March 16, 1900.] 



SCIENCE. 



431 



same. I pointed out {Am. Journ. Math. TV. 86, 

 but whether I first made the suggestion or not I 

 do not know) that a finite collection diflfers from 

 an infinite collection in nothing else than that 

 the syllogism of transposed quality is appli- 

 cable to it (and by the consequences of this 

 logical property). For that reason, the char- 

 acter of being finite seemed to me a positive ex- 

 tra determination which an infinite collection 

 does not possess. Dr. Dedekind defines an in- 

 finite collection as one of which every echter 

 Theil is similar to the whole collection. It ob- 

 viously would not do to say a, part, simply, for 

 every collection, even if it be infinite, is com- 

 posed of individuals ; and these individuals are 

 parts of it, differing from the whole in being 

 indivisible. Now I do not believe that it is pos- 

 sible to define an echter Theil without substan- 

 tially coming to my definition. But, however 

 that may be, Dedekind's definition is not of the 

 kind of which I was in search. I sought to de- 

 fine a finite collection in logical terms. But a 

 'part,' in its mathematical, or collective, sense, 

 is not a logical term, and itself requires defini- 

 tion. 



2. Professor Royce remarks that my opinion 

 that differentials may quite logically be con- 

 sidered as true infinitesimals, if we like, is 

 shared by no mathematician ' outside of Italy.'' 

 As a logician, I am more comforted by cor- 

 roboration in the clear mental atmosphere of 

 Italy than I could be by any seconding from a 

 tobacco-clouded and bemused land (if any such 

 there be) where no philosophical eccentricity 

 misses its champion, but where sane logic has 

 not found favor. Meantime, I beg leave briefly 

 to submit cer.tain reasons for my opinion. 



In the first place, I proved in January, 1897, 

 in an article in the Monist (VII. 215), that the 

 multitude of possible collections of members of 

 any given collection whatever is greater than the 

 multitude of the latter collection itself. That 

 demonstration is so simple, that, with your per- 

 mission, I will here repeat it. If there be any 

 collection as great as the multitude of possible 

 collections of its members, let the members of one 

 such collection be called the A's. Then, by Can- 

 tor's definition of the relation of multitude, there 

 must be some possible relation, r, such that 

 every possible collection of ^'s is r to some A, 



while no two possible collections of ^'s are r to 

 the same A. But now I will define a certain 

 possible collection of ^'s, which I will call the" 

 collection of B's, as follows : Whatever A there 

 may be that is not included in any collection 

 of j4's that is r to it, shall be included in the 

 collection of B'a, and whatever A there may be 

 that is included in a collection of .4's that is r 

 to it, shall not be included in the collection of 

 i?'s. If there is any A to which no collection of 

 ^'s stands in the relation r, I do not care 

 whether it is included among the -B's or not. 

 Now I say the collection of S's is not in the re- 

 lation r to any A. For evei-y A is either an A to 

 which no collection of A's stands in the relation 

 r, or it is included in a collection of A'& that is 

 r to it, or it is excluded from every collection of 

 ^'s that is r to it. Now the collection of JS's, 

 being a collection oi A's, is not r to any A to 

 which no collection of ^'s is r ; and it is not r 

 to any A that is included in a collection of -d's 

 that is r to it, since only one collection of ^'s is r 

 to the same A, so that were that the case the A 

 in question would be a B, contrary to the defini- 

 tion which makes the collection of -B's exclude 

 every A included in a collection that is r to it ; 

 and finally, the collection of B's is not r to any 

 A not included in any collection of ^'s that is r 

 to it, since by definition every such A is & B, so 

 that, if the collection of B's were r to that A, 

 that A would be included in a collection of A's 

 that was r to it. It is thus absurd to say that 

 the collection of B's is r to any J. ; and thus 

 there is always a possible collection of .4's not r 

 to any A ; in other words, the multitude of pos- 

 sible collections of ^'s is greater than the mul- 

 titude of the A'a themselves. That is, every 

 multitude is less than a multitude ; or, there is 

 no maximum multitude. 



In the second place I postulate that it is an 

 admissible hypothesis that there may be a some- 

 thing, which we will call a line, having the fol- 

 lowing properties : 1st, points may be deter- 

 mined in a certain relation to it, which relation 

 we will designate as that of ' lying on ' that 

 line ; 2d, four different points being so deter- 

 mined, each of them is separated from one of 

 the others by the remaining two ; 3d, any three 

 points. A, B, C, being taken on the line, any 

 multitude whatever of points can be deter- 



