The proof still holds if we only assume that all events co-punctual with a given quintet can be well ordered. Given any class of events , let be the class of those events which are co-punctual with a. Then by definition a is a point if . The necessary and sufficient condition that all the members of a should have a point in common is that a should be contained in . This condition is necessary, for, if is a point and is contained in , it follows that is contained in , and that , so that is contained in . The proof that the condition is sufficient is longer; it is as follows. If , is a point. If not, let denote the part of which is outside . Using again the -series of all events, put and so on, as long as possible. If , precedes in the -order. Hence, as before, there must come a stage when no fresh 's can be constructed. If is the class consisting of a together with all the 's yielded by the method, is a point. For (1) all the quintets in are co-punctual, by the construction; (2) a term co-punctual with all the quartets of cannot be later than all the 's, because if there were such a term we could construct more 's; (3) such a term cannot be[Pg 302] earlier than some member of because, if it were, it would have been chosen as the of that stage in the construction; hence no event outside is co-punctual with every quartet of . Hence is a point. To say that a collection of events have a point in common is to say that the collection is part (or the whole) of the class which is the point. Conversely, a collection of events may contain a sub-class which is a point; the necessary and sufficient condition for this is that should be contained in , where is the collection in question. The proof proceeds exactly as before, if we now make mean the part of which is not contained in . A group of events a is "co-punctual" if is contained in , and a "point" is a co-punctual group which cannot be enlarged without ceasing to be co-punctual. A few purely logical properties of points may be noted. Given any two classes and , if is contained in , then is contained in . Hence if and are points and is contained in , and are identical; for in that case and are respectively identical with and , and therefore if is contained in , is contained in , so that and are identical. Every co-punctual group of events contains at least one point. This has already been proved, since to say that a is a co-punctual group is to say that a is contained in . It may be taken that, in general, there are a number of points of which any given event is a member. Such a set of points will fill a "region," but not every region will be the set of points to which some one event belongs. This topic, however, cannot be dealt with until we have discussed space-time order. FOOTNOTES: