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Incompleteness I
7th February 2014

       The importance of the role of actual measurements in providing a complete and consistent description of a physical system can not be underestimated. The completeness requires a measurement to be a zero entropy measurement.  The consistency requirement for a measurement results in a measurement algorithm which can be used to verify the completeness of the measurements being performed.

    Consider for example a circle, being measured by two observers of different capabilities. The human observer will look at it and determine it to be a circle which is essentially a single measurement, whereas the bug crawling on the perimeter does not know what a circle is and may not even complete the measurements before its life-cycle is ended. Therefore for a human observer entropy for the problem of determination of the circle is kBloge1 = 0, whereas for the bug the entropy may be infinite as the number of measurements is very large. (kB is the Boltzmann constant.)

      Herein lies the importance of the entropy in characterizing a system. For same result, different observers will have different entropies depending upon the system they have designed to make measurements based on their capabilities. The result or the event being measured is independent of the entropy and is considered absolute or a PE1 event.

      When it comes to the universe, it is an ongoing measurement with human observer in the position of the bug. We have already identified the problem of the determination of the origin is the critical one. Therefore we can state the following:

"A system as designed by humans to measure the universe, can never have the complete information about the origin and therefore can not be absolute."

Also it is important to note that,
  • The nature requires a single-step or a zero-entropy solution.
  • If an observer is making more than one measurement, but less than infinite, or in other words finite number of measurements greater than one, then the solution represents an unstable or a finite life-time event.
  • If an observer is able to make infinite number of measurements, it is possible only if the observer has no entropy and hence it is equivalent to saying that the solution is a single-step solution.
Any system developed in the physical world to measure an event will be riddled with an inherent entropy and hence can not be perfect, or it can not measure a PE1 event.

      It also implies that any event which is determined to be measured complete can not represent a infinite life-time event or a stable structure. Therefore the life-times for the particles such as protons and electrons may be very long, but they can not be infinite. Similarly with the given capabilities a human observer can not completely measure an event of infinite life time. Quite a dilemma we are in, what we measure does not matter eventually and what really matters we can not measure.
Thanks go to JB for suggesting that we should look into Gödel's work. As crazy as it may sound we have a feeling that the metric suggested by Gödel for GTR may be in the right direction. 

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Previous Blogs:

Discrete Measurement Space 26Jan 2014
KoopMandooka 8Jan 2014
Visual Complex Analysis 22Dec 2013
The Einstein Theory of Relativity 7Dec 2013