300 SCIENCE PROGRESS 



points that approach the limiting point. There are analogies with such a point x„ 

 in the point at infinity used in the theory of functions of a complex variable 

 and in the various elements at infinity used in modern geometry. Cantor went 

 much farther than this in his construction of what he called " transfinite numbers," 

 and for a fuller description of what he did we must refer to his Contributions to 

 the Founding of the Theory of Transfinite Numbers (English translation, Chicago 

 and London, 191 5). For the present purpose, we must remember that the trans- 

 finite numbers thus introduced are ordinal numbers which apply to series, and not 

 cardinal numbers which apply to classes ; and also that Cantor almost wholly 

 restricted his investigations to transfinite cardinal and ordinal numbers of what he 

 called the " first and second number-classes." The first number-class consists of 

 all the finite ordinal numbers ; the cardinal number of this class is the least 

 transfinite cardinal number ; and the ordinal numbers of the " well-ordered " 

 series in which can be arranged the various transfinite parts of a class of the 

 cardinal number just spoken of form the various ordinal numbers of the second 

 number-class. In this the idea of " well-ordering " comes in : a " well-ordered " 

 series is a series in which every part has a first term, though not necessarily a last 

 one. Any finite class arranged as a series in linear order is a well-ordered series, 

 but the series of rational numbers, for example, is not well-ordered (see the 

 Contributions just quoted, pp. 56-61, 137-8, 159, 169). 



Now Cantor, in his memorable work published in 1883 (cf. ibid. pp. 62-3), 

 stated his belief that any class whatever can be well-ordered. Of course this does 

 not mean that any class can be well-ordered in such a way that it has an ordinal 

 number of the first or second number-class ; we have to contemplate number- 

 classes of higher order, even of transfinite order. It is extremely probable that 

 what lay in Cantor's mind when he made this assertion was the process which 

 immediately suggests itself for bringing a class into a well-ordered form : the 

 process of selecting some one member from the class, then another, and so on, 

 repeating the series of selections over and over again, not only to infinity but also 

 beyond infinity — transfinitely. This process involves at every step a choice among 

 an infinity of members, and the validity of the process is made questionable by 

 the fact that it is not supposed to break off after a finite number of steps. Indeed, 

 it seems that, except in particular cases, we cannot strictly define such a process 

 at all, for the simple reason that we cannot " give " an infinity of members unless 

 we can give a law which decides which members are selected and which not. 

 For example, if the given class is the totality of rational numbers between o and 1, 

 we can define a law which determines an infinity of these members which can 

 be put in a well-ordered form by fixing that " all those numbers of the form 2*, 

 where n is a finite integer, are to be chosen." This phrase expresses a law 

 determining a certain class of an infinity of members. 



The principle of arbitrary selection applied an infinity of times has been used 

 by many people, explicitly or implicitly ; while Cantor seems implicitly to have 

 doubted it as a means of rigid proof, and Peano first expressed (1889) the fact that 

 it was not such a means ; but this remark seems to have been neglected. The 

 first published implicit use of it for the purpose of well-ordering a class seems to 

 have been made by myself in 1903, and published in the Philosophical Magazine 

 for January 1904. At that time I did not realise that any assumption which could 

 not be justified was made in this process : indeed the process only occupies a 

 secondary place in what I wanted to prove, because I relied on some work by 

 G. H. Hardy (1903) for an important result obtained by this process. In 1904 and 

 1908 E. Zermelo started a logical principle which he showed to lie at the bottom 



