The most important step in Frege’s logicist programme was to define arithmetical notions, such as that of number, in terms of purely logical notions, such as that of class. Frege achieves this by treating the cardinal numbers as classes of equivalent classes, that is to say, of classes with the same number of members. Thus the number two is the class of pairs, and the number three the class of trios. Such a definition at first sight appears circular, but in fact it is not since the notion of equivalence between classes can be defined without making use of the notion of number. Two classes are equivalent to each other if they can be mapped onto each other without residue. Thus, to take an example of Frege’s, a waiter may know that there are as many knives as there are plates on a table without knowing how many of each there are. All he needs to do is to observe that there is a knife to the right of every plate and a plate to the left of every knife.
Thus, we could define four as the class of all classes equivalent to the class of gospel-makers. But such a definition would be useless for the logicist’s purpose since the fact that there were four gospel-makers is no part of logic. Frege has to find, for each number, not only a class of the right size, but one whose size is guaranteed by logic. He does this by beginning with zero as the first of the number series. This can be defined in purely logical terms as the class of all classes equivalent to the class of objects that are not identical with themselves: a class that obviously has no members (‘the null class’). We can then go on to define the number one as the class of all classes equivalent to the class whose only member is zero. In order to pass from these definitions to definitions of the other natural numbers Frege needs to define the notion of ‘succeeding’ in the sense in which three succeeds two, and four succeeds three, in the number series. He defines ‘n immediately succeeds m’ as ‘There exists a concept F, and an object falling under it x, such that the number of Fs is n and the number of Fs not identical with x is m’. With the aid of this definition the other numbers can be defined without using any notions other than logical ones such as identity, class, and class-equivalence.