As the fundamental relation in the construction of points, we take a five-term relation of "co-punctuality," which holds between five events when there is a region common to all of them. A group of five or more events is called "co-punctual" when every quintet chosen out of the group has the relation of co-punctuality. A "point" is a co-punctual group which cannot be enlarged without ceasing to be co-punctual. In order to demonstrate the existence of points so defined, it is sufficient to assume that all events (or at least all events co-punctual with a given co-punctual quintet) can be well ordered. If Zermelo's axiom is true, this must be the case; if not, it may involve some limitation as to the number of events. I have been led by the arguments, first of Dr H. M. Sheffer, and then of Mr F. P. Ramsey, to the view that Zermelo's axiom is true; I am therefore less reluctant than I[Pg 300] should have been formerly to assume that events can be well ordered. To prove that every event is a member of at least one point, we proceed as follows—assuming that there are co-punctual quintets. Let be a well-ordered series whose field consists of all events; put Let , , , , be a co-punctual quintet. If is the only event co-punctual , , , , then the class whose only members are , , , , is a point according to the definition. If, on the other hand, there are 's other than which are co-punctual with , , , , , let be the first of them. If no other than and is co-punctual with , , , , and , then , , , , and form a point. Otherwise, let be the first other than and and co-punctual with , , , , , , then must be later in the -series than . If this process comes to an end with , then , , , , , , ... together form a point. If it does not come to an end with any finite , it may happen that no outside the series (, , ... ,...) is co-punctual with , , , and all the 's; in that case, , , , and these 's form a point. But if there are 's other than the 's and co-punctual with all of them, let be the first of them. Then is later in the -series than any of the finite 's. We proceed in this way as long as possible, using two principles: (1) given a series of 's ending with , let be the first in the -series after and co-punctual with the group of all the previous 's; (2) given a series of 's having no last term, take as the next the first in the -series which is after all the 's hitherto selected and co-punctual with all of them. If, at any stage, there is no such , the 's already selected form a point. Now this process must end sooner or later; for the 's (other than ) form an ascending series[Pg 301] selected from , and therefore, sooner or later, there will be no 's later than all the 's previously selected. At this stage, if not before, , , , and the 's already selected will form a point. Hence if all events can be well ordered, every event is a member of at least one point, provided every event is a member of a co-punctual quintet.