Ultimate Event Number of a Connected Event Chain

It follows from the Axiom of Finitude (Axiom: “Every event chain is made up of a finite number of ultimate events”) and the Definition of an ‘ultimate event’ (Definition: “An ultimate event is an event that cannot be further decomposed”), that every ‘connected’ event-chain is made up of a certain number of ultimate events, i.e. has associated with it an Ultimate Event Number which is an integer.
Currently, we have no means of ascertaining this number, but it is not inconceivable that one day, for simple event chains at any rate, we will actually be able to do just this. When Democritus first proposed the atomic hypothesis some two thousand and five hunded years ago, he may well have imagined that the number of atoms in a given mass of air would never actually be known : today, we can put a specific whole number to the number of gas molecules in a given container (6.022 × 10 exp. 23 for 1 kg of gas at standard temperature and pressure).
Secondly, even if we never do work out this number, we may be able to deduce certain results which can be checked experimentally, or explain certain known effects in a more satisfactory manner. And even if none of this is feasible, as far as I am concerned the principle is worth stating since truth is better than error and what is plausible to be preferred to what is utterly fantastic. The only alternative to assuming that there is a finite number of events in a given event-chain, is to assume that there is an ‘infinite’ number of events in such a  chain — which is indeed more or less what is assumed in the current mathematical treatment. But mathematics is not reality and if events are indeed real, i.e. take place, which I honestly believe they do, there must be a finite number of them between any two specific events, A and B where A and B are the first and last events in a recongizable chain.

In these days of Relativity, one might get the impression that there is nothing one can be sure of under the sun, especially in physics. However, in Ultimate Event Theory, the occurrence of an event is not  relative, has absolutely nothing to do with an observer’s ‘state of motion’ or suchlike considerations : every ‘macroscopic event’, if it is a genuine, bona fide event, is made up of a certain number of ultimate events which are ineradicable and that is the end of the matter.
By ‘connected’ event chain I have in mind a chain of events which we consider to be ‘continuous’, ‘all of a piece’ (even though no chains are strictly speaking continuous), for example, the trajectory of a bullet or the life and death of a star. One could pick out ultimate events from anywhere you like on the Locality and consider them to be a chain : but they would not usually constitute a valid ‘event-chain’ because there would be nothing tying them together except my mental attention. Valid event-chains must subscribe to certain conditions such as not being too far apart, not being  too heterogeneous and so forth : these are the same sort of criteria that we employ when we decide whether certain events are ‘causally connected’. Einstein, in his early work, made it clear that the finite speed of light, a speed which no transmission procedure could exceed, imposed serious restrictions on the operation of causality and it was because of this that he never completely accepted Quantum Mechanics (see Note 1).
It is also assumed here that an event-chain, or portion of an event-chain, has an ascertainable  first and last member. An ‘open-ended’ event-chain such as the successive states of the ‘universe’ we live in, is (perhaps) an open-ended event-chain, in the sense that it may be indefinitely extendable, can ‘go on for ever’ (though I doubt this very much). Such event-chains which do not have a last member are ‘not properly defined’ as a mathematician would say and so are excluded from consideration.
Suppose a ball that I drop onto the floor from a height of one metre. The ball is very elastic and after each bounce rises to half of its original height. After a certain ‘time’ it will stop bouncing and remain fixed to a certain spot indefinitely (though its atoms keep on vibrating). This is a clearcut causally connected event-chain which has a first and last member defined by the two states of rest. Such a chain is, according to the principles of Ultimate Event Theory, made up of a certain number of ultimate events which distinguish it from another, similar, bouncing ball chain. The Event Number for this chain is large but certainly not infinite, whatever infinite means.
One might think this hardly worth saying. But in the current idealized mathematical treatment, there are supposedly an ‘infinite’ number of bounces involved and, if we add up the upward distances, we have the well-known series  1 + 1/2 + 1/4 + 1/8 + …….  But, despite having bounced an ‘infinite’ number of times, the ball will nonetheless at most only have traversed 2 metres !  This nonsense is supposed to show that “an infinite series can nonetheless add to a finite sum” — a fatuous statement if ever there was one. In fact, even  mathematically speaking, the above is false : all that can be shown by the mathematics is that a series of this type, no matter how far you extend it by  taking partial sums, has a finite limit which is quite another matter (Note 2). In reality, of course, any actual ball would only bounce a certain, not very large, number of times and the total distance traversed could easily be measured. But even an idealized, perfectly elastic ball,  if it ever came to rest at all (which it would have to), would have only bounced a certain number of times and have traversed a definite distance which, by calculation we can see will be less than 2 metres. I am merely extending this common sense consideration to all valid, clearly defined, event chains. (In Ultimate Event Theory what we call the ‘ball’ will itself be a compact cluster of identically repeating ultimate events but, for the moment, we will leave this aside.)
I am not bothered at all by how the principles of Ultimate Event Theory accord with Cantor’s Transfinite Set Theory which is a pure fantasy, but I am somewhat worried about how all this ties in with Relativity, Special Relativity anyway. In books on Relativity, we sometimes come across the notion of an “event which all observers would agree took place”, and a typical example given is that of the explosion of a supernova. But in Ultimate Event Theory every event is like the explosion of a supernova — there is no intrinsic difference.  How we actually perceive  the ‘duration’ of an event, or its precise location with respect to other events or compact bundles of events we call stars, does seemingly depend on our own relative state of motion but I cannot see that the occurrrence or not of an event can possibly depend on such things. I considered at one point introducing the Axiom “If an event has occurrence for one observer, it has occurrence for all possible observers“, but I think this is in effect implied by the Axioms we already have. It is sometimes difficult or impossible to arrange certain remote events in a coherent sequence which is true for all possible observers. In the classic example cited in books on Relativity, for me in one part of the town, a space fleet may  already be on its way here from Andromeda while, for someone in another part of the town, the decision to invade the Earth has not even been taken. So for someone else, the event ‘Departure of Space Fleet from Andromeda’ is in the ‘future’ while for me it is in the present or recent past. But according to the principles I am developing here, the event has occurrence full stop, whether or not it is, temporarily, in someone else’s ‘future’ or not.
What about the Twins Paradox, where one of them goes on a flight to Andromeda and returns to Earth while the other stays put? The question is by no means so straightforward as most popular books make out, since the flight to Andromeda and back involves accelerations which are not covered by the Special Theory of Relativity. Even taking this and other considerations from General Relativity into account, the consensus is that the space-voyager, Jack the Nimble, “will have aged less” than his stay-at-home twin. (This discrepancy effect has, allegedly, even been observed : the so-called Mössboauer effect.)
Now, form the point of view of Ultimate Event Theory, we have two extended event-sequences which we are going to compare. If we assume for the sake of argument that both twins have absolutely regular heart beats, we can simply measure the ‘time’, and the aging, by the number of beats of each twin, disregarding other factors. The number of heart-beats of the stay-at-home twin will, thus, according to Relativity Theory, be less than the corresponding number of heart-beats of his twin, Jack the Nimble. Thisn is perfectly conceivable but it is quite unacceptable to me to imagine that a perfectly clearcut event such as a heart-beat should ‘exist in one person’s co-ordinate system’ while not existing in someone else’s : it either takes place or it does not.
In this case, how do I account for the so-called ‘time dilatations’ and ‘space-contractions’ of relative motion that a large number of observations apparently confirm?  The only option open to me is to assume that although the number of ultimate events in a chain is fixed, is ‘absolute’ if you like, the relative spacing of these events, as perceived by certain observers, is not fixed but is variable. The next post will exmine this issue.

Notes  (1) : Einstein’s original idea was actually quite sensible and, though surprising because of what he deduced from it, perfectly easy to comprehend. If I am to make something happen, I have to do something first, fire a gun, push a rock, pull a lever, or, more indirectly, by send a message to someone else to do whatever it is I want doing. In every case, however, there is inevitably a slight delay, a transmission time. But if the speed of light is fixed once and for all and cannot be exceeded, this definitely puts a limit on the operation of causality at any rate as it is normally understood : there will be places and persons that I cannot affect because even a message at the speed of light would not get there. In the usual diagram, an imaginary cone encloses that portion of Space/Time that can be reached from a particular point. It turns out that the perceived order of events taking place at points outside the cone is variable in the sense that, for some hyp[othetical observer, event A may preceded event B, while, for another, it will be the other way round. Because Andromeda is so far away, what is going on there might well be in a different order for two people not so far apart on this Earth, but the whole argument is rather academic since, once the event-chain initiated in Andromeda has consequences for us down here, i.e. the Space fleet has arrived in the solar system, there is no further ambiguity.
Einstein was seriously bothered by Quantum Mechanics because, arguably, QM violates the conditions laid down by Einstein for the operation of causality. To date, experiment has, surprisingly, confirmed QM rather than Einstein, see articles “Reality Check” in New Scientist  26 Feb 2011 and “Quantum Entanglement” 20 Nov 2010.

Note 2 :    Mathematical analysis juggles unending series — I call them ‘indefinitely exptendable’ rather than ‘infinite’ — where various terms are specified and one takes ‘partial sums’, i.e. adds them up to a certain point.
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ……     is a typical series. Each term is the reciprocal of a power of 2. I can ‘add this up’ by taking so many terms, and, for practical purposes, this is what I would do. But, clearly, I cannot add up ‘all’ the terms since the series ‘goes on for ever’ (can always be extended). Now, such series fall into two classes, those that have a ‘finite limit’ (‘converge’) and those that do not (‘diverge’). The fact that each term is smaller than its predecessor does not itself guarantee that there will be a limit that cannot be exceeded. The so-called Harmonic Series, 1 + 1/2 + 1/3 + 1/4 + …..  does not have a finite limit even though the individual terms are decreasing — this is not jiggery-pokery since, no matter what limit you like to mention, I can, with the aid of a very powerful computer, derive a sum that exceeds it strange though this may seem.
How do I know that this is not the case for 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ……     ?    Quite simple. The partial sums are, successively, 1; 3/2; 7/4; 15/8 and so on.  These partial sums can be presented as
2 – 1; 2 – 1/2; 2 – 1/4; 2 – 1/8; 2 – 1/16;  …….
Thus, no matter how many terms one takes, one will never attain 2 which is the limit of this particular series, meaning that one gets arbitrarily close to it but never exceeds or even attains it. In general the limit of such a series, if it exists, is not attained except in very simple cases. I regard the expression “sum to infinity” as nonsensical : no such ‘sum’ exists or can exist.   S.H.