Informally, the "real numbers" are the
rational numbers with all
the holes plugged.
Before we proceed with any formalism, let's exhibit an example of a
"hole" in the rational numbers. We will take, for our "hole", the square root
of 2:
Lemma: √2 is not a rational number.Suppose
x^{2} = 2. If x is a rational
number then
we can express it as a fraction, and what's more, we can express it as
a
reduced fraction. Let
represent the set of
integers. Then:
(L.1)
and
(
L.2)
Then we must have
(L.3)
And so
n^{2} must be
even. But since √2
obviously isn't an integer, while
n
certainly
is an integer, we see that
n must
also
be even.
If n is even, then clearly
n^{2} must be divisible by
4. In that case,
n^{2}/2 must also be even,
and since we have
(L.4)
we see that
m^{2} is even also. By the same argument,
therefore, m must be even. But if both n and m are even, we must be
able to express
n/m as
(L.5)
But this contradicts (
L.2), and we conclude that √2 must not be
expressible as a rational number.
The real numbers satisfy the
axioms of the rational numbers, with one important change. We remove the "
rationality axiom", and add the
continuum axiom in its place:
(
C.1)
Any nonempty
subset of the real numbers
which is bounded
above has a least upper bound.
There are actually several equivalent ways to state the continuum
axiom. The statement I've just given, though perhaps not maximally
intuitive, is convenient for use in the construction of the real
numbers.
We've just shown, above, that the rational numbers
do
not satisfy the continuum axiom. For consider the set:
(
1)
This is certainly nonempty: 1 ε
S. It's certainly bounded
above: 3 is larger than any element of S. But the least upper bound
of the set is √2, which is not a rational number.
Very well, so the rationals don't satisfy (A.1) -- but how do we
know
any set will satisfy it? How do we know the real numbers
exist? We will construct them ... or, rather, we will sketch the
construction of the reals; the details actually fill a slim book [
Landau:1].
We will start with the rational numbers, which we call
. We first
define a
cut (also called a "Dedekind cut"). If a nonempty subset
of the rationals,
Γ, is a
cut, then it divides ("cuts") the rational
numbers in two, and has these properties:
(C.1)
(C.2)
(C.3)
Property (C.1) says a cut contains a continuous block of numbers,
extending to the "left". Property (C.2) says that the numbers that are
not in the cut are also a continuous block, extending to the
right.
Property (C.3) says that, if a cut has a rational least upper bound,
then that bound is a member of the cut set. However, that may not be obvious, so let's talk a little about it.
The term
says
x is a
lower bound on
Γ^{c}, the complement of the cut. So property (C.3) says that any lower bound of
Γ^{c} must be contained in the cut set. If
x is the
least upper bound on
Γ^{}, then
x is also the greatest lower bound on
Γ^{c}. Since it's a lower bound for
Γ^{c}, it must be a member of the cut set.
We observe immediately that each rational number, q, corresponds to a
cut:
(
2)
From here on, we will use rational numbers interchangeably with the
cuts corresponding to them. In particular, we'll use "0", "1", and
"-1" to refer to Γ(0), Γ(1), and Γ(-1), as the need
arises.
At this point, we observe that the set
S, defined above in (
1), is also a
cut, and it corresponds to √2. If we view the
rational numbers as being contained in the set of cuts, via the
correspondence
q ↔ Γ(
q) as defined in (
2), then the
set of cuts must therefore be a
proper superset of the rational
numbers.
The set of all cuts will be our model of the real numbers. But we're
not done yet: we still need to define comparisons, and we need to
define multiplication and addition.
Comparisons are easy. It's simpler to define ≤ than
<, so that's what we'll do. For two cuts, G and H,
(3)
The "complementary" relations are then defined in the obvious way:
Addition is easy, too:
(4)
Multiplication is a bit trickier. The problem is that all our cuts
have "tails" extending arbitrarily far to the left, and if we just
multiply all the members of two cuts we'll get something that has a
tail extending arbitrarily far to the
right. So we need to be
cleverer than that.
We can define the product of two
non-negative cuts:
(5)
In other words, to find the product of two non-negative cuts, just
ignore the negative "tail" of each cut (remove the tail). Take the set containing the products of all the
non-nonegative members of the cuts, and then just add all the negative numbers to the set (i.e., put the "tail" back on).
Now, before we define the product of two
general cuts, we need a few
"helper definitions". The first of these is
negation:
(6)
We can define absolute value in terms of negation:
(7)
We need to know how to multiply by
-1:
(8)
We'll define one "helper function":
(9a)
(9b)
And finally, we can define multiplication of two general cuts:
(10)
This completes the creation of the model. We have defined comparisons,
addition, and multiplication. The rationals are embedded in the new
model, and the comparison operation is clearly an extension of the
comparison operation for the rationals. The definitions for addition
and multiplication are pretty clearly the appropriate extensions from
the rationals but rigorous proofs of those claims, as well as the claim
that the model satisfies the axioms of the reals, would take more work
than what I have displayed here.
The continuum axiom, as I stated it here, just describes a cut of the
rationals. In our model for the real numbers, the least upper bound
of such a set is the cut itself. So the model does indeed satisfy the
continuum axiom.
The Axiom of Choice
It's worth calling some attention to one of the steps I glossed over.
I rather casually said we would form the set of all cuts of the
rationals. That's actually an enormous set, and it's formed as a
subset of an equally enormous set, which is the
power set of the
rational numbers. It actually requires an
uncountably infinite
number of operations to form the set of all cuts. The assertion that
we can form that set depends on the
axiom of choice. This leads
to the real numbers having some rather peculiar properties, and not
everyone feels it's entirely legitimate.
It is possible to develop a version of analysis that doesn't depend on
the axiom of choice, based on what are called the "constructible reals".
Page not complete. We
would like to add the proof that the model satisfies the axioms, some
comments on alternative statements of the continuum axiom, and a few
additional small theorems and
properties which can be derived easily from the axioms, which will show
that we really have constructed the familiar real numbers here.
As
of 11/2007, however, these are not done, even on paper, and I don't
know when, if ever, they'll be added to this page.
References:
See
Edmund Landau,
Foundations of Analysis, for a thorough exposition
of the construction of the real numbers. I've glossed over a lot and
left out a great deal on this page. Landau does it "right". If you
can find it in the library, that's the way to go -- it's well worth
reading once, but it's not a book you'll refer to a lot.
This material was originally posted on Anarchopedia, in somewhat different form, on 03:39, 19 Nov 2004 (UTC)
Page
created on 11/18/2007