Thus, so far as the abstractive sets of events are concerned, an abstractive set converges to nothing. There is the set with its members growing indefinitely smaller and smaller as we proceed in thought towards the smaller end of the series; but there is no absolute minimum of any sort which is finally reached. In fact the set is just itself and indicates nothing else in the way of events, except itself. But each event has an intrinsic character in the way of being a situation of objects and of having parts which are situations of objects and—to state the matter more generally—in the way of being a field of the life of nature. This character can be defined by quantitative expressions expressing relations between various quantities intrinsic to the event or between such quantities and other quantities intrinsic to other events. In the case of events of considerable spatio-temporal extension this set of quantitative expressions is of bewildering complexity. If e be an event, let us denote by q(e) the set of quantitative expressions defining its character including its connexions with the rest of nature. Let e1, e2, e3, etc. be an abstractive set, the members being so arranged that each member such as en extends over all the succeeding members such as en+1, en+2 and so on. Then corresponding to the series
e1, e2, e3, …, en, en+1, …,
there is the series
q(e1), q(e2), q(e3), …, q(en), q(en+1), ….
Call the series of events s and the series of quantitative expressions q(s). The series s has no last term and no events which are contained in every member of the series. Accordingly the series of events converges to nothing. It is just itself. Also the series q(s) has no last term. But the sets of homologous quantities running through the various terms of the series do converge to definite limits. For example if Q1 be a quantitative measurement found in q(e1), and Q2 the homologue to Q1 to be found in q(e2), and Q3 the homologue to Q1 and Q2 to be found in q(e3), and so on, then the series
Q1, Q2, Q3, …, Qn, Qn+1, …,
though it has no last term, does in general converge to a definite limit. Accordingly there is a class of limits l(s) which is the class of the limits of those members of q(en) which have homologues throughout the series q(s) as n indefinitely increases. We can represent this statement diagrammatically by using an arrow (→) to mean ‘converges to.’ Then
e1, e2, e3, …, en, en+1, … → nothing,
and
q(e1), q(e2), q(e3), …, q(en), q(en+1), … → l(s).