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LII. On the Trans-finite Numbers. 

 By A. E. Harward, Indian Civil Service*. 



I HAVE read with great interest Mr. Jourdain's articles 

 on the transfinite cardinal numbers in recent numbers 

 o£ the Phil. Mag., which constitute a most important addition 

 to our knowledge o£ the subject. But as there appears to be 

 considerable logical confusion about the subject at present, 

 and as Mr. Jourdain's proofs of Schroder and Bernstein's 

 theorem and the theorems that & = tf & and that & 2 = & 

 appear to me to be open to serious criticism, I have at- 

 tempted in this article to give a brief restatement of the 

 whole subject with rigorous proofs of the above theorems. 



1. The term " class " cannot be defined, but it implies that 

 individuals belonging to a class can be distinguished in some 

 way from individuals which do not belong to that class. It 

 does not imply that the individuals belonging to a class can 

 be thought of collectively as a whole. The term " aggregate " 

 does imply that the individuals spoken of can be thought 

 of collectively as a whole, f Any class which is such that 

 the individuals belonging to it cannot be thought of collec- 



* Communicated by the Author. 



I must apologise for absence of references to earlier works on the 

 subject, but I have not had any opportunity of referring- to any works on 

 the subject except .Russell's ' Principles of Mathematics ' (Cambridge, 

 1903), Mr. Jourdain's two papers (Phil. Mag. vol. vii. p. 61 & p. 294 ; 1904), 

 and a paper by Mr. Hardy (Quarterly Journal of Mathematics, 1903, 

 pp. 87-94). The extent of my indebtedness to ' The Principles of 

 Mathematics ' will be obvious to all who have read that book. As I 

 have ventured to criticise some of Mr. Jourdain's reasoning adversely, 

 I feel bound to say that I owe entirely to him the notion that every aggre- 

 gate can be arranged in a well-ordered series. I had previously arrived 

 independently at the conclusion that some distinction must be drawn 

 between aggregates and unlimited classes, but I could not see how the 

 distinction could be practically brought into evidence in mathematics. 

 Mr. Jourdain's paper solved the difficulty for me. I am also indebted 

 to the same paper for the proposition that there cannot be a progression 

 of ordinals in inverse order of magnitude. 



[This article was originally written in December 1904. The manuscript 

 was submitted to Mr. Jourdain, and I am indebted to him for some 

 valuable comments which have enabled me to correct some errors due to 

 imperfect acquaintance with the literature of the subject. — A. E. H. 

 May 17, 1905.] 



t This definition should be regarded as merely provisional, videos ote B. 

 It should be observed that I use the term aggregate to denote what 

 Mr. Jourdain calls "consistent aggregate" or "manifold." The term 

 aggregate by itself implies that the elements are to be taken together as 

 a whole. Unlimited classes (inconsistent multitudes) ought not to be 

 called aggregates. The term manifold has, I think, a more restricted 

 meaning, and should not be applied to aggregates generally. 



