Axioms for the Reals

The practical understanding of real magnitudes and their uses in approxi­

mations leads eventually to the idea of characterizing the field R of real numbers by a suitable list of axioms. It is not enough to require just that R be an ordered field, for there are many such fields, including the field Q of rational numbers and the formal power series field R«/» . The crucial feature is a completeness axiom, stated in §I.4: Every non-empty set of reals with an upper bound has a least upper bound. An ordered field with

5. Axioms for the Reals 1 03

this property is called a complete ordered field. The real numbers form such a field. The formal power series do not.

This completeness axiom implies the Archimedean law. For suppose to the contrary that there were positive reals

a >

0 and

b >

0 such that no multiple na exceeded

b.

Then the set S of all multiples na, for

n

a natural number, has an upper bound, namely

b.

By the completeness axiom, S then has some least upper bound, call it

b *;

thus

b * �

na for all

n.

This also means that

b * � (n + l )a

for all

n,

and so that

b * -a �

na for all

n.

Thus

b * - a

is less than

b *

and is also an upper bound for S, a contra­

diction to the choice of

b *

as a least upper bound for S.

The force of the completeness axiom is to insure that all the real numbers that ought to be there are there. For example, the irrational V2 must be there, as the least upper bound of the set 1 , 1 .4, 1 .4 1 , 1 .4 1 4, . . . of rationals (those approximating V2). Similarly ." must be there, as the least upper bound of the set of decimals 3, 3 . 1 , 3. 1 4, 3 . 1 4 1 , 3 . 1 4 1 5, 3 . 1 4 1 59, . . . . Indeed, since there is a rational between any two reals, any real can be expressed as the least upper bound of a set of rationals. The completeness axiom is also used in more sophisticated ways, for example in the proof of Rolle's theorem and of the mean value theorem of the calculus (Chapter V).

There are other equivalent forms of the completeness axiom. Instead of requiring that every bounded set of reals have a least upper bound, one may require any one of the following:

Dedekind Cut Axiom. If the set R of reals is the union of two disjoint non-empty subsets L and U such that

x

E L and y E U imply

x <

y, then there is a real number ^r such that

x <

^r for all

x

E L and ^r

<

y for all y E U.

Cauchy Condition. Every Cauchy sequence of real numbers has a limit.

Weierstrass Condition. If a series

C( + C2 + . . . + Cn + .

. ^. of posi­

tive real numbers

Cn >

0 is such that there is an upper bound

b

for all partial sums

Sn

⁼

C( + . . . + Cn ,

then the series converges.

It is illuminating to establish the equivalence of the different complete­

ness axioms. The Dedekind cut axiom is perhaps the more geometric (and has already appeared (§III.2) in the completeness axioms for the geometry of the plane): Any cut of the real line into a lower part L and an upper part U must be a cut

at

some real number ^r. Indeed, each real number determines j ust two such cuts, one with L consisting of all

x <

^r, the other with L consisting of all

x <

^r. In this way, real numbers can be completely described by cuts.

The completeness axioms also serves to determine the real numbers uniquely, up to an isomorphism of ordered fields. We sketch the proof.

1 04

IV. Real Num bers Suppose that R ' is any complete ordered field, with unit element

I '

for multiplication. Then

0 < I ',

for otherwise

I ' < 0,

which would give

o <

- I ' ^{and so}

I '

⁼

( - I '

)(

- I ' ) > 0,

a contradiction. For each natural number

n > 0

the multiple

n I '

must then be positive, and all these elements

n I '

are different elements in our field R '. Therefore

cp ( n ) = n

_{I '} defines an injective map

cp :

N ^... R

'

. Because R ' is a field, this

cp

can be extended to a map

cp : Q

^... R

'

by setting

cp ( n/m ) = cp ( n )jcp ( m )

whenever

m =1= 0;

in other words, R ' contains a copy

cp ( Q )

of the ordinary rational numbers

Q.

Also,

cp

preserves the order. By the Archimedean law, each element

r '

in R ' must then be a least upper bound of a set L ' of these rationals, indeed it is the least upper bound of the image

cp

(L) of the set

L =

{n lm 1 m =1= 0, cp (n lm) < r ' } .

This set L has the special property that ^x

<

y and y E L imply ^x E L, so it is the lower half of a "cut" in the rationals. Since it is bounded, it has an (ordinary) real number

r

as least upper bound, and

r

in turn determines L. The one-to-one correspondence

r 1-+ r '

then maps the ordinary real numbers on the given complete ordered field R '. One can show that it preserves sums, products and order, hence it is the desired order isomor­

phism R == R '.

Note that this argument makes essential use of sets, such as the sets L of rationals. In this it is like the use of sets in the induction axioms to prove that the Peano postulates uniquely determine the natural numbers.

We now have two different axiomatic descriptions of the reals: Here, the real

field,

described as a complete ordered field, in §I.4 the real

contin­

uum,

described as an unbounded ordered set with a denumerable dense subset. Each description determines the set of reals uniquely, up to an iso­

morphism of the structure concerned. However the structures are drasti­

cally different. The real continuum has only the order structure, and there are many automorphisms of this structure. The real field has both order and algebraic structure, and its only automorphism (by a proof like that just above) is the identity automorphism. These differing structures on the same "thing" (here the reals) are much like the differing structures on the plane (without and with orientations, as in §III.8). These differing struc­

tures furthermore reflect practical differences. Thus the ordered contin­

uum handles comparisons of many items where one knows only which of two items is the larger, with no measure of "how much" larger.

In either structure, the real numbers form a "continuous" scale. Physi­

cists (and others) sometimes suggest that an "atomic" or "discrete" scale would be more "real"; finitists propose finite scales of magnitude. These proposals turn out to be hard to execute-the continuous scale of reals works smoothly!

In document Mathematics Form and Function (Page 113-116)