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Updated: May 5, 2025
You will remember that in my last lecture I defined the concept of an abstractive set of durations. This definition can be extended so as to apply to any events, limited events as well as durations.
I will therefore use the name 'event-particles' for the ideal minimum limits to events. Thus an event-particle is an abstractive element and as such is a group of abstractive sets; and a point namely a point of timeless space will be a class of event-particles.
It is this character of being an absolute minimum which we want to get at and to express in terms of the extrinsic characters of the abstractive sets which make up a point. Furthermore, points which are thus arrived at represent the ideal of events without any extension, though there are in fact no such entities as these ideal events.
We can then conceive all the intricacies of tangency. In particular we can conceive an abstractive set of which all the members have point-contact at the same event-particle. It is then easy to prove that there will be no abstractive set with the property of being covered by every abstractive set which it covers.
Accordingly there are different types of extrinsic character of convergence which lead to the approximation to different types of intrinsic characters as limits. We now pass to the investigation of possible connexions between abstractive sets. One set may 'cover' another.
The first thing to do is to get hold of the class of abstractive elements which are in some sense the points of space. Such an abstractive element must in some sense exhibit a convergence to an absolute minimum of intrinsic character. Euclid has expressed for all time the general idea of a point, as being without parts and without magnitude.
It will be noted that it is the infinite series, as it stretches away in unending succession towards the small end, which is of importance. The arbitrarily large event with which the series starts has no importance at all. We can arbitrarily exclude any set of events at the big end of an abstractive set without the loss of any important property to the set as thus modified.
Also if P be any moment, either every abstractive element belonging to a given punct lies in P, or no abstractive element of that punct lies in P. Position is the quality which an abstractive element possesses in virtue of the moments in which it lies.
Then the definition of the event-particle associated with the punct is that it is the group of all the σ-primes, where σ has this particular meaning. It is evident that with this meaning of σ every abstractive set equal to a σ-prime is itself a σ-prime.
Accordingly an instantaneous plane in the instantaneous space of a moment will be called a 'level, an instantaneous straight line will be called a 'rect, and an instantaneous point will be called a 'punct. Thus a punct is the assemblage of abstractive elements which lie in each of four moments whose families have no special relations to each other.
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