NATURE 
193 
THURSDAY, DECEMBER 27, 1906. 
THE THEORY OF AGGREGATES. 
The Theory of Sets of Points. By W. H. Young and 
G. C. Young. Pp. xii+316. (Cambridge: Uni- 
versity Press, 1906.) Price 12s. net. 
ONDESCENDING to a pun, Gauss once remarked 
that he was more interested in notions than in 
notations. The theory of aggregates is so independent 
of the ordinary symbolism of mathematics that it re- 
quires hardly any previous acquaintance with other 
branches of the science from those who proceed to the 
study of it. At the same time, it is full of peculiar diffi- 
culties: it abounds in seeming paradoxes; and some 
of its fundamental problems are at the present 
time the subject of’ keen research and controversy. 
A hearty welco:ne is therefore due to a work composed 
‘by authors who are familiar with all that has been 
published about aggregates, and have themselves 
made important contributions to the subject. 
It is impossible to go into detailed criticism of this 
treatise without the use of technical terms which 
would convey no meaning to the ordinary reader; but 
an attempt may be made to show the general nature 
of this novel theory and the influence it has had, and 
will extend, over the first principles of other parts of 
mathematics. 
The names of Cantor and Dedekind will always be 
associated with the first truly logical definition of the 
arithmetical continuum, or, which comes to the same 
thing, of the range of a real arithmetical variable. 
It is hardly possible to lay too much emphasis on 
the fact that all strictly arithmetical operations are 
connected with the elements (rational and irrational) 
of this continuum. Cantor’s transcendental numbers 
obey laws of operation different from those of ordinary 
arithmetic, and the calculus associated with them 
ought to have another name. 
With the help of a postulate, the necessity of which 
was first realised by Cantor and Dedekind, we can 
establish a one-one correspondence between the values 
of a real positive variable and the points of a finite 
straight line, exclusive of one end, if the variable is 
not allowed to be infinite. If we extend the postulate 
so as to include both ends of the segment, we have 
to include values 0 and o for the variable, which, 
from this point of view, are equally definite. For 
‘convenience, we speak of a point x, instead of saying 
““a point corresponding to the number x.” 
From the arithmetical side we have to investigate 
the properties of the continuum and of its parts; and 
the special interest of the subject begins when we 
consider parts which contain more than a finite set of 
elements. The simplest of these is the natural scale 
I, 2, 3, &c.; its characteristic properties are that it 
has a natural order with a first element, each element 
being succeeded by the next higher in magnitude, and 
there being no last element. 
The set of rational numbers differs from the natural 
scale in some very important respects. As repre- 
sented by points on a line, they have an order of 
position, corresponding to their order of size; but we 
NO. 1939, VOL. 75] 
| cannot say that in this order any element is followed 
by a “‘next’’ element. In fact, between any two 
distinct rational points lie an infinite set of other 
rational points. But Cantor was the first to point 
out that the rational set may be brought into a 
one-one correspondence with the natural scale, 
example, in the order :— 
for 
2 3. Usa See eT 
3) 3523) a Sakis ve 
2 29 D> 
where the fractions, in their lowest terms, are ar- 
ranged so that the sum of numerator and denominator 
never diminishes, while those with the same sum are 
placed in descending order of magnitude. Every set 
which can be thus brought into correspondence with 
the natural scale is said to be countably infinite, or 
to be of potency a. Examples of such sets are: (1) the 
set of all algebraic numbers, (2) all points with 
rational coordinates in a space of n dimensions, where 
n is any assigned positive integer. 
On the other hand, the arithmetical continuum is 
not countably infinite, and its potency, denoted by c, 
is of a higher order than a. One of the outstanding 
difficulties of the subject is the question whether any 
set exists with a potency between a and c. The most 
important theorem in this connection is that every 
perfect set is of potency c (p. 234). A remarkable 
illustration of this is that all the points within a sphere 
may be brought into one-one correspondence with the 
points on a definite straight line. As might be ex- 
pected, the correspondence is not continuous. 
One of the most troublesome questions. discussed in 
this volume is that of Cantor’s potency ‘‘ aleph-one ”’ 
(p. 135). Here we have a set the elements of which are 
sets of points: the element-sets recur over and over 
again, qua sets, and are treated as being all distinct 
in virtue of a criterion of order connected with a 
process of ‘‘derivation’’ and ‘‘deduction.’? Bern- 
stein hopes to show that ¥, =c. 
The potency of an infinite set is analogous to the 
number of elements in a finite set. Cantor intro- 
duces symbols for potencies, calls them transfinite 
cardinal numbers, and investigates a calculus for 
them. As pointed out on p. 150, there is still a 
difficulty in the way of establishing comparisons be- 
tween potencies really corresponding to comparisons of 
size between finite numbers; and, in any case, the 
calculus of transfinite cardinals must greatly differ 
from ordinary arithmetic. 
Besides these ideas of order and potency, which are 
discussed in chapters i.—v., vi, xi, there is the addi- 
tional one of content, which dominates most of the 
rest of the book. It must suffice to give a few illus- 
trations to show the general nature of these inquiries. 
If on a line of finite length 1 we construct an infinite 
set of non-overlapping segments, their content, that 
is, the sum of their lengths, cannot exceed 1. If it 
is less than 1, there may be left over a set of com- 
plementary segments, but this is not necessarily the 
case, as is shown by the example on p. 78. Here 
we have a set of segments whose content is less than 
0.2 of the original line, yet no segment exists on 
the line which does not lie wholly, or in part, within 
one of the selected set. The points not within any of 
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