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. |
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 of sets (A,B), where A and B are nonempty subsets of Q,
A is the set of all lower bounds of B (in Q), and B is the set of all upper
bounds of A (in Q). An example of such a cut is
A = {r Î Q: r < Ö5}, B = {r Î Q: r >
Ö5}.
It is possible to describe such cuts without mentioning square roots
(mention squares of rational numbers instead). 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. |
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. |
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. |
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. |