THE POSITIVE THEORY OF INFINITY 203 



considerations becomes of use in showing how infinite 

 classes can be amenable to number in spite of being in- 

 capable of enumeration. 



Frege next asks the question : When do two collections 

 have the same number of terms ? In ordinary life, we 

 decide this question by counting ; but counting, as we saw, 

 is impossible in the case of infinite collections, and is not 

 logically fundamental with finite collections. We want, 

 therefore, a different method of answering our question. 

 An illustration may help to make the method clear. I 

 do not know how many married men there are in England, 

 but I do know that the number is the same as the number 

 of married women. The reason I know this is that the 

 relation of husband and wife relates one man to one 

 woman and one woman to one man. A relation of this 

 sort is called a one-one relation. The relation of father 

 to son is called a one-many relation, because a man can 

 have only one father but may have many sons ; conversely, 

 the relation of son to father is called a many-one relation. 

 But the relation of husband to wife (in Christian countries) 

 is called one-one, because a man cannot have more than 

 one wife, or a woman more than one husband. Now, 

 whenever there is a one-one relation between all the 

 terms of one collection and all the terms of another 

 severally, as in the case of English husbands and English 

 wives, the number of terms in the one collection is the 

 same as the number in the other ; but when there is not 

 such a relation, the number is different. This is the 

 answer to the question : When do two collections have 

 the same number of terms ? 



We can now at last answer the question : What is 

 meant by the number of terms in a given collection ? 

 When there is a one-one relation between all the terms of 

 one collection and all the terms of another severally, we 



