ESSAYS 3di 



of this process, and stated it as an axiom which cannot be proved, but is used, so 

 he said, "everywhere " in mathematics. It can take the form of a " principle of 

 arbitrary selection repeated an infinity of times." Practically all the work of 

 mathematicians and logicians relating to this subject during the last fourteen 

 or fifteen years has been in the exact determination of the regions of mathematics 

 in which this principle is, or is not, used ; no attempt — or, at least, no successful 

 attempt — has been made to prove this " axiom " or principle. 



In what follows I shall, by using a perfectly definite process of determining 

 members of the given class M without any arbitrary selections at all, construct 

 a class of well-ordered parts of M each of which exhausts the given class M. A 

 proof of Zermelo's principle follows as a corollary from this ; for, if M can be put 

 in the form of a well-ordered series M', any part of M which remains after some 

 members have been removed determines uniquely the corresponding part of M' 

 and the latter has one and only one first member, which can then be used to con- 

 tinue, without any new choice being made, the series of members removed from 

 M. It is evident that, even if the series removed from M is transfinite — so that 

 its ordinal number is, say, one of Cantor's second number-class— and there is a 

 part of M left, there is determined one and only one member of the part, which 

 follows next in the process of removal. 



We will return for a moment to the question of the extension of the theory of 

 combinations to transfinite classes. One of the most interesting results obtained 

 by Cantor was that the cardinal number of the collection of sub-classes of a given 

 transfinite class is greater than the cardinal number of the class itself. If n is the 

 cardinal number last spoken of, 2" is the cardinal number of the collection of sub- 

 classes mentioned, and we have thus a generalisation of the theorem on combina- 

 tions mentioned at the beginning of this article. The theory of combinations has 

 been extended to transfinite classes by A. N. Whitehead (Amer. Journ. of Math. 

 1902, 24), who obtained some interesting results. But far more important was the 

 fact that a product of an infinity of cardinal numbers was defined in this paper by 

 Whitehead, though it had been hinted at previously by Schoenflies (1900), and 

 had, without any doubt, been considered by Cantor himself as a preliminary to the 

 definition he gave (1895, cf. his Contributions, pp. 94-5, 205) of the exponentiation 

 of any cardinal number by any — even transfinite— cardinal number. From the 

 present point of view, the most interesting thing about this is that the necessary 

 proof that an infinite product of cardinal numbers such that no one of them is zero 

 is not itself zero can only be given if we assume Zermelo's principle. Whitehead 

 did not recognise this until about 1904. This is one out of many instances which 

 led many of us to believe that the theory of combinations of an infinity of members 

 depends on the theory of permutations of an infinity of members. The first stage 

 in the following proof is to define a permutation of an infinity of members of a 

 class M. 



Consider the class of all those parts of M which can be well-ordered, and 

 suppose these parts to be well-ordered in all possible ways. Thus we arrive at 

 considering all possible well-ordered parts of M ; of course, the members of M in 

 these parts have not necessarily the. same order with respect to one another as 

 they have in the original M, if M should be given as ordered at all. We will call 

 a part of M which is well-ordered so that it is of ordinal number y — that is, what 

 Cantor and others also expressed as " well-ordered in type y " — a " chain of M of 

 type y," provided that the same part in different orders forms different "chains" ; 

 the part in all these orders may be of the same ordinal type. Thus, a " chain " is 

 a class of couples (m, a), where m is a member of M and a is an ordinal number, 

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