b ∈ B
- A has no highest element.
The set B
might or might not have a lowest element. Here are some examples of cuts in
which B has a lowest element:
A-2 = {x ∈
Q: x < -2}, B-2 = {x ∈
Q: x > -2} A3.7 = {x ∈
Q: x < 3.7}, B3.7 = {x ∈
Q: x > 3.7}
and here
is an example of a cut in which B has no lowest element:
A = {r ∈ Q: r < 0 or r2 <
5}, B = {r ∈ Q: r >0 and r2 > 5}.
(That cut
would be called (A√5, B√5) if we had a √5.) The set of all
cuts can be made into a complete ordered field, if we define addition and
multiplication the right way. Again, it's tedious; you can find some of the
details worked out in W. Parzynski and P. Zipse, Introduction to
Mathematical Analysis.
- Proof using Cauchy sequences. Again start from the rational numbers. Say that a
sequence r1, r2, r3, ... of rational
numbers is a Cauchy sequence if it has the property that
for each positive integer p there
exists a positive integer m (which may depend on p and on the particular
sequence being studied) such that, whenever i and j are greater than m, then
|ri − rj| < 1/p.
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- Now, say that two Cauchy sequences r1, r2,
r3, ... and s1, s2, s3, ... of
rationals are equivalent if they have the property that
for each positive integer p there
exists a positive integer m (which may depend on p and on the particular
sequences being studied) such that, whenever i is greater than m, then |ri
− si| < 1/p.
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- By an equivalence class we mean the set of all
the sequences that are equivalent to some particular sequence. Now, it can
be shown that the set of all equivalence classes is a complete
ordered field, if we define addition and multiplication on it in the right
fashion. This proof, due to Cantor, is a slight modification of a proof
that can be found in many analysis or topology books, showing that every
metric space has a metric completion.
The other theorem is harder to
prove, and I won't even sketch a proof here. In fact, this theorem is even
difficult to state:
Theorem 2. Any two Dedekind-complete ordered fields are isomorphic
i.e., there exists a one-to-one correspondence between them that preserves,
in both directions, the orderings and the arithmetical operations. Thus, any
two Dedekind-complete ordered fields are essentially "the same";
one is simply a relabeled copy of the other.
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In particular, the decimal
expansions, the Dedekind cuts, and the equivalence classes of Cauchy sequences,
though they appear to be entirely different, all turn out to have the same
arithmetic and algebraic structure -- they are really the "same"
object. It is that object which we call the real number system.
Finally,
the real definition of the reals
(No pun intended.)
Definition. The real number system is that unique algebraic structure
represented by all Dedekind-complete ordered fields.
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You might wonder why mathematicians
want to use such a complicated definition. Wouldn't it be easier to simply
define the real numbers to be the Dedekind cuts, or define the real numbers to
be the decimal expansions, or something like that? That is the approach taken
in some elementary textbooks, but ultimately it is less productive. When we
actually use the real number system in proofs, the properties that we
need are not specifically the properties of (for instance) Dedekind cuts or of
decimal expansions. Rather, the properties we need are the axioms of a Dedekind
complete ordered field. It is much simpler to think in terms of those axioms.
To think of "numbers" as being cuts or expansions would just encumber
us with extra baggage. The cuts or expansions are models -- they are
useful for the job proving Theorem 1, but they are useful for little else. Once
they've done that job, we can discard and forget them.
If you wish, you can now think of
the points on a line as representing the members of a Dedekind-complete
ordered field. It is then correct to say that the real numbers are the points
on a line.
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