The relation of to , confined to inductive numbers, is one-one, has the whole of the inductive numbers for its domain, and all except 0 for its converse domain. Thus the whole class of inductive numbers is similar to what the same class becomes when we omit 0. Consequently it is a "reflexive" class according to the definition, and the number of its terms is a "reflexive" number. Again, the relation of to , confined to inductive numbers, is one-one, has the whole of the inductive numbers for its domain, and the even inductive numbers alone for its converse domain. Hence the total number of inductive numbers is the same as the number of even inductive numbers. This property was used by Leibniz (and many others) as a proof that infinite numbers are impossible; it was thought self-contradictory that [Pg 80] "the part should be equal to the whole." But this is one of those phrases that depend for their plausibility upon an unperceived vagueness: the word "equal" has many meanings, but if it is taken to mean what we have called "similar," there is no contradiction, since an infinite collection can perfectly well have parts similar to itself. Those who regard this as impossible have, unconsciously as a rule, attributed to numbers in general properties which can only be proved by mathematical induction, and which only their familiarity makes us regard, mistakenly, as true beyond the region of the finite. Whenever we can "reflect" a class into a part of itself, the same relation will necessarily reflect that part into a smaller part, and so on ad infinitum. For example, we can reflect, as we have just seen, all the inductive numbers into the even numbers; we can, by the same relation (that of to ) reflect the even numbers into the multiples of 4, these into the multiples of 8, and so on. This is an abstract analogue to Royce's problem of the map. The even numbers are a "map" of all the inductive numbers; the multiples of 4 are a map of the map; the multiples of 8 are a map of the map of the map; and so on. If we had applied the same process to the relation of to , our "map" would have consisted of all the inductive numbers except 0; the map of the map would have consisted of all from 2 onward, the map of the map of the map of all from 3 onward; and so on. The chief use of such illustrations is in order to become familiar with the idea of reflexive classes, so that apparently paradoxical arithmetical propositions can be readily translated into the language of reflexions and classes, in which the air of paradox is much less.