ON ISOID RELATIONS AND THEORIES OF
IRRATIONAL NUMBER
BY
P H I L I P E.
B.
JOURDAIN.
I call a relation which is reflexive, symmetrical, and transitive an isoid relation.
Such relations come prominently forward in t h e work of P e a n o * and his school. The
closely connected subject of "definition by a b s t r a c t i o n " has been especially treated
by Burali-Forti and more recently (1907) by P a d o a ; the writer last named develops
some of the conclusions previously reached by Russell.
Russell f pointed out a means whereby, given t h a t a stands in an isoid relation
to b, an entity can always be defined which has a one-many relation to all such b's as
stand in the isoid relation spoken of to a. Russell called the principle which is based
on this construction t h e " principle of abstraction." I wish to point out t h a t Russell
uses, in a manner apparently not quite free from arbitrariness, different methods for
the construction of such entities according as he deals with integer numbers (cardinal
or ordinal) or real numbers, and to initiate a discussion as to why these particular
forms should be used in preference to others. I n connexion with this, I will discuss
certain points in t h e theories of irrational number.
I. Consider t h e case of cardinal numbers. W h e r e u, v, ... are classes which
have the isoid relation of what Cantor called " equivalence," and what Dedekind and
Russell called " similarity," to each other, Russell constructed a many-one relation
between t h e similar classes and an entity, z, which is, in conformity with the views of
Frege and Weber, " t h e cardinal n u m b e r " of all these classes. B u t Russell also
remarked t h a t other particular $'s are possible : "...la démonstration donne une façon
dont ceci p e u t se faire, sans prouver qu'il n'y a pas d'autres façons de la faire."
I n fact, we shall, in t h e case of real numbers, meet with a different construction of
S and z.
B u t Russell's proof of his proposition 6. 2 does not make t h e n a t u r e of t h e
existence-theorem clear. T h e n a t u r e of existence-theorems is, in essentials, t h e
same throughout mathematics ; in t h e present case what we do is J : W e show that,
* Cf., for example, my paper on " T h e Development of the Theories of Mathematical Logic and the
Principles of Mathematics," Quart. Journ. of Math. 1912, pp. 272, 280, 282-3, 284, 293-4, 302-4.
t " S u r la logique des relations," Rev. de Math, vu, 1902, propositions 6. 1, 6. 2 ; The Principles of
Mathematics, Cambridge, 1903, pp. 166, 219-20, 305, 314, 499, 519; cf. L. Couturat, Les Principes des
Mathématiques, Paris, 1905, pp. 49—51, 77.
X I use Russell's notation of the paper referred to.
ON ISOID RELATIONS AND THEORIES OF IRRATIONAL NUMBER
493
if xRy, by choosing z to be px (which = py), the relation e is an example of an S.
In Russell's proof this does not seem as clear as it might be.
II. Russell's second construction of an entity which justifies the principle
of abstraction is in his theory of real numbers. Here he takes a definite member
out of the class of classes between which an isoid relation subsists, so that the relation
between the classes in isoid relation and the new entity obtained by the principle of
abstraction is the same isoid relation as that which correlates the classes spoken of,
and is not e, as it was before. In the case of real numbers, the isoid relation is the
relation which is practically the same as that which Cantor has called " Zusammengehörigkeit" and which Russell* has translated "coherency." The entity chosen by
Russell is what Pasch and Peano called a " segment " of rational numbers.
III. It may be remarked, by the way, that the chief advantages of the
principle of abstraction seem to be the avoidance of the necessity (1) of introducing
new indefinables and (2) of redefining equality ; and the advantage, mentioned in the
book of 1903f, of enabling the existence-theorems to be proved is comparatively
unimportant.
If we apply this second method to the theory of cardinal numbers, we need,
among all the classes similar to u, one specialized one. Thus we should have the
results that u is similar to its cardinal number, instead of, as in the theory of Russell,
u is a member of its cardinal number. We will return to the consideration of this
theory.
IV. If, now, we attempt to define the real number belonging to a class of
rational numbers by a method analogous to that which Russell has employed to
define the cardinal number of a class, we would arrive at the definition as the class of
those classes of rational numbers which are coherent to u. It seems to me that
there is no particular advantage in defining a real number as a class of such classes
over the definition as a segment, or vice versa.
V. Let us now consider if there is, among all classes which are similar to
a given class u, a specialized one which can therefore be used as a definition of " the
cardinal number of u." The method which immediately suggests itself is to start
with the null-class and define the cardinal number 1 as the class whose only member
is the null-class. Then 2 may be defined as the class whose members are the nullclass and 1 ; and so on. It is visible that, in this way, we would get a one-one
correlation of cardinal numbers with ordinal numbers ; but, on the other hand, there
is a rise of type at each step, so that it would appear that we would be unable
to define any of the transfinite cardinal or ordinal numbers. However, it must be
remarked that the rise of type is of a different nature from the rise of type which is
treated at length by Russell. There would appear to be no objection to the definition
of types whose order is a limit of lower orders.
The distinction between the hierarchy of types which I have indicated and the
hierarchy which Russell has treated is, in a way, analogous to the distinction that
there is between limits in the case of ordinal numbers and those in the case of
* Principles, p. 283.
t P. ix. Cf. Whitehead and Russell, Principia Mathematica, Cambridge, 1912, p. 4.
494
P. E. B. JOURDAIN
cardinal numbers ; or again between û)W in Cantor's notation and the cardinal number
Ko exponentiated by ü0, or the ordinal type a exponentiated by the ordinal type ß
(Hausdorff and myself). It seems that there would be no difficulty, then, in
extending the series of Alephs beyond Xw and the series of ordinal numbers beyond
û)W. It is known that Russell's theory of types puts a limit to the series of Alephs
and ordinal numbers at these points.
VI. It has appeared from somewhat careful historical investigations of mine
that irrational "numbers" were based by Cauchy on geometrical foundations, but
that later there grew up, in some text-books, a would-be arithmetical theory in w^hich
the irrational numbers are defined as the limits of series of rational numbers which
have no rational limits and which are convergent. Since the " convergence " of
a series was usually defined by means of a presupposed limit, the circle was obvious,
and it was the avoidance of this circle which seems to have given rise to the theories
of Weierstrass and Cantor*, and perhaps also of Méray. When, however, the
"convergence" of a sequence (sv) is defined without assuming the existence of
a limit, by means of the well-known criterion
there is no circle, but the assumption of existence appears even more clearly. In the
analogous case of the Dedekind-Peano theory, this assumption was rightly emphasized
by Russell f. On the other hand, Russell \ seems to neglect the advance made in the
avoiding of the circle. This advance lay, perhaps, rather in the fact that mathematicians began to see the necessity of attending to fundamental questions than in
the fact that an obvious logical error was not committed. Still, almost certainly the
error did not seem so obvious about 1870 as it does now.
In Russell's theory§, which was anticipated by Paschi|, the selection of the entity
defined as the real number appears somewhat arbitrary. There is, quite ready to
hand, another entity, which is formed in a manner analogous to that in which the
entity which Frege and Russell called " the cardinal number " of classes which are in
the isoid relation of similarity is formed. It would, further, appear that this was the
entity which Frege fl indicated as the real number belonging to certain classes of
rationals. In any case, there is a difficulty in admitting that Russell's entity or this
entity is the real number corresponding to a given segment of rationals. Many
entities, no one of which seems to have any special advantage over the others, can be
defined by " the principle of abstraction."
VII. It may be remarked that theories of irrational numbers do not depend
upon limits** in the sense that irrationals are defined as limits. If limits are defined
* See Cantor, Math. Ann. xxi, 1883, pp. 566, 568.
f Principles, 1903, pp. 270, 280—285.
î See Ibid. pp. 278, 280—285.
§ Ibid. pp. 271—275, 285, 286.
|| Einleitung in die Differential- und Integralrechnung, Leipzig, 1882 ; see especially p. 1 1 . It is a
curious fact that Pasch seemed so unconscious of the merits of this definition that later (Grundlagen der
Analysis, Leipzig and Berlin, 1909, p. 98) he abandoned his definition by defining (or postulating) an
irrational number as a new "thing," somewhat as Peano did.
IT Die Grundlagen der Arithmetik, Breslau, 1884, pp. 114—115.
** Cf. Russell, op. cit. pp. 277-278.
ON ISOID RELATIONS AND THEORIES OF IRRATIONAL NUMBER
495
before irrationals are introduced (as they usually are, for pedagogical reasons, because
some limits are rational), to define an irrational as a limit would be to commit
the circle referred to. In Peano's introduction of the conception of a limit, on the
other hand, a " limit " and a " real number " are merely two names for the same thing ;
thus limits are not introduced at first for rationals alone, but are "defined by
abstraction," as Peano and Burali-Forti would say, as new entities*, and this
" definition " leads to Russell's definition or to the one given in the fourth section.
VIII. In his remarks on Dedekind's axiom, Russell *f quoted his criticisms J on
Couturat's account of Dedekind's axiom, given in Couturat's De l'infini mathématique^.
Now these criticisms are very much to the point à propos of Couturat's mistaken
version of the axiom, in which the point dividing the Dedekindian sections is said to
be both greater than all the numbers of the lower section, and less than all the
numbers of the upper section. But it does not seem always to be relevant to
Dedekind's own statement.
IX.
The following points seem worthy of notice in the theories of irrationals :
(1) Dedekind's theory had not for its object to prove the existence of irrationals I] : it showed the necessity, as Dedekind thought, for the mathematician to
create them. In the idea of the creation of numbers, Dedekind was followed by
Stolz; but Weber and Pasch showed how the supposition of this creation could
be avoided: Weber defined real numbers as sections (Schnitte) in the series of
rationals; Pasch (like Russell) as the segments which generate these sections;
(2) In Weierstrass's theory, irrationals were defined as classes of rationals.
Hence Russeli'sIT objections do not hold against it, nor does Russell seem to credit
Weierstrass and Cantor with the avoidance of quite the contradiction that they did
avoid ;
(3) The real objection to Weierstrass's theory, and one of the objections** to
Cantors theory, is that equality has to be re-defined;
(4) In the various arithmetical theories of irrational numbers there are three
tendencies : (a) the number is defined as a logical entity—a class or an operation—,
as with Weierstrassft> Weber, Pasch, Russell and Pieri; (b) it is "created," or, more
frankly, postulated, as with Dedekind, Stolz, Peano, and Méray; (c) it is defined
as a sign (for what, is left indeterminate), as with Heine, Cantor^, Thomae,
* Cf. Archiv der Math, und Phys. (3), xvi, pp. 36—37.
f Op. cit. pp. 279—280.
% Mind, N. S., vi, 1897, p. 117.
§ Paris, 1896, p. 416.
|i Cf. Russell, op. cit. p. 280.
IT Ibid. p. 282.
** Not made by Russell, ibid. p. 284 ; cf. p. 285.
f t I have referred elsewhere (Archiv der Math, und Phys. (3), xiv, pp. 306, 311) to the merits of
Weierstrass's theory, which were unappreciated by Frege (Grundgesetze der Arithmetik, II, Jena, 1903,
pp. 149—154).
XX It is a remarkable fact that Cantor was a nominalist as regards real numbers (Math. Ann. xxi, 1883,
pp. 589—590), and rejected with scorn the nominalistic thesis in the case of integers (Zur Lehre vom
Transfiniten, Halle, 1890, pp. 16—18).
496
P. E. B. JOURDAIN
Pringsheim. I will not consider here the geometrical theories, in which, as with
Paul du Bois-Reymond, a real number is a sign for a length ;
(5) In Russell's theory it appears to be equally legitimate to define a real
number in various ways. I do not propose the alternative form I have suggested as
in any way superior to Russell's own, but merely to show the need of some further
discussion.
Finally, I would remark that Russell's later work seems to show that the
advantage claimed in 1903 for the definition of certain numbers as classes, viz. that
an " existence-theorem " can be asserted of them, cannot be maintained. Russell's
" existence," in fact, which does not apply to relations such as the rationals unless
a relation is a class of couples, does not relate to the question of what mathematicians
usually call the " existence " of real numbers.