Slanders of t he Exponential Form. 55 



conception). There is, however, so I contend, no reason for 

 thus denying existence to the totality of Alephs, but only 

 for denying the existence of the cardinal number of this 

 aggregate. This indicates the difference between my 

 conception of •'•inconsistency" and that of Hilbert. 



Cantor* has defined a " consistent " aggregate (consistente 

 Vielheit) as such that the supposition of a collection by the 

 mind of all its elements to one thing leads to no contradiction. 

 Since this collection was considered by Cantor as the essential 

 thing in his definition of " Menge," and hence of cardinal 

 number f. this definition tends to agree with mine, in 

 opposition to Hilbert's. But Cantor's definition is not of the 

 nature of the (nominal) definitions in the symbolic logic of 

 Peano and Russell, but rather a ; ' phrase indicating what is 

 to be spoken of" *. 



So T replaced Cantor's definition, in my first paper §, by a 

 formal definition, and I contend that the necessary limitation, 

 noticed by Russell ||, but not discovered by him, in the 

 notion of a "class" is supplied by introducing the postulate 

 of " consistency." For Russell's contradiction seems to arise 

 solely from the use of Cantor's inequality 



2 a >a, 



where ft i? supposed to be the cardinal number of an 

 inconsistent class, such as the class of all propositions H. 



Although we have thus arrived at the formulation of the 

 restricted concept of " class,'' the " search with a mental 

 telescope"** for this. concept appears difficult, and Cantor's 

 ••definitions" are, I think, to be regarded as attempts in 

 such a search. 



The idea of an inconsistent aggregate as an absolutely 

 infinite one (§ 8) — a term also used by Cantor — appears to 

 me to be suggestive. For then finite and transfinite aggregates 

 'which are now both subject to mathematical operations) 

 appear, after suitable rearrangement, as segments of an 

 infinite whole (which is not thus subject). And thus the 

 relation of this infinite to the transfinite aggregates has a 



a Letter of November 4th, L903, referred to in Phil. Mag. Jan. 

 1904, pp. <)7 70. 

 f See Math. Ann. lid. xlvi. L895, pp. 481-482, 497. 

 X Russell, • The Principles of Mathematics/ vol. i.. Cambridge, 1903, 

 p. 304. Cf. Russell's definition of a cardinal number as a class, pp. .*J0">, 

 111-116. 



j Phil. Mag. .Jan. 1904, p. 67. 



Op. cit. p. 20; cf. pp. 36ft 368, L01-107. 

 - This is also the opinion of Prof. Cantor (letter of July 9th, 1904). 

 Russell, op. cit. preface, p. v. 



