If there is a limit, say , to the series of 's, we require: (1) That should be beyond all the 's, i.e. that for every and we should have contained in i.e. that we should have contained in ; (2) That there should be no point beyond all the 's but between them and , i.e. that, if is any point such that is contained in , then is contained in . [Pg 310] A sufficient condition is, therefore, . If there is a point fulfilling this condition, it is the required limit. If there is an event such that every quartet of is co-punctual with and every quartet of which is co-punctual with is a part of , then there is a point which contains and has for a member, and this point will be such that , so that it will be the required limit. But if there is no such event as , we must proceed differently. In this case we need a new axiom, namely: If is between and , and is a member of but not of , then there is a quartet which is contained in and but is not co-punctual with . In the figure, represents a member of such a quartet. Given this axiom, we proceed as follows. Since is between and , if is a member of but not of , there is a quartet which is contained in and , but is not co-punctual with . Now is contained in ; therefore there is a quartet which is a part of but is not co-punctual with . It follows by transposition that if is a member of and every quartet of is co-punctual with , then is a member of . It follows that is a member of , , ... so that is a member of . Hence, since may be any member of , it follows that any member of which is co-punctual with the whole of is a member of . Now the terms co-punctual with the whole of constitute the class . Hence the common part of and is contained in , and is therefore equal to , since is contained in and in . [Pg 311] Now if is a point which contains , it follows that is contained in ; hence is contained in , and is therefore equal to , since is contained in and in . Hence is the required limit. It follows from this that a compact series of points contained within a stretch of collinear points is continuous. It does not follow that there are compact series of points; this would require existence-axioms which there is no object in introducing, since we do not know whether space-time is continuous or not. It is, however, interesting to observe that an initial apparatus of events suffices to generate a continuous space-time of points, by means of the relations of co-punctuality and logical inclusion.