What we have been trying to do is to describe a given system without having to use the conventional 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 capacity.  

    One may also think of it as an introduction of a minimum quantum (observer1-observer2), whose value is based on the maximum of the observer's capacity.  What does this quantum mean we do not know yet.

      We have established the origin in the discrete measurement space, we called j-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.
 

      Remember that our origin is infinite and a PE1 event, therefore the next PE1 event also has to be infinite.  We can think of universe as an ongoing measurement which is yet to be completed.

  The Bug Measures a Circle   
     
      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 capacities, 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 on its measurements 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,

This criterion leads to optimum use of the resources while making a measurement, an optimization which is a fundamental requirement in nature.

       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 non-existent 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 capacity.  We have already mentioned that observers with with different capacities, will be measuring the values of  0's,  ∞'s and the interval [0, ∞] differently.  We will attach a subscript to the numbers to signify the measurement space specific to the observer's capacity.

      For example an observer who can measure "null" will have infinite capacity.  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 capacity 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 less or more details for an observer of capacity greater or less respectively than that of Obs_j, because the definitions of  origin will be different for each observer.  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 capacity which is equivalent to saying that the observer of finite capacity 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 "true-point".  A true-point 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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* i-space represents an infinite information space, an infinite source, or an exact source. 

** For example, we can think of second universe only when we have completely measured the first one, a PE1 event.   We keep in mind that natural numbers begin with "1". The concepts of "0" and negative numbers apply only when the time-axis has been initiated due to entropy.