2. DEDEKIND CUTS
2. Nested Intervals and Completeness. The introduction of real num- bers by means of nested intervals is motivated by the following situation
We consider a sequence of intervals
'2,..
., on the arithmetical line continuum (or real axis) each of which is contained within the one which precedes it, and such that the length of 1,,, the nth interval, tends to zero as n increases. (In the particular case of decimal intervals the length of I,, is and the endpoints of are integral multiples of 10".) We require that corresponding to every such sequence of nested intervals there should exist one and only one point on the real axis which is contained in all the intervals of the sequence:I I
I I
A rational sequence of nested intervals, or more shortly a rational net, is a sequence of closed intervals [r1,, with Q,such that D for
=0. A is said to be finer than (Ia),
c
I,,
for alln. We say that (Ia) and
are equivalent if there is a net (Jo) which is finer than each, and we say that is a refinement ofand only if = =
4)
because= 4']
is then a common refinement. We can now define real numbers as equivalence classes of nets. The rational numbers are embedded in these real numbers inasmuch as to every r E Q, corresponds the equivalence class containing the (constant) net (Ia) defined by I, : [r, r) for all n.An example of a net of nested intervals is where := (1 +
and 4
(1+
This net defines the real number e = 2.71828 introduced by EULER, which is of fundamental importance in analysis in the theory of the logarithmic and exponential functions (see also Chapter 5).At this point addition, multiplication and an ordering for these equiva- lence classes of nets ought to be defined and the axioms (R1)—(R3) stated at the beginning of §2 ought to be verified. We shall not adopt this course, how- ever, but instead set up a direct correspondence between nets and Dedekind cuts on the one hand, and between nets and fundamental sequences
on the other.
Corresponding to a given net ([re, sn]) we form the sets a := (z: x Q,
andz for all n} and /3' := {y: y Qand
y>
for all n). 11/3' contains a least element, we remove it and form the set 13 /3' — {min8'}. Then (a,8) has the properties (D1)—(D4) of the Dedekind cut (see 2.1). If we refine the net, the cut remains unchanged. Conversely, to every Dedekind cut (a,fl) there corresponds a net witha and s, E /3. We
begin with any r0a,
/3 and proceed recursively: having obtainedSn we form the arithmetic mean =
+
and define—
f if 4 G,
[rn+I,sn+1) — jf €13.
All nets [rn, with a and /3 are equivalent. We associate (a,$)
with the equivalence class. The two correspondences that have thus been defined are mappings inverse to one another. If the rational numbers are
regarded firstly as equivalence classes of constant nets, and secondly as rational cuts, then the former is the image of the latter and vice versa in the correspondence which has just been described.
The direct relationship between nets and fundamental sequences rests on the following facts: (1) every bounded, monotone sequence is a fundamen- tal sequence. (2) to every rational fundamental sequence (an) corresponds a monotonically increasing rational sequence (re) and a monotonically de- creasing rational sequence (sn), such that — and — are null sequences. Now if ([rn, is a given net of nested intervals, (rn) and (se) are fundamental sequences, and — is a null sequence. If the net is refined to — r,,,) is a null sequence. The correspondence ((rn, .— (re) therefore induces a well defined mapping of equivalence classes of rational nets of nested intervals into the Cantor field F/N of fun- damental sequences modulo the null sequences. Conversely, corresponding to any given fundamental sequence (an) one can choose a monotonically increasing sequence (re) and a monotonically descreasing sequence by the rule (2), and then ([rn, sn]) will be a net. If one had started from an- other fundamental sequence instead of from (an) so that — an)
were a null sequence, and had then chosen (r,) and (4) by the rule (2), then clearly ([rn, sn]) would be equivalent to
4]).
We therefore have a well defined mapping of the fundamental sequences modulo the null se- quence into the set of equivalence classes of nets of nested intervals. This mapping is inverse to the one described above.The practical advantages of nested intervals over cuts or fundamental sequences are as follows. If the real number z is described by (la) the position of x on the number axis is fixed within defined bounds by each 4,. On the other hand with a fundamental sequence (re), the knowledge of one still tells us nothing about the position of z. Again, the description of z as a cut (a, /3) can result from a definition of the set a by means of statements which say nothing directly about the position of z.
The theoretical disadvantage of using the nested interval approach is that introducing the relation between equivalence classes of nets of nested intervals and verifying the field properties for addition and multiplication is somewhat troublesome.
§5. AxIoMATIc DEFINITION OF REAL NUMBERS
While axiomatic methods were at first used only in geometry (see, EUCLID'S Elements), it was not until comparatively recently with the publication of HILBERT'S Grundlagen der Geometric [13] [Foundations of geometry] that they were also used for real numbers. The axiomatic treatment that follows
will however be based not on the system of axioms proposed by HILBERT (in
§13 of [13], where it is called "the theory of ratios," following the tradition set by EUCLID in his Elements), but on the axioms (Rl)—(R3) of §2.