The Probability Abacus 
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SUMMARY: We will describe how to count in the discrete measurement space in terms of P(E) = 1 events. Our Abacus counts only P(E) = 1 events.
What we have been trying to do is to describe a given system without
having to use the traditional coordinate frames, such as Cartesian,
Spherical, or Cylindrical. We suspect that origin is not
accurately measurable due to the observer's limitations. And therefore
we introduced a discrete measurement space based on an observer's
capability. You may also think of it as an introduction of a minimum
quantum (observer1observer2), whose value is based on the maximum of
the observer's capability. What does this quantum mean we do not know
yet.
We have established the origin in the discrete measurement space. Then we should be very specific how the count is progressed from the origin or “zero”, based on an established criterion. We define the criterion for progression of the count as P(E) = 1 or PE1 event, where P(E) = 1 implies that the probability P(E) of an event E taking place is equal to 1. Only when a PE1 measurement is completed by an observer we should progress the count to next integer, i.e. "0" a PE1 event, is incremented to "1" which has to be another PE1 event. We can not increment the count from a PE1 event to a PE<1 event. Therefore we have two
important considerations before the description
of a system is developed, first we have to define the conditions
leading to establishment of the origin and then we have to increment
the count based on PE1 events only.
Within the discrete measurement space we have to appreciate the importance of a PE1 measurement. An example of such case will be the determination of a circle by two observers of different capabilities, one is a bug constrained to crawl on the circumference of the circle and making measurements and other is a human. In the case of an unlikely event the bug may have completed all the measurements, i.e. all the outcomes are determined, a PE1 event is completed, and the bug can develop an algorithm based to interpret the result whatever it could be for the bug. The human at the other hand knows that it is a circle and draws conclusions from it without making measurements. We note that
the human
observer does not apply a measurement force to make a determination
of circle whereas bug has to. This is an important consideration for an
observer to be efficient that is, "an efficient observer does not
apply a measurement force to make a measurement or an observation".
This criterion leads to optimum use of the resources while making a measurement.
We have another concern. What if
the circle being
measured, is drawn infinitesimally thin? Then neither bug nor human
observer will be able to measure it at their existing resolutions and
therefore they will both arrive at the same conclusion that the
circle does not exist, but circle does exist. Therefore a PE1 event
has taken place but it is determined to be nonexistent because of
the measurement limitations, represented by observer's inability to
determine the origin with absolute accuracy. The possibility that a
PE1 event has occurred but can not be measured due to observer's
limitations is quite worrisome.
Measured
Numbers We are going to
make out description more specific to the observer's
capability. We have already mentioned that observers with with
different capabilities will be measuring the values of 0's, ∞'s and the intervals [ 0, ∞
] differently. We will attach a subscript to the numbers to signify the
measurement space specific to the observer's capability.
For example an observer who can measure "null" will have infinite
capability. We will call such observer Obs_{i} (i for infinite)
and we denote the interval measured by Obs_{i} as [ 0_{i}, ∞_{i}
]. Similarly we denote the observer with finite capability as Obs_{j}
and the interval measured by Obs_{j} as [ 0_{j}, ∞_{j} ]. We have to
keep in mind that Obs_{j} represents a pair of observers. It is important to note that numbers 0_{j}, 1_{j}, 2_{j},.........,∞_{j} represent the "measured numbers". The measured numbers will represent a picture of more or less details for an observer of capability greater or less respectively than that of Obs_{j}. At the same time 0_{j} will be measured as 0_{j} and 1_{j} will be measured as 1_{j} and so on, by the observers of identical capabilities. We will get used to these notations. In order to measure the null an observer requires infinite capability which is equivalent to saying that the observer of finite capability will have to measure an infinite information space, where the term "information" represents an abstract notion we will try to resolve. We further represent this infinite information space by a truepoint. A truepoint can be thought of as a disc without a boundary, an abstract region whose characteristics can not be measured. We will need a quantum toy based on the particle in a box analogy to play with this concept.
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1 orange consists
of billions of cells, 1 cell consists of billions of atoms, 1 atom contains billions of elementary states......., 1 universe contains billions of oranges, 1 source contains billions of universes, Have we counted "one" yet?  A limerick from the Bugworld, Unknown Author. 