Transfinite Numbers. 445 



(a) Since every aggregate of ordinals in order of magni- 

 tude has a first term, it follows that there cannot be a 

 progression of ordinals in inverse order of magnitude. That 

 is to say, if we take any ordinal 0i and then any ordinal 

 S 2 (</3 1 ) and any ordinal /? 3 (</3 2 ) and so on, we must 

 after some finite number of steps arrive at the first ordinal. 



(l>) Any part of a well-ordered series is ordinally similar 

 either to the whole series or to some initial segment of it. 

 For we can correlate the first term of the part with the first 

 term of the series and so on, and in this way every term of 

 the part is correlated either with itself or with some term 

 which precedes it in the original series. Therefore if a and 

 /3 be ordinals and if a series of type a. be similar to a part o£ 

 a series of type /3, it follows that «</3. 



From this it follows that ot/3>/3 (it is necessarily >«), also 

 a + /3">/3 (it is necessarily >a). 



5. We are now in a position to tackle the cardinal numbers. 

 Since every aggregate can be arranged in a well-ordered 

 series, it follows that of two dissimilar aggregates one must 

 be similar to a part of the other, for when they are arranged 

 in well-ordered series, one must be similar to an initial 

 segment of the other. 



" It is not possible for two dissimilar aggregates to be each 

 similar to a part of the other *. 



For let A be an aggregate of cardinal number R and B an 

 aggregate of cardinal number U and let A be similar to a 

 part of B and B be similar to a part of A. 



Let A be arranged in a well-ordered series, and let a denote 

 the ordinal type of this. This contains a part B x similar to- 

 B, let /?! denote the ordinal type of this, B x contains a part 

 A t similar to A, let a x denote the ordinal type of this ; A x 

 contains a part B 2 similar to B, let /3 2 denote the ordinal type- 

 of B 2 and so on. 



We thus get a progression f of parts of the aggregate A 

 similar to B and A alternately. It follows from § 4 b that 



* This is Schroder & Bernstein's theorem. I have not seen their 

 proofs, but the proof which Mr. Jourdain gives (Phil. Mag-. Jan. 1904, 

 pp. 71-73) appears to me defective. See Note A at the end of this 

 article. 



t It should he observed that this argument is not open to the objection 

 which I have urged against the argument used in the proof given by 

 Mr. Jourdain. We do not require any step by step process of selection 

 to get the series of parts of A which are similar" to B and to A alternately. 

 The whole series is completely determinate when once we have defined 

 the correlation of all the elements of A with some of the elements of B 

 and of all the elements of B with some of the elements of A. 



