618 SCIENCE PROGRESS 



(Rend. circ. mat. di Palermo, 1897, 11, 260 ; the famous paper 

 being on pp. 154-64); Burali-Forti then assumed that all 

 perfectly ordered classes were ordinally comparable, and thus 

 that any perfectly ordered class is ordinally similar to a 

 segment of an aggregate n. But the perfectly ordered aggre- 

 gate formed by adding on an element to the end of n cannot 

 be ordinally similar to a segment of ft. Hagstrom now shows 

 by an example that, though a well-ordered aggregate is also 

 perfectly ordered, a perfectly ordered one is not necessarily 

 well-ordered, but may contain a part of inverse type. Such 

 an aggregate is not ordinally comparable with a well-ordered 

 aggregate, and so certain perfectly ordered aggregates are not 

 ordinally comparable. This, now, is what Burali-Forti 

 assumed. But even granting that Burali-Forti made a slip 

 in thinking that his aggregates were well-ordered, the inter- 

 esting fact remains that the antinomy subsists when we consider 

 well-ordered aggregates instead of perfectly ordered ones, 

 and Hagstrom, at the end of his paper, indicates this as a 

 problem which must be investigated. Since the author, unlike 

 all mathematical logicians, has missed the interesting part of 

 Burali-Forti 's discovery, it is unfortunate that he should 

 condemn mathematical logic for obscurity. We may add 

 that Burali-Forti (ibid. 156-7) originally misunderstood what 

 Cantor meant by " well-ordered " in quite another way and 

 thought that a " perfectly ordered class " was well-ordered 

 but not necessarily vice versa, that ft is the type (ibid. 163), 

 defined by abstraction, of the class of ordinal numbers (types 

 of perfectly ordered classes) and that this class is perfectly 

 ordered. Since, then, ft + 1 could be proved to be both 

 greater and less than n, it seemed to follow that there are 

 at least two types which are not ordinally comparable. Later 

 (ibid. 260) Burali-Forti corrected his mistake as to Cantor's 

 meaning, pointed out that his reasoning was concerned with 

 perfectly ordered aggregates and not with what he thought 

 that Cantor called " well-ordered aggregates," and concluded : 

 " It easily follows that every well-ordered class is perfectly 

 ordered, but not vice versa. The reader can verify that the 

 propositions of my note are true for well-ordered classes." 



In the twelfth volume (191 5) of the Periodico di Matematica 

 there are papers on the concept of real number by C. Mineo 

 (1— 15) and A. Maccaferri (87-9), articles by A. Palomby on 



