ABSTRACT
A compact series we defined as one in which there is a term between any two. But in such a series it is always possible to find two classes of terms which have no term between them, and it is always possible to reduce one of these classes to a single term. For example, if P be the generating relation and x any term of the series, then the class of terms having to x the relation P is one between which and x there is no term.† The class of terms so defined is one of the two segments determined by x; the idea of a segment is one which demands only a series in general, not necessarily a numerical series. In this case, if the series be compact, x is said to be the limit of the class; when there is such a term as x, the segment is said to be terminated, and thus every terminated segment in a compact series has its defining term as a limit.
But this does not constitute a definition of a limit. To obtain the general definition of a limit, consider any class u contained in the series generated by P. Then the class u will in general, with respect to any term x not belonging to it, be divisible into two classes, that whose terms have to x the relation P (which I shall call the class of terms preceding x), and that whose terms have to x the relation P˘ (which I shall call the class of terms following x). If x be itself a term of u, we consider all the terms of u other than x, and these are still divisible into the above two classes, which we may call πu x and π˘u x respectively. If, now, πu x be such that, if y be any term preceding x, there is a term of πu x following y, i.e. between x and y, then x is a limit of πu x. Similarly if π˘u x be such that, if z be any term after x, there is a term of π˘u x between x and z, then x is a limit of π˘u x. We now define that x is a limit of u if it is a limit of either πu x or π˘u x. It is to be observed that u may have many limits, and that all the limits together form a new class contained in the series generated by P. This is the class (or rather this, by the help of certain further assumptions, becomes the class) which Cantor designates as the first derivative of the class u.
