 Perhaps the best example of this phenomenon is Dedekind, who in giving his ‘definition’ of an “infinite class” as “a class which is similar to a proper subclass of itself” (PG 464), “tried to describe an infinite class” (PG 463). If, however, we try to apply this ‘definition’ to a particular class in order to ascertain whether it is finite or infinite, the attempt is ‘laughable’ if we apply it to a finite class, such as “a certain row of trees”, and it is ‘nonsense’ if we apply it to “an infinite class”, for we cannot even attempt “to co-ordinate it” (PG 464), because “the relation \(m = 2n\) [does not] correlate the class of all numbers with one of its subclasses” (PR §141), it is an “infinite process” which “correlates any arbitrary number with another”. So, although we can use \(m = 2n\) on the rule for generating the naturals (i.e., our domain) and thereby construct the pairs (2,1), (4,2), (6,3), (8,4), etc., in doing so we do not correlate two infinite sets or extensions (WVC 103). If we try to apply Dedekind’s definition as a criterion for determining whether a given set is infinite by establishing a 1–1 correspondence between two inductive rules for generating “infinite extensions”, one of which is an “extensional subset” of the other, we can’t possibly learn anything we didn’t already know when we applied the ‘criterion’ to two inductive rules. If Dedekind or anyone else insists on calling an inductive rule an “infinite set”, he and we must still mark the categorical difference between such a set and a finite set with a determinate, finite cardinality.